13 th Nov, 2007Kings College, London Break Even Volatilities Dr Bruno Dupire Dr Arun Verma...

13th Nov, 2007 King’s College, London

Break Even Volatilities

Dr Bruno Dupire

Dr Arun Verma

Quantitative Research, Bloomberg LP


Theoretical Skew from Prices

Problem : How to compute option prices on an underlying without options?

For instance : compute 3 month 5% OTM Call from price history only.

1) Discounted average of the historical Intrinsic Values.

Bad : depends on bull/bear, no call/put parity.

2) Generate paths by sampling 1 day return re-centered histogram.

Problem : CLT => converges quickly to same volatility for all strike/maturity; breaks auto-correlation and vol/spot dependency.

?

=>


Theoretical Skew from Prices (2)

3) Discounted average of the Intrinsic Value from re-centered 3 month histogram.

4) Δ-Hedging : compute the implied volatility which makes the Δ-hedging a fair game.


Theoretical Skewfrom historical prices (3)

How to get a theoretical Skew just from spot price history?

Example:

3 month daily data

1 strike – a) price and delta hedge for a given within Black-Scholes

model– b) compute the associated final Profit & Loss: – c) solve for– d) repeat a) b) c) for general time period and average– e) repeat a) b) c) and d) to get the “theoretical Skew”

1TSkK

PL 0/ kPLk

t

S

1T 2T

K1TS


Zero-finding of P&L


Strike dependency

• Fair or Break-Even volatility is an average of returns, weighted by the Gammas, which depend on the strike


Alternative approachesShifting the returnsA simple way to ensure the forward is properly priced is to

shift all the returns,. In this case, all returns are equally affected but the probability of each one is unchanged. (The probabilities can be uniform or weighed to give more importance to the recent past)


Alternative approaches

Entropy method• For those who have developed or acquired a taste for

equivalent measure aesthetics, it is more pleasant to change the probabilities and not the support of the measure, i.e. the collection of returns. This can be achieved by an elegant and powerful method: entropy minimization. It consists in twisting a price distribution in a minimal way to satisfy some constraints. The initial histogram has returns weighted with uniform probabilities. The new one has the same support but different probabilities.

• However, this is still a global method, which applies to the maturity returns and does not pay attention to the sub period behavior. Remember, option pricing is made possible thanks to dynamic replication that grinds a global risk into a sequence of pulverized ones.


Alternate approaches: Fit the best log-normal


Implementation detailsTime windows aggregation• The most natural way to aggregate the results is to simply average

for each strike over the time windows. An alternative is to solve for each strike the volatility that would have zeroed the average of the P&Ls over the different time windows. In other words, in the first approach, we average the volatilities that cancel each P&L whilst in the second approach, we seek the volatility that cancel the average P&L. The second approach seems to yield smoother results.

Break-Even Volatility Computation• The natural way to compute Break-Even volatilities is to seek the

root of the P&L as a function of . This is an iterative process that involves for each value of the unfolding of the delta-hedging algorithm for each timestep of each window.

• There are alternative routes to compute the Break-Even volatilities. To get a feel for them, let us say that an approximation of the Break-Even volatility for one strike is linked to the quadratic average of the returns (vertical peaks) weighted by the gamma of the option (surface with the grid) corresponding to that strike.


Strike dependency for multiple paths


SPX Index BEVL <GO>


New Approach: Parametric BEVL

• Find break-even vols for the power payoffs• This gives us the different moments of the

distribution instead of strike dependent vol which can be noisy

• Use the moment based distribution to get Break even “implied volatility”.

• Much smoother!

