Financial (and Commodity)...
Transcript of Financial (and Commodity)...
1
Financial (and Commodity) Derivatives
RNDr. Jiří
Witzany, Ph.D.([email protected], NB 178, Tuesday 10-12)
2
Literature
Requirement ISBN Title Author Year of Publication
Required1 0-13-149908-4
Options, Futures, and Other Derivatives, 789 p.
Hull, John C. 2006, 6th edition
Optional 978-80-245-1274-7
International Financial Markets, Oeconomia VŠE, 180 p.
Witzany, J. 2007
Optional 80-245-1033-2
Deriváty, 297 s. Dvořák, Petr 2006
Optional 80-247-1099-4
Finanční a komoditní deriváty v praxi, 630 s.
Jílek, Josef 2005
Optional 0-471-96717-3
Financial derivatives in theory and practice, 393 p.
P.J.Hunt, J.E. Kenedy
2000
1) The course should cover Chapters 1-15 from John Hull
3
Content
•
Introduction –
principles of financial derivatives
•
Future and forward markets•
Determination of forward and futures prices
•
Interest rate and currency swaps•
Mechanics of options markets
4
Content - continued
•
Modeling of market rates and valuation of options
•
Options on stock indices, currencies, and futures (Greek letters, Delta-hedging etc., volatility smiles etc.)
•
(Credit risk and credit derivatives, exotic, weather, energy, and insurance derivatives if time allows)
6
Introduction Principles of Derivatives
•
A derivative contract can be defined as a financial instrument whose value depends (is derived from) the values of other, more basic underlying variables (financial assets -
FX rates, interest rates,
stock prices, bond prices, commodities, weather,..)•
Derivatives are settled at a certain future time
•
Derivative markets –
trading with pure risk – physical settlement of underlying assets often
eliminated•
Derivative markets are
often
more liquid than the
spot markets (e.g. commodity futures) –
spot prices „derived“ from derivative prices
7
Interest Rate Derivatives Market Development
•
Derivative markets have become increasingly important during the last 25 years
9
Exchange-Traded Markets versus OTC Markets
•
Standardized
derivative
contracts
defined by the
exchange
•
The
contracts
are always
between
the exchange
and
a participant
•
OTC derivatives
generally
between
any
two subjects
•
Derivative
Exchanges
exist
since
19th century
(Chicago Board
of
Trade
–
futures
like
contracts,…)
10
OTC versus Exchange Market Development
Source: John Hull, Options, Futures, and
Other
Derivatives, 6th edition
15
Basic Derivative Types – Forward Contracts•
Forward contract –
an agreement to buy or
sell an asset at a certain future time for a certain price
Source: John Hull, Options, Futures, and
Other
Derivatives, 6th edition
Payoff=K-STPayoff=ST
-K
16
Forward contracts •
Arbitrage-less market … any possible arbitrage opportunity quickly disappears in an efficient market
•
FX forward transaction can be equivalently achieved as a combination of a reverse FX spot and two deposits
–
relatively straightforward
pricing•
Generally for FX direct quotations FC/DC:
360)(1
3601
360)(
1
3601
3601 drrdr
drr
dr
dr
spotFXforwardFX
FCDC
FC
FCDC
FC
DC
17
Domestic
Currency
Foreign
Currency
T+2 T+2+d
2. Buy
FC at
S
1. Borrow
at
rDC
3. Deposit at
rFC
4. Sell
FC at
F
FX Forward Arbitrage Argument
5. Repay
the
loan
18
FX Forward pricing
•
Example: Assume that the spot rate EUR/PLN equals to 3,76. Estimate the 6 M (months) forward exchange rate if the 6 M interest rates in EUR and PLN are 4,35% and 4,85%.
