Financial risk management

Financial Risk Management 2010-11 Topics T1 Stock index futures

Duration, Convexity, Immunization

T2 Repo and reverse repo Futures on T-bills Futures on T-bonds Delta, Gamma, Vega hedging

T3 Portfolio insurance Implied volatility and volatility smiles

T4 Modelling stock prices using GBM Interest rate derivatives (Bond options, Caps, Floors, Swaptions)

T5 Value at Risk

T6 Value at Risk: statistical issues Monte Carlo Simulations Principal Component Analysis Other VaR measures

T7 Parametric volatility models (GARCH type models) Non-parametric volatility models (Range and high frequency models) Multivariate volatility models (Dynamic Conditional Correlation DCC models)

T8 Credit Risk Measures (credit metrics, KMV, Credit Risk Plus, CPV)

T9

Credit derivatives (credit options, total return swaps, credit default swaps) Asset Backed Securitization Collateralized Debt Obligations (CDO)

* This file provides you an indication of the range of topics that is planned to be covered in the module. However, please note that the topic plans might be subject to change.

Financial Risk Management

Topic 1

Managing risk using Futures Managing risk using Futures Reading: CN(2001) chapter 3

� Futures Contract: � Speculation, arbitrage, and hedging

�Stock Index Futures Contract:

� Futures Contract: � Speculation, arbitrage, and hedging

�Stock Index Futures Contract:

Topics

�Stock Index Futures Contract: � Hedging (minimum variance hedge ratio)

�Hedging market risks

�Stock Index Futures Contract: � Hedging (minimum variance hedge ratio)

�Hedging market risks

Futures Contract

� Agreement to buy or sell “something” in the future at

a price agreed today. (It provides Leverage.)

� Speculation with Futures: Buy low, sell high

� Futures (unlike Forwards) can be closed anytime by taking

an opposite position

Arbitrage with Futures: Spot and Futures are linked � Arbitrage with Futures: Spot and Futures are linked

by actions of arbitragers. So they move one for one.

� Hedging with Futures: Example: In January, a farmer

wants to lock in the sale price of his hogs which will

be “fat and pretty” in September.

� Sell live hog Futures contract in Jan with maturity in Sept

Speculation with Futures


� Purchase at F0 = 100

� Hope to sell at higher price later F1 = 110

� Close-out position before delivery date.

� Obtain Leverage (i.e. initial margin is ‘low’)

� ExampleExampleExampleExample: : : : Nick Leeson: Feb 1995

� Long 61,000 Nikkei-225 index futures (underlying

value = $7bn).

� Nikkei fell and he lost money (lots of it)

� - he was supposed to be doing riskless ‘index

arbitrage’ not speculating

$10


Profit/Loss per contractLong future

Futures price

-$10

0F1 = 110

F1 = 90

F0

= 10

0

Short future

+1


� Profit payoff (direction vectors)

-1

F increasethen profit increases

F increasethen profit decrease

Profit/Loss Profit/Loss

Underlying,S

or Futures, F+1

+1

Long Futures

or, Long Spot

Short Futures

or, Short Spot

-1

-1

Arbitrage with Futures


� At expiry (T), FT = ST . Else we can make riskless profit (Arbitrage).

� Forward price approaches spot price at maturity

Forward price ‘at a premium’ when : F > S (contango)

Forward price, F

Stock price, St

T

Forward price ‘at a premium’ when : F > S (contango)

Forward price ‘at a discount’, when : F < S (backwardation)

0

At T, ST = FT


� General formula for non-income paying security: F0 = S0erT or F0 = S0(1+r)T

� Futures price = spot price + cost of carry

� For stock paying dividends, we reduce the ‘cost of � For stock paying dividends, we reduce the ‘cost of

carry’ by amount of dividend payments (d)

� F0 = S0e(r-d)T

� For commodity futures, storage costs (v or V) is

negative income

� F0 = S0e(r+v)T or F0 = (S0+V)erT


� For currency futures, the ‘cost of carry’ will be

reduced by the riskless rate of the foreign currency

(rf)

� F0 = S0e(r-rf)T

For stock index futures, the cost of carry will be � For stock index futures, the cost of carry will be

reduced by the dividend yield

� F0 = S0e(r-d)T


� Arbitrage at t<T for a non-income paying security:

� If F0 > S0erT then buy the asset and short the futures

contract

� If F0 < S0erT then short the asset and buy the futures

contract

� Example of ‘Cash and Carry’ arbitrage: S=£100, � Example of ‘Cash and Carry’ arbitrage: S=£100, r=4%p.a., F=£102 for delivery in 3 months.

� We see £� Since Futures is over priced,

0.04 0.25100 101F e ×= × =ɶ

time = Now time = in 3 months

•Sell Futures contract at £102

•Borrow £100 for 3 months and buy stock

•Pay loan back (£101)

•Deliver stock and get agreed price of £102

Hedging with Futures


� F and S are positively correlated

� To hedge, we need a negative correlation. So we

long one and short the other.

� Hedge = long underlying + short Futures


Simple Hedging Example:Simple Hedging Example:Simple Hedging Example:Simple Hedging Example:

� You long a stock and you fear falling prices over the

next 2 months, when you want to sell. Today (say

January), you observe S0=£100 and F0=£101 for

April delivery.

� Today: you sell one futures contractso r is 4%

� Today: you sell one futures contract

� In March: say prices fell to £90 (S1=£90). So

F1=S1e0.04x(1/12)=£90.3. You close out on Futures.

� Profit on Futures: 101 – 90.3 = £10.7

� Loss on stock value: 100 – 90 =£10

� Net Position is +0.7 profit. Value of hedged portfolio

= S1+ (F0 - F1) = 90 + 10.7 = 100.7

so r is 4%


� F1 value would have been different if r had changed.

� This is Basis Risk Basis Risk Basis Risk Basis Risk (b1 = S1 – F1)

� Final Value = S1 + (F0 - F1 ) = £100.7

= (S1 - F1 ) + F0

= b + F= b1 + F0

where “Final basis” b1 = S1 - F1

� At maturity of the futures contract the basis is zero

(since S1 = F1 ). In general, when contract is closed

out prior to maturity b1 = S1 - F1 may not be zero.

However, b1 will usually be small in relation to F0.

Stock Index Futures ContractStock Index Futures Contract

Hedging with SIFs

Stock Index Futures Contract

� Stock Index Futures contract can be used to

eliminate market risk from a portfolio of stocks

� If this equality does not hold then index arbitrage

(program trading) would generate riskless profits.

( )0 0

r d TF S e −= ×

(program trading) would generate riskless profits.

� Risk free rate is usually greater than dividend yield

(r>d) so F>S

Hedging with Stock Index Futures

� Example: A portfolio manager wishes to hedge her

portfolio of $1.4m held in diversified equity and

S&P500 index

� S0 = 1400 index point

� Number of stocks, Ns = TVS0/S0 = $1.4m/1400

Total value of spot position, TVS0=$1.4m

� Number of stocks, Ns = TVS0/S0 = $1.4m/1400

=1000 units

� We want to hedge Δ(TVSt)= Ns . Δ(St)

� Use Stock Index Futures, F0=1500 index point, z=

contract multiplier = $250

� FVF0 = z F0 = $250 ( 1500 ) = $375,000


� The required number of Stock Index Futures contract

to short will be 3

� In the above example, we have assumed that S and

0

0

$1, 400, 0003.73

$375, 000F

TVSN

FVF

= − = − = −

� In the above example, we have assumed that S and

F have correlation +1 (i.e. )

� In reality this is not the case and so we need

minimum variance hedge ratiominimum variance hedge ratiominimum variance hedge ratiominimum variance hedge ratio

S F∆ = ∆


� Minimum Variance Hedge RatioMinimum Variance Hedge RatioMinimum Variance Hedge RatioMinimum Variance Hedge Ratio

∆∆∆∆V = change in spot market position + change in Index Futures position

= Ns . (S1-S0) + Nf . (F1 - F0) z

= Ns S0. ∆∆∆∆S /S0 + Nf F0. (∆∆∆∆F /F0) z= TVS0 . ∆∆∆∆S /S0 + Nf . FVF0 . (∆∆∆∆F /F0)

where, z = contract multiple for futures ($250 for S&P 500 Futures); ∆∆∆∆S =

FFSSFVFTVSf

N

FFFVFfNSSTVSV

/,/.002

2/

2)

0(

2)(

2/)

20(

2

∆∆+

∆+∆=

σ

σσσ

where, z = contract multiple for futures ($250 for S&P 500 Futures); ∆∆∆∆S =S1 - S0, ∆∆∆∆F = F1 - F0

The variance of the hedged portfolio is


� To obtain minimum, we differentiate with respect to NNNNffff

( ) and set to zero

( )

( )0

2 2/ / , /0 0

2/ , / /

0N f F F S S F F

N f S S F F F F

F V F T V S F V F

T V S

σ σ

σ σ

⋅∆ ∆ ∆

∆ ∆ ∆

= − ⋅

= −

2/ 0NV f

σ∂ ∂ =

where Ns = TVS0/S0 and beta is regression coefficient of the

regression

( )0

0

0

0

/ , / /

/ , /

N f S S F F F F

S S F F

F V F

T V S

F V F

σ σ

β

∆ ∆ ∆

∆ ∆

= −

= −

( ) ( )0 // /S FS S F Fα β ε∆ ∆∆ = + ∆ +


� SUMMARYSUMMARYSUMMARYSUMMARY

pf FVF

TVSN β.0−=

impliesf

NV 0/2

=∂∂σ

� If correlation = 1, the beta will be 1 and we just have

=−=

0tatfuturesofFaceValue

PositionSpotofValue

pf FVF0

pβ

0

0f

TVSN

FVF= −


Application: Changing beta of your portfolio: “Market Application: Changing beta of your portfolio: “Market Application: Changing beta of your portfolio: “Market Application: Changing beta of your portfolio: “Market

Timing Strategy”Timing Strategy”Timing Strategy”Timing Strategy”

� Example: βp (=say 0.8) is your current ‘spot/cash’ portfolio of stocks

).(0

0phf FVF

TVSN ββ −=

But

• You are more optimistic about ‘bull market’ and desire a higher exposure of

βh (=say, 1.3)

• It’s ‘expensive’ to sell low-beta shares and purchase high-beta shares

• Instead ‘go long’ more Nf Stock Index Futures contracts

Note: If βh= 0, then Nf = - (TVS0 / FVF0) βp


� If you hold stock portfolio, selling futures will place a hedge and reduce the beta of your stock portfolio.

�If you want to increase your portfolio beta, go long futures.

�Example: Suppose β = 0.8 and Nf = -6 contracts would �Example: Suppose β = 0.8 and Nf = -6 contracts would make β = 0.

�If you short 3 (-3) contracts instead, then β = 0.4

�If you long 3 (+3) contracts instead, then β = 0.8+0.4 = 1.2


Application: Stock Picking and hedging market riskApplication: Stock Picking and hedging market riskApplication: Stock Picking and hedging market riskApplication: Stock Picking and hedging market risk

You hold (or purchase) 1000 undervalued shares of Sven plc

V(Sven) = $110 (e.g. Using Gordon Growth model)

P(Sven) = $100 (say)

Sven plc are underpriced by 10%.

Therefore you believe Sven will rise 10% more than the market over the next

3 months.

But you also think that the market as a whole may fall by 3%.

The beta of Sven plc (when regressed with the market return) is 2.0


Can you ‘protect’ yourself against the general fall in the market and hence any

‘knock on’ effect on Sven plc ?

Yes . Sell Nf index futures, using:

pf FVF

TVSN β.0−=

If the market falls 3% then

Sven plc will only change by about 10% - (2x3%) = +4%

But the profit from the short position in Nf index futures, will give you an

additional return of around 6%, making your total return around 10%.

FVF 0


Application: Future stock purchase and hedging market Application: Future stock purchase and hedging market Application: Future stock purchase and hedging market Application: Future stock purchase and hedging market

riskriskriskriskYou want to purchase 1000 stocks of takeover target with βp = 2, in 1

month’s time when you will have the cash.

You fear a general rise in stock prices.

Go long Stock Index Futures (SIF) contracts, so that gain on the futures will Go long Stock Index Futures (SIF) contracts, so that gain on the futures will

offset the higher cost of these particular shares in 1 month’s time.

SIF will protect you from market risk (ie. General rise in prices) but not from

specific risk. For example if the information that you are trying to takeover

the firm ‘leaks out’ , then price of ‘takeover target’ will move more than that

given by its ‘beta’ (i.e. the futures only hedges market risk)

pf FVF

TVSN β.

0

0=


Topic 2

Managing interest rate risksManaging interest rate risksReference: Hull(2009), Luenberger (1997), and CN(2001)

�Duration, immunization, convexity

�Repo (Sale and Repurchase agreement) and Reverse Repo

�Duration, immunization, convexity

�Repo (Sale and Repurchase agreement) and Reverse Repo

Topics

�Hedging using interest rate Futures

�Futures on T-bills

�Futures on T-bonds

�Hedging using interest rate Futures

�Futures on T-bills

�Futures on T-bonds

Readings

� Books

� Hull(2009) chapters 6

� CN(2001) chapters 5, 6

� Luenberger (1997) chapters 3

� Journal Article

Fooladi, I and Roberts, G (2000) “Risk Management with Duration Analysis” � Fooladi, I and Roberts, G (2000) “Risk Management with Duration Analysis” Managerial Finance, Vol 25, no. 3

Hedging Interest rate risks: Duration

Duration measures sensitivity of price changes (volatility) withchanges in interest rates

1 (1 )(1 )

T

TtTt

t

BParValueCP

rr=

= +++∑

Lower the couponsfor a given time to maturity, greater change in price to change in interest

Duration

1

For a given percentage change in yield, the actual price increase isgreater than a price decrease

(1 )r+

1 (1 )(1 )

T

TtTt

t

BParValueCP

rr=

= +++∑

rates

Greater the time to maturity with a given coupon, greater change in price to change in interest rates

2

3

Duration (also called Macaulay Duration)

� Duration of the bond is a measure that summarizes

approximate response of bond prices to change in yields.

� A better approximation could be convexity of the bond .

1

i

ny t

ii

ny t

B c e −

=

−

=

⋅

∑

∑

weight

� Duration is weighted average of the times when payments

are made. The weight is equal to proportion of bond’s total

present value received in cash flow at time ti.

� Duration is “how long” bondholder has to wait for cash flows

1

1

i

i

ny t

y ti i ni i

ii

t c ec e

D tB B

−−

=

=

⋅

= =

∑∑

Macaulay Duration

� For a small change in yields

� Evaluating :

d BB y

d y∆ = ∆

/y d y∆

d B

d y1

i

ny t

i ii

B t c e y−

=

∆ = − ∆ ∑

� D measures sensitivity of percentage change in bond

prices to (small) changes in yields

� Note negative relationship between Price (B)

and yields (Y)

1i

B D y

BD y

B

=

= − ⋅ ⋅ ∆∆ = − ⋅ ∆

Modified Duration and Dollar Duration

� For Macaulay Duration, y is expressed in continuous

compounding.

� When we have discrete compounding, we have Modified Modified Modified Modified

DurationDurationDurationDuration (with these small modifications)

� If yyyy is expressed as compounding mmmm times a year, we divide DDDD

by (1+y/m)(1+y/m)(1+y/m)(1+y/m)

(1 / )

DB B y

y m∆ = − ⋅ ⋅∆

+

� Dollar Duration, DDollar Duration, DDollar Duration, DDollar Duration, D$$$$ = B.D

� That is, D$ = Bond Price x Duration (Macaulay or Modified)

�

� So D$ is like Options Delta

*

(1 / )B B y

y m

B B D y

∆ = − ⋅ ⋅∆+

∆ = − ⋅ ⋅∆

$B D y∆ = − ⋅∆$

BD

y

∆= −∆

Duration

� Example: Consider a trader who has $1 million in

bond with modified duration of 5. This means for

every 1 bp (i.e. 0.01%) change in yield, the value of

the bond portfolio will change by $500.

� A zero coupon bond with maturity of n years has a

( )$1, 000, 000 5 0.01% $500B∆ = − × ⋅ = −

� A zero coupon bond with maturity of n years has a

Duration = n

� A coupon-bearing bond with maturity of n years will

have Duration < n

� Duration of a bond portfolio is weighted average of

the durations of individual bonds

( )/p o r t fo l io i ii

D B B D= ⋅∑

Duration -example

Example: Consider a 7% bond with 3 years to maturity. Assume that the bondis selling at 8% yield.

××

A B C D E

Year Payment Discount factor 8%

Present value = B C

Weight = D/Price

A E

0.5 3.5 0.962 3.365 0.035 0.0170.5 3.5 0.962 3.365 0.035 0.0171.0 3.5 0.925 3.236 0.033 0.0331.5 3.5 0.889 3.111 0.032 0.0482.0 3.5 0.855 2.992 0.031 0.0612.5 3.5 0.822 2.877 0.030 0.0743.0 103.5 0.79 81.798 0.840 2.520

Sum Price = 97.379 Duration = 2.753Here, yield to maturity = 0.08, m = 2, y = 0.04, n = 6, Face value = 100.

Qualitative properties of duration

� Duration of bonds with 5% yield as a function of

maturity and coupon rate.

Coupon rate

Years tomaturity

1% 2% 5% 10%

1 0.997 0.995 0.988 0.9771 0.997 0.995 0.988 0.9772 1.984 1.969 1.928 1.8685 4.875 4.763 4.485 4.15610 9.416 8.950 7.989 7.10725 20.164 17.715 14.536 12.75450 26.666 22.284 18.765 17.384100 22.572 21.200 20.363 20.067Infinity 20.500 20.500 20.500 20.500

Properties of duration

1. Duration of a coupon paying bond is always less than its maturity. Duration decreases with the increase of coupon rate. Duration equals bond maturity for non-coupon paying bond.

2. As the time to maturity increases to infinity, the 2. As the time to maturity increases to infinity, the duration do not increase to infinity but tend to a finite limit independent of the coupon rate.

Actually, where λ is the yield to maturity

per annum, and m is the number of coupon

payments per year.

λ

λmD

+→

1

Properties of Duration

3. Durations are not quite sensitive to increase in coupon rate (for bonds with fixed yield). They don’t vary huge amount since yield is held constant and it cancels out the influence of coupons.

4. When the coupon rate is lower than the yield, the 4. When the coupon rate is lower than the yield, the duration first increases with maturity to some maximum value then decreases to the asymptotic limit value.

5. Very long durations can be achieved by bonds with very long maturities and very low coupons.

Changing Portfolio Duration

Changing Duration of your portfolio:� If prices are rising (yields are falling), a bond

trader might want to switch from shorter duration bonds to longer duration bonds as longer duration bonds have larger price longer duration bonds have larger price changes.

� Alternatively, you can leverage shorter maturities. Effective portfolio duration = ordinary duration x leverage ratio.

Immunization (or Duration matching)

� This is widely implemented by Fixed Income Practitioners.

� You want to safeguard against interest rate increases.