32 ,SS


Discrete Local Volatility

Or

Regional Volatility


Local Volatility Model

Given smooth, arbitrage free , there is a unique :

Given by

tS , TKTKC ,, 0

0

,

,TKT CKSE

dWtSdS

2)

1)

TKKC

TKTC

TK,

,2,

2

22

GOOD

BAD• Requires a continuum of strikes and maturities

• Very sensitive to interpolation scheme

• May be compute intensive

(r=0)


Market facts


S&P Strikes and Maturities

T

KS

ept 0

7

Oct

07

Dec

07

Mar

08

Jun

08

Dec

08

Mar

09

Jun

09

Aug

07


Discrete Local Volatilities

002211 ,,, TKTTiTK CC

i

1TSK

Price at T1 of :2,TK

C

Can be replicated by a PF of T1 options: of known price 1,TKi i

C

K 1TS

f

iK

K

1T 2T

00 ,TS

21 ,TT


Discrete Local Volatilities

f

02,TK

C

DTK ,

Discrete local vol: that retrieves market priceDTK ,


Taking a position

• Local vol = 5%

• User thinks it should be 10%


• Buy , Sell

P&L at T1

1,

%5 TKi iC

2,TKC


P&L at T2

• Buy , Sell 1,

%10 TKi iC

2,TKC


Link Discrete Local Vol / Local Vol

is a weighted average of

with the restriction of the Brownian

Bridge density between T1 and T2

Assume real model is: dWtSdS ,

DTK ,

1T

K

00 ,TS

2T

Market prices tell us about some averages of local volatilities - Regional Vols


Numerical example


Crude approximation:

for instance constant volatility

(Bachelier model)

does not give constant discrete local

volatilities:

Price stripping

Finite difference approximation:

2,,,

,,

2

22

2

22

,

,2,

K

CCCT

CC

C

C

TKKC

TKTC

TKTKTKKTKK

TKTTK

KK

T

dWdS

TK


Cumulative Variance

TV TKTK2,,

• Naïve idea:

dttKVVT

T

TKTK 2

1

12,2

,,

1T 2T

00 ,TS

K K

1T 2T

00 ,TS

K'K

• Better approximation:

dttKKTT

TtKVV

dttSKT

tSV

T

T

TKTK

T

TK

2

1

12,''

,

12

12,',

0

002

,


Vol stripping

• The approximation leads to

• Better: following geodesics:

dttSKT

tSV

T

TK

0

002

, ,

K

V

T

SK

T

V

u

VTK

02 ,

T

SKu 0

1where

dtttfVT

TKTK 0

,2

, ,

K

VTf

T

V

u

VTK TK

',

2 , where

Tfu

TK',

1

Anyway, still first order equation


Vol stripping

The exact relation is a non linear PDE :

2

222

002

2

1

4

1

21,

K

V

K

V

VV

KS

K

V

V

KSTK

T

V

• Finite difference approximation:

• Perfect if

VVVV

KSV

VKS

VTK

KKKK

T

21

41

21

,2

2

00

2

FDTKdWdS ,:

TK


Price Stripping Vol Stripping

BS prices (S0=100; =20%, T=1Y) stripped with Bachelier formula th=.K

Numerical examples

thestimated

K


estimatedth

1

2

3

VT

SKVTK KT 02 ,

1

2

3 VV

VVKS

VVKS

VTK

KKKK

T

21

41

21

,2

2

00

2

V

V

KSV

TK

K

T

0

2

1,

KT

Accuracy comparison

(linearization of )3


Interpolate from

with

Local Vol Surface constructionFinite difference of Vol PDE gives averages of 2, which we use to build

a full surface by interpolation.

TK ,2

jT 1jT

iK

1iK

2iK

1iK

2iK

1jT

2

1,1,11,1

2

,,1,1

1,11,1,1,1

,1,

22

2

1

222

1

K

VVV

K

VVVV

K

VV

K

VVV

T

VVV

jijijijijijiKK

jijijijiK

jijiT

VVVV

KSV

VKS

V

TTK

KKKK

T

jji

21

41

21

2,

22

00

12

jiji TKVV ,, (where )


Reconstruction accuracy

• Use FWD PDE to recompute

option prices

• Compare with initial market price

• Use a fixed point algorithm to correct for convexity bias

2

22

2 K

C

T

C


Conclusion

• Local volatilities describe the vol information and correspond to forward values that can be enforced.

• Direct approaches lead to unstable values.

• We present a scheme based on arbitrage principle to obtain a robust surface.

13 th Nov, 2007Kings College, London Break Even Volatilities Dr Bruno Dupire Dr Arun Verma...

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