•
Solution:
769,31,02191,0244*76,3
3601810,04351
3601810,04851
*76,3
360dr1
360dr1
SFEUR
PLN
19
Other Basic Derivative Types
•
Futures contracts –
similar to forwards but traded on an exchange (Chicago Board of Trade,…), standardized, margins to cover daily P/L
•
Options –
right to buy/sell certain asset (stocks, currencies,…) by (at) a certain date for a certain price –
call/put options, strike/exercise price,
expiration/maturity, European/American options, traded OTC (FX,..) or on an exchange (Chicago Board Options Exchange,…), binary, barrier,…
20
Options • Example: profit/loss
on call/put options
Source: John Hull, Options, Futures, and
Other
Derivatives, 6th edition
21
Other Derivative types•
Swaps: two parties agree to a periodic exchange of certain cash flows –
interest rates, equity returns,
nominals
in different currencies etc., always OTC•
FRA –
Forward Rate Agreements –
payoff is
defined as the difference between agreed and future interest rates
-
OTC
•
Credit derivatives: the payoff on the creditwothiness
of one or more companies or
countries –
credit default swaps, CDO,…
-
OTC•
Other underlyings: weather derivatives (daily temperature), energy derivatives -
crude oil,
natural gas, electricity, …
22
Types of Traders
•
Hedgers –
fundamental need to reduce
(insure) risks
•
Speculators –
use derivatives as an easy way to take a position/speculate on the market –
for
example the „hedge funds“•
Arbitrageurs –
combine different products in
different markets to lock in a risk-less profit –
see e.g. the relationship between spot and forward prices –
only small arbitrage opportunities exist
23
Speculation example•
$2000 to speculate with: The Amazon.com stock $20, 2-month call option with strike $22,50 is sold for $1
Source: John Hull, Options, Futures, and
Other
Derivatives, 6th edition
24
Derivatives Risk Management•
Derivatives can be used to take huge risk with minimal initial investment
•
Future derivatives settlement may allow to hide transactions for some time
•
Nick Leeson
–
Barrings
Bank –
Singapore office –
1 billion dollar loss
•
Jerome Kerviel
–
Societte
General –
Equity Index Derivatives –
5 bln
EUR loss
•
UBS, Merill
Lynch, Morgan Stanley, AIG… –
CDOs
– 100s bln
USD losses
•
Derivatives can be compared to electricity –
very useful but dangerous
Need for high quality risk management –
risk limits on exposures, products, counterparties etc.
26
Futures Markets•
Financially
similar
to forwards
but
exchange
traded
•
Futures exchange/clearinghouse stands between the market participants
(compare to forwards)
•
Majority of future contracts do not lead to physical delivery
Futures Exchange
Counterparties
witha long
position
Counterparties
witha short
position
+N -N
28
Specification of a Futures Contract
•
The asset: commodities must be exactly specified•
The contract size: depend on average user and delivery costs
•
Delivery arrangements: cash or physical, important for commodities (cattle, lumber, cotton…). The party with short position usually files a notice of intention to deliver –
selection of
location etc.•
Delivery time: usually end of month, or a longer delivery period
•
Price quotes conventions: e.g. $/barrel with two decimal places; price and position limits
29
Daily Settlement and Margins•
Settlement/counterparty risk: counterparty does not deliver
•
Margin mechanism –
daily financial settlement •
Initial Margin –
set so that daily losses are covered with
high probability (e.g.99,5%)•
Daily P/L (variation in futures price) is debited/credited on the margin account
•
Maintenance
margin (around 75% of the initial margin) must be always maintained
•
If the balance falls bellow the maintenance margin there is a margin call, the account must be topped to the initial margin level
•
The account balance is release when the position is closed out or settled
30
Daily Settlement and Margins•
Example: +2 (100 ounces) gold futures
contracts,
price
in $ per troy oz.
