Question

In this question we assume the Black-Scholes model. We denote interest rate by r, drift rate pi and volatility by o. A European power put option is an option with the payoff function below, Ka – rº, ha if x <K, 0, if x > K, for some a > 0. In particular, it will be a standard European put option when a = 1. (a) Derive the pricing formula for the time t, 0 <t< T, price of a European power put option in the Black-Scholes model. [20] (b) In the Black-Scholes model with S(0) = 100, r = 2% and o = 15% compute the price at time t = 0 of a European power put option with the power a = 2 and strike price K = 100 expiring at time T = 1 year. (Values of the cumulative normal distribution function are given in the attached table.) [5]

Answer 1

The Black-Scholes formula established by Fischer Black and Myron Scholes in 1973 was innovative in its effect on the financial industry. Today, many of the techniques and pricing models used in finance are deep-rooted in the ideas and methods presented by these two men. This paper will help as an exposition of the formula with extensions to more exotic options with barriers and will also highlight two different methods for solving the options pricing problem. We will first derive the formula by determining the expected value of the option, a different method than the one originally employed by Black and Scholes. This method, although it is somewhat less rigorous, gives the same result, namely that the price of a European call option is given by C = S0N ! rT + ν2T 2 + ln S0 K ν √ T " − Ke−rT N ! rT − ν2T 2 + ln S0 K ν √ T " , where S0 is the initial price of the stock, N(x) represents the cumulative distribution function of a standard normal variable, r is the risk-free interest rate, K is the strike price of the option, T is the amount of time until the option expires, and ν is the annual volatility of the stock price

The derivation of this formula requires some non-intuitive assumptions. As a result, we will define some basic terminology about risk, and then we will invoke Ito’s Lemma to derive the Black-Scholes equation, named so because it was used by Black and Scholes in their original derivation. The basic idea here is that, by hedging away all risk in our portfolio, it becomes perfectly reasonable to assume that people are risk-neutral. This is a very necessary step, though, for most people are naturally risk-averse. This section will essentially follow the methods employed by Black and Scholes and, along with the derivation for barrier options, will highlight the basic method that they used and a different approach to the problem than that of expected value. We will introduce the concept of no-arbitrage, also known as the no-free-lunch principle, in order to develop the idea of put-call parity. This method of solving for the European put option price is much simpler than repeating the original derivation and provides insight into basic ideas in financial mathematics. Our last task will be to extend the basic principles of the Black-Scholes equation (not the formula above) to price barrier options, which are options whose validities are contingent upon hitting some pre-determined stock price. Intuitively, because they have an extra imposition, barrier options should be worth less than a regular option. In order to price them, we will use the same technique employed by Black and Scholes in which they transformed the Black-Scholes equation into the heat equation. The key difference will be in the boundary conditions, a fact that emphasizes the versatility of this technique in the pricing of more exotic options.

We begin with a review of some basic terminology in probability theory.

Definition 2.1.

The cumulative distribution function, F, of the random variable X is defined for all real numbers b, by F(b) = P{X ≤ b} We say X admits a probability density function or density f if P{X ≤ b} = F(b) = # b −∞ f(x) dx for some nonnegative function f.

Definition 2.2.

X is a normal random variable with parameters µ and σ2 > 0 if the density of X is given by f(x) = 1 √2πσ e −(x−µ)2 2σ2 − ∞ <x< ∞ Thus, the cumulative distribution function of a standard normal random variable, i.e. one with mean 0 and variance 1, is given by N(x) = 1 √2π # x −∞ e −y2 2 dy

∞ −∞ xf(x) dx