Generalized Put-Call Parity: Practice Problems and Solutions

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov.

This is Section 6 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here.

It is not necessary for options on an asset to be paid for with cash at expiration. Rather, the strike asset (the asset which is being offered in exchange for the asset on which the option is written) can be something else - a different stock, bond, widget, or hippopotamus. The generalized put-call parity equation expresses the relationship between puts and calls where the underlying asset and the strike asset can possibly be anything.

The equation for generalized put-call parity is

C(S_t, Q_t, T-t) - P(S_t, Q_t, T-t) = F^P_t,T(S) - F^P_t,T(Q)

Here, the assets under our consideration are

Asset A - the underlying asset - the asset on which the options are written and

Asset B - the strike asset - which we give up in return for the underlying asset.

Explanation of variables:
F^P_t,T(S) = the time t price of a prepaid forward on Asset A, paying S_T at time T.

F^P_t,T(Q) = the time t price of a prepaid forward on Asset B, paying Q_T at time T. Note that this is the analog of PV_0,T(K) in our prior put-call parity formula, where K was the strike price of the underlying asset and the strike asset was cash.

C(S_t, Q_t, T-t) = price of a European call option with underlying asset A, strike asset B, and time to expiration T-t.

P(S_t, Q_t, T-t) = price of a European put option with underlying asset A, strike asset B, and time to expiration T-t.

At time T, the call payoff is C(S_T, Q_T, 0) = max(0, S_T - Q_T).

At time T, the put payoff is P(S_T, Q_T, 0) = max(0, Q_T - S_T).

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9, p. 287.