� A few ideas:

time 0 time 1 time 2 time 3

0 pay $ pay $ pay $

� A few ideas:

1. Buy zero coupon bond with maturities matching timing of

cash flows (*Not available) [Rolling hedge has reinv. risk]

2.Keep portfolio of assets and sell parts of it when cash is

needed & reinvest in more assets when surplus (* difficult as

Δ value of in portfolio and Δ value of obligations will not

identical)

3. Immunization - matching duration and present values

of portfolio and obligations (*YES)

Immunization� Matching present values (PV) of portfolio and obligations

� This means that you will meet your obligations with the cash

from the portfolio.

� If yields don’t change, then you are fine.

� If yields change, then the portfolio value and PV will both change

by varied amounts. So we match also Duration (interest rate risk)

PV PV PV+ =� Matching duration

� Here both portfolio and obligations have the same sensitivity to

interest rate changes.

� If yields increase then PV of portfolio will decrease (so will the PV

of the obligation streams)

� If yields decrease then PV of portfolio will increase (so will the PV

of the obligation streams)

1 2 obligationPV PV PV+ =

1 1 2 2 obligation obligationD PV D PV D PV+ =

Immunization

Example

Suppose Company A has an obligation to pay $1 million in 10 years. How to invest in bonds now so as to meet the future obligation?obligation?

• An obvious solution is the purchase of a simple zero-coupon bond with maturity 10 years.

* This example is from Leunberger (1998) page 64-65. The numbers are rounded up by the author so replication would give different numbers.

Immunization

Suppose only the following bonds are available for its choice.

coupon rate maturity price yield durationBond 1 6% 30 yr 69.04 9% 11.44Bond 2 11% 10 yr 113.01 9% 6.54Bond 3 9% 20 yr 100.00 9% 9.61

• Present value of obligation at 9% yield is $414,642.86.• Present value of obligation at 9% yield is $414,642.86.

• Since Bonds 2 and 3 have durations shorter than 10 years, it is notpossible to attain a portfolio with duration 10 years using these two bonds.

Suppose we use Bond 1 and Bond 2 of amounts V1 & V2,V1 + V2 = PV

P1V1 + D2V2 = 10 × PV

giving V1 = $292,788.64, V2 = $121,854.78.

ImmunizationYield

9.0 8.0 10.0Bond 1Price 69.04 77.38 62.14Shares 4241 4241 4241Value 292798.64 328168.58 263535.74

Bond 2

Observation: At different yields (8% and 10%), the value of the portfolio almost agrees with that of the obligation.

Bond 2Price 113.01 120.39 106.23Shares 1078 1078 1078Value 121824.78 129780.42 114515.94Obligation value 414642.86 456386.95 376889.48Surplus -19.44 1562.05 1162.20

Immunization

Difficulties with immunization procedure

1. It is necessary to rebalance or re-immunize the portfolio from time to time since the duration depends on yield.

2. The immunization method assumes that all yields 2. The immunization method assumes that all yields are equal (not quite realistic to have bonds with different maturities to have the same yield).

3. When the prevailing interest rate changes, it is unlikely that the yields on all bonds change by the same amount.

Duration for term structure� We want to measure sensitivity to parallel shifts in the spot

rate curve

� For continuous compounding, duration is called FisherFisherFisherFisher----Weil Weil Weil Weil

durationdurationdurationduration.

� If x0, x1,…, xn is cash flow sequence and spot curve is st where

t = t0,…,tn then present value of cash flow isn

∑

� The Fisher-Weil duration is

0

t ii

i

ns t

ti

PV x e− ⋅

=

= ⋅∑

0

1 t ii

i

ns t

FW i ti

D t x ePV

− ⋅

=

= ⋅ ⋅∑

Duration for term structure

� Consider parallel shift in term structure:

� Then PV becomes

� Taking differential w.r.t ∆y in the point ∆y=0 we get

( ) ( )0

t ii

i

ns y t

ti

P y x e− + ∆ ⋅

=

∆ = ⋅∑

( )i it ts changes to s y+ ∆

( ) ndP y∆∑

� So we find relative price sensitivity is given by DFW

1 (0)

(0) FW

dPD

P d y⋅ = −

∆

( )0

0

| t ii

i

ns t

y i ti

dP yt x e

d y− ⋅

∆ ==

∆= − ⋅

∆ ∑

Convexity

� Duration applies to only small changes in y

� Two bonds with same duration can have different

change in value of their portfolio (for large changes

in yields)

� First order approximation cannot capture this, so we

take second order approximation (convexity)

Convexity

� Convexity for a bond is

� Convexity is the weighted average of the ‘times squared’

22

212

1

1i

i

ny t

y ti i ni i

ii

t c ec ed B

C tB d y B B

−−

=

=

⋅

= = =

∑∑

Convexity is the weighted average of the ‘times squared’

when payments are made.

� From Taylor series expansion

� So Dollar convexity is like Gamma measure in

options.

( )

( )

22

2

2

1

2

1

2

d B d BB y y

d y d y

BD y C y

B

∆ = ∆ + ∆

∆ = − ⋅ ∆ + ⋅ ∆

REPO and REVERSE REPOREPO and REVERSE REPO

Short term risk management using Repo� RepoRepoRepoRepo is where a security is sold with agreement to buy it back at

a later date (at the price agreed now)

� Difference in prices is the interest earned (called repo raterepo raterepo raterepo rate)

� It is form of collateralized short term borrowing (mostly overnight)

� Example: a trader buys a bond and repo it overnight. The

money from repo is used to pay for the bond. The cost of this

deal is repo rate but trader may earn increase in bond prices deal is repo rate but trader may earn increase in bond prices

and any coupon payments on the bond.

� There is credit risk of the borrower. Lender may ask for

margin costs (called haircut) to provide default protection.

� Example: A 1% haircut would mean only 99% of the value of

collateral is lend in cash. Additional ‘margin calls’ are made if

market value of collateral falls below some level.

Short term risk management using Repo� Hedge funds usually speculate on bond price differentials

using REPO and REVERSE REPO

� Example: Assume two bonds A and B with different prices (say price(A)<price(B)) but similar characteristics. Hedge Fund (HF) would like to buy A and sell B simultaneously. This can be financed with repo as follows:

� (Long position) Buy Bond A and repo it. The cash obtained is used to pay for � (Long position) Buy Bond A and repo it. The cash obtained is used to pay for the bond. At repo termination date, sell the bond and with the cash buy bond back (simultaneously). HF would benefit from the price increase in bond and low repo rate

� (short position) Enter into reverse repo by borrowing the Bond B (as collateral for money lend) and simultaneously sell Bond B in the market. At repo termination date, buy bond back and get your loan back (+ repo rate). HF would benefit from the high repo rate and a decrease in price of the bond.

Interest Rate FuturesInterest Rate Futures

(Futures on T-Bills)(Futures on T-Bills)

Interest Rate FuturesIn this section we will look at how Futures contract written on a

Treasury Bill (T-Bill) help in hedging interest rate risks

Review - What is T-Bill?

� T-Bills are issued by government, and quoted at a discount

� Prices are quoted using a discount rate discount rate discount rate discount rate (interest earned as % of

face value)face value)

� Example: 90-day T-Bill is quoted at 0.080.080.080.08. This means annualized

return is 8% of FV. So we can work out the price, as we know FV.

� Day Counts convention Day Counts convention Day Counts convention Day Counts convention (in US)

1. Actual/Actual (for treasury bonds)

2. 30/360 (for corporate and municipal bonds)

3. Actual/360 (for other instruments such as LIBOR)

9 01

1 0 0 3 6 0

dP F V

= −

Interest Rate Futures

So what is a 3-month T-Bill Futures contract?

At expiry, (T), which may be in say 2 months time

the (long) futures delivers a T-Bill which matures at

T+90 days, with face value M=$100.

As we shall see, this allows you to ‘lock in’ at t=0, the forward As we shall see, this allows you to ‘lock in’ at t=0, the forward rate, f12

� T-Bill Futures prices are quoted in terms of quoted index,quoted index,quoted index,quoted index, QQQQ

(unlike discount rate for underlying)

Q = $100 – futures discount rate (df)

� So we can work out the price as

9 01

1 0 0 3 6 0fd

F F V = −

Hedge decisions

When do we use these futures contract to hedge?

Examples:

1) You hold 3m T-Bills to sell in 1-month’s time ~ fear price fall~ sell/short T-Bill futures

2) You will receive $10m in 3m time and wish to place it on a Eurodollar bank deposit for 90 days ~ fear a fall in interest rates ~ go long a Eurodollar futures contract

3) Have to issue $100m of 180-day Commercial Paper in 3 months time (I.e. borrow money) ~ fear a rise in interest rates~ sell/short a T-bill futures contract as there is no commercial bill futures contract (cross hedge)

Cross Hedge: US T-Bill FuturesExample: � Today is May. Funds of $1m will be available in August to

invest for further 6 months in bank deposit (or commercial bills)~ spot asset is a 6-month interest rate

� Fear a fall in spot interest rates before August, so today BUY T-bill futures

Assume parallel shift in the yield curve. (Hence all interest rates move by the same amount.)~ BUT the futures price will move less than the price of the commercial bill - this is duration at work!

� Use Sept ‘3m T-bill’ Futures, ‘nearby’ contract~ underlying this futures contract is a 3-month interest rate

higher the maturity, more sensitive are changes in prices to interest rates

Cross Hedge: US T-Bill Futures

3 month exposure period

Desired investment/protection period = 6-months

May Aug. Sept. Feb.Dec.

Maturity of ‘Underlying’

Purchase T-Bill future with Sept. delivery date

Known $1m cash receipts

Maturity date of Sept.T-Bill futures contract

Maturity of ‘Underlying’ in Futures contract

Question: How many T-bill futures contract should I purchase?


� We should take into account the fact that:

1. to hedge exposure of 3 months, we have used T-bill futures

with 4 months time-to-maturity

2. the Futures and spot prices may not move one-to-one

� We could use the minimum variance hedge ratio:


� We could use the minimum variance hedge ratio:

� However, we can link price changes to interest rate

changes using Duration based hedge ratioDuration based hedge ratioDuration based hedge ratioDuration based hedge ratio

pf FVF

TVSN β.

0

0=

Duration based hedge ratio

� Using duration formulae for spot rates and futures:

� So we can say volatility is proportional to Duration:

S s F F

S FD y D y

S F

∆ ∆= − ⋅ ∆ = − ⋅ ∆

( ) ( )2 2 2 2 2 2S s F F

S FD y D y

S Fσ σ σ σ∆ ∆ = ⋅ ∆ = ⋅ ∆

( )( )

( )

, S s F F

S F s F

S FCov D y D y

S F

D D y yσ

∆ ∆ = Ε − ⋅ ∆ − ⋅ ∆

= ⋅ ⋅ ∆ ∆


� Expressing Beta in terms of Duration:

0

0

.

,

f p

TVSN

FVF

S FCov

β

=

∆ ∆

We can obtain last term by

( )( )

0

20

02

0

,

s Fs

F F

CovTVS S F

FFVFF

y yTVS D

FVF D y

σ

σσ

= ∆

∆ ∆ = ∆

last term by regressing

0S y Fy yα β ε∆ = + ∆ +


� Summary:

where beta is obtained from the regression of yields

0

0

. sf y

F

TVS DN

FVF Dβ

=

α β ε∆ = + ∆ +0S y Fy yα β ε∆ = + ∆ +


3 month exposure period

Desired investment/protection period = 6-months

May Aug. Sept. Feb.Dec.

Maturity of ‘Underlying’

Example REVISITED

Purchase T-Bill future with Sept. delivery date

Known $1m cash receipts

Maturity date of Sept.T-Bill futures contract

Maturity of ‘Underlying’ in Futures contract



May (Today). Funds of $1m accrue in August to be invested for 6- monthsin bank deposit or commercial bills( Ds = 6 )

Use Sept ‘3m T-bill’ Futures ‘nearby’ contract ( DF = 3)

Cross-hedge.Here assume parallel shift in the yield curveHere assume parallel shift in the yield curve

Qf = 89.2 (per $100 nominal) hence:F0 = 100 – (10.8 / 4) = 97.30

F

FVF0 = $1m (F0/100) = $973,000

Nf = (TVS0 / FVF0) (Ds / DF )= ($1m / 973,000) ( 0.5 / 0.25) = 2.05 (=2)

Cross Hedge: US T-Bill Futures� Suppose now we are in August:

3 month US T-Bill Futures : Sept Maturity

Spot Market(May)

(T-Bill yields)

CME Index

Quote Qf

Futures Price, F

(per $100)

Face Value of $1m

Contract, FVF

May y0 (6m) = 11% Qf,0 = 89.2 97.30 $973,000

August y1(6m) = 9.6% Qf,1 = 90.3 97.58 $975,750

� Key figure is F1 = 97.575 (rounded 97.58)

� Gain on the futures position

= TVS0 (F1 - F0) NF = $1m (0.97575 – 0.973) 2 = $5,500

August y1(6m) = 9.6% Qf,1 = 90.3 97.58 $975,750

Change -1.4% 1.10 (110 ticks) 0.28 $2,750

(per contract)

Durations are : Ds = 0.5, Df = 0.25

Amount to be hedged = $1m. No. of contracts held = 2


� Invest this profit of $5500 for 6 months (Aug-Feb) at y1=9.6%:

= $5500 + (0.096/2) = $5764

� Loss of interest in 6-month spot market (y0=11%, y1=9.6%)

= $1m x [0.11 – 0.096] x (1/2) = $7000

� Net Loss on hedged position $7000 - $5764 = $1236

(so the company lost $1236 than $7000 without the hedge)

Potential Problems with this hedge:1. Margin calls may be required2. Nearby contracts may be maturing before September. So we may have to roll over the hedge

3. Cross hedge instrument may have different driving factors of risk

Interest Rate FuturesInterest Rate Futures

(Futures on T-Bonds)(Futures on T-Bonds)

US T-Bond Futures

� Contract specifications of US T-Bond Futures at CBOT:

Contract size $100,000 nominal, notional US Treasury bond with 8% coupon

Delivery months March, June, September, December

Quotation Per $100 nominal

Tick size (value) 1/32 ($31.25)

Last trading day 7 working days prior to last business day in expiry month

Delivery day Any business day in delivery month (seller’s choice)

� Notional is 8% coupon bond. However, Short can choose to

deliver any other bond. So Conversion Factor Conversion Factor Conversion Factor Conversion Factor adjusts “delivery

price” to reflect type of bond delivered

� T-bond must have at least 15 years 15 years 15 years 15 years time-to-maturity

� Quote ‘98‘98‘98‘98----14’14’14’14’ means 98.(14/32)=$98.4375 per $100 nominal

Delivery day Any business day in delivery month (seller’s choice)

Settlement Any US Treasury bond maturing at least 15 years from the contract month (or not callable for 15 years)

US T-Bond Futures

� Conversion Factor (CF)Conversion Factor (CF)Conversion Factor (CF)Conversion Factor (CF): : : : CF adjusts price of actual bond to be

delivered by assuming it has a 8% yield (matching the bond to

the notional bond specified in the futures contract)

� Price = (most recent settlement price x CF) + accrued interest

� Example: Example: Example: Example: Possible bond for delivery is a 10% coupon (semi-� Example: Example: Example: Example: Possible bond for delivery is a 10% coupon (semi-

annual) T-bond with maturity 20 years.

� The theoretical price (say, r=8%):

� Dividing by Face Value, CF = 119.794/100 = 1.19794 (per

$100 nominal)

40

401

5 100119.794

1.04 1.04ii

P=

= + =∑

If Coupon rate > 8% then CF>1If Coupon rate < 8% then CF<1

US T-Bond Futures

� Cheapest to deliverCheapest to deliverCheapest to deliverCheapest to deliver::::

In the maturity month, Short party can choose to deliver any

bond from the existing bonds with varying coupons and

maturity. So the short party delivers the cheapest one.

� Short receives:

(most recent settlement price x CF) + accrued interest(most recent settlement price x CF) + accrued interest

� Cost of purchasing the bond is:

Quoted bond price + accrued interest

� The cheapest to deliver bond is the one with the smallest:

Quoted bond price - (most recent settlement price x CF)

Hedging using US T-Bond Futures

� Hedging is the same as in the case of T-bill Futures (except

Conversion Factor).

� For long T-bond Futures, duration based hedge ratio is given

by:

0 . sTVS DN CFβ

= ⋅

where we have an additional term for conversion factor for

the cheapest to deliver bond.

0

0

. sf y CTD

F

TVS DN CF

FVF Dβ

= ⋅


Topic 3a

Managing risk using Options Managing risk using Options Readings: CN(2001) chapters 9, 13; Hull Chapter 17

� Financial Engineering with Options

� Black Scholes

�Delta, Gamma, Vega Hedging

� Financial Engineering with Options

� Black Scholes

�Delta, Gamma, Vega Hedging

Topics

�Portfolio Insurance �Portfolio Insurance

Options Contract - Review

� An option (not an obligation), American and European

-

-

Put Premium

Financial Engineering with options

� Synthetic call option

� Put-Call Parity: P + S = C + Cash

� Example: Pension Fund wants to hedge its stock holding

against falling stock prices (over the next 6 months) and

wishes to temporarily establish a “floor value” (=K) but also

wants to benefit from any stock price rises.

Financial Engineering with options

� Nick Leeson’s short straddle

You are initially credited with the call and put pr emia C + P (at t=0) but if at expiry there is either a large fall or a large rise in S ( relative to the strike price K ) then you will make a loss

(.eg. Leeson’s short straddle: Kobe Earthquake which led to a fall in S (S = “Nikkei-225”) and thus large losses).

Black Scholes

� BS formula for price of European Call option

0 1 2( ) ( )r Tc S N d K e N d−= −

Probability of call option being in-the-money and getting stock

D2=d12 1d d Tσ= −

� c = expected (average) value of receiving the stock in the event of exercise MINUS cost of paying the strike price in the event of exercise

Present value of the strike price

Probability of exercise and paying strike price

Black Scholes

where 2 20 0

1 2 2 1

ln ln2 2

; or

S Sr T r T

K Kd d d d T

T T

σ σ

σσ σ

+ + + − = = = −

Sensitivity of option prices

Sensitivity of option prices (American/European nonSensitivity of option prices (American/European nonSensitivity of option prices (American/European nonSensitivity of option prices (American/European non----

dividend paying)dividend paying)dividend paying)dividend paying)

� c = f ( K, S0, r, T, σ )

- + + + +This however can be negative for

dividend paying European options.

Example: stock pays dividend in

� p = f ( K, S0, r, T, σ )

� Call premium increases as stock price increases (but less than one-for-one)

� Put premium falls as stock price increases (but less than one-for-one)

+ - - + +

Example: stock pays dividend in 2 weeks. European call with 1 week to expiration will have more value than European call with 3 weeks to maturity.


