Problem 18.10 pg 95 PQ set Hull Chapter 19: The Greek Letters (Hull Text Q & A)

[Dr. Jayanthi Sankaran]

Hi David,

As referenced above, problem 18.10 of Hull is:

What is the delta of a short position in 1,000 European call options on silver futures? The options mature in eight months, and the futures contract underlying the option matures in nine months. The current nine-month futures price is $8 per ounce, the exercise price of the options is $8, the risk-free interest rate is 12% per annum, and the volatility of silver is 18% per annum.

The answer given is:

The delta of a European futures call option is usually defined as the rate of change of the option price with respect to the futures price (not the spot price). It is

exp(-rT)*N(d1)

In this case F(0) = 8, K = 8, r = 0.12, sigma = 0.18, T = 0.6667

d1 = [ln(8/8) + (0.18^2/2)*0.6667]/0.18*SQRT(0.6667) = 0.0735

My question is why is the risk-free rate not included in the above formula - should it not be

d1 = [ln(8/8) + (.12 + 0.18^2/2)*0.6667]/0.18*SQRT(0.6667) = 0.6178?

Thanks!
Jayanthi

 

[David Harper CFA FRM]

Hi @Jayanthi Sankaran It does look like a typo at first glance, but in the case of an option on a forward (future) contract, as opposed to our typical use case of an option on a spot asset, the riskfree rate is omitted, as shown in Hull's answer. The reason is that we're inputing the forward price; our formula includes LN(Forward/Strike) rather than the more typical LN(Spot/Strike). Per cost of carry, F= S*exp(rT), such that the forward price is already an expected future value, so the formula does not need to "build in" the risk-free drift. Adding riskfree rate would be a kind of double-counting of the risk free rate which is already implicitly in the forward price. I hope that explains, thanks!

 

[Dr. Jayanthi Sankaran]

Thanks David - that explains it - Much appreciated!

Jayanthi