(Initial
Margin
$2000, Maintenance Margin $1500 per contract)
Source: John Hull, Options, Futures, and
Other
Derivatives, 6th edition
32
Hedging Using Futures
•
If future P/L depends linearly on the future price of an asset then futures contracts can be used for a perfect hedge
•
P/L = N x P, P future price of a unit of the asset, N positive or negative / the sensitivity
•
Take futures position equivalent to -N units of the asset (short for N positive, long otherwise)
•
Then
P/L = N x P -
N x P = 0 for the total position after the futures contract settlement
•
There are arguments for and against hedging
33
Hedging Using Futures
•
Example: A farmer plans to sell his cattle on a local market one year from now, let us say in August 2008. The market prices of live cattle are quite volatile so the farmer decides to use ten live cattle futures contracts to fix his selling price. The table on the next slide shows quoted live cattle futures (trade unit is 40 000 pounds and the price is in cents per pound). Propose an effective hedging strategy for the farmer.
35
Basis Risk•
Basis = spot price of the asset –
futures price of
the contract = S –
F = b … due
to: time, not exactly identical assets
• The
effective
hedging
price
= F1
+b2
= F1
+(S2
-F2
)
Source: John Hull, Options, Futures, and
Other
Derivatives, 6th edition
36
Cross Hedging
•
The hedged asset and the futures asset are not same, but the prices are correlated
•
For example hedging the price of jet fuel using heating oil futures, or hedging FX P/L from on equity investment using FX futures
•
Find the OLS regression coefficient: S= h* F + , i.e. h=
* S
/ F•
The hedge ratio h minimizes
the variance of P/L
caused by S
37
Cross Hedging Example
h = 0,78… using
elementary
statistics
formulas
Source: John Hull, Options, Futures, and
Other
Derivatives, 6th edition
40
Hedging an Equity Portfolio Example
•
Value of S&P 500 index = 1000•
Value of the hedged portfolio = $5
mil.
•
Risk-free interest rate = 10% p.a.•
Index dividend yield = 4% p.a.
•
Beta of the hedged portfolio = 1,5•
Use 4-month S&P 500 futures (currently valued at 1020,20) to hedge the value of the portfolio.
•
Simulate the outcome using CAPM given different values of the index in 3 month (800, 900, 1100
etc.).
41
Hedging an Equity Portfolio Example
Source: John Hull, Options, Futures, and
Other
Derivatives, 5th edition
42
Hedging an Equity Portfolio•
Index futures can be used in case the portfolio manager assumes a temporary decline of the market,
•
or wants to speculate against the benchmark (index),
•
or wants to adjust the beta of the portfolio, •
or wants to take a temporary position without investing into the stocks etc.
•
Index futures can be also used to hedge a single stock against the market volatility
43
Rolling the Hedge Forward•
Example of April 2004 –
June 2005 hedge using three
6 month futures rolled forward (short positions)
Source: John Hull, Options, Futures, and
Other
Derivatives, 6th edition
44
Rolling the FX Hedge Forward
•
Example: You run a CZK denominated money market fund for US investors. Show how to use 6 months forwards for a continuous FX hedging of the portfolio. The initial value of the portfolio is 1.7 bln
CZK, and the expected growth is 4%
annually. Give an example if the exchange rate development is
e.g.