The Greek LettersThe Greek LettersThe Greek LettersThe Greek Letters

� Delta, Delta, Delta, Delta, ∆∆∆∆ measures option price change when stock

price increase by $1

� Gamma, Gamma, Gamma, Gamma, ΓΓΓΓ measures change in Delta when stock

price increase by $1price increase by $1

� Vega, Vega, Vega, Vega, υυυυ measures change in option price when there

is an increase in volatility of 1%

� Theta, Theta, Theta, Theta, Θ measures change in option price when

there is a decrease in the time to maturity by 1 day

� Rho, Rho, Rho, Rho, ρ measures change in option price when there

is an increase in interest rate of 1% (100 bp)


� Using Taylor series,

2

2; ; ; ;

f f f f f

S S T rυ ρ

σ∂ ∂ ∂ ∂ ∂∆ = Γ = = Θ = =∂ ∂ ∂ ∂ ∂

Read chapter 12 of McDonald text book “Derivative Markets” for more about GreeksRead chapter 12 of McDonald text book “Derivative Markets” for more about GreeksRead chapter 12 of McDonald text book “Derivative Markets” for more about GreeksRead chapter 12 of McDonald text book “Derivative Markets” for more about Greeks

( )21

2d f d S d S d t d r dρ υ σ≈ ∆ ⋅ + Γ ⋅ + Θ ⋅ + ⋅ + ⋅

Delta

� The rate of change of the option price with respectto the share price

� e.g. Delta of a call option is 0.6

� Stock price changes by a small amount, then the optionprice changes by about 60% of that

Optionprice

S

CSlope = ∆ = ∂c/ ∂ S

Stock price

Delta

� ∆ of a stock = 1

�

�

( )1 0call

CN d

S

∂∆ = = >∂

( )1 1 0put

PN d

S

∂∆ = = − <∂

(for long positions)

� If we have lots of options (on same underlying) then

delta of portfolio is

where Nk is the number of options held. Nk > 0 if long

Call/Put and Nk < 0 if short Call/Put

S∂

portfolio k kk

N∆ = ⋅∆∑

Delta

� So if we use delta hedging for a short call position, wemust keep a long position of N(d1) shares

� What about put options?

� The higher the call’s delta, the more likely it is that theoption ends up in the money:

� Deep out-of-the-money: Δ ≈ 0

� At-the-money: Δ ≈ 0.5

� In-the-money: Δ ≈ 1

� Intuition: if the trader had written deep OTM calls, itwould not take so many shares to hedge - unlikely thecalls would end up in-the-money

Theta

� The rate of change of the value of an option withrespect to time

� Also called the time decay of the option

� For a European call on a non-dividend-paying stock,

� Related to the square root of time, so the relationship isnot linear

2

0 1 22

'( ) 1( ) where '( )

2 2

xrTS N d

rKe N d N x eT

σπ

−−Θ = − − =

Theta

� Theta is negative: as maturity approaches, the optiontends to become less valuable

� The close to the expiration date, the faster the value ofthe option falls (to its intrinsic value)

� Theta isn’t the same kind of parameter as delta� Theta isn’t the same kind of parameter as delta

� The passage of time is certain, so it doesn’t makeany sense to hedge against it!!!

� Many traders still see theta as a useful descriptive statisticbecause in a delta-neutral portfolio it can proxy forGamma

Gamma

� The rate of change of delta with respect to theshare price:

� Calculated as

2

2

f

S

∂∂

1'( )N d

S TσΓ =

� Sometimes referred to as an option’s curvature

� If delta changes slowly → gamma small → adjustmentsto keep portfolio delta-neutral not often needed

0S Tσ

Gamma

� If delta changes quickly → gamma large → risky to leave an originally delta-neutral portfolio unchanged for long periods:

Optionpriceprice

S

C

Stock price

C'C''

S'

Gamma

MakingMaking aa PositionPosition GammaGamma--NeutralNeutral

� We must make a portfolio initially gamma-neutral as well as delta-neutralif we want a lasting hedge

� But a position in the underlying share can’t alter the portfolio gammasince the share has a gamma of zero

� So we need to take out another position in an option that isn’t linearlySo we need to take out another position in an option that isn’t linearlydependent on the underlying share

� If a delta-neutral portfolio starts with gamma Γ, and we buy wT optionseach with gamma ΓT, then the portfolio now has gamma

Γ + wT Γ T

� We want this new gamma to = 0:

Γ + wT Γ T = 0

� Rearranging,

TT

w−Γ=Γ

Delta-Theta-GammaFor any derivative dependent on a non-dividend-paying stock,

Δ , θ, and Г are related

� The standard Black-Scholes differential equation is

where f is the call price, S is the price of the underlying

share and r is the risk-free rate

22 2

2

1

2

f f frS S rf

t S Sσ∂ ∂ ∂+ + =

∂ ∂ ∂

share and r is the risk-free rate

� But

� So

� So if Θ is large and positive, Γ tends to be large and negative,

and vice-versa

� This is why you can use Θ as a proxy for Γ in a delta-neutral

portfolio

2

2, and

f f f

t S S

∂ ∂ ∂Θ = ∆ = Γ =∂ ∂ ∂

2 21

2rS S rfΘ + ∆ + Θ Γ =

Vega

� NOTNOTNOTNOT a letter in the Greek alphabet!

� Vega measures, the sensitivity of an option’s the sensitivity of an option’s the sensitivity of an option’s the sensitivity of an option’s

price to volatilityprice to volatilityprice to volatilityprice to volatility:

f

συ ∂=

∂ 0 1'( )S T N dυ =

� High vega → portfolio value very sensitive to

small changes in volatility

� Like in the case of gamma, if we add in a traded

option we should take a position of – υ/υT to

make the portfolio vega-neutral

συ

∂ 0 1

Rho

� The rate of change of the value of a portfolio ofoptions with respect to the interest rate

� Rho for European Calls is always positive and Rho for

2( )rTKTe N dρ −=f

rρ ∂=

∂� Rho for European Calls is always positive and Rho for

European Puts is always negative (since as interest ratesrise, forward value of stock increases).

� Not very important to stock options with a life of a fewmonths if for example the interest rate moves by ¼%

� More relevant for which class of options?

Delta Hedging

� Value of portfolio = no of calls x call price + no of stocks x stock price

� V = NC C + NS S

� 1 0C S

V CN N

S S

∂ ∂= ⋅ + ⋅ =∂ ∂

C∂

� So if we sold 1 call option then NC = -1. Then no of stocks to buy will be NS = ∆call

� So if ∆call = 0.6368 then buy 0.63 stocks per call option

S C

S C c a l l

CN N

SN N

∂= − ⋅∂

= − ⋅ ∆

Delta Hedging

� Example: Example: Example: Example: As a trader, you have just sold (written) sold (written) sold (written) sold (written)

100 call options100 call options100 call options100 call options to a pension fund (and earned a

nice little brokerage fee and charged a little more

than Black-Scholes price).

� You are worried that share prices might RISEshare prices might RISEshare prices might RISEshare prices might RISE, hence

the call premium RISE, hence showing a loss on your the call premium RISE, hence showing a loss on your

position.

� Suppose ∆ of the call is 0.4. Since you are short,

your your your your ∆ = = = = ----0.40.40.40.4 (When S increases by +$1 (e.g. from

100 to 101), then C decrease by $0.4 (e.g. from 10

to 9.6)).

Delta Hedging

Your 100 written (sold) call option (at C0 = 10 each option)

You now buy 40-shares

Suppose S FALLS by $1 over the next month

THEN fall in C is 0.4 ( = “delta” of the call)

So C falls to C1 = 9.6So C falls to C1 = 9.6

To close out you must now buy back at C1 = 9.6 (a GAIN of $0.4)

Loss on 40 shares = $40

Gain on calls = 100 (C0 - C1 )= 100(0.4) = $40

Delta hedging your 100 written calls with 40 shares means that the value of your ‘portfolio is unchanged.

.

Delta Hedging

Call Premium

∆ = 0.4 .

∆ = 0.5

B

A

� As S changes then so does ‘delta’ , so you have to rebalance your portfolio. E.g. ‘delta’ = 0.5, then you now have to hold 50 stocks for every written call.This brings us to ‘Dynamic Hedging’ , over many periods.

� Buying and selling shares can be expensive so instead we can maintain the hedge by buying and selling options.

.0

Stock Price100 110

A

(Dynamic) Delta Hedging

� You’ve written a call option and earned C0 =10.45 (with K=100, σ = 20%, r=5%, T=1)

� At t = 0: Current price S0 = $100. We calculate ∆ 0 = N(d1)= 0.6368.

� So we buy ∆0 = 0.6368 shares at S0 = $100 by borrowing debt.

Debt, D0 = ∆0 x S0 = $63.68

� At t = 0.01: At t = 0.01: At t = 0.01: At t = 0.01: stock price rise S1 = $100.1. We calculate ∆ 1 = 0.6381.

� So buy extra (∆ 1 – ∆ 0) =0.0013 no of shares at $100.1.

Debt, D1 = D0 ert + (∆ 1 – ∆ 0) S1 = $63.84

� So as you rebalance, you either accumulate or reduce debt

levels.

Delta Hedging

� At t=T, if option ends up well “in the money”At t=T, if option ends up well “in the money”At t=T, if option ends up well “in the money”At t=T, if option ends up well “in the money”

� Say ST = 163.3499. Then ∆ T = 1 (hold 1 share for 1 call).

� Our final debt amount DT = 111.29 (copied from Textbook page 247)

� The option is exercised. We get strike $100 for the share.

Our Net Cost: NC = D – K = 111.29 – 100 = $11.29� Our Net Cost: NCT = DT – K = 111.29 – 100 = $11.29

How have we done with this hedging?How have we done with this hedging?How have we done with this hedging?How have we done with this hedging?

� At t = 0t = 0t = 0t = 0, we received $10.45 and at t = T t = T t = T t = T we owe $11.29

� % Net cost of hedge, % NCT = [ (DT – K )-C0 ] / C0 = 8%

(8% is close to 5% riskless rate)

Delta Hedging

One way to view the hedge:One way to view the hedge:One way to view the hedge:One way to view the hedge:The delta hedge is supposed to be riskless (i.e. no change in value of portfolio of “One written call + holding ∆ shares” , over any very small time interval )

Hence for a perfect hedge we require:

dV = N dS + (N ) dC ≈ N dS + (-1) [ ∆ dS ] ≈ 0dV = NS dS + (NC ) dC ≈ NS dS + (-1) [ ∆ dS ] ≈ 0

If we choose NS = ∆ then we will obtain a near perfect hedge

(ie. for only small changes in S, or equivalently over small time intervals)

Delta Hedging

Another way to view the hedge:Another way to view the hedge:Another way to view the hedge:Another way to view the hedge:The delta hedge is supposed to be riskless, so any money we borrow (receive) at t=0 which is delta hedged over t to T , should have a cost of r

Hence: For a perfect hedge we expect: NDT / C0 = erT so, NDT e-r T - C0 ≈ 0

If we repeat the delta hedge a large number of times then:

% Hedge Performance, HP = stdv( NDT e-r T - C0) / C0

HP will be smaller the more frequently we rebalance the portfolio (i.e. buy or sell stocks) although frequent rebalancing leads to higher ‘transactions costs’ (Kurieland Roncalli (1998))

Gamma and Vega Hedging

� Long Call/Put have positive and

� Short Call/Put have negative and

2

2

f f

Sυ

σ∂ ∂Γ = =∂ ∂

Γ υΓ υ

� Gamma /Vega Neutral: Stocks and futures have

So to change Gamma/Vega of an existing options

portfolio, we have to take positions in further (new)

options.

, 0υΓ =

Delta-Gamma Neutral

� Example: Suppose we have an existing portfolio of options, with a value of Γ = - 300 (and a ∆ = 0)

� Note: Γ = Σi ( Ni Γi )

� Can we remove the risk to changes in S (for even large changes in S ? )Can we remove the risk to changes in S (for even large changes in S ? )

� Create a “Gamma-Neutral” Portfolio

� Let ΓZ = gamma of some “new” option (same ‘underlying’)

� For Γport = NZ ΓZ + Γ = 0

� we require: NZ = - Γ / ΓZ “new” options

Delta-Gamma Neutral

� Suppose a Call option “Z” with the same underlying (e.g. stock) has a delta =

0.62 and gamma of 1.5

� How can you use Z to make the overall portfolio gamma and delta neutral?

� We require: Nz Γz + Γ = 0

Nz = - Γ / Γz = -(-300)/1.5 = 200Nz = - Γ / Γz = -(-300)/1.5 = 200

implies 200 long contracts in Z (ie buy 200 Z-options)

� The delta of this ‘new’ portfolio is now ∆ = Nz.∆z = 200(0.62) = 124

� Hence to maintain delta neutrality you must short 124 units of the underlying -this will not change the ‘gamma’ of your portfolio (since gamma of stock is zero).

Delta-Gamma-Vega Neutral

� Example: You hold a portfolio with

� We need at least 2 options to achieve Gamma and Vega neutrality. Then we rebalance to achieve Delta neutrality of the ‘new’ Gamma-Vega neutral portfolio.

Suppose there is available 2 types of options:

500, 5000, 4000port port portυ∆ = − Γ = − = −

� Suppose there is available 2 types of options:� Option Z with

� Option Y with

� We need

0.5, 1.5, 0.8Z Z Zυ∆ = Γ = =0.6, 0.3, 0.4Y Y Yυ∆ = Γ = =

0

0Z Z Y Y port

Z Z Y Y port

N N

N N

υ υ υ+ + =

Γ + Γ + Γ =

Delta-Gamma-Vega Neutral

� So

� Solution:

( ) ( )( ) ( )0.8 0.4 4000 0

1.5 0.3 5000 0

Z Y

Z Y

N N

N N

+ − =

+ − =

2222.2 5555.5Z YN N= =� Go long 2222.2 units of option Z and long 5555.5 units of option Y to attain Gamma-Vega neutrality.

�New portfolio Delta will be:

�Therefore go short 3944 units of stock to attain Delta neutrality

2222.2 5555.5Z Y

2222.2 5555.5 3944.4Z Y port× ∆ + × ∆ + ∆ =

Portfolio Insurance

Portfolio Insurance

� You hold a portfolio and want insurance against

market declines. Answer: Buy Put options

� From put-call parity: Stocks + Puts = Calls + T-bills

Stock+Put = {+1, +1} + {-1, 0} = {0, +1} = ‘Call payoff’

� This is called Static Portfolio Insurance .� This is called Static Portfolio Insurance .

� Alternatively replicate ‘Stocks+Puts’ portfolio price movements with

‘Stocks+T-bills’ or

‘Stocks+Futures’. [called Dynamic Portfolio Insurance ]

� Why replicate? Because it’s cheaper!

Dynamic Portfolio Insurance

Stock+Put (i.e. the position you wish to replicate)

N0 = V0 /(S0 +P0) (hold 1 Put for 1 Stock)

N0 is fixed throughout the hedge:

At t > 0 ‘Stock+Put’ portfolio:

Vs,p = N0 (S + P)

Hence, change in value:

This is what we wish to replicate

( )pps N

S

PN

S

V∆+=

+= 11 00

,

∂∂

∂∂

Dynamic Portfolio InsuranceReplicate with (N0*) Stocks + (Nf) Futures:

N0* = V0 / S0 (# of index units held in shares)N0* is also held fixed throughout the hedge. Note: position in futures costs nothing (ignore interest cost on margin funds.)

At t > 0: VS,F = N0* S + Nf (F zf) ( )r T tF S e −= ⋅At t > 0: VS,F = N0* S + Nf (F zf)

Hence:

Equating dV of (Stock+Put) with dV(Stock+Futures) to get Nf :

( )[ ]f

tTr

tptf z

eNNN

)(*00 1

−−

−∆+=

+=

S

FNzN

S

Vff

FS

∂∂

∂∂ *

0,

F( )r T tF

eS

−∂ =∂


Replicate with ‘Stock+T-Bill’

VS,B = NS S + NB B

∂∂V

SS B,

= sN

ttStps SNV )()( , −

Equate dV of (Stock+Put) with dV(Stock+T-bill)

tsN )( = ( )tpN ∆+10 = tcN )(0 ∆

NB,t = t

ttStps

B

SNV )()( , −


� Example:Example:Example:Example:Value of stock portfolio V0 = $560,000

S&P500 index S0 = 280Maturity of Derivatives T - t = 0.10

Risk free rate r = 0.10 p.a. (10%)Compound\Discount Factor er (T – t) = 1.01Standard deviation S&P σ = 0.12Standard deviation S&P σ = 0.12

Put Premium P0 = 2.97 (index units)Strike Price K = 280Put delta ∆p = -0.38888(Call delta) (∆c = 1 + ∆p = 0.6112)

Futures Price (t=0) F0 = S0 er(T – t ) = 282.814Price of T-Bill B = Me-rT = 99.0


Hedge Positions

Number of units of the index held in stocks = V0 /S0 = 2,000 index units

Stock-Put Insurance

N0 = V0 / (S0 + P0) = 1979 index units

Stock-Futures Insurance

Nf = [(1979) (0.6112) - 2,000] (0.99/500) = - 1.56 (short futures)

Stock+T-Bill Insurance

No. stocks = N0 ∆c = 1979 (0.612) = 1,209.6 (index units)NB = 2,235.3 (T-bills)


1) Stock+Put Portfolio

Gain on Stocks = N0.dS = 1979 ( -1) = -1,979Gain on Puts = N0 dP = 1979 ( 0.388) = 790.3Net Gain = -1,209.6

2) Stock + Futures: Dynamic Replicatin

Gain on Stocks = Ns,o dS = 2000 (-1) = -2,000Gain on Futures = Nf.dF.zf = (-1.56) (-1.01) 500 = +790.3

Net Gain = -1,209.6


3) Stock + T-Bill: Dynamic Replication

Gain on Stocks = Ns dS = 1209.6 (-1) = -1,209.6Gain on T-Bills = 0

(No change in T-bill price)

Net Gain = -1,209.6

The loss on the replication portfolios is very close to thaton the stock-put portfolio (over the infinitesimally small time period).

Note: We are only “delta replicating” and hence, if there are large changes in S or changes in σσσσ, then our calculations will be inaccurate

When there are large market falls, liquidity may “dry up” and it may not be possible to trade quickly enough in ‘stocks+futures’ at quoted prices (or at any price ! e.g. 1987 crash).


Topic 3b

Option’s Implied VolatilityOption’s Implied Volatility

�Option’s Implied Volatility

�VIX

�Volatility Smiles

�Option’s Implied Volatility

�VIX

�Volatility Smiles

Topics

Readings

Books

� Hull(2009) chapter 18

� VIX

� http://www.cboe.com/micro/vix/vixwhite.pdf

Journal Articles

� Bakshi, Cao and Chen (1997) “Empirical Performance of Alternative

Option Pricing Models”, Journal of Finance, 52, 2003-2049.

Options Implied Volatility

Itō’s Lemma: The Lognormal Property

� If the stock price If the stock price If the stock price If the stock price SSSS follows a GBM (like in the BS model), follows a GBM (like in the BS model), follows a GBM (like in the BS model), follows a GBM (like in the BS model),

then then then then lnlnlnln((((SSSSTTTT////SSSS0000) is normally distributed.) is normally distributed.) is normally distributed.) is normally distributed.

