Security Markets X

1. Security MarketsX Miloslav S. Vosvrda Theory of Capital Markets

2. A continous-time version of the crown valuation The problem solved by the Black-Scholes Option Pricing Formula is a special case of the following continuous-time version of the crown valuation problem, treated in a binomial random walk setting. We are given the riskless security defined by a constant interest rate r and a risky security whose price process S is described by with dividend rate

3. We are interested in the value of a security, say a crown, that pays a lump sum of at a future time T , where is g sufficiently well behaved to justify the fillowing calculations. (It is certainly enough to know that g is bounded and twice continuosly differentiable with a bounded derivative.) In the case of an option on the stock with exercise price K and exercise date T , the payoff function is defined by which is sufficiently well behaved. We will suppose that the value of the crown at any time is ,where C is a function that is twice continuously differentiable for

4. In particular, For convenience, we use the notation

5. We can solve the valuation problem by explicity determining the function C . For simplicity, we supose that the riskless security is a discount bond maturing after T, so that its market value at time t is Suppose an investor decides to hold the portfolio of stock and bond at any time t , where and

6. This particular trading strategy has two special properties. First, it is self-financing, meaning that it requires an initial investment of but neither generates nor requires any further funds after time zero. To see this fact, one must only show that The left hand side is the market value of the portfolio at time t; the right hand side is the sum of its initial value and any interim gains or losses from trade.

7. This equation can be verified by an application of Ito’s Lemma in the following form, If is twice continuously differentiable and X defined by the stochastic differential equation , then for any time where

8. The second important property of the trading strategy is the equality which follows immediately from the definitions of and . From Ito’s Lemma, we have (*) Using and

9. we can collect the terms in and separately. If (*) holds, the integrals involving and must separately sum to zero. Collecting the terms in alone, for all But then this expresion implies that C must satisfy the partial differential equation

10. for We have the boundary condition The partial differential equation with boundary condition can be shown to have the solution where Z is normally distributed with mean and variance

11. For the case of the call option payoff function, we can quickly check that is precisely the Black-Scholes Option Pricing Formula. More generally, can be solved numerically by standard Monte Carlo simulation and variance reduction methods.