option prices and the black-scholes-merton formula

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Option prices and the Black-Scholes-Merton formula

Gabor Molnar-Saska

3 October 2006

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Morgan Stanley

Morgan Stanley is a leading global financial services firm, offering a wide variety of products and services. A partial list of these products and services includes:

Investment banking services such as advising, securities underwriting

Institutional sales and trading, including both equity and fixed income investments

Research services

Individual investor services such as credit, private wealth management, and financial and estate planning

Traditional investments such as mutual funds, unit investment trusts and separately managed accounts

Alternative investments such as hedge funds, managed futures, and real estate

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Morgan Stanley

Morgan Stanley is an industry leader in underwriting Initial public offerings of stock worldwide.

Morgan Stanley reported net revenues of $52.498 billion in 2005.

Morgan Stanley ranks as the 30th largest U.S. corporation in 2005.

In 2004, Morgan Stanley held the #1 industry rank for the following categories: Global Equity and Equity-Related Underwriting Market Share, Global IPO Market Share, and Global Equity Trading Market Share.

Morgan Stanley had 53,760 total employees worldwide as of August 31, 2005.

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The binomial model

1

2

0.5

S0 is the initial stock price (at time t=0)S1 is the stock price at time t=1

Assume P(S1=2)=0.5 and P(S1=0.5)=0.5

r: continuously compounded interest rate, i.e. 1$ at time zero will grow to exp(rt)(assume now r=0)

Call option: strike K=1payout (S1-K)+

What is the price of this option?

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The binomial model

1

2

0.5

Buy a portfolio consisting of 2/3 of a unit of stock and a borrowing of 1/3 of a unit of bond.

The cost at time zero: 2/3*1$-1/3*1$=0.33$

After an up-jump: 2/3*2$-1/3*1$=1$After a down-jump: 2/3*0.5$-1/3*1$=0$

The correct price: 0.33$

X=1

X=0

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The binomial model

s0

s1up

s1down

Consider a general portfolio (a,b)

The cost at time 0: as0+bB0

After an up-jump: as1up+bB0exp(rt)=Xup

After a down-jump: as1down+bB0exp(rt)=Xdown

Xup

Xdown

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The binomial model

The value of the portfolio:

downup

downup

ss

XXa

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downup

updownupup

ss

sXXXrtBb

11

110 )exp(

downup

updownupup

downup

downup

ss

sXXXrt

ss

XXsV

11

1

110 )exp(

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The binomial model

0<q<1 and the value of the portfolio is

downup

down

ss

srtsq

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10 )exp(

updown qXXqrtV 1)exp(

Expectation under a new measure!

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The binomial tree model

Si is the value of stock at time i (binomial tree model)

Bi is the value of the bond at time i.

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The binomial tree model

Find a new measure under which Zi = Bi-1Si is a martingale (Q),

where Bi-1 is the discount process.

Binomial representation theorem: Suppose Q is such that the binomial price process Z is a Q-martingale. If N is any other Q-martingale, then there exists a previsible process such that

i

kkki ZNN

10

where is the change in Z from tick time i-1 to i, and

is the value of at the appropriate node at tick-time i. 1 iii ZZZ i

)( 1iTQi FXBEN Let

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The binomial tree model

Consider the following construction strategy: at tick-time i, buy the portfolio Hi, consisting of

• ai+1 units of the stock

• bi+1=Ni-ai+1Bi-1Si units of the cash bond

At time zero the portfolio worth a1S0+b1B0=N0=EQ(BT-1X)

One tick later: a1S1+b1B1=B1(N0+a1(B1-1S1-B0

-1S0))=B1N1

Self-financing strategy,

At the end we have BTBT-1X=X, whatever actually happened to S.

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The binomial tree model

Conclusion:

The price of the claim X is

)( 1XBE TQ

where Q is the risk-neutral measure

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Continuous time models

Let be deterministic

where r is the riskless interest rate, is the stock volatility and is the stock drift. Both instruments are freely and instantaneously tradable either long or short at the price quoted.