,2

)/ln(lnln 22

0

−≈=− TTSSSS TTT σσµφ

Estimating Volatility

� The volatility is the standard deviation of the

continuously compounded rate of return in 1 year

� The standard deviation of the return in time ∆t

is

� Estimating Volatility: Historical & Implied Estimating Volatility: Historical & Implied Estimating Volatility: Historical & Implied Estimating Volatility: Historical & Implied –––– How?How?How?How?

t∆σ

Estimating Volatility from Historical Data

� Take observations S0, S1, . . . , Sn at intervals of t years

(e.g. t = 1/12 for monthly)

� Calculate the continuously compounded return in each

interval as:

� Calculate the standard deviation, ssss , of the ui´s)/ln( 1−= iii SSu

i

� The variable ssss is therefore an estimate for

� So:

∑=

−−

=n

ii uu

ns

1

2)(1

1

t∆σ

τσ /ˆ s=

Price Relative Daily Return

Date Close St/St-1 ln(St/St-1)

03/11/2008 4443.3

04/11/2008 4639.5 1.0442 0.0432

05/11/2008 4530.7 0.9765 -0.0237

06/11/2008 4272.4 0.9430 -0.0587

07/11/2008 4365 1.0217 0.0214

10/11/2008 4403.9 1.0089 0.0089

11/11/2008 4246.7 0.9643 -0.0363

12/11/2008 4182 0.9848 -0.0154

13/11/2008 4169.2 0.9969 -0.0031

14/11/2008 4233 1.0153 0.0152

� For volatility estimation (usually) we assume that there are 252 trading days within one year

mean -0.13%

Estimating Volatility from Historical Data

14/11/2008 4233 1.0153 0.0152

17/11/2008 4132.2 0.9762 -0.0241

18/11/2008 4208.5 1.0185 0.0183

19/11/2008 4005.7 0.9518 -0.0494

20/11/2008 3875 0.9674 -0.0332

21/11/2008 3781 0.9757 -0.0246

24/11/2008 4153 1.0984 0.0938

25/11/2008 4171.3 1.0044 0.0044

26/11/2008 4152.7 0.9955 -0.0045

27/11/2008 4226.1 1.0177 0.0175

28/11/2008 4288 1.0146 0.0145

01/12/2008 4065.5 0.9481 -0.0533

02/12/2008 4122.9 1.0141 0.0140

03/12/2008 4170 1.0114 0.0114

04/12/2008 4163.6 0.9985 -0.0015

05/12/2008 4049.4 0.9726 -0.0278

08/12/2008 4300.1 1.0619 0.0601

7

� Back or forward looking volatility measure?

mean -0.13%

stdev (s) 3.5%

ττττ 1/252

σ(yearly) s / sqrt(ττττ) = 55.56%

BS Parameters BS Parameters BS Parameters BS Parameters

S: S: underlying index value

X: X: options strike price

T: T: time to maturity

Unobserved Parameters:Unobserved Parameters:

Black and Scholes

σ:σ: volatility

Observed Parameters:Observed Parameters:

Implied Volatility

r: r: risk-free rate

q: q: dividend yield

• Traders and brokers often quote implied volatilities

rather than dollar prices

How to estimate it?

� The implied volatility of an option is the volatility

for which the Black-Scholes price equalsequalsequalsequals (=) (=) (=) (=) the

market price

� There is a one-to-one correspondence between

prices and implied volatilities (BS price is

Implied Volatility

prices and implied volatilities (BS price is

monotonically increasing in volatility)

� Implied volatilities are forward looking and price

traded options with more accuracy

� Example: If IV of put option is 22%, this means

that pbs = pmkt when a volatility of 22% is used in

the Black-Scholes model.9

� Assume c is the call price, f is an option pricing

model/function that depends on volatility σ and other

inputs:

� Then implied volatility can be extracted by inverting the

formula:

( )σ,,,, TrKSfc =

( )mrkcTrKSf ,,,,1−=σ

Implied Volatility

formula:

where cmrk is the market price for a call option.

� The BS does not have a closed-form solution for its inverse

function, so to extract the implied volatility we use root-

finding techniques (iterative algorithms) like NewtonNewtonNewtonNewton----

RaphsonRaphsonRaphsonRaphson methodmethodmethodmethod

10

( )mrkcTrKSf ,,,,1−=σ

( ) 0,,,, =− mrkcTrKSf σ

� In 1993, CBOE published the first implied

volatility index and several more indices later on.

� VIXVIXVIXVIX: 1-month IV from 30-day options on S&P

� VXNVXNVXNVXN: 3-month IV from 90-day options on S&P

� VXDVXDVXDVXD: volatility index of CBOE DJIA

Volatility Index -VIX

� VXDVXDVXDVXD: volatility index of CBOE DJIA

� VXNVXNVXNVXN: volatility index of NASDAQ100

� MVXMVXMVXMVX: Montreal exchange vol index based on

iShares of the CDN S&P/TSX 60 Fund

� VDAXVDAXVDAXVDAX: German Futures and options exchange vol

index based on DAX30 index options

� Others: VXI, VX6, VSMI, VAEX, VBEL, VCACVXI, VX6, VSMI, VAEX, VBEL, VCACVXI, VX6, VSMI, VAEX, VBEL, VCACVXI, VX6, VSMI, VAEX, VBEL, VCAC11

Volatility Smile

What is a Volatility Smile?

� It is the relationship between implied

volatility and strike price for options with a

certain maturity

Volatility Smile

certain maturity

� The volatility smile for European call

options should be exactly the same as

that for European put options

13

� Put-call parity p +S0e-qT = c +Ke–r T holds for market

prices (pmkt and cmkt) and for Black-Scholes prices

(pbs and cbs)

� It follows that the pricing errors for puts and calls

are the same: p −p =c −c

Volatility Smile

are the same: pmkt−pbs=cmkt−cbs

� When pbs=pmkt, it must be true that cbs=cmkt

� It follows that the implied volatility calculated from a

European call option should be the same as that

calculated from a European put option when both

have the same strike price and maturity14

� In addition to calculating a volatility

smile, traders also calculate a

volatility term structure

� This shows the variation of implied

Volatility Term Structure

� This shows the variation of implied

volatility with the time to maturity of

the option for a particular strike

15

IV Surface

16

IV Surface

17

IV Surface

18

Also known as:

Volatility smirk

Volatility skew

Implied Volatility Surface (Smile) from Empirical

Studies (Equity/Index)

Volatility Smile

19Bakshi, Cao and Chen (1997) “Empirical Performance of Alternative Option Pricing Models ”, Journal of Finance, 52, 2003-2049.

Implied vs Lognormal Distribution

Volatility Smile

20

� In practice, the left tail is heavier and the right tail is less heavy than the lognormal distribution

� What are the possible causes of the Volatility Smile anomaly?� Enormous number of empirical and theoretical papers

to answer this …

Volatility Smile

to answer this …

21

Possible Causes of Volatility Smile� Asset price exhibits jumps rather than continuous changes (e.g. S&P 500 index)

Date Open High Low Close Volume Adj Close Return

04/01/2000 1455.22 1455.22 1397.43 1399.42 1.01E+09 1399.42 -3.91%

18/02/2000 1388.26 1388.59 1345.32 1346.09 1.04E+09 1346.09 -3.08%

20/12/2000 1305.6 1305.6 1261.16 1264.74 1.42E+09 1264.74 -3.18% -ve Price jumps

Volatility Smile

22

20/12/2000 1305.6 1305.6 1261.16 1264.74 1.42E+09 1264.74 -3.18%

12/03/2001 1233.42 1233.42 1176.78 1180.16 1.23E+09 1180.16 -4.41%

03/04/2001 1145.87 1145.87 1100.19 1106.46 1.39E+09 1106.46 -3.50%

10/09/2001 1085.78 1096.94 1073.15 1092.54 1.28E+09 1092.54 0.62%

17/09/2001 1092.54 1092.54 1037.46 1038.77 2.33E+09 1038.77 -5.05%

16/03/00 1392.15 1458.47 1392.15 1458.47 1.48E+09 1458.47 4.65%

15/10/02 841.44 881.27 841.44 881.27 1.96E+09 881.27 4.62%

05/04/01 1103.25 1151.47 1103.25 1151.44 1.37E+09 1151.44 4.28%

14/08/02 884.21 920.21 876.2 919.62 1.53E+09 919.62 3.93%

01/10/02 815.28 847.93 812.82 847.91 1.78E+09 847.91 3.92%

11/10/02 803.92 843.27 803.92 835.32 1.85E+09 835.32 3.83%

24/09/01 965.8 1008.44 965.8 1003.45 1.75E+09 1003.45 3.82%

Trading was suspended

-ve Price jumps

+ve price jumps

� Asset price exhibits jumps rather than continuous changes

� Volatility for asset price is stochastic� In the case of equities volatility is negatively related to stock prices because of the impact of leverage. This is consistent with the skew (i.e., volatility smile) that is observed in practice

Volatility Smile

Possible Causes of Volatility Smile

the skew (i.e., volatility smile) that is observed in practice

23

Volatility Smile

24

� Asset price exhibits jumps rather than continuous changes

� Volatility for asset price is stochastic� In the case of equities volatility is negatively related to stock prices because of the impact of leverage. This is consistent with the skew that is observed in practice

Volatility Smile

Possible Causes of Volatility Smile

the skew that is observed in practice

� Combinations of jumps and stochastic volatility

25

Alternatives to Geometric Brownian Motion

� Accounting for negative skewness and excess kurtosis by generalizing the GBM� Constant Elasticity of Variance

� Mixed Jump diffusion

� Stochastic Volatility

Volatility Smile

� Stochastic Volatility

� Stochastic Volatility and Jump

� Other models (less complex → ad-hoc)� The Deterministic Volatility Functions (i.e., practitioners Black and Scholes)

26

(See chapter 26 (sections 26.1, 26.2, 26.3) of Hull for these alternative specifications to Black-Scholes)

Topic # 4: Modelling stockprices, Interest rate

derivatives

Financial Risk Management 2010-11

February 7, 2011

FRM c Dennis PHILIP 2011

1 Modelling stock prices 2

1 Modelling stock prices

� Modelling the evolution of stock prices isabout introducing a process that will explainthe random movements in prices. This ran-domness is explained in the E¢ cient MarketHypothesis (EMH) that can be summarizedin two assumptions:

1. Past history is re�ected in present price

2. Markets respond immediately to anynew information about the asset

� So we need to model arrival of new infor-mation that a¤ects price (or much more re-turns).

� If asset price is S. Suppose price changesto S + dS in a small time interval (say dt).Then we can decompose returns dS

Sinto de-

terministic/anticipated part � and a randompart where prices changed due to some ex-ternal unanticipated news.

dS

S= �dt+ �dW



� The randomness in the random part is ex-plained by a Brownian Motion process andscaled by the volatility � of returns.

� We can introduce time subscripts and re-arrange to get

dSt = �Stdt+ �StdWt

This process is called the Geometric Brown-ian Motion.

� Why have we used Brownian Motion processto explain randomness?

� In practice, we see that stock prices be-have, atleast for long stretches of time,like random walks with small and fre-quent jumps

� In statistics, random walk, being thesimplest form, have limiting distribu-tions and since BM is a limit of therandom walk, we can easily understandthe statistics of BM (use of CLT)



Next we see, what is thisW (and in turnwhat is dW)?

� Brownian motion is a continuous time (rescaled)random walk.

� Consider the iid sequence "1; "2; ::: with mean� and variance �2: Consider the rescaled ran-dom walk model

Wn(t) =1pn

X1�j�nt

"j

� The interval length t is divided into nt equalsubintervals of length 1=n and the displace-ments / jumps "j; j = 1; 2; :::; nt in nt stepsare mutually independent random variables.

� Then for large n; according to Central LimitTheorem: W (t) � N (�t; �2t) :



� Special cases: Standard Brownian Motionarises when we have � = 0; and � = 1.

� W is a Standard Brownian Motion if

1. W (0) = 0

2. W has stationary (for 0 � s � t;Wt �Ws and Wt�s have the same distrib-ution. That is, Wt � Ws

D= Wt�s �

N (0; t� s))3. W has independent increments (for s �t;Wt �Ws is independent of past his-tory of W until time s)

4. Wt � N (0; t)



� For a BrownianMotion only the present valueof the variable is relevant for predicting thefuture [also called Markov property]. There-fore BM is a markov process.

� It does not matter how much you zoom in,it just looks the same. That is, the random-ness does not smooth out when we zoom in.

� BM�ts the characteristics of the share price.Imagine a heavy particle (share price) thatis jarred around by lighter particles (trades).Trades a¤ect the price movement.

what is this dW?

� Consider a small increment in W

W (t+�t) =W (t) + "(t+�t)

where "(t+�t) � iidN(0;�t) [Std BM].



� Taking limit as � ! 0; the change in W (t)is

dWt = limdt!0

W (t+ dt)�W (t)

= limdt!0

"(t+ dt)

� iidN(0; dt)

So in the di¤erential form, we can write theStandard Brownian motion process as

dWt = etpdt where et � N(0; 1)



Stochastic processes used in Finance

Arithmetic Brownian Motion for a shareprice

� A stock price does not generally have a meanzero and atleast would grow on average withthe rate of in�ation. Therefore we can write

dSt = � (St; t) dt+ �(St; t)dWt

= drift term+ diffusion term

= E(dS) + Stddev(dS)

� When the drift function � (St; t) = � anddi¤usion function �(St; t) = �; both con-stants, we have the Arithmetic BM.

dSt = �dt+ �dWt

= �dt+ �etpdt

� In the case of ABM, S may be positive ornegative. Since prices cannot be negative,we generally use the Geometric BM for assetprices and made the drift and volatility asfunctions of the stock price.



Geometric Brownian Motion


� If S starts at a positive value, then it will re-main positive. The solution of the SDE St isan exponential function which is always pos-itive. Also, note that S will be lognormallydistributed.

� GBM is related to ABM according to

dStSt

= �dt+ �dWt

where � is the instantaneous share price volatil-ity, and � is the expected rate of return

� The Hull andWhite (1987) Model uses GBM.



Ornstein-Uhlenbeck (OU) Process

� The Arithmetic Ornstein-Uhlenbeck processis given by

dSt = � (�� St) dt+ �dWt

where � is the long run mean and � (� > 0)is the rate of mean reversion. The drift termis the mean reversion component, in thatthe di¤erence between the long run meanand the current price decides the upward ordownward movement of the stock price to-wards the long run mean �: Over time, theprice process drifts towards its mean � andthe speed of mean reversion is determinedby �:

� This is an important process to model in-terest rates that show mean reversion whereprices are pulled back to some long-run av-erage level over time.

� The Vasicek Model uses this kind of process.



� A special case is when the mean is zero.Then we can write the OU process as

dSt = ��Stdt+ �dWt

� In the OU process, the stock price can benegative. Therefore we can introduce theGeometric OU process

� The Geometric OU process is given by

dSt = � (�� St)Stdt+ �StdWt

where the asset prices St would always bepositive.

� So we can model asset prices using the Geo-metric OU process and their log returns willthen follow an Arithmetic OU process.

dSt = � (�� St)Stdt+ �StdWt

dStSt

= � (�� St) dt+ �dWt



Square Root Process

A square root process satis�es the SDE

dSt = �Stdt+ �pStdWt

� This type of process generates positive pricesand used for asset prices whose volatilitydoes not increase too much when St increases.

Cox-Ingersoll-Ross (CIR) process

� The CIR combines mean reversion and squareroot process and satis�es the SDE

dSt = � (�� St) dt+ �pStdWt

� This process was introduced in the Hull andWhite (1988), and Heston (1993) stochas-tic volatility models. This class of mod-els generated strictly non-negative volatilityand accounted for the clustering e¤ect andmean reversion observed in volatility.



� Also used to model short rates features pos-itive interest rates, mean reversion, and ab-solute variance of interest rates increases withinterest rates itself.

Solving the Stochastic Di¤erential Equa-tions

� Consider the GBM


In the integral form

TZ0

dSt =

TZ0

�Stdt+

TZ0

�StdWt

ST = S0 +

TZ0

�Stdt+

TZ0

�StdWt

= reimann integ + Ito integ

� So we have to solve the intergrals to get aclosed form solutions to this SDE.



� We use Ito-lemma to solve this problem.

� Not all SDE�s have closed form solutions.When there are no solutions, we have todo numerical approximations for these in-tegrals.

Examples:

� Geometric Brownian Motion


has the solution

St = S0e(�� 1

2�2)t+�Wt

� Ornstein-Uhlenbeck (OU) Process

dSt = � (�� St) dt+ �dWt

has the solution

St = S0e��t+�

�1� e��t

�+�

tZ0

e��(t�s)dWs



� Consider the following process

dSt = �Stdt+ �dWt

has the solution

St = S0e�t + �

tZ0

e�(t�s)dWs

Simulating Geometric Brownian Motion

� We can write St = S0e(�� 1

2�2)t+�Wt in dis-

crete time intervals and substituting for Wt

asSt = St�1e

(�� 12�2)�t+�et

p�t

where et � N(0; 1)

� So we randomly draw et and �nd the valueof St


2 Interest Rate Derivatives 16

2 Interest Rate Derivatives

� The payo¤ of interest rate derivatives woulddepend on the future level of interest rates.

� The main challenge in valuing these deriv-atives are that interest rates are used bothfor discounting and for de�ning payo¤s.

� For valuation, we will need a model to de-scribe the behavior of the entire yield curve.

Black�sModel to price European Options

� Consider a call option on a variable whosevalue is V:

� To calculate expected payo¤, the model ass-sumes:

1. VT has lognormal distribution with

V ar(lnVT ) = �2T

2. E(VT ) = F0



� The payo¤ is max(VT �K; 0) at time T .

� We discount the expected payo¤ at time Tusing the risk-free rate given by P (0; T )

We will use the key result that you knowfrom Derivatives:

If V is lognormally distributed and stan-dard deviation on ln(V ) is s, then

E[max ( V�K; 0)] = E[ V ] N(d1)�KN(d2)

where

d1 =ln(E[ V ]=K) + s2=2

s

d2 =ln(E[ V ]=K)� s2=2

s

� Therefore, value of the call option is givenby

c = P (0; T ) [F0N(d1)�KN(d2)]



where

d1 =ln(F0=K) + �

2T=2

�pT

and

d2 =ln(F0=K)� �2T=2

�pT

= d1 � �pT

where

�F is forward price of V for a contractwith maturity T

�F0 is value of F at time zero

�K is strike of the option

� � is volatility of forward contract

� Similarly, for a put option

p = P (0; T ) [KN(�d2)� F0N(�d1)]



European Bond Options

� Bond option is an option to buy or sell aparticular bond by a certain date for a par-ticular price.

� Callable bonds and Puttable bonds are ex-amples of embedded bond options.

� The payo¤ is given by max(BT � K; 0) fora call option.

� To price an European Bond Option:

�we assume bond price at maturity ofoption is lognormal

�we de�ne � such that standard devia-tion of ln(BT ) = �

pT

�F0 can be calculated as

F0 =B0 � IP (0; T )

where B0 is bond (dirty) price at timezero and I is the present value of couponsthat will be paid during the life of op-tion



�Then using Black�s model we price of abond option

Interest Rate Caps and Floors

� An interest rate Cap provides insurance againstthe rate of interest on a �oating-rate noterising above a certain level (called Cap rate).

� Example:Principal amount = $10 million

Tenor = 3 months (payments made everyquarter)

Life of Cap = 5 years

Cap rate = 8%

� If the �oating-rate exceeds 8 %, then youget cash of the di¤erence.

� Suppose at a reset date, 3-month LIBOR is9%, the �oating-rate note would have to pay

0:25� 0:09� $10million = $225; 000



and with the Cap rate at 8%, the paymentwould be

0:25� 0:08� $10million = $200; 000

� Therefore the Cap provides a payo¤of $25,000to the holder.

