Generalized Put-Call Parity: Practice Problems and Solutions

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.

Problem GPCP1. Bard Co. and Drab Co. have publicly traded stock, and option contracts are written on the stock of Bard Co. with the stock of Drab Co. as the strike asset. (One share of Drab Co. is the strike for an option written on one share of Bard Co.) Neither companies' stock pays any dividends. Bard Co. trades at $34 per share, and Drab Co. trades at $56 per share. The price of this kind of put option on Bard Co. is currently $25. Find the price of a call option on Bard Co. Both options expire in 2 years.

Solution GPCP1. Neither companies' stock pays any dividends, so the stocks' prepaid forward prices equal the stock prices themselves.

Thus, we can use the formula C(S_t, Q_t, T-t) - P(S_t, Q_t, T-t) = F^P_t,T(S) - F^P_t,T(Q),

where F^P_t,T(S) is the price of Bard Co. stock, i.e., 34, and F^P_t,T(Q) is the price of Drab Co. stock, i.e., 56, while P(S_t, Q_t, T-t) = 25. We note that it does not matter when the options expire in this case, because the stocks do not pay dividends, so their prepaid forward prices will always equal the stock prices.

Hence, we rearrange the equation as follows:
C(S_t, Q_t, T-t) = F^P_t,T(S) - F^P_t,T(Q) + P(S_t, Q_t, T-t) = 34 - 56 + 25 = C(S_t, Q_t, T-t) = 3

Problem GPCP2. The stock of Tram Law, Inc., currently trades at $356 per share. One hippopotamus can be purchased for $1000. Options on Tram Law stock are written, with hippopotami as the strike asset. (I told you it could be done!) The contracts stipulate that, at expiration, one hippopotamus can be exchanged for three shares of Tram Law, Inc. Neither Tram Law, Inc. nor a hippopotamus pays any dividends. A call option of this kind has time to expiration of 5 years and costs $95. What is the price of a put option of this kind?

Solution GPCP2. Likewise, because neither the underlying asset nor the strike asset pay any dividends, the assets' prepaid forward prices equal their current prices. Furthermore, for this reason, time to expiration of these options does not matter - unless you are the person having to hold on to hippopotami for 5 years. We note, however, that the underlying asset is actually three shares of Tram Law, Inc., and this asset has a price of 356*3 = 1068 = F^P_t,T(S). F^P_t,T(Q) is 1000, the price of one hippopotamus. C(S_t, Q_t, T-t) = 95.

We arrange the generalized put-call parity equation as follows.

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

Thus, 95 - 1068 + 1000 = P(S_t, Q_t, T-t) = 27

Problem GPCP3. Profligate, Inc., pays dividends on its stock; its continuously compounded dividend yield is 0.02. The price of a share of Profligate, Inc., stock is currently $67. Misers and Associates pays no dividends on its stock; its current stock currently trades for $95/share. A call option on Profligate, Inc., stock with Misers and Associates stock as the strike asset expires 13 months from now. The price of this call option is $45. Find the price of a put option on Profligate, Inc., stock with the same time to expiration. (One share of Misers and Associates is the strike for an option written on one share of Profligate, Inc.)

Solution GPCP3. Since Profligate, Inc., pays dividends on its stock, its prepaid forward price will differ from the stock price. Namely, since time to expiration is 13/12 years, F^P_t,T(S) = e^-0.02*13/12*67 = 65.56394676. F^P_t,T(Q) is still 95, since Misers and Associates pays no dividends. We are given that C(S_t, Q_t, T-t) = 45.

Thus, we can use the formula C(S_t, Q_t, T-t) - P(S_t, Q_t, T-t) = F^P_t,T(S) - F^P_t,T(Q),

rearranging it as follows:
P(S_t, Q_t, T-t) = C(S_t, Q_t, T-t) - F^P_t,T(S) + F^P_t,T(Q)

Thus, 45 - 65.56394676 + 95 = P(S_t, Q_t, T-t) = 74.43605324 (Profligate, Inc., stock is quite a volatile asset, indeed!)

Problem GPCP4. Hippopotami multiply at an annual continuously compounded rate of 0.1, whereas ostriches multiply at an annual continuously compounded rate of 0.25. One hippopotamus currently costs $2000, whereas one ostrich currently costs X. Call and put options are written on hippopotami, with one ostrich as the strike asset per one hippopotamus option. The call option currently costs $543, whereas the put option currently costs $324. Both options expire in 12 years. How much does one ostrich currently cost? (That is, find X).

Solution GPCP4. Did you think that hippopotami and ostriches could not pay dividends? It turns out that they can; here, their rate of multiplication is analogous to a continuously compounded dividend yield.

We can use the formula C(S_t, Q_t, T-t) - P(S_t, Q_t, T-t) = F^P_t,T(S) - F^P_t,T(Q), noting that

C(S_t, Q_t, T-t) = 543 and P(S_t, Q_t, T-t) = 324; thus, C(S_t, Q_t, T-t) - P(S_t, Q_t, T-t) = 219.

Hippopotami are the underlying asset, so F^P_t,T(S) = e^-0.1*12*2000 = 602.3884238.

Ostriches are the strike asset, so F^P_t,T(Q) = e^-0.25*12*X = 0.0497870684X

Thus, 219 = 602.3884238 - 0.0497870684X

So 602.3884238 - 219 = 383.3884238 = 0.0497870684X

383.3884238/0.0497870684 = X = 7700.562337

Problem GPCP5. Superwidgets spontaneously replicate themselves at an annual effective rate of 0.05. One superwidget costs $356. Gold does not spontaneously replicate itself and currently costs $567 per ounce. A call option on superwidgets with gold as the strike asset currently costs $34. A put option on superwidgets with the same time to expiration costs $290. One ounce of gold is the strike for an option written on one superwidget. Determine the time to expiration of the call and put options.

Solution GPCP5. We can use the formula

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

C(S_t, Q_t, T-t) = 34 and P(S_t, Q_t, T-t) = 290; so C(S_t, Q_t, T-t) - P(S_t, Q_t, T-t) = -256

Gold is the strike asset and pays no dividends, so the prepaid forward price is equal to the asset price. Thus, F^P_t,T(Q) = 567.

Superwidgets are the underlying asset, with time Y used in the present value factor. This is our time to expiration. F^P_t,T(S) = (1.05)^-Y*356

Thus, -256 = (1.05)^-Y*356 - 567

311 = (1.05)^-Y*356

(1.05)^Y = 356/311 = 1.144694534

Y = ln(1.144694534)/ln(1.05) = Y = 2.769775855 years

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.