Let X be a payout at time T.

)exp(rtBt

)exp(0 tWSS tt

,,r

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Continuous time models

Discrete approximation:

tts exp if up jump

tts exp if down jump

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Continuous time models

Three steps to replication

Find a measure Q under which the discounted stock price exp(-rt)St is a martingale

Form the process Nt=exp(-rT)EQ(X|Ft)

Find a previsible process such that

Price of the claim: exp(-rT)EQ(X)

Price of the call option: (X=(ST-K)+) is

exp(-rT)EQ((ST-K)+)

t ttt dZdN

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Continuous time models

What is the dynamics under the risk neutral measure?

Ito’s formula: If X is a stochastic process, satisfying and f is a deterministic twice continuously differentiable function, then Yt=f(Xt) is also a stochastic process and is given by

dtdWdX tt

dtXfXfdWXfdY ttttt

)(''

2

1)(')(' 2

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Continuous time models

)exp(0 tWSS tt

Let Xt=log(St). Then we have dtdWSd tt )(log

Using the Ito’s lemma we get

dtSdWSdS tttt )2

1( 2

Under the risk neutral measure

ttt WdSdS ˆ

)2

1ˆexp( 20 tWSS tt

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Continuous time models

Thus, we know that ST is log-normally distributed under the risk-neutral measure Q.

The price of the call option: (if X=(ST-K)+) is

T

Trk

s

rTKT

Trk

s

sKSErT TQ

22

0

2

1log

)exp(2

1log

)exp(

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Connection with partial differential equations

Consider an agent who at each time t has a portfolio valued at X(t). This portfolio invests in a money market account paying a constant rate of interest r and in a stock modeled by the geometric Brownian motion:

Suppose the investor holds shares of stock and the remainder of the portfolio is invested in the money market account.

Then

)()()()( tdWtSdttStdS

)(t)()()( tSttX

)()()()())(()())()()(()()()( tdWtStdttSrtdttrXdttSttXrtdSttdX

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Connection with partial differential equations

Using the Ito formula we have

)()()()())(( tdWtSedttSertSed rtrtrt

and

))(()())(( tSedttXed rtrt

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Connection with partial differential equations

Let c(t,x) denote the value of the call option at time t if the stock price at that time is S(t)=x.

According to the Ito formula we have

)())(,()())(,()(2

1))(,()())(,())(,( 22 tdWtStctSdttStctStStctStStctStdc xxxxt

)())(,()())(,()(2

1))(,()())(,())(,()))(,(( 22 tdWtStctSedttStctStStctStStctStrcetStced x

rtxxxt

rtrt

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Connection with partial differential equations

A hedging portfolio starts with some initial capital X(0) and invests in the stock and money market account so that the portfolio value X(t) at each time agrees with c(t,S(t)). This happens if and only if

))(,()( tStcetXe rtrt

for all t. One way to ensure this is to make sure that

)))(,(())(( tStcedtXed rtrt

and X(0)=c(0,S(0)).

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Connection with pde

We get

)()()()())(( tdWtStdttSrt

)())(,()())(,()(2

1))(,()())(,())(,( 22 tdWtStctSdttStctStStctStStctStrc xxxxt

Equate the dW(t) terms: ))(,()( tStct x

Equate the dt terms: ))(,()(2

1))(,()())(,())(,( 22 tStctStStctrStStctStrc xxxt

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Connection with pde

In conclusion we should seek a continuous function c(t,x) that is a solution to the Black-Scholes-Merton partial differential equation

for all and that satisfies the terminal condition0,0 xTt

)(),( KxxTc

),(),()(2

1))(,(),( 22 xtrcxtctxtStrxcxtc xxxt

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Challenges

Volatility is stochastic

Interest rate r is stochastic

Claim is path dependent (exotic options)

The dynamics of the stock process is not geometric Brownian

Correlation between the dynamics of different market processes