� Consider a Cap with total life of Tn; a Prin-cipal of L, Cap rate of RK based on a refer-ence rate (say, on LIBOR) with a ��monthmaturity denoted by R(t) at date t.

� The contract follows the schedule:�� t � T0 � T1 � T2 � � � � � � Tn� � � � C1 � C2 � � � � � � Cn

�� T0 is the starting date. For all j = 1; :::; n,we assume a constant tenor Tj � Tj�1 = �

� On each date Tj; the Cap holder receives acash �ow of Cj

Cj = L� � �max [R(Tj�1)�RK ; 0]

� The Cap is a portfolio of n such options andeach call option is known as the caplets.



� Lets now consider a Floor with the samecharacteristics. On each date Tj; the Floorholder receives a cash �ow of Fj

Fj = L� � �max [RK �R(Tj�1); 0]

� The Floor is a portfolio of n such optionsand each put option is known as the �oor-lets.



� Interest rate Caps can be regarded as a port-folio of European put options on zero-couponbonds.

� Put-Call parity relation:Consider a Cap and Floor with same strikeprice RK . Consider a Swap to receive �oat-ing and pay a �xed rate of RK , with no ex-change payments on the �rst reset date. ThePut-Call parity states:

Cap price = Floor price + value of Swap



Collar

� A Collar is designed to guarantee that theinterest rate on the underlying �oating-ratenote always lie between two levels.

� Collar = long position in Cap + short posi-tion in Floor

� It is usually constructed so that the price ofCap is equal to price of the �oor. Then thecost of entering into a Collar is zero.

Valuation of Caps and Floors

� If the rate R(Tj) is assumed to be lognormalwith volatility �j, the value of the caplettoday (t) for maturity Tj is given by

Caplett = L��P (t; Tj)�FTj�1;Tj �N(d1)�RKN(d2)

�where

d1 =ln(FTj�1;Tj=RK) + �

2j (Tj�1 � t) =2

�p(Tj�1 � t)



andd2 = d1 � �j

q(Tj�1 � t)

where FTj�1;Tj is the forward rate underlyingthe Caplet from Tj�1 to Tj:

� Similarly,Floorlett = L��P (t; Tj)

�RKN(�d2)� FTj�1;Tj �N(�d1)

�



Lecture 5

Value at RiskValue at Risk

Readings: CN(2001) chapters 22,23; Hull_RM chp 8

1

�Value at Risk (VaR)

�Forecasting volatility

�Back-testing

�Value at Risk (VaR)

�Forecasting volatility

�Back-testing

Topics

�Risk Grades

�VaR: Mapping cash flows

�Risk Grades

�VaR: Mapping cash flows

2

Value at Risk

� Example:

If at 4.15pm the reported daily VaR is $10m (calculated at 5%

tolerance level) then:

I expect to lose more than $10m only 1 day in every 20 days I expect to lose more than $10m only 1 day in every 20 days I expect to lose more than $10m only 1 day in every 20 days I expect to lose more than $10m only 1 day in every 20 days

((((ieieieie. 5% of the time). 5% of the time). 5% of the time). 5% of the time)

� The VaR of $10m assumes my portfolio of assets is fixed

� Exactly how much will I lose on any one day?

� Unknown !!!

3

Value at Risk

� Statement (how bad can things get?):

“We are x% certain that we will not loose more than V dollars in

the next N days”

� V dollars = f(x%, N days)

� Suppose asset returns is niid, then risk can be measured by

variance/S.D.

� From Normal Distribution critical values table, we can work out

the VaR.

� Example: For 90% certainty, we can expect actual return to be

between the range { 1 .6 5 , 1 .6 5 }µ σ µ σ− +

4

Value at Risk � Normal Distribution (N(0,σ))

5% of the area

Mean = 0 Probability

5% of the area

Mean = 0

Only 5% of the time will the actual % return R be below:

“ R = µ - 1.65 σ1” where µ = Mean (Daily) Return.

If we assume µ=0, VaR = $V (1.65 σσσσ1)

-1.65σ 0.0 +1.65σ

5% of the area

5% of the area

-1.65σ 0.0 +1.65σ

5% of the area

5% of the area

Return

5

VaR for single assetExample:

Mean return = 0 %. Let σ1 = 0.02 (per day)

Only 5% of the time will the loss be more than 3.3% (=1.65 x 2%)

VaR of a single asset (Initial Position V0 =$200m in equities)

VaR = V0 (1.65 σσσσ1111 ) = 200 ( 0.033) = $6.6mVaR = V0 (1.65 1111 ) = 200 ( 0.033) = $6.6m

That is “(dollar) VaR is 3.3% of $200m” = $6.6m

VaR is reported as a positive number (even though it’s a loss)

Are Daily Returns Normally Distributed? - NO • Fat tails (excess kurtosis), peak is higher and narrower, negative skewness, small (positive) autocorrelations, squared returns have strong autocorrelation, ARCH.• But niid is a (reasonable) approx for portfolios of equities, long term bonds, spot FX , and futures (but not for short term interest rates or options)6

VaR for portfolio of assets

7


8


9


10


� Summary: Variance Variance Variance Variance –––– Covariance methodCovariance methodCovariance methodCovariance method

If VVVVp is the market value of your portfolio of nnnn assets and wwwwiiii is

the proportionate weight in each asset iiii then

where

[ ]1/2'p pVaR V zCz=

where

( ) ( ) ( )1 1 2 21.65 , 1.65 , , 1.65n nz w w wσ σ σ= …

12 1

21

1

1

1

n

n

C

ρ ρρ

ρ

=

…

⋱

⋮

11

ForecastingForecasting

12

Forecasting σσσσ� Simple Moving Average ( Assume Mean Return = 0 )

σ2t+1|t = (1/n) Σi R2t-i

� Exponentially Weighted Moving Average EWMA

σ2t+1|t = Σi wi R

2t-i wi = (1-λ) λi

� It can be shown that this may be re-written:

σ2t+1|t = λ σ2

t| t-1 + (1- λ) Rt2

� Longer Horizons: -rule - for iid returns.σΤ = σ

T

T

13

Forecasting σσσσ� Exponentially Weighted Moving Average (EWMA)

σ2t+1|t = λ σ2

t| t-1 + (1- λ) Rt2

� How to estimate λ?1. Use GARCH models to estimate λ2. Minimize forecast error Σ (Rt+12 – σ2 t+1|t) where the sum is

over all assets, and say 100 days

3. λ = 0.94 as by JPMorgan

� Suppose λ = 0.94 then weights decline as 0.94, 0.88, 0.83,….

and past observations are given less weight than current

forecast of variance.

14

Back-testing

� In back-testing, we compare our (changing) daily

forecast of VaR with actual profit or loss over some

historic period.

� Example: For a portfolio of assets,• forecast all the individual VaR = V1.65 σ , • forecast all the individual VaRi = Vi1.65 σt+1|t ,

• calculate portfolio VaR for each day:

VaRp = [Z C Z’]1/2

• then see if actual portfolio losses exceed this only 5% of the time (over some historic period, e.g. 100 days).

15

Back-testingDaily $m profit/loss

= forecast= actual

Days

Only 6 violations out of 100 = just ‘OK’16

VaR and Capital Adequacy-BasleBasle uses a more ‘conservative’ measure of VaR than J. P. Morgan

Calc VaR for worst 1% of losses over 10 days

Use at least 1-year of daily data to estimate σt+1|t

VaRi = 2.33 σ( 2.33 = 1% left tail critical value, σ = daily vol )

10

Internal Models ( 2.33 = 1% left tail critical value, σ = daily vol )

Capital Charge KC

KC = Max ( Avg. of previous 60-days VaR x M, previous day’s VaR)

M = multiplier (min = 3)

• KC set equal to max. forecast loss over 20 day horizon = pre-announced $VaR• If losses exceed VaR, more than 1 day in 20, then impose a penalty.

Pre-commitment approach

Internal Models approach

17

VaR and Coherent Risk Measures� Risk measures that satisfy all the following 4 conditions are called as a

Coherent Risk Measure.

� Monotonicity:

(higher the riskiness of the portfolio, higher should be risk capital)

� Translation invariance:

( ) ( )1 2 1 2X X R X R X≤ ⇒ ≤

( ) ( )R X k R X k k+ = − ∀ ∈ℝ� Translation invariance:

(if cash k is added to portfolio, risk should go down by k)

� Homogeneity:

(if you change portfolio by a factor of λ, risk is proportionally increased)

� Subadditivity:

(diversification leads to less risk)18

( ) ( )R X k R X k k+ = − ∀ ∈ℝ

( ) ( ) 0R X R Xλ λ λ= ∀ ≥

( ) ( ) ( )R X Y R X R Y+ ≤ +

VaR and Coherent Risk Measures

� VaR violates the subadditivity condition and therefore not coherent. VaR cannot capture the benefits of diversification.

� VaR can actually show negative diversification benefit!

� VaR only captures the frequency of default but not the size of default. Even if the largest loss is doubled, the VaR figure could remain the same.

� Other measures such as Expected Shortfall are coherent measures.

19

Risk Grades

20

Risk Grades

� RG helps to calculate changing forecasts of risks (volatilities)

� RG quantifies volatility/risk (similar to variance, std. deviation,

beta, etc)

� RG can range from 0 to over 1000, where 100100100100 corresponds to

the average risk of a diversified market-cap weighted index of the average risk of a diversified market-cap weighted index of

global equities.

� So if two portfolio’s have RG1 = 100 and RG2 = 400, portfolio 2

is four times riskier than portfolio 1

� RG scales all assets to a common scale and so it is able to

compare risk across all asset classes.

21

Risk Grades

RG of a single asset

σi is the DAILY standard deviation

σbase is fixed at 20% per annum (= 5 yr. av. for international portfolio of

252 252100 100

0.20i iRGi base

σ σσ × ×= =

σbase is fixed at 20% per annum (= 5 yr. av. for international portfolio of stocks)

Formula looks complex but RG is just a “scaled” daily standard deviation

e.g. If RG = 100% then asset has 20% p.a. risk

RG of a portfolio of assets

∑ ∑ ∑+= jRGiRGjwiwiRGiwPRG ρ22222

Risk Grades

� Risk Grades in 2009Risk Grades in 2009Risk Grades in 2009Risk Grades in 2009

www.riskgrades.com

23

Risk GradesRisk GradesRisk GradesRisk Grades

in 2009 ofin 2009 ofin 2009 ofin 2009 of

indicesindicesindicesindices

heating up heating up heating up heating up

and coolingand coolingand coolingand cooling

offoffoffoff

Risk Grades

offoffoffoff

24

VaRMapping

(VaR for different assets)(VaR for different assets)

25

VaR for different assetsPROBLEMS

STOCKS : Too many covariances [= n(n-1)/2 ]

FOREIGN ASSETS : Need VaR in “home currency”

BONDS: Many different coupons paid at different times BONDS: Many different coupons paid at different times

DERIVATIVES: Options payoffs can be highly non-linear (ie. NOT normally distributed)

SOLUTIONS = “Mapping”

(RiskMetricsTM produce volatility & correlations for various assets across 35 countries and useful for “Mapping”)26

VaR for different assets

� STOCKS

Within each country use “single index model” SIM

� FOREIGN ASSETS

Treat one asset in foreign country as = “local currency risk”+ spot FX risk (like 2-assets, with equal weight)

� BONDS

Consider each bond as a series of “zeros”

� OTHER ASSETS

Forward-FX, FRA’s, Swaps: decompose into ‘constituent parts’

� DERIVATIVES(non-linear)27

Mapping Stocks

� Problem : Too many covariances to estimate

� Soln. All n(n-1)/2 covariances “collapse or mapped” into σmand the asset betas (“n” of them)

Consider ‘p’ = portfolio of stocks held in one country with (Rm , σm) (for e.g. S&P500 in US)

and the asset betas (“n” of them)

� Single Index Model:Ri = ai + bi Rm + εi

Rk = ak + bk Rm + εk

� assume Eεi εk = 0 and cov (Rm , ε ) = 0

� All the systematic variation in Ri AND Rk is due to Rm28

1) In a portfolio idiosyncratic risk εi is diversified away = 0

2) Then each return-i depends only on the market return (and beta). Hence ALL variances and covariances also only depend on these 2 factors. It can be shown that

σp = bp σm

(i.e. Calculation of portfolio beta requires, only n-beta’s and σ )

Mapping Stocks

(i.e. Calculation of portfolio beta requires, only n-beta’s and σm )

3) Also, ρ = 1 because (in a well diversified portfolio) each return moves only with Rm

4) We end up with VaRp = VP 1.65 ( bP σm )or equivalently VaRp = (Z C Z’ )1/2

where Z = [ VaR1, VaR2 …. } C is the unit matrix 29

Mapping Foreign Assets

(Mapping foreign stocks into domestic currency VaR)(Mapping foreign stocks into domestic currency VaR)

30

Mapping Foreign assets

Example:Example:Example:Example:

US resident holds a diversified portfolio of German stocks

equivalent to German stocks + Euro-USD, FX risk

� Use SIM to obtain stdv of foreign (German) portfolio returns,

σσσσGGGGσσσσGGGG

� Then treat ‘foreign portfolio’ as (two) equally weighted

assets:

= $V in German asset + $V foreign currency position

� Then use standard VaR formula for 2-assets

31

Mapping Foreign assets� US based investor: with €100m in a German stock portfolio

σσσσG = ββββP σσσσDAX

Sources of risk:a) Stdv of the German portfolio (‘local currency’ portfolio) b) Stdv of €/$ exchange rate ( σσσσFX ) c) one covariance/correlation coefficient ρ (between DAX and FX rate)

e.g. Suppose when German stock market falls then the € also falls -‘double whammy’ for the US investor, from this positive correlation, soforeign assets are very risky (in terms of their USD ‘payoff’)

� Let : S = 1.2 $/ €

� Dollar initial value Vo$ = 100m x 1.2 = $120m

� Linear dVP = V0$ (RG + RFX)

above implies wi = Vi / V0$ = 1 32

Mapping Foreign assets

� Dollar-VaRp = Vo$ 1.65 σp

� σp =

� No ‘relative weights’ appear in the formula

� Matrix Representation: Dollar VaR

( ) 21

222FXGFXG σρσσσ ++

� Matrix Representation: Dollar VaR

� Let Z = [ V0,$ 1.65 σσσσG , V0,$ 1.65 σσσσFX ]

= [ VaR1 , VaR2 ]

� V0,$ = $120m for both entries in the Z-vector (i.e. equal amounts)

Then VaRp = (Z C Z’ )1/2

33

Mapping Bonds

(Mapping coupon paying bonds)(Mapping coupon paying bonds)

34

Mapping Coupon paying BondsExample:Coupons paid at t=5 and t=7Treat each coupon as a separate zero coupon bond

77

55 )1(

100

)1(

100

yyP

++

+=

P V V= +

P is linear in the ‘price’ of the zeros, V5 and V7

We require two variances of “prices” V5 and V7 andcovariance between these prices.

Note: σσσσ5(dV5 / V5) = D σσσσ(dy5) but Risk Metrics provides the price volatilities, σ5(dV5 / V5)

5 7P V V= +

35

Mapping Coupon paying Bonds

Treat each coupon as a zeroCalculate:price of zero, e.g. V5 = 100 / (1+y5)5

VaR5 = V5 (1.65 σσσσ5)

VaR7 = V7 (1.65 σσσσ7)

VaR (both coupon payments):VaR (both coupon payments):

VaRp = (Z C Z’ )1/2

=

r = correlation: bond prices at t=5 and t=7(approx 0.95 - 0.99 )

[ ] /VaR VaR VaR VaR52

72

5 71 22+ + ρ

36

Mapping FRA

37

Mapping FRA

� To calculate VaR for a FRA, we break down cash flows into equivalent synthetic FRA and use spot rates only (since we do not know the forward volatilities)

� Example: Consider an FRA on a notional of $1m that involves lending $1m in 6 months time for a future of 6 months.

Receipt of $1m + Interest

� Let y6 = 6.39%, y12 = 6.96% and there are 182 days in the first leg and 183 days in the second leg (day count: actual/365).

� The implied f6,12 = 7.294% and therefore the 12 month investment will give $1,036,572 return (with round off error).

Lend $1m

Receipt of $1m + Interest

6m 12m0

38

Mapping FRA

� The original FRA

� Synthetic FRALend $1m

Receipt of $1,036,572

6m 12m0

Receipt of $1,036,572 from 12 month lendingBorrow at 6 month rate

� So at time 0, we borrow $969,121 [=1m / 1+(y6*182/365)] and lend this money at a 12 month rate leading to $1,036,572 [=$969,121*(1+y12)]

Repay 6 month loanof $1m

from 12 month lending

6m 12m0

Lend at 12 month rate

39

Mapping FRA

� Suppose the standard deviation of the prices for 6 month asset is 0.1302% and for 12 month asset is 0.2916%. Suppose ρ = 0.7

� To calulate the VaR for each of these positions:

� VaR6 = $969,121 (1.65) (0.1302%) = $2082

� VaR12 = $969,121 (1.65) (0.2916%) = $4663� VaR12 = $969,121 (1.65) (0.2916%) = $4663

( )2 2 1/ 26 12 6 12[ 2 ]

$3534

VaR VaR VaR VaR VaRρ= + + −=

40

Mapping FX Forwards

41

Mapping FX Forwards

� Consider a US resident who holds a long forward contract to purchase €10million in 1 year.

� What is the VaR for this contract?

� We map Forward into � two spot rates and one spot FX rate.

� Then we calculate VaR from the VaR of each individual mapped asset.asset.

Mapping a forward contract

42

Mapping FX Forwards

43

Mapping FX Forwards

44

Mapping Options

45

Mapping Options

46

Mapping Options

47

Mapping Options

48

Mapping Options

49


Topic 6

Statistical issues in VaRStatistical issues in VaRReadings: CN(2001) chapters 24, Hull_RM chp 8,

Barone-Adesi et al (2000)

RiskMetrics Technical Document (optional)

1

�Value at Risk for options

�Monte Carlo Simulation

�Historical Simulation

�Value at Risk for options

�Monte Carlo Simulation

�Historical Simulation

Topics

�Bootstrapping

�Principal component analysis

�Other related VaR measures� Marginal VaR, Incremental VaR, ES

�Bootstrapping

�Principal component analysis

�Other related VaR measures� Marginal VaR, Incremental VaR, ES

2

MCS -VaR of Call option

� Option premia are non-linear (convex) function of underlying

� Distribution of gains/losses is not normally distributed

� Therefore dangerous to use Var-Cov method

� Assets Held: One call option on stock

Here, Black-Scholes is used to price the option during the Monte Carlo Simulation (MCS).

� Problem

Find the VaR over a 5-day horizon

3

MCS -VaR of Call option � If V is price of the option (call or put) and P is price of

underlying asset in the option contract (stock)

� V will change from minute to minute as P changes. For anequal change in P of +1 or -1 ,the change in call premia areNOT equal: this gives “non-normality” in distribution of thechange in call premium4


� To find VaR over a 5-day horizon:1) Given P0 calculate the option price, V0 = BS(P0, K, T0 …..)