17.00, 17.50, 18.00, 18.70,
18.50 CZK/USD in 6 months periods.
46
Interest Rates
•
Present
value
–
cashflow discounting
–
key concept
in derivatives
valuation
•
Risk free rate
x credit
margin•
Treasury rates
and
LIBOR rates
•
Different
interest
rate
conventions•
Different
compounding
frequency
•
Continuous
compounding
FV=AeRt
•
Bond pricing
and
zero
rates
-
bootstrapping
48
Bootstrapping Example
•
Calculate
EUR 6M, 1Y, 2Y, and
3Y zero coupon
discount
rates
in continuous
compounding
given
the
following
bid
rates:
49
Forward interest rates
•
Forward
rates
can
be
derived
from
the
zero
rates•
eR(T)T
= eR(t)t eR(t,T)(T-t)
… solve
for
R(t,T), the
forward
interest
rate
from
t to T•
FRA –
forward
rate
agreement
–
cash settlement
of
the
difference
between
agreed
rate
RK
and
the actually
observed
rate
RM
(Libor) –
usually
at
the beginning
of
the
interest
period
•
Valuation
of
FRA: V=L(RK
-RF
)(T2
-T1
)e-R2.T2, if RK
is
the
contracted
interest
earned, RF
is
the forward
rate
from
T1
to T2
at
the
time
of
valuation
52
Forward and Futures Pricing
•
Forwards
and
Futures have
different
settlement, however
the
price
are very
close
•
Investment
x consumption
assets
… arbitrage argument can
be
used
only
for
investment
assets
•
Short
selling
–
possible
if
the
asset
can
be borrowed
–
in case of
stock
dividends
must
be
paid
to the
owner•
A forward
contract
can
be
replaced
by
shortselling/investing
into
the
asset
and depositing/borrowing
the
corresponding
amount
54
Forward Pricing
•
If
the
current
price
of
an
investment
asset with
no income
is
S0
then
the
forward
price F0
=S0
erT
•
If
the
investment
asset
provides
income with
a present
value
I, then
F0
=(S0
–I)erT
•
If
the
investment
asset
provides
known yield
q with
continuous
compounding
then
F0
=S0
e(r-q)T, r-q is
also
called
the cost of carry
55
Domestic
Currency
Underlying
Asset
Spot Market Forward Market
Buy
the
asset
at
S
Financing
Cost
Storage
Cost
-
Income
Sell
the
asset
at
F
General
Forward
Arbitrage Argument
57
Market Value of a Forward
•
Initially
the
value
of
a forward
contract
is
zero, later
it
may
be
positive or
negative
•
f=(F0
-K)e-rT, where
K is
the
contracted
price, F0 the
actual
forward
price, and
T time
to delivery
from
today.•
Notice
that
futures contracts
settle
F0
-K, not (F0
- K)e-rT
•
It
can
be
shown
however
that
forward
and
futures prices
are theoretically
same
if
the
margin
account
yields
the
market rate
58
Futures on Stock Indices
•
The
underlying
index is
assumed
to provide an
expected
yield
q (which
should
represent
the
average
annualized
dividend yield during
the
life
of
the
contract)
•
Then
F0
=S0
e(r-q)T
•
The
average
dividend yield
q is
usually lower than then r (F0
>S0
), but
may
be
also higher
in some
periods
(F0
<S0
)
59
Forwards and Futures on Currencies
•
F0
=S0
e(r1-r2)T, if
the
exchange
rate
is
quoted
as currency1 for
one
unit of
currency2, i.e. currency2
plays
the
role of
an
investible
asset
with
the
yield
r2
.•
Example:
61
Futures on Commodities
•
Commodities
generally
have
storage
costs
and
do not provide
income, except
gold
and
silver.
•
Storage
costs
U (present
value) can
be
treated
as negative income, hence F0
=(S0
+U)erT
•
Some
assets
like
crude
oil
may
provide
so
called convenience
yield
y, then
F0
=(S0
+U)e(r-y)T= S0
e(r+u-y)T
•
Futures prices
of
oil
tend
to decrease, i.e. r+u<y, conevenience
yield
–
possible
shortages
62
Futures price versus expected spot price•
Long
futures position
is
equivalent
to investment
into
the
asset•
An
investor generally
require
extra return
for
systematic
risk•
If
k is
the
required
rate
of
return
on the
asset
then
it
follows
F0
=E(ST
)e(r-k)T, hence F0
<E(ST
) if
k>r•
This is the case in particular for futures on stocks or indexes
–
positive beta –
systematic
risk
•
If the asset has negative systematic
risk, then
F0
> E(ST
)
65
Interest
Rate
futures Eurodollar
futures
•
Eurodollars
are dollars
deposited
outside
of
the United
States
•
Three
month
Eurodollar
futures (CME) contracts 3-month
interest
rate. Delivery
March-June-
September-December
up
to 10 years
in the
future•
The
quote
= 100 –
r, where
r is
the
annualized
3-
month
rate
in the
Act/360 convention. One
basis point is
equivalent
to $25 settlement amount.
