Question 1: Let St denote a stock modeled by the Black Scholes model dSt = μStdt+ σStdWt.
a) Now assume the stock starts paying dividends continuously at a rate of D times the stock price St.
(This is a reasonable assumption for derivatives on stock indexes.) Argue that the SDE for St becomes
dSt = (μ - D)Stdt + σStdWt
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 Πt = ΦtSt + ψtBt 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
![796_Write down what replaces the self-financing condition.png](https://secure.tutorsglobe.com/CMSImages/796_Write%20down%20what%20replaces%20the%20self-financing%20condition.png)
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
![873_Write down what replaces the self-financing condition1.png](https://secure.tutorsglobe.com/CMSImages/873_Write%20down%20what%20replaces%20the%20self-financing%20condition1.png)
(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
![650_Write down what replaces the self-financing condition2.png](https://secure.tutorsglobe.com/CMSImages/650_Write%20down%20what%20replaces%20the%20self-financing%20condition2.png)
(c) Let h(S) denote the payoff function of some derivative. Then we can synthesize the derivative by
![2297_Write down what replaces the self-financing condition3.png](https://secure.tutorsglobe.com/CMSImages/2297_Write%20down%20what%20replaces%20the%20self-financing%20condition3.png)
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) =
![91_Write down what replaces the self-financing condition4.png](https://secure.tutorsglobe.com/CMSImages/91_Write%20down%20what%20replaces%20the%20self-financing%20condition4.png)
(d) Using Taylor's theorem, show that
![1784_Write down what replaces the self-financing condition5.png](https://secure.tutorsglobe.com/CMSImages/1784_Write%20down%20what%20replaces%20the%20self-financing%20condition5.png)
(e) Show that V(S, t) is equal to
![1777_Write down what replaces the self-financing condition6.png](https://secure.tutorsglobe.com/CMSImages/1777_Write%20down%20what%20replaces%20the%20self-financing%20condition6.png)
Question 3: Carry out a similar analysis to the above using the utility function Up(x) = xp/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.