Stochastic Calculus for Finance

Michaelmas Term 1998: Problems for solution

1. Consider the following simple model of stock price movement. The value of the stock at time zero is . At time , the price has moved to either , or . The risk free interest rate is such that $1 now will be worth $ at time .

1. Suppose that . Show that the market price of a European call option which matures at time with strike price K is

   

2. What happens if we drop the assumption that ?

2. Suppose that at current exchange rates, 100 is worth 280DM. A speculator believes that by the end of the year there is a probability of 2/3 that the pound will have fallen to 2.60DM, and a 1/3 chance that it will have gained to be worth 3.00DM. He therefore buys a European put option that will give him the right (but not the obligation) to sell 100 for 290DM at the end of the year. He pays 20DM for this option. Assume that the risk free interest rate is zero. Using a single period binary model, either construct a strategy whereby one party is certain to make a profit or prove that this is the fair price.

3. (Put-Call parity.) Let us denote by and respectively the prices of a European call and a European put option, each with maturity T and strike K. Assume that the risk free rate of interest is constant, r (so the cost of borrowing $1 for s units of time is $). Show that for each ,

4. Suppose that the price of a certain asset has the lognormal distribution. That is is normally disrtibuted with mean and variance . Calculate .

5. Find the risk-neutral probabilities for the model in Question 2. That is, find the probabilities p, 1-p for upward/downward movement of the pound, under which `'. Check that the fair price of the option is then (since r=0) where the expectation is calculated with respect to these probabilities..

6. Consider two dates with . A forward start option is a contract in which the holder receives at time , at no extra cost, an option with expiry date and strike price equal to (the asset price at time ). Assume that the stock price evolves according to a two-period binary model, in which the asset price at time is either or , and at time is one of , and with

where r denotes the risk free interest rate. Find the fair price of such an option at time zero.

7. A digital option is one in which the payoff depends in a discontinuous way on the asset price. The simplest example is the cash-or-nothing option, in which the payoff to the holder at maturity T is where X is some prespecified cash sum.

Suppose that an asset price evolves according to the Cox Ross Rubinstein model (CRR model). That is, a multiperiod binary model in which, at each step, the asset price moves from its current value to one of and . As usual, if denotes the length of each time step, .

Find the time zero price of the above option. You may leave your answer as a sum.

8. Suppose that an asset price evolves according to the CRR model described in Question 7. For simplicity suppose that the risk free interest rate is zero and is 1. Suppose that under the probability , at each time step, stock prices go up with probability p and down with probability 1-p.

The conditional expectation

is a stochastic process. Check that it is a -martingale and find the distribution of ?

9. Show how to derive the put-call parity relationship of Question 3 from Theorem 3.3 of lectures.

10. Let be standard Brownian motion. Which of the following are Brownian motions?

1. 
2. , where c is a constant,
3. ,
4. 

Justify your answers.

11. Suppose that is standard Brownian motion. Prove that conditional on , the probability density function of is

This tells us that the conditional distribution is a normally distributed random variable. What are the mean and variance?

12. Let be standard Brownian motion. Let be the `hitting time of level a', that is

Then we shall prove in lectures that

Use this result to calculate

1. ,
2. .

13. Let denote standard Brownian motion and define by

Suppose that . Calculate

1. ,
2. .

14. Let be standard Brownian motion. Let denote the hitting time of the sloping line a+bt. That is,

We show in lectures that

The aim of this question is to calculate the distribution of , without inverting the Laplace transform. In what follows, and

1. Find .
2. Using the fact that has the same distribution as , show that

   

3. By conditioning on the value of , use the previous part to show that

   

4. Substitute for the probability in the integral and deduce that

   

15. Let denote the natural filtration associated to a standard Brownian motion . Which of the following are -martingales?

1. ,
2. , where c is a constant,
3. ,

16. Let denote the natural filtration associated to a standard Brownian motion . Define the process by

For which values of is the process an -martingale?

17. A function, f, is said to be Lipschitz continuous on if there exists a constant C>0 such that for any

Show that a Lipschitz continuous function has bounded variation and zero quadratic variation.

