:: Collaborations ::

:: Options on Futures ::

Here we touch on compound derivative products.  Unlike normal derivative products that are written on a standard underlying assets (cash products, stocks, shares, etc), compound derivatives are essentially 'derivatives on derivatives'.

As you may presume, the pricing of such exotic products may become highly complex (as if it wasn't complex enough already!) and are subject to even more leverage than standard derivatives.  In this article we are not going to consider the innumerous exotic derivative products can be constructed.  Instead, we shall consider a very simple product to exemplify a compound derivative.

One of the simplest examples one can take is an option on a future contract. The pricing of futures contracts israther straight forward (see article on 'Pricing Forwards and Futures by Arbitrage Argument').  This is a commonly traded product since futures contracts are often more liquid than standard assets and carry lower transaction costs.

Using arbitrage principles we have previously derived the price of a future contract F(t) at time 't', on an underlying asset of price S(t) with maturity at time T (t<T) to be

where r is the rate or risk-free return.

From the original Black-Scholes equation in partial differential form we have

for an option with value V(S,t).

Now we just need to transform the Black-Scholes in terms of and t so we can solve for V(F,t).  This can be done by direct substitution and transforming the differential operators.  Namely,

and

The transformed Black-Scholes equation reads

Alternatively, one can also derive this equation directly from a Black-Scholes treatment.  For this we must create a risk-free portfolio and perform a delta hedge.  It is assumed that the future contract value also follows a geometric random walk (similar to the underlying asset)

Note that a future contract costs nothing to enter (i.e. =0), therefore, the portfolio is worth only V(F,t).

Using the delta hedging principle one can recover the transformed Black-Scholes equation as written above.  This modified Black-Scholes equation is solved in a similar way to a european call option to give

where and take the usual form.

Naturally, the option must expiry before the maturity of the future contract, since the future is settled (closed) shortly after expiry.

Written by Alessio Farhadi

Back for more