black scholes on futures question

[xiaoxin]

If the futures period T_2 is greater than the option period T_1, why do we scale F, the futures price, by e^(-rT_1) instead of e^(-rT_2). You need F*e^(-r * T_2) in order to have F by the end of the futures period.

Can someone help explain? Thanks!

[xiaoxin]

I take that back--I don't think I fully understand this. Help?

[xiaoxin]

If anyone knows the answer to this, feel free to comment.

[Academic Actuary]

Originally Posted by xiaoxin

If the futures period T_2 is greater than the option period T_1, why do we scale F, the futures price, by e^(-rT_1) instead of e^(-rT_2). You need F*e^(-r * T_2) in order to have F by the end of the futures period.

Can someone help explain? Thanks!

I'm not sure what you mean by scaling the futures price. Are you refering to the Black model for pricing an option on future. If so e^(-rT_1) is a discounting factor from the time of potential payoff.

I don't how far you are in the material, but any derivative price (assuming a complete market which means at least in theory the option payoff can be replicated with traded assets) can be valued as a "nisk neutral EPV". Under such an assumption, the futures price is a martingale, which means F(0,T_2) = E[F(T_1, T_2)] so e^(-T_1) F(0,T_2) is a discounted expected value under risk neutrality.

[xiaoxin]

Maybe I should ask this question.

Why is the payoff of a call option at time T on a futures contract delivering at time T_F going to be F_{T, T_F} - K? If anything, shouldn't it be F_{T, T_F} * e^{-r(T_F - T)} - K?

I thought F_{T, T_F} is the amount you would pay at time T_F, not at time T.

[Academic Actuary]

Originally Posted by xiaoxin

Maybe I should ask this question.

Why is the payoff of a call option at time T on a futures contract delivering at time T_F going to be F_{T, T_F} - K? If anything, shouldn't it be F_{T, T_F} * e^{-r(T_F - T)} - K?

I thought F_{T, T_F} is the amount you would pay at time T_F, not at time T.

If its an option on a future, the underlying is the futures price, not the prepaid futures price. The fact that the futures delivers at a later date would be reflected in the value of the option.

[xiaoxin]

So the fact that a futures delivers at a later date will be reflected in the value of the option but not in the payoff?

[Academic Actuary]

Consider two one year options, one where the underlying is a two year futures and the second where the underlying is a three year futures. At expiration of the option the payoff of the first option is based upon the one year futures price and the second on the two year futures price. These will generally not be the same.

[xiaoxin]

That wasn't what I was asking. Why does it matter that the underlying is the futures price? With a futures price of F_{T, T_F} at time T, you're still paying e^(r(T_F - T)) * F_{T, T_F} at the end...

I really don't understand when you said "the fact that the futures delivers at a later date would be reflected in the value of the option." Could you please explain that in some detail?

[Academic Actuary]

Originally Posted by xiaoxin

That wasn't what I was asking. Why does it matter that the underlying is the futures price? With a futures price of F_{T, T_F} at time T, you're still paying e^(r(T_F - T)) * F_{T, T_F} at the end...

I really don't understand when you said "the fact that the futures delivers at a later date would be reflected in the value of the option." Could you please explain that in some detail?

First with any option on a futures, there is always cash settlement, never a delivery of an underlying. An option on a future as opposed to an actual facilitates hedging as transactions costs are lower in futures markets. When the option and future mature at the same date, an option on a future would be equivalent to an option on the actual underlying in terms of payoff.

When the futures maturity date is later the underlying becomes the futures price which will track the underlying asset but it is not the same thing. If the option was a call maturing at T, with the future maturing at TF, the payoff function for the call is Max[ F(T,TF) - K ,0]. Different values of TF will lead to different payoffs and therefore be reflected in the time 0 option value.