This is fixed throughout the MCS

2) MCS = Simulate the stock price and calculate P5

3) Calculate the new option price, V5= BS(P5, K, T0 – 5/365, …..)

4) Calculate change in option premium

∆V(1) = V5 – V0

5) Repeat steps 2-4, 1000 times and plot a histogram of the change in the call premium. We can then find the 5% lower cut-off point for the change in value of the call (i.e. it’s VaR).

5


10

12

14

16

18

20F

req

uen

cy

5% of area

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10 More

Fre

qu

ency

Change in Call Premium

$-VaR is $5m

6


� Simulate stock prices using

P(t+1) = P(t) exp[ (µ - σ2/2) ∆t + σ (∆t)1/2 εt+1 ]

� To generate 5-day prices from daily prices, you can use the root-T rule

∆t = 5/365 (or 5/252) ( ie. five day)∆t = 5/365 (or 5/252) ( ie. five day)

� Alternatively, we can generate Tt+5 directly. If P0 is today’s known price. Use a ‘do-loop’ over 5 ‘periods’ to get P5 (using 5 - ‘draws’ of εt+1 )

7


� Stock price paths generated

8

Monte Carlo Simulation for Two AssetsMonte Carlo Simulation for Two Assets

(two call options on different underlying assets)(two call options on different underlying assets)

9

Simulating correlated random variables

10


11


12

MCS and VaR: two asset example

13


14


15


16


17


18


19

Comparing VaR forecasts

20

Comparing VaR forecasts

21

Historical SimulationHistorical Simulation

(Historical simulation + bootstrapping)(Historical simulation + bootstrapping)

22

Historical Simulation (HS)Suppose you currently hold $100 in each of 2 assets

Day = 1 2 3 4 5 6 …1000

R1(%) +2 +1 +4 -3 -2 -1 +2

R2(%) +1 +2 0 -1 -5 -6 -5

_____________________________________________________

∆Vp($) +3 +3 +4 -4 -7 -7 -5∆Vp($) +3 +3 +4 -4 -7 -7 -5

Order ∆Vp in ascending order (of 1000 numbers) e.g.

-12, -11, -11 -10, -9, -9, -8, -7, -7, -6 | -5, -4, -4, …. +8, ……. +14

VaR forecast for tomorrow at 1% tail (10th most negative)

= -$6

Above is equivalent to the histogram23

Historical Simulation (HS)

� This is a non parametric method since we do not estimate any variances or covariances or assume normality.

� We merely use the historic returns, so our VaR estimates encapsulate whatever distribution the returns might embody ( e.g. Student’s t) as well as any autocorrelation in individual returns.well as any autocorrelation in individual returns.

� Also, the historic data “contain” the correlations between the returns on the different assets, their ‘own volatility’ and their own autocorrelation over time

� It does rely on ‘tomorrow’ being like ‘the past’.

24

HS + Bootstrapping

� Problems:

Is data >3 years ago useful for forecasting tomorrow?

Use most recent data - say last 100 days ?

1% tail: Has only one number in this tail, for the actual data !

Extreme case !

Actual data might have largest negative (for 100 days ago) of minus 50%

- this would be your forecast VaR for tomorrow using historic simulation approach. Is this okay or not?

25

HS + Bootstrapping

� You have “historic” daily data on each of 10 stock returns (i.e. your portfolio )

� But only use last 100 days of historic daily returns, So we have a data matrix of 10 x 100.

� We require VaR at the 1st percentile (1% cut off)� We require VaR at the 1st percentile (1% cut off)

� We sample “with replacement” from these 100 observations, giving equal probability to each ‘day’ , when we sample.

� This allows any one day’s returns to be randomly chosen more than once (or not at all).

� It is as if we are randomly ‘replaying’ the last 100 days of history, giving each day equal probability26

HS + BootstrappingThe Bootstrap� Draw randomly from a uniform distribution with an equal probability of

drawing any number between 1 and 100.

� If you draw “20” then take the 10-returns in column 20 and revalue the portfolio. Call this $-value, ∆VP

(1)

� Repeat above for 10,000 “trials/runs” (with replacement), obtaining � Repeat above for 10,000 “trials/runs” (with replacement), obtaining 10,000 possible (alternative) values ∆VP

(i)

� “With replacement” means that in the 10,000 runs you will “choose” some of the 100 columns more than once.

� Plot a histogram of the 10,000 values of ∆VP(i) - some of which will be

negative

� Read off the “1% cut off” value (=100th most negative value). This is VaRp

27

Filtered Historical Simulation (FHS)Filtered Historical Simulation (FHS)

28

FHS� HS assumes risk factors are i.i.d however this is usually not the case.

� HS assumes that distribution of returns are stable and that the past and present moments of the density function of returns are constant and equal.

� The probability of having a large loss is not equal across different days. There are periods of high volatility and periods of low volatility (volatility

29

There are periods of high volatility and periods of low volatility (volatility clustering).

� In FHS, historical returns are first standardized by volatility estimated on that particular day (hence the word Filtered).

� The filtering process yields approximately i.i.d returns (residuals) that are suited for historical simulation.

read Barone-Adesi et al (2000) paper in DUO on FHS

Principal Component AnalysisPrincipal Component Analysis

(Estimating risk factors using PCA)(Estimating risk factors using PCA)

30

Estimating risk factors using PCA

31


32


33


34


35


36


37

PCA and risk management

38

PCA and risk management

39

Other related VaR measuresOther related VaR measures

40

Other related VaR measures

41


42


43


44


45


46


47

Topic # 7: Univariate andMultivariate Volatility

Estimation


February 28, 2011


1 Volatility modelling 2

1 Volatility modelling

� Volatility refers to the spread of all likelyoutcomes of an uncertain variable.

� It can be measured by sample standard de-viation

� =

vuut 1

T � 1

TXt=1

(rt � �)2

where rt is the return on day t, and � is theaverage return over the T� day period.

� But this statistic only measures the spreadof a distribution and not the shape of a dis-tribution (except normal and lognormal).

� Black Scholes model assumes that asset pricesare lognormal (which implies that returnsare normally distributed).

� In practice, returns are however non-normaland also the return �uctuations are time vary-ing.


1 Volatility modelling 3

� Example: daily returns of S&P 100 showfeatures of volatility clustering

� Therefore Engle (1982) proposed Autoregres-sive Conditional Heteroscedasticity (ARCH)models for modelling volatility

� Other characteristics documented in litera-ture


1.1 Parametric volatility models 4

�Long memory e¤ect of volatility (au-tocorrelations remain positive for verylong lags)

� Squared returns proxy volatility

�Volatility asymmetry / leverage e¤ect(volatility increases if the previous dayreturns are negative)

1.1 Parametric volatility models

ARCH model

� �Autoregressive�because high/low volatilitytends to persist, �Conditional�means time-varying or with respect to a point in time,and �Heteroscedasticity�is a technical jargonfor non-constant volatility.

� Consider previous t day�s squared returns("2t�1; "

2t�2; :::) that proxy volatility.

� It makes sence to give more weight to recentdata and less weight to far away observa-tions.



� Suppose we are assuming that previous q ob-servations a¤ect today�s returns. So today�svolatility can be

�2t =

qXj=1

�j"2t�j

where �i < �j for i > j andPq

j=1 �j = 1

� Also we can include a long run average vari-ance that should be given some weight aswell

�2t = V0 +

qXj=1

�j"2t�j

where V0 is average variance rate and +Pqj=1 �j = 1

� The weights are unknown and needs to beestimated.

� This is the ARCH model introduced by En-gle (1982)



ARCH (1)

� An ARCH(1) model is given by

rt = �+ "t "t � N�0; �2

��2t = �0 + �1"

2t�1

� Since �2t is variance and has to be positive,we impose the condition

�0 � 0 and �1 � 0

� Generalization: ARCH(q) model

�2t = �0 + �1"2t�1 + :::+ �q"

2t�q

where shocks up to q periods ago a¤ect thecurrent volatility of the process.



EWMA

� Exponentially weighted moving average (EWMA)model is the same as the ARCH models butthe weights decrease exponentially as youmove back through time.

� The model can be written as

�2t = ��2t�1 + (1� �)u2t�1

where � is the constant decay rate, say 0.94.

� To see that the weights cause an exponentialdecay, we substitute for �2t�1:

�2t = ��2t�2 + (1� �)u2t�2

�+ (1� �)u2t�1

= (1� �)u2t�1 + � (1� �)u2t�2 + �2�2t�2

� With �=0.94

�2t = (0:06) u2t�1 + (0:056) u2t�2 + (0:883) �2t�2

� Substituting for �2t�2 now:



�2t = (1� �)u2t�1 + � (1� �)u2t�2 + �2��2t�3 + (1� �)u2t�3

�= (1� �)u2t�1 + � (1� �)u2t�2 + �2(1� �)u2t�3 + �3�2t�3

� With �=0.94

�2t = (0:06) u2t�1+ (0:056) u2t�2+ (0:053) u2t�3+ (0:83) �

2t�3

� Risk Metrics uses EWMA model estimatesfor volatility with � = 0.94.



Generalized ARCH (GARCH) model

� This model generalizes the ARCH speci�ca-tion.

� As one increase the q lags in an ARCHmodelfor capturing the higher order ARCH e¤ectspresent in data, we loose parsimonity.

� Bollerslev (1986) proposed GARCH(p; q)

�2t = �0 +

qXj=1

�j"2t�j +

pXj=1

�j�2t�j

where the weights �0 � 0; �j � 0 and �j �0: Further, for stationarity of this autore-gressive model, we need the condition

qXj=1

�j +

pXj=1

�j

!< 1

� In this model, today�s volatility is explainedby the long run variance rate, the past squaredobservations, and the past volatility history.



� Special case: GARCH(1,1)

�2t = �0 + �1"2t�1 + �1�

2t�1

where 0 � (�1 + �1) < 1:

� To see that the weights cause an exponentialdecay, consider a GARCH(1,1) process

�2t = �0 + �1"2t�1 + �1�

2t�1

� We substitute for �2t�1:

�2t = �0 + �1"2t�1 + �1

��0 + �1"

2t�2 + �1�

2t�2�

= �0 + �0�1 + �1"2t�1 + �1�1"

2t�2 + �21�

2t�2

� Substituting for �2t�2 now:

�2t = �0 + �0�1 + �1"2t�1 + �1�1"

2t�2 +

�21��0 + �1"

2t�3 + �1�

2t�3�

= �0 + �0�1 + �0�21 + �1"

2t�1 + �1�1"

2t�2 +

�1�21"2t�3 + �31�

2t�3

� So the weight applied to "2t�i is �1�i�11 : Theweights decline at rate �:



� GARCH is same as EWMA in assigning ex-ponentially declining weights to past obser-vations. However, GARCH also assigns someweight to the long-run average variance rate.

� When the intercept parameter �0 = 0 and�1 + �1 = 1; then GARCH reduces to aEWMA.

� A GARCH(1,1) model can be interpreted asan 1� order ARCH.

� To see this, consider a ARCH(1)

�2t = �0 + �1"2t�1 + �2"

2t�2 + :::+ �1"

2t�1

� Problem of how to estimate this

�Use the Koyck transform

� Assume �1 is declining

�E¤ect of lagged residuals falls as timegoes by

�1 > �2 > �3 > ::: > �1



� Also assume a geometric decline in �1 suchthat

�k = a�k

where 0 < � < 1

� Gives conditional variance equation as:

�2t = �0 + a�"2t�1 + a�2"2t�2 + :::+ a�1"2t�1= �0 + a

��"2t�1 + �2"2t�2 + :::+ �1"2t�1

�� Now consider one lag of the variance equa-tion �2t�1

�2t�1 = �0+a��"2t�2 + �2"2t�3 + :::+ �1"2t�1

�� Multiply by � through

��2t�1 = �0�+a��"2t�2 + �2"2t�3 + :::+ �1"2t�1

�� Now derive �2t��2t�1 using the above equa-tions

�2t � ��2t�1 = �0 + a��"2t�1 + �2"2t�2 + :::+ �1"2t�1

��0�+ a�

��"2t�2 + �2"2t�3 + :::+ �1"2t�1

�



� Solving

�2t � ��2t�1 = (�0 � �0�) + a�"2t�1�2t = (�0 � �0�) + a�"2t�1 + ��2t�1

GJR model or TARCH model

� Introduced by Glosten, Jagannathan, Run-kle (1993). Hence called GJR model.

� Also called Threshold ARCH (TARCH)model.

� Asymmetries in conditional variances couldalso be introduced by distinguishing the signof the shock.

� We can therefore separate the positive andnegative shock and allow for di¤erent coef-�cients in a GARCH framework.

�2t = �0+

qXj=1

�+j�"+t�j

�2+

qXj=1

��j�"�t�j

�2+

pXj=1

�j�2t�j

� If �+j = ��j for j = 1; 2; :::; q then this re-duces to a GARCH(p,q) model.



� GJR(1993) proposed keeping the original GARCHframework and adding an extra componentthat captures the negative shocks. This isbecause bad news usually has a greater im-pact on volatility than good news.

� The GJR(1,1) model is

�2t = �0 + �+1 "2t�1 + �T "

2t�1Dt�1 + �1�

2t�1

where

Dt�1 =

�1 for "t�1 < 00 for "t�1 � 0

� Remarks:

�For testing symmetry, we consider test-ing H0 : �T = 0: The leverage e¤ect isseen in �T > 0:

��+1 is the coe¢ cient for positive shocks.

��+1 + �T�= ��1

�is the coe¢ cient for

negative shocks.

�For positivity of the conditional vari-ances, we require �0 > 0; �+1 � 0; and�+1 + �T � 0. Hence, �T is allowed tobe negative provided �+1 > j�T j :



Exponential GARCH (EGARCH)

� Recall that in the case of (G)ARCH models,we provided certain coe¢ cient restrictions inorder to ensure �2t (conditional variance of"t) is non-negative with probability one.

� An alternative way of ensuring positivity isspecifying an EGARCH framework for �2t :

� Another characteristic of the model is thatit allows for positive and negative shocksto have di¤erent e¤ects on conditional vari-ances (unlike GARCH).

� The model re�ects the fact that �nancialmarkets respond asymmetrically to good newsand bad news.

� EGARCH(p,q) model is

ln��2t�= �0 +

qXj=1

�j

�"t�j�t�j

�+

qXj=1

��j

�� "t�j�t�j

��

�

+

pXj=1

�j ln��2t�j

�FRM c Dennis PHILIP 2011


where if the error terms "t=�t � N (0; 1)

then � = Eh�� "t�t ��i =q 2

�

� Remarks:

�We can use other fat failed distribu-tions such as student t; or GED in thecase of non-normal errors. In this case,� will take other forms.

� Specifying the model as a logarithm en-sures positivity of �2t : Therefore the lever-age e¤ect is exponential rather than quadratic.

�We divide the errors by the conditionalstandard deviations, "t

�t: Therefore we

standardize (scale) the shocks.

� "t�tcaptures the relative size of the shocks

and �j captures the sign of the relativeshocks

�The magnitude is captured by the vari-able that substracts the mean from theabsolute value of the scaled shocks.

� Example:



� Suppose we specify a EGARCH(0,1)

ln��2t�= �0 + �1�t�1 + ��1 [j�t�1j � �]

where �t�1 = "t�1=�t�1:

�Consider the estimated component:

�1�t�1 + ��1 [j�t�1j � �]

where �1 = 0:3; ��1 = 0:6 and � = 0:85:

�Case 1: impact of positive scaled shock+1:0

0:3 (1) + 0:6 [j1j � 0:85] = 0:39

Case 2: impact of negative scaled shock�1:0

0:3 (�1) + 0:6 [j�1j � 0:85] = �0:21

� We see that positive shock has a greater im-pact than negative shock for �1 positive. Ifwe have �1 negative, say �1 = �0:3; a +1:0shock will have an impact of �0:21 and a�1:0 shock will have an impact of 0.39.


1.2 Non-parametric volatility models 18

� Thus, �1 allows for the sign of the shock tohave an impact on the conditional volatility;over and above the magnitude captured by��1:

1.2 Non-parametric volatility mod-els

Range-based estimators

� Suppose log prices of assets follow a Geo-metric Brownian Motion (GBM). The vari-ous variance estimators have been proposedin literature.

� Notation:

� � volatility to be estimated

�Ct closing price on date t

�Ot opening price on date t

�Ht high price on date t

�Lt low price on date t



� ct = lnCt � lnOt, the normalized clos-ing price

� ot = lnOt � lnCt�1, the normalizedopening price

�ht = lnHt� lnOt, the normalized highprice

� lt = lnLt � lnOt, the normalized lowprice

� The classical sample variance estimator ofvariance �2 is

�2 =1

T � 1

TXi=1

[(oi + ci)� (o+ c)]2

where

(o+ c) =1

T

TXi=1

(oi + ci)

and T is the total number of days consid-ered. So this is the average volatility over Tdays.



� Parkinson (1980) introduced a range estima-tor of daily volatility based on the highestand lowest prices on a particular day.

� He used the range of log prices to de�ne

�2t =1

4 ln 2(ht � lt)

2

since it can be shown that E�(ht � lt)

2� =4 ln(2)�2t

� Garman and Klass (1980) extended Parkin-son�s estimator where information about open-ing and closing prices are incorporated asfollows:

�2t = 0:5 (ht � lt)2 � [2 ln 2� 1] c2t

� Parkinson (1980) and Garman and Klass (1980)assume that the log-price follows a GBMwith no drift term. This means that theaverage return is assumed to be equal tozero. Rogers and Satchell (1991) relaxes thisassumption by using daily opening, high-est, lowest, and closing prices into estimat-ing volatility.



� Rogers and Satchell (1991) estimator is givenby

�2t = ht (ht � ct) + lt (lt � ct)

� This estimator performs better than the es-timators proposed by Parkinson (1980) andGarman and Klass (1980).

� Yang and Zhang (2000) proposed a re�ne-ment to Rogers and Satchell (1991) estima-tor for the presence of opening price jumps.

� Due to overnight volatility, the opening priceand the previous day closing price are mostlynot the same. Estimators that do not incor-porate opening price jumps underestimatevolatility.

� Yang and Zhang (2000) estimator is givenby

�2 = �2open + k�2close + (1� k)�2RS

where �2open and �2close are the classical sam-

ple variance estimators with the use of dailyopening and closing prices, respectively. �2RSis the average variance estimator introducedby Rogers and Satchell (1991).



� The constant k is set to be

k =0:34

1:34 + (T + 1)=(T � 1)

where T is the number of days.

Realized Volatility

� Realized volatility is referred to volatility es-timates calculated using intraday squaredreturns at short intervals such as 5 or 15minutes.

� For a series that has zero mean and no jumps,the realized volatility converges to the con-tinuous time volatility.

� Consider a continuous time martigale processfor asset prices

dpt = �tdWt

where dWt is a standard brownian motion.



� Then the conditional variance for one-periodreturns, rt+1 � pt+1 � pt isZ t+1

t

�2sds

which is called the integrated volatility (orthe true volatility) over the period t to t+1:

� We don�t know what �2t is. So we estimateit.

� Let m be the sampling frequency such thatthere are m continuously compounded re-turns in one unit of time (say, one day).