•
Example: Use futures contract
to lock
3-month interest
rate
earned
on $1 mil. one
year
later
68
Long Term Interest Rate Futures
•
Day count
conventions: Act/Act
(Treasuries), 30/360 (Corporate
bonds), Act/360 (Money
market)•
Bond prices: cash price
= clean
price
+ accrued
interest•
Treasury bond futures are quoted
in the
same
way
as the
bonds
(110-03 means
110 and
3/32
of the face value
$100 000)
•
Any Treasury
bond with
at
least
15 years
to maturity can
be
delivered
using so called
conversion factors (6% YTM convention)
72
Hedging Example
•
We
hold 100 T-Bonds
currently
priced
at 95% (Nominal=$100 000), with 20 years to
maturity and duration 15 years. We need to sell the bonds in 2 months.
•
Propose an appropriate number of 3 months futures contracts on 30Y T-Bonds to hedge the price provided the quoted futures price is 90%, the current conversion factor for the CTD is 1.3, and its duration is 10.
73
Are forward
and
futures prices equal?
•
Yes, if
we
assume
that
the
interest
rates
are constant, or
at
least
sufficiently
independent
on underlying
asset
prices•
This
assumption
does
not hold for
interest
rate
futures and
forwards•
IR futures rates
are higher
that
IR forward
(FRA) rates
due
to the
discounting difference
and
the
convexity
adjustment
•
Forward
rate
= Futures rate
-
2T1
T2
/2
75
Swaps
•
Exchange of
cash flows
in the
future•
„plain
vanilla“ interest
rate
swaps
•
Currency
swaps
Source: John Hull, Options, Futures, and
Other
Derivatives, 5th edition
76
Interest Rate Swap Example
Source: John Hull, Options, Futures, and
Other
Derivatives, 5th edition
77
Interest Rate Swap Example•
Intel transforms
fixed
5,2% payments
to Libor +
0,2%•
Microsoft transforms
Libor + 0,1% payments
to
5,1% fixed
payments•
Assets
can
be
transformed
as well
78
Role of Financial Intermediary
•
Banks
play the
role of
intermediaries•
The
banks
moreover
exchange
„the
positions“
mutually, some
play the
role of
market-makers•
Banks
use so
called
ISDA master agreements
81
Interest Rate Swap Example – Valuation of IRS
Source: John Hull, Options, Futures, and
Other
Derivatives, 5th edition
• Can
be
regarded
as an
exchange
of
a fixed-coupon
bond (FCB) fora floating
rate
note, MV = Value
of
the
FRN –
Value
of
the
FCB
85
Currency Swaps (CCS)•
Exchange of
interest
payments
(usually
fix-fix) in two
currencies•
Exchange of
principals
at
the
beginnig
and
at
the
end
of
the
contract•
Valuation
similar
to IRS
•
Credit
risk of
CCS is
higher
than
that
of
IRS
87
Underlying assets•
Call/Put, long/short
position, European/American,
in/at/out
of
the
money, intrinsic/time
value•
Stock
options
–
mostly
exchange
traded
–
CBOE,
PHLX, AMEX, PACIFEX, EUREX•
Options
on indices
–
exchange
traded
–
cash
settlement•
Currency
options
-
exchange
traded
and
OTC
•
Options
on futures contracts
–
exchange
traded
– options
to acquire
long
or
short
position
in a
futures contract•
Expiration
date
and
strike price
is
defined
–
options
are traded
at
a premium
91
Margins