18. Let denote standard Brownian motion. For a partition of , write for the mesh of the partition and for the number of intervals. We write to denote a generic subinterval in the partition. Calculate

1. 

2. 

   This is the Stratonovich integral of with respect to itself.

19. Suppose that is a function of bounded quadratic variation on , and is a Lipschitz continuous function on . Using to denote the quadratic variation of a function f over the interval , show that

20. If f is a simple function, prove that the process given by the Itô integral

is a martingale.

21. Verify that

(If you need the moment generating function of , you may assume the result of Question 23.)

22. Use Itô's formula to write down stochastic differential equations for the following quantities. (As usual, denotes standard Brownian motion.)

1. ,
2. ,
3. .

23. Let denote Brownian motion and define . Use Itô's formula to write down a stochastic differential equation for . Hence find an ordinary (deterministic) differential equation for , and solve to show that

24. (The Ornstein-Uhlenbeck process). Let denote standard Brownian motion. The Ornstein-Uhlenbeck process, is the unique solution to Langevin's equation,

This equation was originally introduced as a simple idealised model for the velocity of a particle suspended in a liquid. Verify that

and use this expression to calculate the mean and variance of .

25. Suppose that under the probability measure , is a Brownian motion with constant drift . Find a measure , equivalent to , under which is a Brownian motion with drift .

26. Suppose that an asset price is such that , where is, as usual, standard Brownian motion. Let r denote the risk free interest rate. The price of a riskless asset then follows . We write for the portfolio consisting of units of the riskless asset, , and units of at time t. For each of the following choices of , find so that the portfolio is self-financing. (Recall that the value of the portfolio at time t is , and that the portfolio is self-financing if .)

1. ,
2. ,
3. .

27. Let be the natural filtration associated with Brownian motion (as in the proof of Theorem 8.4 of lectures). Show that if X is an -measurable random variable, then if is a probability measure equivalent to that of the Brownian motion, then the process

is a -martingale.

28. Use the Black-Scholes model to value a forward start option (described in Question 3).

29. Suppose that the value of a European call option can be expressed as (as we prove in Proposition 9.2). Then , and we may define by

Under the risk-neutral measure, the discounted asset price follows , where (under this probability measure) is a standard Brownian motion.

1. Find the stochastic differential equation satisfied by .
2. Using the fact that is a martingale under the risk-neutral measure, find the partial differential equation satisfied by , and hence show that

   

   This is the Black-Scholes equation.

30. (Delta-hedging). The following derivation of the Black-Scholes equation is very popular in the finance literature. We will suppose, as usual, that an asset price follows a geometric Brownian motion. That is, there are parameters , , such that

Suppose that we are trying to value a European option based on this asset. Let us denote the value of the option at time t by . We know that at time T, , for some function f.

1. Using Itô's formula express V as the solution to a stochastic differential equation.
2. Suppose that a portfolio, whose value we denote by , consists of one option and a (negative) quantity of the asset. Assuming that the portfolio is self-financing, find the stochastic differential equation satisfied by .
3. Find the value of for which the portfolio you have constructed is `instantaneously riskless', that is for which the stochastic term vanishes.
4. An instantaneously riskless portfolio must have the same rate of return as the risk free interest rate. Use this observation to find a (deterministic) partial differential equation for the . Notice that this is the Black-Scholes equation obtained in Question 29.
5. Now for the crunch: is the portfolio that you have constructed self-financing?
   1. Write , where v is a solution of the Black-Scholes equation. Use Itô's formula to write down a stochastic differential equation for U.
   2. Find an expression for .
   3. If the trading strategy is self-financing then it must satisfy . Use the stochastic differential equation for and the Black-Scholes differential equation (differentiated with respect to the x variable) to show that this is equivalent to

      

      Here is a Brownian motion under the risk neutral measure.

   4. Deduce that the portfolio is not self-financing.

You have shown that the delta-hedging argument is mathematically unsatisfactory in the continuous time setting. It should be emphasized that it is unquestionable in the discrete time setting. All the same, in continuous time it does lead to the right value of the option. The reason is that the additional cost that might be associated with this trading strategy is a martingale under the risk neutral measure.

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