Question

) Consider a one-year European call option on a stock when the stock price is $30, the strike price is $30, the risk-free rate is 5%, and the volatility is 25% per annum.

(a) Use the DerivaGem software to calculate the price, delta, gamma, vega, theta, and rho of the option.

(b) Verify that delta is correct by changing the stock price to $30.1 and recomputing the option price. Verify that gamma is correct by recomputing the delta for the situation where the stock price is $30.1. Carry out similar calculations to verify that vega, theta, and rho are correct.

(c) Suppose that over the course of one day, the stock price increases by $0.5. Find the approximate price of the call option on the next day.

Answer 1

The price, delta, gamma, vega, theta, and rho of the option are 3.7008, 0.6274, 0.050, 0.1135, −0.00596, and 0.1512.

When the stock price increases to 30.1, the option price increases to 3.7638. The change in the option price is 3.7638− 3.7008 = 0.0630.

Delta predicts a change in the option price of 0.6274 × 0.1 = 0.0627 which is very close. When the stock price increases to 30.1, delta increases to 0.6324. The size of the increase in delta is 0.6324 − 0.6274 = 0.005.

Gamma predicts an increase of 0.050 × 0.1 = 0.005 which is the same.

When the volatility increases from 25% to 26%, the option price increases by 0.1136 from 3.7008 to 3.8144. This is consistent with the vega value of 0.1135. When the time to maturity is changed from 1 to 1−1/365 the option price reduces by 0.006 from 3.7008 to 3.6948. This is consistent with a theta of −0.00596.

Finally, when the interest rate increases from 5% to 6%, the value of the option increases by 0.1527 from 3.7008 to 3.8535. This is consistent with a rho of 0.1512.

value of the option increases by 0.1527 from 3.7008 to 3.8535. This is consistent with a rho of

0.1512.