•
Investors
writing
an
option
(short
position) must maintain
a margin
•
Naked
option
is
an
option
written
without
the offsetting
position
in the
underlying
stock
–
the
margin
than
must
be
at
least
10%, resp. 20% of
the underlying
price
•
Writing
covered
calls
–
the
underllying
is
already owned
•
An
Option
Clearing Corporation
usually guarantees
the
settlement
92
Factors affecting option prices
•
The
current
stock
price, S0
•
The
strike price, K•
The
time
to expiration, T
•
The
volatility of
the
stock
price, •
The
risk-free interest
rate, r
•
The
dividends
expected
during
the
life
of the
option
96
Factors affecting option prices
Source: John Hull, Options, Futures, and
Other
Derivatives, 5th edition
97
Assumptions and notation•
No transaction
cost, equal
taxes, borrowing
and
lending
at
the
risk-free rate, no arbitrage
opprotunities, no bid-ask spreads
•
Notation:
98
Upper and lower bounds for option prices•
A call option
can
never
be
worth
more than
the
stock
S0•
A European
put option
can
never
be
worth
more than
the
discounted
strike price
Ke-rT
•
A European
call option
on non-dividend stock
is
worth
at
least
S0 - Ke-rT
•
A European
put option
on non-dividend stock
is
worth
at
least
Ke-rT
- S0
99
Put-Call Parity
•
Portfolio A: one
European
call option
plus cash Ke-rT
•
Portfolio B: one
European
put option
plus one
stock
•
The
value
of
both
portfolios
at
T is max(K,ST
)•
Hence c+Ke-rT
= p+S0
, otherwise
there
is
an arbitrage
opportunity
100
Put-Call Parity Example
• Strategy: short
B, buy
A
Source: John Hull, Options, Futures, and
Other
Derivatives, 5th edition
101
Trading strategies involving options•
Different
strategies
combining
a long/short
position
in a stock
and
a long/short
position in a put/call option
•
More complex
strategies
involve
two different
options
•
Bull/Bear/Butterfly/Calendar/Diagonal spreads, straddles, strips, straps, strangles
107
Pricing of options – Binomial trees•
Example: European
call option
to buy
a
stock
in three
months
for
$21
Source: John Hull, Options, Futures, and
Other
Derivatives, 5th edition
108
One step binomial tree example
•
It
is
possible
to set up
a riskless
portfolio combining
a position
in the
stock
and
in the
option
•
The
portfolio is
riskless
if
the
profit/loss
is
the same
in all
(both) scenarios
•
Long: 0,25 shares•
Short: 1 option
•
The
value
of
the
portfolio is
$4,5 in three
months in both
cases
•
Hence the
present
value
of
the
option
is
calculated as $5 minus discounted
$4,5
110
Risk-neutral Valuation
•
Note, that
the
result
does
not depend
on probabilities
of
the two
scenarios
•
However
if
we
set up
p, probability of
the
movement
up, so
that
E(ST
)=S0
eRT (the
stock
return
equls
to the
risk-free rate) then
it
turns
out
that
the
price
of
the
option
equals
to
the
discounted
expected
pay-off•
Risk neutral valuation principle: we can assume that the world is risk neutral when pricing an option. The result is valid also in the real world which is not risk-neutral!