� The jth return is given by

rt+j=m � pt+j=m � pt+(j�1)=m

� The realized volatility (in one unit of time)can be de�ned as

RVt+1 =X

j=1;:::;m

r2t+j=m



� Then from the theory of quadratic variation,if sample returns are uncorrelated,

p limm!1

Z t+1

t

�2sds�X

j=1;:::;m

r2t+j=m

!= 0

� As we increase sampling frequency, we get aconsistent estimate of volatility.

� In the presence of jumps, RV is no longer aconsistent estimator of volatility.

� An extension to this estimator is the stan-dardized Realized Bipower Variation mea-sure de�ned as

BV[a;b]t+1 =

�1

m

�[1�(a+b)=2]�mXj=1

��rt+j=m��a ��rt+(j�1)=m��bfor a; b > 0:

� When jumps are large but rare, the simplestcase where a = b = 1 captures the jumpswell.



� High frequency returns measured below 5minutes are a¤ected by market microstruc-ture e¤ects including nonsynchronous trad-ing, discrete price observations, intraday pe-riodic volatility patterns and bid�ask bounce.


2 Multivariate Volatility Models 26

2 Multivariate VolatilityMod-els

� Multivariate modelling of volatilities enableus to study movements across markets andacross assets (co-volatilities).

� Applications in �nance: asset pricing andportfolio selection, market linkages and in-tegration between markets, hedging and riskmanagement, etc.

� Consider a n-dimensional process fytg : If wedenote � as the �nite vector of parameters,we can write

yt = �t (�) + "t

where �t (�) is the conditional mean vectorand

"t = H1=2t (�) � zt

where H1=2t (�) is an N �N positive de�nite

matrix.

� The N � 1 vector zt is such that

zt � iidD (0; IN)



where IN is the identity matrix of order N:

� The matrix Ht is the conditional variancematrix of yt

� How do we parameterize Ht?

Vech Representation

� Bollerslev, Engle andWooldridge (1988) pro-pose a natural multivariate extension of theunivariate GARCH(p; q) models where

vech (Ht) =W+

qXi=1

A�i vech�"t�i"

0

t�i

�+

pXj=1

��j vech (Ht�j)

where vech is the vector-half operator, whichstacks the lower triangular elements of anN �N matrix into a [N (N + 1) =2]� 1 vec-tor.

� The challenge in this parameterization is toensure Ht is positive de�nite covariance ma-trix. Also, as the number of assets N in-crease, the number of parameters to be esti-mated is very large.



� f"tg is covariance stationary if all the eigen-values of A� and B� are less than 1 in mod-ulus.

� Bollerslev, Engle andWooldridge (1988) pro-posed a "Diagonal vech" representation whereA�i and �

�j are diagonal matrices.

� Example: For N = 2 assets and a single-period lag model (p = q = 1),24 h11;th21;th22;t

35 =

24 w1w2w3

35+24 a�11

a�22a�33

3524 "21;t�1"2;t�1; "1;t�1

"22;t�1

35+

24 b�11b�22

b�33

3524 h11;t�1h21;t�1h22;t�1

35� The diagonal restriction reduces the num-ber of parameters but the model is not al-lowed to capture the interactions in vari-ances among assets (copersistence, causalityrelations, asymmetries)

� The diagonal vech is stationary i¤Pq

i=1 a�ii+Pp

j=1 b�jj < 1



BEKK Representation

� Engle and Kroner (1995) propose a BEKKrepresentation where

Ht = cc0+

qXi=1

Ai

�"t�i"

0

t�i

�A0i+

pXj=1

�jHt�j�0j

where c is a lower triangular matrix andtherefore cc0 will be positive de�nite. Also,by estimating A and B rather than A� andB�; we ensure positive de�niteness.

� In the case of 2 assets:

h11;t h12;th21;t h22;t

= cc0+�a11 a12a21 a22

� �"21;t�1 "1;t�1; "2;t�1

"2;t�1; "1;t�1 "22;t�1

��a11 a12a21 a22

�0+

�b11 b12b21 b22

��h11;t�1 h12;t�1h21;t�1 h22;t�1

� �b11 b12b21 b22

�0� To reduce the number of parameters to beestimated, we can impose a "diagonal BEKK"model where Ai and Bj are diagonal.



� Alternatively, we can have Ai and Bj asscalar times a matrix of ones. In this case,we will have a "scalar-BEKK" model.

� Diagonal BEKK and Scalar-BEKK are co-variance stationary if

Pqi=1 a

2nn;i+

Ppj=1 b

2nn;j <

1 8n = 1; 2; :::; N andPq

i=1 a2i +Pp

j=1 b2j < 1

respectively.

Constant Conditional Correlation (CCC)Model

� Bollerslev (1990), assuming conditional cor-relations constant, proposed that conditionalcovariances (Ht) can be parameterized as aproduct of corresponding conditional stan-dard deviations.

Ht = DtRDt

=��ijphii;t hjj;t

�

whereDt =

264ph11;t

. . . phNN;t

375 ; R =FRM c Dennis PHILIP 2011


266641 �12 � � � �1N

�21 1...

.... . .

�N1 � � � 1

37775� Each conditional standard deviations can bein turn de�ned as any univariate GARCHmodel such as GARCH(1,1)

hii;t = wi+�i"2i;t�1+�ihii;t�1 i = 1; 2; :::; N

� Ht is positive de�nite i¤ all N conditionalcovariances are positive and R is positivede�nite.

� In most empirical applications, the condi-tional correlations are not constant. There-fore Engle (2002) and Tse and Tsui(2002)propose a generalization of the CCC modelby allowing for conditional correlation ma-trix to be time-varying. This is the DCCmodel.



Tests for Costant Correlations

� Tse (2000) proposes testing the null that

hijt = �ijphiithjjt

against the alternative that

hijt = �ijtphiithjjt

where the conditional variances, hiit and hjjtare GARCH-type models. The test statisticis an LM statistic which is asymptotically�2 (N (N � 1) =2) :

� Engle and Sheppard (2001) propose anothertest with the null hypothesis

H0 : Rt = �R for all t

against the alternative

H1 : vech (Rt) = vech��R�+��1vech (Rt�1)+

:::+ ��pvech (Rt�p)

� The test statistic employed is again chi-squareddistributed.



Dynamic Conditional Correlation (DCC)model

� Tse (2000) and Engle and Sheppard (2001)propose tests of constant conditional corre-lation hypothesis.

� In most applications, we see the hypothe-sis of constant conditional correlation is re-jected.

� Engle (2002) propose the DCC framework,

Ht = DtRtDt

where Dt is the matrix of standard devia-tions (as de�ned in the case of CCC), hii;tcan be any univariate GARCH model andRt is the conditional correlation matrix.

� We then standardize each return by the dy-namic standard deviations to get standard-ized returns. Let

ut = "t

hdiag

�ph11t � � � hNNt

�i�1



be the vector of standardized residuals of NGARCH models. These variables now havestandard deviations of one.

� We now model the conditional correlationsof raw returns ("t) by modelling conditionalcovariances of standardized returns (ut).

� We de�ne Rt as

Rt = diag (Qt)�1=2Qtdiag (Qt)

�1=2

where Qt is an N � N symmetric positivede�nite matrix given by

Qt=

1�

qXi=1

�i �pXj=1

�j

!�Q+

qXi=1

�i�ut�iu

0t�i�+

pXj=1

�jQt�j

where

��i � 0; �j � 0;Pq

i=1 �i +Pp

j=1 �j < 1

� �Q = 1T

PTt=1 utu

0t is the standardized

unconditional covariance matrix



Example:

� Consider the following model:

rt = "t

and

"t � N (0; Ht) where Ht = var (rtjt�1)

� We can write Ht as

Ht = DtRtDt

where

�Dt =

264ph11;t

. . . phNN;;t

375 is di-agonal matrix of conditional standarddeviations. Each hii;t follows a GARCH(1,1)process

� ut = D�1t "t

� Rt=

264pq11

. . . pqNN

375�1

t

�Qt�

264pq11

. . . pqNN

375�1

twhere



�Qt =h1�

PLl=1 �l �

PSs=1 �s

i�Q+PL

l=1 �l�ut�lu

0t�l�+PS

s=1 �sQt�s

�Qt = �Q +PL

l=1 �l��ut�lu

0t�l�� Q

�+PS

s=1 �s�Qt�s � �Q

��For the case of one lags,Qt = �Q+�1

��ut�1u

0t�1�� Q

�+�1

�Qt�1 � �Q

�� Remarks:

�Qt is a function of:

1. unconditional covariance matrix �Qof standardized residuals

2. covariance matrix of standardizedresiduals u for L lags

3. Qt�s i.e s lags of itself

�Q at time t =

8>>>>>>><>>>>>>>:

�uncond. component

(of resid)

�+

�addl. persistent componentfrom past L lags (of resid)

�+

�include past S period addl.persistent component (of Q)

�



Two-step Estimation of DCC models

� Under the assumption of normality of inno-vations, Engle and Sheppard show that theDCC can be estimated in 2 steps.

� Let "t � N (0; Ht) : Let � be vector of un-known parameters in matrix Ht:

� The log-likelihood function is given by

l (�) = �12

TXt=1

�N log (2�) + log jHtj+ r

0

tH�1t rt

�= �1

2

TXt=1

�N log (2�)+ log jDtRtDtj+r

0

tD�1t R�1t D�1

t rt

�= �1

2

TXt=1

�N log (2�)+2 log jDtj+ log jRtj+u

0

tR�1t ut

�



where ut = D�1t rt is the standardized returns.

Adding and substracting u0tut we get

l (�) = �12

TXt=1

(N log (2�) + 2 log jDtj+ r0

tD�1t D�1

t rt

�u0tut + log jRtj+ u0

tR�1t ut)

� Remarks:

�The representation allows us to decom-pose the log-likelihood l (�) as a sumof the volatility part (lV (�)) contain-ing the parameters in matrixD and thecorrelation part (lC ( )) containing theparameters in matrix R:

�That is, we partition vector of parame-ters into 2 subsets: � = f�; g : Thelog-likelihoods can be written as

l (�) = lV (�) + lC ( j�)

where

lV (�) = �1

2

TXt=1

�N log (2�) + log jDtj2 + r

0

tD�2t rt

�FRM c Dennis PHILIP 2011


lC ( j�) = �1

2

TXt=1

�log jRtj+ u

0

tR�1t ut � u0tut

��The volatility term lV (�) is apparentlythe sum of the individual GARCH like-lihoods, which is jointly maximized byseperately maximizing each term. Thisgives us the parameters �:

�Then the correlation term lC ( j�) ismaximized conditional over the volatil-ity parameters that were estimated be-fore.

� In the correlation term, Rt takes theDCC form diag (Qt)

�1=2�Qt�diag (Qt)�1=2

The two-step estimation procedure:

1. Estimate the conditional variances (volatil-ity terms �) using MLE. That is, maximizethe likelihood to �nd �

� = argmax flV (�)g

2. Then we compute the standardized returnsut = D�1

t rt and we estimate the correlations



among the returns "t of several assets. Herewe maximize the likelihood function of thecorrelation term

= argmaxnlC

� j��o

� The DCC estimation method employs theassumption of Normality in conditional re-turns, which is generally not the case �nan-cial assets.

� One can use alternative fat-tailed distribu-tions such as student-t, laplace, logistic inorder to represent the data.



Univariate Volatility Models References:

� Hull (2010) Risk management and �nancialinstitutions, Chapter 9

� Christo¤ersen (2003) Elements of �nancialrisk management, Chapter 2

� Andersen, T. G. And L. Benzoni (2008), �Re-alized volatility�, Chapter in Handbook ofFinancial Time Series, Springer Verlag.

Additional references:

� Andersen, T. G., Bollerslev, T., Diebold,F. X. and P. Labys (1999), �(Understand-ing, optimizing, using and forecasting) real-ized volatility and correlation�, Manuscript,Northwester University, Duke University andPennsylvania University. Published in re-vised form as �Great realizations� in Risk,March 2000, 105-108.

� Andersen, T., Bollerslev, T., Diebold, F.X.and Ebens, H. (2001), "The distribution ofstock return volatility" Journal of FinancialEconomics, 61, 43-76.



� Andersen, T. G., Bollerslev, T., Diebold,F. X. and P. Labys (2003), �Modelling andforecasting realized volatility�, Economet-rica, 71, 529-626.

� Andersen, T. G., Bollerslev, T., Diebold,F. X. and J. Wu (2004), �Realized beta:persistence and predictability�, Manuscript,Northwester University, Duke University andPennsylvania University.

Multivariate VolatilityModels Readings:

� Christoggersen (2003) Elements of �nancialrisk management, Chapter 3

� Engle, R (2002) Dynamic Conditional Cor-relation: A Simple Class of Multivariate Gen-eralized Autoregressive Conditional Heteroskedas-ticity Models Journal of Business & Eco-nomic Statistics. vol. 20, no. 3

� Bauwens, L; Laurent, S and Rombouts, J(2006) Multivariate GARCHmodels: A Sur-vey Journal of Applied Econometrics. vol.21 pp. 79�109



� Engle, R (2009) Anticipating Correlations:A New Paradigm for Risk Management Prince-ton University Press



Topic 8a

Credit Risk MeasuresCredit Risk MeasuresReadings: CN(2001) chapters 25

Hull’s (Risk Management) book chapters 14, 15

�Credit Risk Measures

�Credit Metrics

�KMV Credit Monitor (and Merton

�Credit Risk Measures

�Credit Metrics

�KMV Credit Monitor (and Merton

Topics

(1974) model)

�CSFP Credit Risk Plus

�McKinsey’s Credit Portfolio View, CPV

(1974) model)

�CSFP Credit Risk Plus

�McKinsey’s Credit Portfolio View, CPV

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4 models

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C-VaR

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credit rate change

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-> find VaR

Credit Metrics Approach

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Credit Metrics

� CreditMetrics (J.P. Morgan 1997) measures risk, associated with credit events, for a portfolio of rated exposures (such as corporate bonds)

� We will cover the following topics:� We will cover the following topics:

� Transition probabilities

� Valuation

� Joint migration probabilities

� Many Obligors: Mapping and MCS

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AAA bond -> AA -> value bond reduce

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-> find final figure of C-Var

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1

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2

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what's the value chage (V1 -> V2) how much am i losing?

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bc have 3 4 bonds -> joint

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not normal

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Transition probabilities

� Credit ratings measure credit risk.

� Practical issue: How do we measure changes in credit risk (i.e. credit ratings)?

ProblemsProblems

� Lack of historical data to measure proportion of firms that migrate between each credit ratings

� The distribution of credit returns are highly skewed (see graph on next slide)

� Therefore ‘change in value’ cannot be explained by std. dev.

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change in rating

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not much data

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0.075

0.100

0.900BBB

A

Frequency Distribution for a 5 year BBB bond after 1 year

50 60 70 80 90 100 1100.000

0.025

0.050

Default CCCB

BBA

AAAAA

Revaluation at Risk HorizonSkewed:prob(BBB to Default) is low but change in value is largewhereas prob(BBB to A) relatively high but change in value is relatively small

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histogram

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prob

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prob that BBB jump to A

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thickness:

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fall

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this is assumption that can known by lookig at this

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C-Var can use standart deviation


To calculate Transition Prob:

� Suppose we have a sample of 1000 firms and their bond credit ratings from 1980-2000.

� Consider bonds initially rated CCC.

For each year, 1-year marginal mortality rate (MMR ) is� For each year, 1-year marginal mortality rate (MMR1) is

� Transition probability is average MMR:

1

1

1

value of CCC bonds defaulting in yearMMR

total value of CCC bonds at beginning of year=

2000

1,1980

( ) i iP CCC to default w MMR= ∑

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of every rating bonds

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how rate going to change

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default only

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historical

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(

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weight

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by number of firms

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(tong Wi =1)

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historical for all teh bonds

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ALL bonds

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do it for all others


� We repeat this exercise for CCC-rated bonds moving to all possible ratings.

� Also, we repeat this for all other bonds.

� This gives us the Empirical Transition Probabilities

This model assumes probability of transition in any � This model assumes probability of transition in any year is independent of probability in earlier years (called first-order Markov process).

� Survival rates: 1i iSR MMR= −

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C-VaR

� We intend to calculate Credit Value at Risk (C-VaR)

� Aim is to establish the $-value for 1% tail cut-off, from histogram of all possible values of firm’s bonds at end-year, after credit migration

This requires:

� calculating the probability of migration between different � calculating the probability of migration between different credit ratings and the calculation of the expected value of a single bond in different potential credit ratings.

� deriving the possible values of portfolio of n-bonds at year-end, after all migrations

� Calculating the probabilities (likelihood) of joint migration of n-bonds,. between credit ratings.

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want to do MSC at the end

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question: if BAD case -> what's the max loss?

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money

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under VaR setting

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C-VaR for One Bond � First consider calculations for a Single Bond

� Possible Transitions

� {A stays at A} or { A to B} or {A to D} for example

� Suppose a bank holds “senior unsecured” A-rated bond with � Suppose a bank holds “senior unsecured” A-rated bond with 6% coupon and 7-year maturity.

� Credit risk horizon is 1 year ahead (assumed).

� Transition probabilities, calculated using historical data, are:Transition Matrix (Single Bond)

Initial Probability : End-Year Rating (%) Sum Rating A B D

A pAA = 92 pAB = 7 pAD = 1 100

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e.g

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3 possible transition

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want to calculate

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%

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C-VaR for One Bond� We calculate market value of the bond at the end of 1-year.

� Consider a set of forward ratesOne Year Forward Zero Curves

Credit Rating f12 f13 f14 …

A 3.7 4.3 4.9 … B 6.0 7.0 8.0 …

Notes : f12 = one-year forward rate applicable from the end of year-1 to the end of year-2 etc.

67,1

32, )1(

106$...

)049.1(

6$

)043.1(

6$

)037.1(

6$6$

fV AA +

+++++=

� Forward rates for end-year A-rated are lower than for B-rated ~ reflects different credit risk

� If A-rated bond stays A-rated, the value of bond at the end of year 1 is:

(assume 6% annual coupons paid at end of the year)

year-2 etc.

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want to calculate market value

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discount back

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using different forward rate

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(get in the market)

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1yr starting from 1y

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4yr start from 1

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A better rate than B

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discount using different rates

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value

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C-VaR for One Bond � If A-rated bond migrates to B rating, the value of bond at the

end of year 1 is:

� Suppose calculations yield VA,A = $109 and VA,B = $107

67,1

22, )1(

106$...

)08.1(

6$

)07.1(

6$

)06.1(

6$6$

fV BA +

+++++=

� If A-rated bond catastrophically defaults, the value of the bond is the recovery value calculated from the recovery rates given below. VA,D = 51% of face value = $51

Recovery Rates After Default (% of par value)

Seniority Class Mean (%) Standard Deviation (%) Senior Secured 53 27 Senior Unsecured 51 25 Senior Subordinated 38 24 Subordinated 33 20 Junior Subordinated 17 11

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use B rate

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D: default

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C-VaR for One Bond� To calculate C-VaR, we have the ingredients:

� Transition probabilities

� Bond values associated to different transitions

C-VaR (using standard deviation method): =1.65 σvC-VaR (using standard deviation method): =1.65 σv

� Mean and Standard Deviation of end-year Value is

(we can calculate σ around the mean Vm or around VA,A)

3

1

0.92($109) 0.07($107) 0.01($51) $108.28m i ii

V p V=

= = + + =∑

( ) ( )3 3

2 22

1 1

$5.78v i i m i i mi i

p V V p V Vσ= =

= − = − =∑ ∑

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1

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2

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formular for volatility

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different b/w V & mean

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lay cac gia tri tu slide tren

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mean

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waighted

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second moment - mean square

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multiply to get VaR

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C-VaR for One Bond� This calculation assumes value of bond in the various

states is known with no uncertainty.

