Please only answer (a) and (b).

The definition of Delta is defined as the rate of change of the option price with respect to the price of the underlying asset. It is the slope of the curve that relates the option price to the underlying asset price. The following picture gives the definition of delta

The theta () of a portfolio of options is the rate of change of the value of the portfolio with respect to the passage of time with all else remaining the same. Theta is sometimes referred to as the time decay of the portfolio. The following picture also defines Theta

The gamma (T) of a portfolio of options on an underlying asset is the rate of change of the portfolios delta with respect to the price of the underlying asset. It is the second partial derivative of the portfolio with respect to asset price. The following picture also defines Gamma

5. This equation relates to the Greeks A, T and . Consult Chapter 19 of book. This question requires you to use simple computer programming). Consider a stock whose price follows a Geometric Brownian Motion representing the dynamics of the stock price. The initial price of the stock is $60. Assuming that the volatility of the price is 20%, the drift is 14%, the risk free rate is 0.2% and the strike price is $61, and the exercise time T = 2 months. (a) Compute the three Greeks A, I and O. (b) Simulate the price of the stock at the end of each day, for the next 60 days (= 2 months). Use this data to compute the A for the 60 days, and apply dynamic A hedging for 5 sets of simulated prices, and report the mean and variance of the cost of the hedge. Assume you are A hedging a portfolio of 100,000 call options (described before), and resources are available interest free for the hedging. (c) For this option plot a graph of the I as a function of the stock price. (d) Is there a relation between the three Greeks A, I and O? Specify this for a portfolio P which is A-neutral. For this P, show how the graph for can be derived from one found in part (d). slope of the curve that relates the option price to the underlying asset price. Suppose that the delta of a call option on a stock is 0.6. This means that when the stock price changes by a small amount, the option price changes by about 60% of that amount. Figure 19.2 shows the relationship between a call price and the underlying stock price. When the stock price corresponds to point A, the option price corresponds to point B, and A is the slope of the line indicated. In general, ac Aras where c is the price of the call option and S is the stock price. 19.5 THETA The theta (O) of a portfolio of options is the rate of change of the value of the portfolio with respect to the passage of time with all else remaining the same. Theta is sometimes referred to as the time decay of the portfolio. For a European call option on a non- dividend-paying stock, it can be shown from the Black-Scholes-Merton formula (see Problem 15.17) that SON'(dio (call) =- --rKe-T Nd) 2T where d, and d, are defined as in equation (15.20) and N'(x) = - e-**/2 (19.2) 27 is the probability density function for a standard normal distribution. For a European put option on the stock, (put) = 10 +rke-T N(-d2) 2.VT Because N(-d) = 1 - N(d), the theta of a put exceeds the theta of the corresponding call by rke-T. In these formulas, time is measured in years. Usually, when theta is quoted, time is measured in days, so that theta is the change in the portfolio value when 1 day passes 19.6 GAMMA The gamma (T) of a portfolio of options on an underlying asset is the rate of change of the portfolio's delta with respect to the price of the underlying asset. It is the second partial derivative of the portfolio with respect to asset price:/ a 11 Ias? If gamma is small, delta changes slowly, and adjustments to keep a portfolio delta neutral need to be made only relatively infrequently. However, if gamma is highly negative or highly positive, delta is very sensitive to the price of the underlying asset. It is then quite risky to leave a delta-neutral portfolio unchanged for any length of time. Figure 19.7 illustrates this point. When the stock price moves from S to S', delta hedging assumes that the option price moves from C to C', when in fact it moves from C to C". The difference between C' and C" leads to a hedging error. The size of the error depends on the curvature of the relationship between the option price and the stock price. Gamma measures this curvature. 5. This equation relates to the Greeks A, T and . Consult Chapter 19 of book. This question requires you to use simple computer programming). Consider a stock whose price follows a Geometric Brownian Motion representing the dynamics of the stock price. The initial price of the stock is $60. Assuming that the volatility of the price is 20%, the drift is 14%, the risk free rate is 0.2% and the strike price is $61, and the exercise time T = 2 months. (a) Compute the three Greeks A, I and O. (b) Simulate the price of the stock at the end of each day, for the next 60 days (= 2 months). Use this data to compute the A for the 60 days, and apply dynamic A hedging for 5 sets of simulated prices, and report the mean and variance of the cost of the hedge. Assume you are A hedging a portfolio of 100,000 call options (described before), and resources are available interest free for the hedging. (c) For this option plot a graph of the I as a function of the stock price. (d) Is there a relation between the three Greeks A, I and O? Specify this for a portfolio P which is A-neutral. For this P, show how the graph for can be derived from one found in part (d). slope of the curve that relates the option price to the underlying asset price. Suppose that the delta of a call option on a stock is 0.6. This means that when the stock price changes by a small amount, the option price changes by about 60% of that amount. Figure 19.2 shows the relationship between a call price and the underlying stock price. When the stock price corresponds to point A, the option price corresponds to point B, and A is the slope of the line indicated. In general, ac Aras where c is the price of the call option and S is the stock price. 19.5 THETA The theta (O) of a portfolio of options is the rate of change of the value of the portfolio with respect to the passage of time with all else remaining the same. Theta is sometimes referred to as the time decay of the portfolio. For a European call option on a non- dividend-paying stock, it can be shown from the Black-Scholes-Merton formula (see Problem 15.17) that SON'(dio (call) =- --rKe-T Nd) 2T where d, and d, are defined as in equation (15.20) and N'(x) = - e-**/2 (19.2) 27 is the probability density function for a standard normal distribution. For a European put option on the stock, (put) = 10 +rke-T N(-d2) 2.VT Because N(-d) = 1 - N(d), the theta of a put exceeds the theta of the corresponding call by rke-T. In these formulas, time is measured in years. Usually, when theta is quoted, time is measured in days, so that theta is the change in the portfolio value when 1 day passes 19.6 GAMMA The gamma (T) of a portfolio of options on an underlying asset is the rate of change of the portfolio's delta with respect to the price of the underlying asset. It is the second partial derivative of the portfolio with respect to asset price:/ a 11 Ias? If gamma is small, delta changes slowly, and adjustments to keep a portfolio delta neutral need to be made only relatively infrequently. However, if gamma is highly negative or highly positive, delta is very sensitive to the price of the underlying asset. It is then quite risky to leave a delta-neutral portfolio unchanged for any length of time. Figure 19.7 illustrates this point. When the stock price moves from S to S', delta hedging assumes that the option price moves from C to C', when in fact it moves from C to C". The difference between C' and C" leads to a hedging error. The size of the error depends on the curvature of the relationship between the option price and the stock price. Gamma measures this curvature