Question 1: Let S_t denote a stock modeled by the Black Scholes model dSt = μS_tdt+  σS_tdWt.

a) Now assume the stock starts paying dividends continuously at a rate of D times the stock price S_t.

(This is a reasonable assumption for derivatives on stock indexes.) Argue that the SDE for St becomes

dS_t = (μ - D)S_tdt + σS_tdW_t

b) Consider a derivative on S. Derive the PDE that V(S, t) must satisfy

∂V/∂t + σ²/2 s²∂²V/∂S² + (r - D) S ∂V/∂S -rv = 0

The thing that changes in the derivation from the ordinary Black-Scholes equa-tion is the self-financing condition; the portfolio Π_t = Φ_tS_t + ψ_tB_t is no longer self-financing. Write down what replaces the self-financing condition.

c) Solve the equation for a call, in a manner similar to how we solved the Black Scholes equation on a non-dividend paying stock.

Question 2:

In this problem, you will show how to synthesize any derivative (whose pay-off only depends on the final value of the stock) in terms of calls with different strikes. This allows us to express the value of any deriva¬tive in terms of the Black-Scholes formula.

(a) Consider the following functions: Fix a number h > 0. Define

Show that collection of functions 6h(x) for h > 0 forms a δ- function.

Recall that this implies that for any continuous function f that

(b) Consider a non-dividend paying stock S. Suppose European calls on S with expiration T are available with arbitrary strike prices K. Let C(S, t, K) denote the value at time t of the call with strike K. Thus, for example, C(S, T, K) = max(S - K, 0).

Show

(c) Let h(S) denote the payoff function of some derivative. Then we can synthesize the derivative by

If V(S,t) denotes the value at time t of the derivative with payoff function h if stock price is S (at time t) then why is V(S, t) =

(d) Using Taylor's theorem, show that

(e) Show that V(S, t) is equal to

Question 3: Carry out a similar analysis to the above using the utility function Up(x) = x^p/P P< 1 not equal to 0. Reduce to terms of order (St. You will have to use Taylor's theorem. Your answer should not have a δt in it.