Assignment 8

Assigned 2024-03-13, due 2024-03-20 at noon on Gradescope

Question 1

Consider a market with a bank and one stock. The bank has interest rate $r$ and the stock price is modelled by a geometric Brownian motion with mean return rate~$\alpha$ and volatility~$\sigma$.

  1. Let $K > 0$, $a \in (0, K)$, and consider a butterfly option that matures at time $T$ and pays $S_T - K + a$ if $S_t \in [K-a, K)$, $K + a - S_T$ if $S_T \in [K, K + a)$ and nothing otherwise. Find the arbitrage free price of this option at time $t \leq T$.

  2. A digital option with strike $K$ pays \$1 if $S_T \geq K$, and nothing otherwise. Find the arbitrage free price of this option at time $t \leq T$.

Question 2

Consider the same market as in the previous question. Given $\gamma > 0$ a power option pays $S_T^\gamma$ at maturity $T$. Find the arbitrage free price of this security.

[Hint: Try looking for solutions to the Black–Scholes PDE of the form $f(t, x) = \theta(t) \varphi(x)$.]

Question 3

Let $c(t, x)$ be the arbitrage free price of a European call as given by the Black–Scholes formula.

  1. Find $\lim_{x \to \infty} (c(t, x) - x)$.

  2. Is there some function $f(t)$ such that \begin{equation} \lim_{x \to \infty} x ( c(t, x) - x - f(t) ) \quad \text{exists and is finite?} \end{equation} Justify your answer, and find $f$ if it exists.

    [Note: This question is related to the boundary condition of the Black–Scholes PDE at infinity. There was a typo on this question which was fixed on 2024-03-15. ]