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) Calculate the price and delta of the option.
(b) Suppose an investor sold 50,000 of these call options. What stock position should he take to delta-hedge the call position?
(c) Verify that delta is correct by changing the stock price to $30.1 and recomputing the option price.
(d) Compute the delta when the stock price is $30.1. What changes does the investor need to make to delta-hedge the call position? Assume the investor had already taken the delta-hedging position you found in part b.

Options:
a) Increase the stock position
b) Decrease the stock position
c) Keep the stock position unchanged
d) Sell the stock position

Answer 1

The price and delta of the option can be calculated using the Black-Scholes model. To delta-hedge the call position, the investor should take a stock position equal to the delta multiplied by the number of call options sold. The delta can be verified by changing the stock price and recomputing the option price. The investor can determine the changes needed for delta-hedging by computing the delta at a different stock price.

Step-by-step explanation:

(a) To calculate the price of the option, we can use the Black-Scholes model. The formula for the price of a European call option is:

Call Price = S imes N(d1) - X imes e^{-rT} imes N(d2)

where:

• S = stock price = $30
• X = strike price = $30
• r = risk-free rate = 5% = 0.05
• T = time to expiration = 1 year
• N() = cumulative standard normal distribution
• d1 = (ln(S/X) + (r + (σ^2)/2) imes T) / (σ imes (T))
• d2 = d1 - σ imes (T)

By plugging in the values, we can calculate the price of the option. To calculate the delta, we can use the formula:

Delta = N(d1)

Again, plugging in the values, we can calculate the delta of the option.

(b) To delta-hedge the call position, the investor should take a stock position equal to the delta of the option multiplied by the number of call options sold. In this case, the delta is the same as calculated in part (a), and the number of call options sold is 50,000. Therefore, the stock position should be 50,000 multiplied by the delta.

(c) To verify the delta, we can change the stock price to $30.1 and recompute the option price using the same formula as in part (a). By comparing the new option price with the original option price, we can see if the delta is correct.

(d) To compute the delta when the stock price is $30.1, we can use the same formula as in part (a), but with the new stock price. By comparing the new delta with the original delta, we can determine the changes the investor needs to make to delta-hedge the call position.