Question

Suppose that call options on ExxonMobil stock with time to expiration 3 months and strike price $92 are selling at an implied volatility of 32%. ExxonMobil stock currently is $92 per share, and the risk-free rate is 3%. If you believe the true volatility of the stock is 36%.

a. If you believe the true volatility of the stock is 36%, would you want to buy or sell call options? Buy call options Sell call options

b. Now you need to hedge your option position against changes in the stock price. How many shares of stock will you hold for each option contract purchased or sold?

Answer 1

a]

The price of options is directly related to implied volatility. An increase in implied volatility will result in a rise in the price of options, and a decrease in implied volatility will result in a fall in the price of options.

If the true volatility is higher than the implied volatility, the implied volatility can be expected to rise to match the true volatility. This will result in a rise in the price of the call option (other factors being unchanged).

Therefore, you would want to buy call options as the price of call options can be expected to rise.

b]

Number of shares to hold = delta of option.

Delta of option is N(d₁) in the Black-Scholes formula.

d₁ = (ln(S₀ / K) + (r + σ²/2)*T) / σ√T,

where

S₀ = current spot price

K = strike price

r = risk-free interest rate

t is the time to expiry in years

σ = standard deviation of underlying stock returns. This is taken to be 36%, which is the true volatility.

We calculate d₁ as below :

• ln(S₀ / K) = ln(92 / 92). We input the same formula into Excel, i.e. =LN(92/92)
• (r + σ²/2)*T = (0.03 + (0.36²/2)*0.25
• σ√T = 0.36 * √0.25

d₁ = 0.1317

N(d₁) is calculated in Excel using the NORMSDIST function and inputting the value of d₁ into the function.

N(d₁) = 0.5524.

The delta of the option is 0.5524.

Number of shares to hold for each option = 0.5524