� Usually there is uncertainty in value of the bond and so we associate σ to the three states.

( ) ( )3

22 2v i i i mp V Vσ σ= + −∑

� We know the standard deviation associated to recovery rates in the event of default (σdefault = 25%)

� Another assumption is the distribution of outcomes are normal and so standard deviation is a good measure. Alternatively, one can use a particular percentile value as a measure of C-VaR.

( )1

v i i i mi=∑

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them std (in practice)

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recommended to use this

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historical simulation

C-VaR for Two Bonds� We extend previous calculations to the case of two bonds.

� Table below summarizes transition probabilities and bond values for an A-rated and B-rated bond:

Probabilities and Bond Value (Initial A-Rated Bond)

Year End Rating

Probability %

$Value

A pAA = 92 VAA = 109 A pAA = 92 VAA = 109 B pAB = 7 VAB = 107 D pAD = 1 VAD = 51

Notes : The mean and standard deviation for initial-A rated bond are

Vm,A = 108.28, σV,A = 5.78.

Probability and Value (Initial B-Rated Bond)

Year End Rating Probability $Value A pBA = 3 VBA = 108 B pBB = 90 VBB = 98 D pBD = 7 VBC = 51

Notes : The mean and standard deviation for initial-B rated bond are Vm,B = 95.0, σV,B = 12.19.

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sigma & migration

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A & B are independent

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to make it simpler don't care about Cov

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calculate

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historical

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use forward rate

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C-VaR for Two Bonds

� There are 9 possible values for the two-bond portfolio at the

end of 1-year.

� Joint year-end values for the two bonds is:

Possible Year End Value (2-Bonds)

Obligor-1 (initial-A rated) Obligor-2 (initial-B rated)

1. A 2. B 3. D VBA = 108 VBB = 98 VBD = 51

1. A VAA = 109 217 207 160 2. B VAB = 107 215 205 158 3. D VAD = 51 159 149 102 Notes : The values in the ith row and jth column of the central 3x3 matrix are simply the sum of the values in the appropriate row and column (eg. entry for D,D is 102 = 51 + 51).

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9

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add

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all possible value

C-VaR for Two Bonds

� We also need the Joint Likelihoods or probabilities of credit

migration of the two bonds.

� If we assume independence between credit migrations of the

two bonds, the calculation is straight forward.

� Joint likelihood can be calculated from the transition matrix

Transition Matrix (percent) Transition Matrix (percent)

Initial Rating

End Year Rating Row Sum

1. A 2. B 3. D 1. A 92 7 1 100 2. B 3 90 7 100 3. D 0 0 100 100

Note: If you start in default you have zero probability of any rating change and 100% probability of staying in default.

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%

C-VaR for Two Bonds

� Joint Likelihoods are calculated under independence.

� Example: Prob(‘A remains A’ and ‘B remains B’) = Prob(‘A

remains A’) x Prob(‘B remains B’) = 0.92 x 0.90 = 0.828

Joint Migration Probabilities : ππππij(percent)

Obligor-1 (initial-A rated) Obligor-2 (initial-B rated) 1. A 2. B 3. D 1. A 2. B 3. D p21 = pAB =

3 p22 = pBB =

90 p23 = pBD =

7

1. A p11 = pAA = 92

2.76 82.8 6.44

2. B p12 = pAB = 7

0.21 6.3 0.49

3. D p13 = pAD = 1

0.03 0.9 0.07

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C-VaR for Two Bonds

� C-VaR (using standard deviation method):� The formulae for calculating mean and standard deviation is

the same as in the case of one bond.

3 3

, ,1 1

$203.29m p ij i ji j

V Vπ= =

= =∑∑

( )3 3 22

, , ,1 1

$13.49v p ij i j m pi j

V Vσ π= =

= − =∑∑

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bond 1 and bond 2

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second moment

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C-VaR for Two Bonds

� C-VaR (using percentile method):� Order VA+B in from lowest to highest

� Then add up their joint likelihoods until these reach the 1% value.

VA+B = {$102, $149, $158, $159, …, $217}

π = {0.07, 0.9, 0.49, 0.43, …, 2.76}πi,j = {0.07, 0.9, 0.49, 0.43, …, 2.76}

� $149 is the ‘unexpected loss’, at 1% level

� It is customary to measure the VaR relative to the ‘mean value’ rather than relative to the ’initial value’ of the bond/loan portfolio

� Hence: C-VaR = $54.29 (= mean value – 1% extreme value = $203.29 - $149)

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sort

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sort trang 16

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cumulate -> cut off at the tail

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1% loss

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difference b/w expected and unexpected

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C-VaR - summary• C-VaR of a portfolio of corporate bonds depends on the joint migration

likelihoods (probabilities), the value of the obligor (bond) in default (basedon the seniority class of the bond), and the value of the bond in any newcredit rating

SeniorityCredit Rating Credit Spread

Recovery Rate in Default

MigrationLikelihoods

Value of Bond in new Rating

Standard Deviation or Percentile Level for C-VaR

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history

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Measuring Joint Credit Migration

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Measuring Joint Credit Migration

� Three ways:1. Historic credit migration data- This becomes difficult when one has to measure correlations

between many rating possibilities

- Lack of data

2. Bond spread data2. Bond spread data- Movements in credit yield spread between two firms reflect

changes in credit quality

- Use bond pricing model to extract probabilities of credit rating changes from observed changes in credit spreads

3. Asset value approach - Movements in firm’s stock price reflect changes in credit

quality and hence signal possible change in credit ratings(we discuss point 3 now…)

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(concentrate on this)

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(don't have the data -> limit (problem)

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difference bw yield y(AAA) - y(BB) = spread

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looking at asset (stock) -> use it to calculate credit (bond)

Asset value approach – one stock

One stock example:� Consider initial BB-rated firm.

� Lower the stock returns, lower it’s credit ratings but there are range of return values where ratings remain the same.

� Suppose we assume stock returns are normal and σ is known.known.

� We can ‘Invert’ the normal distribution to obtain cut-off points for stock returns (R) corresponding to the known probability of default (e.g. 1.06% below)

� Then from normal distribution bell:

Pr(default) = Pr(R<ZDef) = Φ(ZDef/σ) = 1.06%

Hence: ZDef = Φ-1(1.06%) σ = -2.30σ

(In Excel: =NORMSINV(0.0106) = -2.30)

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find the threshold (gioi han) de xac dinh luc nao chuyen

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threshold

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find where the bond going to fall -> get the average

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nguoc de tinh gia tri

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Probability

B

BB BBB

A

Probability of default=1.06

Transition probability:

DefCCC

A

AAAAA

-2.301.06

-2.041.00

-1.238.84 80.53

1.377.73

2.39 0.67

2.930.14

3.430.03

Cut-off level:

We assume (for simplicity) that the mean return for the stock of an initial BB-rated firm is zero

Prob ( BB to CCC) = 1.0

Z<- Calc.<- known

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known

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want to calculate this

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(cut off level)

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transition point

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use differnt forward rate


� If the stock return falls by more than -2.30σ (%) then we assume transition of the firm from BB to D (and we revalue the bond using recovery rates in default)

� Now lets consider transition from BB to CCC rating. Suppose 1% is the observed transition probability.

Pr(CCC) = Pr(ZDef<R<Zccc) = Φ(ZCCC/σ) - Φ(ZDef/σ) = 1.00Def ccc CCC Def

Hence: Φ(ZCCC/σ) = 1.0 + Φ(ZDef/σ) = 1.0+1.06 = 2.06

and ZCCC = Φ-1(2.06/ σ) = -2.04σ

� If stock return fall by more than -2.04σ (%) then we assume transition of the firm from BB to CCC (and we revalue the bond using CCC forward rates)

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normal distribution

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cut off level


� In a similar fashion, we obtain cut-off points for all credit rating changes for the initial BB-rated firm, summarized in the table:

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Asset value approach – two stocks

� Consider now two stocks: ‘A-rated’ firm along with the ‘BB-rated’ firm.

� Denote R’ as it’s asset returns, σ’ as it’s standard deviation, and Z’Def, Z’CCC, etc as it’s asset return cut-offs/thresholds.

� Table summarizes the calculations:

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Asset value approach – two stocks

� Until now we have calculated individual obligor credit rating changes.

� To calculate two credit rating changes jointly, we assume two asset returns are correlated and bivariate normal with

Suppose we wish to calculate P(BBB to BB and A to BBB).

2 '

' '2

σ ρσσρσσ σ

Σ =

� Suppose we wish to calculate P(BBB to BB and A to BBB). Let’s call this Y%.

� P(BBB to BB and A to BBB) = Pr(ZB <R<ZBB, Z’BB

<R’<Z’BBB) = dR dR’ = Y%

� We use the same procedure to calculate all joint rating migration πi,j for the two obligors.

),,( ''

'Σ∫∫ RRf

BBB

BB

BB

B

Z

Z

Z

Z

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joint prob

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not known -> use simulate

Asset value approach – using MCS� As number of obligors and number of ratings increase, the

dimensions of joint migration likelihood matrix π explode.

� Instead, we estimate the distribution of portfolio credit values using Monte Carlo Simulation (MCS).

� MCS�We have already found the ‘cut-off points’ Z for each obligor�Now simulate the joint stock returns (with a known correlation) and �Now simulate the joint stock returns (with a known correlation) and associate these outcomes with a JOINT credit rating�Revalue the n-bonds at these new ratings ~ this is the 1st MCS outcome, Vp

(1)

�Repeat above m-times and plot a histogram of Vp(i)

�Order the Vp(i) from lowest to highest and read off the $-value of the

portfolio at the 1% left tail cut-off point

Assumes stock return correlations correctly reflect the economic conditions, that

influence credit migration correlations

TriNamNGO

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EXAM THIS

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KMV’s Credit Monitor

KMV’s Credit Monitor

� As in Credit Metrics model, KMV Credit Monitor model also links stock prices to default probabilities.

� It uses Merton (1974) model to make this link.

� Unlike Credit Metrics which allows for upgrades and downgrades, KMV’s model is a default only model.

Next, we will learn about Merton (1974) model and KMV(see Topic 8b slides)

CSFP Credit Risk Plus

CSFP Credit Risk PlusUses Poisson to give default probabilities and mean default rate µcan vary with the economic cycle.

Assume bank has 100 loans outstanding and estimated default rate= 3% p.a. implying µ = 3 defaults per year (from the 100 firms).

Probability of n-defaults!

),(n

edefaultsnp

nµµ−

=

p(0) = = 0.049, p(1) = 0.149, p(2) = 0.224…p(8) = 0.008 etc ~ humped shaped probability distribution (see figure on next slide).

Cumulative probabilities:p(0) = 0.049, p(0-1) = 0.198, p(0-2) = 0.422, … p(0-8) = 0.996

“p(0-8)” indicates the probability of between zero and eight defaults

Take 8 defaults as approximation to the 99th percentile (1% cut off)


Probability

Expected Loss

0.224

Probability Distribution of Losses

Loss in $’s

Unexpected Loss

Economic Capital

$30,000 $80,000

0.049 99th percentile

CSFP Credit Risk PlusAverage loss given default LGD = $10,000 then:Expected loss = (3 defaults) x $10,000 = $30,000

Unexpected loss (99th percentile) = 8 defaults x $10,000 = $80,000(Note: Line in text p.716 is incorrect)

Capital Requirement = Unexpected loss - Expected Loss= 80,000 - 30,000 = $50,000

TFOLIO OF LOANSTFOLIO OF LOANSSuppose the bank also has another 100 loans in a ‘bucket’ with an average LGD = $20,000 and with µ = 10% p.a.

Repeat the above exercise for this $20,000 ‘bucket’ of loans and derive its (Poisson) probability distribution.

Then ‘add’ the probability distributions of the two buckets (i.e. $10,000 and $20,000) to get the probability distribution for the portfolio of 200 loans (we ignore correlations across defaults here)

McKinsey’s Credit Portfolio View, CPV

McKinsey’s Credit Portfolio View, CPV

Mark-to-market model with direct link to macro variables

Explicitly model the link between the transition probability (e.g. p(C toD)) and an index of macroeconomic activity, y.

pit = f(yt) where i = “C to D” etc.

y is assumed to depend on a set of macroeconomic variables Xit (e.g.GDP, unemployment etc.)GDP, unemployment etc.)

yt = g (Xit, vt) i = 1, 2, … n

Xit depend on own past values plus other random errors εit.(sayVAR(1))It follows that:

pit = k (Xi,t-1, vt, εit)

Each transition probability depends on past values of the macro-variables Xit and the error terms vt, εit. Clearly the pit are correlated.

CPV

Monte Carlo simulation to adjust the empirical (or average) transition probabilities estimated from a sample of firms (e.g. as in CreditMetrics).

Consider one Monte Carlo ‘draw’ of the error terms vt, εit (whichembody the correlations found in the estimated equations for yt andXit above).

This may give rise to a simulated probability pis = 0.25, whereas the

historic (unconditional) transition probability might be pih = 0.20 . This

implies a ratio of

ri = pis / pi

h = 1.25

Repeat the above for all initial credit rating states (i.e. i = AAA, AA, … etc.) and obtain a set of r’s.

CPV

Then take the (CreditMetrics type) historic 8 x 8 transition matrix Tt and multiply these historic probabilities by the appropriate ri so that we obtain a new ‘simulated’ transition probability matrix, T.

T , now embodies the impact of the macro variables and hence the correlations between credit migrations

Then revalue our portfolio of bonds using new simulated probabilities which reflect one possible state of the economy.

This would complete the first Monte Carlo ‘draw’ and give us one new value for the bond portfolio.

Repeating this a large number of times (e.g. 10,000), provides the whole distribution of gains and losses on the portfolio, from which we can ‘read off’ the portfolio value at the 1st percentile.

SUMMARY

A COMPARISON OF CREDIT MODELS

Characteristics J.P.Morgan CreditMetrics

KMV Credit Monitor


McKinsey Credit Portfolio

View Mark-to-Market (MTM) or Default

MTM MTM or DM DM MTM or DM (MTM) or Default Mode (DM) Source of Risk

Multivariate normal stock returns

Multivariate normal stock returns

Stochastic default rate (Poisson)

Macroeconomic Variables

Correlations

Stock prices Transition

probabilities

Option prices Stock price volatility

Correlation between mean default rates

Correlation between macro factors

Solution Method Analytic or MCS Analytic Analytic MCS

Topic # 8b: Merton modeland KMV


March 3, 2011


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Merton (1974) model and KMV

� Assume �rm�s balance sheet looks like this:

� Consider a �rm with risky assets A; thatfollow a GBM.

� Suppose �rm is �nanced by a simple capitalstructure, namely one debt obligation (D)and one type of equity (E)

A0 = E0 +D0

where (Et)t�0 is a GBM that describes evo-lution of equity of the �rm and (Dt)t�0 issome process that describes market value ofdebt obligation of the �rm.


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� Suppose debt holders pay capital D0 at timet = 0 and get F (includes principal and in-terest) at time t = T:

� For debt holders (lending banks), credit riskincrease i¤

P [AT < F ] > 0

� When default probability > 0, we can con-clude that

D0 < Fe�rt

where r is the risk free rate. That is, debtholders would need credit risk premium.

� In other words, D0 is smaller, greater theriskiness of the �rm.

� To hedge this credit risk, debt holders goLong a Put contract on A; with strike Fand maturity T:

� This guarantees credit protection against de-fault of payment:


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� Debt holder�s portfolio consist on a loan andPut contract.

� At t = 0;

D0 + P0 (A0; �A; F; T; r)

� At t = T; portfolio value is F

� Using non-arbitrage principal, at t = 0

D0 + P0 (A0; �A; F; T; r) = Fe�rt

� So the present value of D0 is

D0 = Fe�rt � P0 (A0; �A; F; T; r) (1)


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� The �gure below summarizes this.

� Next, we can think about the value of equityE0 in terms of a Call option.

� Equity holders of the �rm have the right toliquidate the �rm (i.e. paying o¤ the debtand taking over remaining assets)

� Suppose liquidation happens at maturity datet = T:

� Two Scenarios:


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1. AT < F : In this case, there is defaultand there is not enough to pay the debtholders. Moreover, equity holders havea payo¤ of zero.

2. AT � F : In this case, there is a netpro�t for equity holders after paying o¤the debt (AT � F ).

� So the total payo¤ to equity holders is

max(AT � F; 0)

which is the payo¤ of an European call op-tion on A with strike F and maturity T:

� The present value of equity is therefore

E0 = C0 (A0; �A; F; T; r) (2)

� Combining equation 1 and 2, we can obtain

A0 = E0 +D0

= C0 (A0; �A; F; T; r) + Fe�rt

�P0 (A0; �A; F; T; r)A0 + P0 (A0; �A; F; T; r) = C0 (A0; �A; F; T; r) + Fe

�rt

which is nothing but the put-call parity.


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� The above also shows that equity and debtholders have contrary risk preferences.

� By choosing risky investment in some assetA, with higher volatility �A; equity holderswill increase the option premium for bothcall and put.

� This is good for equity holders as they arelong a call and this is bad for debt holdersas they have a short position.

� Equity holders have limited downside riskbut unlimited upside potential.

� The main application of this Merton�s op-tion pricing framework is to estimate defaultprobabilities. This is implemented in theKMV credit monitor system.

� KMV applies Black-Scholes formula in re-verse.

� Conventionally, we observe price of the un-derlying, strike price, etc and we calculate


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the value of the derivative. In this applca-tion, we begin with value of the derivative(value of equity E0 as a call option) andgiven strike F ; and calculate the unobservedvalue of A0:

� Another complication: we need the value of�A but since we do not observe value of A,we have to infer �A from volatility of returnson equity �E:

� The Black-Scholes gives the value of equitytoday as

E0 = A0N(d1)� Fe�rTN(d2) (3)

where

d1 =ln(A0=F ) + (r + �A=2)T

�ApT

andd2 = d1 � �A

pT

� The value of debt today is

D0 = A0 � E0


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� To calculate above we need A0 and �A (un-knowns).

� Using Ito lemma, we yield the relationshipbetween the volatilities at t = 0

�EE0 =@E

@A�AA0 (4)

� Here @E=@A is the delta of the equity. So@E=@A = N(d1)

� So we have two equations (3 and 4) and twounknowns (A0 and �A).

� We can use Excel Solver to obtain the solu-tion to both these equations.

� Now we can calculate the default probabili-ties (called Expected Default frequency, EDF)as

pi = P [AT < F ]

� This can be shown to be

pi = P [Zi < �DD]


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where Zi � N(0; 1) and DD is the �distanceto default�(i.e. number of standard devia-tions the �rm�s assets are away from default)

DD =ln(A0=F ) + (�A � �A=2)T

�ApT

where �A is the mean return/growth rate ofthe assets.


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