The Black-Scholes model assumes continuous portfolio rebalancing, no transaction costs, stable interest rates, and lognormally distributed and continuously changing exchange rates. Each of these assumptions is violated in periods of currency turmoil, such as occurred during the breakup of the exchange-rate mechanism. With foreign exchange markets shifting dramatically from one moment to the next, continuous portfolio rebalancing turned out to be impossible. And with interest rates being so volatile (e.g., overnight interest rates on the Swedish krona jumped from 24% to 500%), the assumption of interest rate stability was violated as well. Moreover, devaluations and revaluations can cause abrupt shifts in exchange rates, contrary to the premise of continuous movements.8
A related point is that empirical evidence indicates that there are more extreme exchange rate observations than a lognormal distribution would predict.9 That is, the distribution of exchange rates is leptokurtic, or fat tailed. Leptokurtosis explains why the typical pattern of implied volatilities is U-shaped (the so-called volatility smile). Finally, although prices depend critically on the estimate of volatility used, such estimates may be unreliable. Users can, of course, estimate exchange rate volatility from historical data, but what matters for option pricing is future volatility, and this is often difficult to predict because volatility can shift.
4 Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, May/June 1973, pp. 637-659.
5 The value of a pure discount bond with a continuously compounded interest rate k and maturity T is e−−−kT. In the examples used in the text, it is assumed that r* and r are the equivalent interest rates associated with discrete compounding.
6 Garman and Kohlhagen, “Foreign Currency Option Values.”
7 N(d) is the probability that a random variable that is normally distributed with a mean of zero and a standard deviation of one will have a value less than d.
8 Other option-pricing models have been developed that allow for discrete jumps in exchange rates. See, for example, David S. Bates, “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in PHLX Deutschemark Options,” Wharton School Working Paper, 1993. This and other such models are based on the original jumpdiffusion model appearing in Robert C. Merton, “Option Pricing When Underlying Stock Returns Are Discontinuous,” Journal of Financial Economics, January/March 1976, pp. 125-144.
9 This is primarily a problem for options that mature in one month or less. For options with maturities of three months or more, the lognormal distribution seems to be a good approximation of reality.
1. Assume that the spot price of the British pound is $1.55, the 30-day annualized sterling interest rate is 10%, the 30-day annualized U.S. interest rate is 8.5%, and the annualized standard deviation of the dollar:pound exchange rate is 17%. Calculate the value of a 30-day PHLX call option on the pound at a strike price of $1.57.
2. Suppose the spot price of the yen is $0.0109, the three-month annualized yen interest rate is 3%, the three-month annualized dollar rate is 6%, and the annualized standard deviation of the dollar:yen exchange rate is 13.5%. What is the value of a three-month PHLX call option on the Japanese yen at a strike price of $0.0099/¥?
Put-Call Option Interest Rate Parity
As we saw in Chapter 4, interest rate parity relates the forward rate differential to the interest differential. Another parity condition—known as put-call option interest rate parity—relates options prices to the interest differential and, by extension, to the forward differential. We are now going to derive the relation between put and call option prices, the forward rate, and domestic and foreign interest rates. To do this, we must first define the following parameters:
C = call option premium on a one-period contract
P = put option premium on a one-period contract
X = exercise price on the put and call options (dollars per unit of foreign currency)
Other variables—e0, e1, f1, rh, and rf—are as defined earlier.
For illustrative purposes, Germany is taken to be the representative foreign country in the following derivation. In order to price a call option on the euro with a strike price of X in terms of a put option and forward contract, create the following portfolio:
1. Lend 1/(1 + rf) euros in Germany. This amount is the present value of €1 to be received one period in the future. Hence, in one period, this investment will be worth €1, which is equivalent to e1 dollars.
2. Buy a put option on €1 with an exercise price of X.
3. Borrow X/(1 + rh) dollars. This loan will cost X dollars to repay at the end of the period given an interest rate of rh.
The payoffs on the portfolio and the call option at expiration depend on the relation between the spot rate at expiration and the exercise price. These payoffs, which are shown pictorially in Exhibit 8B.1, are as follows:
The payoffs on the portfolio and the call option are identical, so both securities must sell for identical prices in the marketplace. Otherwise, a risk-free arbitrage opportunity will exist. Therefore, the dollar price of the call option (which is the call premium, C) must equal the dollar value of the euro loan plus the price of the put option (the put premium, P) less the amount of dollars borrowed.