Option prices and the Black-Scholes-Merton formula Gabor Molnar-Saska 3 October 2006

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The binomial model 2 1 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=1 payout (S1-K)+ What is the price of this option?

The binomial model 2 X=1 1 0.5 X=0 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$

The binomial model s1up Xup s0 s1down Xdown 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

The binomial model The value of the portfolio:

The binomial model 0<q<1 and the value of the portfolio is Expectation under a new measure!

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.

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 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. Let

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.

The binomial tree model Conclusion: The price of the claim X is where Q is the risk-neutral measure

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.

Continuous time models Discrete approximation: if up jump if down jump

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)+)

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

Continuous time models Let Xt=log(St). Then we have Using the Ito’s lemma we get Under the risk neutral measure

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

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

Connection with partial differential equations Using the Ito formula we have and

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

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 for all t. One way to ensure this is to make sure that and X(0)=c(0,S(0)).

Connection with pde We get Equate the dW(t) terms: Equate the dt terms:

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 condition

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