114
Delta
•
Riskless
portfolio: -1 call option, +
stock•
The
change
in the
option
price
is
offset by
* the
change
of
the
stock
price•
Delta equals
to 25% in the
one-step
binomial
tree
example•
Delta changes
in a two-step binomial
tree
115
Volatility•
Volatility
is
measured
as the
standard deviation
of
the
return
(with
expected
value
) normalized
by the
square root
of
the
time•
u and
d are usually
derived
from
those
parameters
•
N-step binomial
tree
can
be
then
used
for
a Monte-Carlo simulation
When
t is
small
116
Behaviour of Stock Prices
•
Stochastic
process
–
variable
whose
value
depends on time
and
changes
in an
uncertain
way
•
Discrete/continuous
time, discrete/continuous variable
•
Markov
process
–
only
the
present
value
of
a variable
is
relevant
for
the
future
•
Market rates
(stock
prices, exchange
rates, interest rates) are assumed
to follow
the
Markov
process
(x technical
analysis)
117
Wiener process
•
Continuous
time
process: any
time
period can
be divided
into
arbitrary
number
of
steps
•
Wiener
process
(Brownian
motion) –
Markov process
where
z(1)-z(0) has distribution
N(0,1) and
the
distributions
are „uniform“ for
smaller
time
steps•
The
square root
of
time
rule: z(T0
+t)-z(T0
) has the distribution
N(0,t)
•
Stochastic
difference
equation: dz=dt, where
is randomply
taken
with
the
distribution
N(0,1)
•
Generalized
Wiener
process: dx=adt
+ bdt, i.e. x(T0
+t)-x(T0
) has the
distribution
N(at,bt)
120
Ito‘s process
•
dx
= a(x,t)dt
+ b(x,t)dz, where
dz
is
the
Wiener process
•
The
drift and
the
variance depend
on x and
t•
The
process
for
Stock
prices
S: normally
distributed
annualized
rate
of
return
with
expected value
and
standard deviation
(observed
for
a
small
time
periods, not one
year)•
dS= Sdt
+ Sdz
(geometric
Brownian
motion)
•
It
turns
out
that
S(1)-S(0), or
S(1)/S(0)-1, are not normally
distributed
… lognormal
distribution
123
Ito‘s Lemma
•
If
G=G(x,t) where
x follows
an
Ito‘s process
dx
= a(x,t)dt
+ b(x,t)dz, then
G
follows
the
Ito‘s process
•
Where
dz
is
the
same
as above
124
Application to forward contracts
•
Forward
price
of
a non-dividend paying stock
F=Ser(T-t)
•
Using
the
Ito‘s lemma it
follows
that
the process
for
F is:
125
Lognormal property
•
Let G=lnS, where
S follows
the
geomentric Brownian
motion, then
applying
the
Ito‘s
lemma we
get:
•
Consequently
lnS(T)-lnS(0) has a normal distribution
N((-2/2)T, T)
129
Black-Scholes differential equation•
Principle: set up
a risk-less
combination
of
an
option
and
the
underlying
stock
–
the price
of
the
option
f depends
on the
underlying
S and
t –
use the
derivative
– delta hedging
•
The
return
on this
portfolio in a short
time period must
be
equal
to the
risk free interest
rate
130
Black-Scholes differential equation
Source: John Hull, Options, Futures, and
Other
Derivatives, 5th edition
134
Black-Scholes differential equationHence the
portfolio is
risk less, as expected, and
so
And we
get
the
Black-Scholes-Merton
differential
equation:
135
Risk-neutral valuation•
The
Black-Scholes-Merton
equation
does
not
depend
on the
expected
return
, i.e. on investors risk preferences!!!
•
We
can
assume
that
we
are in a risk-neutral
world.•
The
result
will
be
the
same
as in the
real
world
with
risk sensitive investors.•
Consequently
we
can
simply
discount
the
expected
pay-off
of
an
option
using
the
risk free rate.
•
The
result
will
be
the
unique
solution
of
the differential
equation
136
The Black-Scholes FormulaConsider
a European
call option, then
in a risk neutral
world
The
value
of
the
the
option
at
expiration
is:
And the
price
of
the
option:
Using
the
model for
the
Stock
price
at
time
T:
137
The Black-Scholes Formulawhere
and
N(x) is
the
cummulative
standardized
normal
distribution
Similarly
fora put option:
138
The Greeks
•
Partial
derivatives
of
an
option
(portfolio) market value
measure
sensitivity with
respect
to the
relevant
variables•
Delta, Gamma –
the
1st and
the
2nd derivatives
w.r.t. the
underlying
asset
price•
Vega –
the
derivative
w.r.t. the
volatility variable
•
Rho –
the
derivative
w.r.t the
interest
rate•
Theta –
the
derivative
w.r.t. to time, ususally
measured
as the
change
of
value
„per day“•
The
Greeks
are used
for
hedging… Delta-hedging,
Vega-hedging, Gamma-hedging e.t.c