One of the problems with derivative pricing models is that it assumes volatility is strictly proportional, so that if the stock price doubles, we assume that the volatility of the asset price will double as well. However, empirical tests suggest that the volatility doesn?t go up as much as our model predicts when prices rise, and doesn?t decline as much as the model predicts when prices fall. There are many ways to model this observation. One simple way is to assume that stock volatility is composed of two parts: a proportional part and a fixed part (the proportional part assumes the lognormal distribution and the fixed part assumes a normally distributed asset ? this means that the actual distribution will be somewhere between the normal and lognormal).

Assume a stock has a price of $40/share and the volatility is given by the following function:


At the current price, the volatility of price (P) and return (R) are given by



If the price of the stock declines, the price volatility will decline, but the return volatility will increase (and the opposite if the asset price rises). An obvious drawback of this model is that stock prices can be negative, as volatility does not go to zero as the stock price approaches zero.

a.The Black-Scholes model will not give exact prices for options on this asset, as a central assumption of BS is violated. Can we still use risk-neutral valuation techniques? Why (or why not).

b.Would a call option on this asset be path-dependent? Why (or why not).

c.Compute the current market price of the following options. You may need to use Monte Carlo simulation to get the correct answer. Compare your prices to a Black-Scholes price using the current volatility of 25%. (Assume risk-free rate is 5% and the expected return of the stock is 12% per year):

2-year call option with a strike of 50
2-year put option with a strike of 30