Given the backdrop of the earlier section, it is important for the reader to understand how the original Black-Scholes option pricing formulae has been adapted by traders to price options on various underlyings (e.g., dividend paying stocks, commodity futures, and so on). In this section, I will discuss the assumptions made by practitioners and the models used to value such options in three different instances.

Pricing Options on Dividend-Paying Stocks

In the previous section, I discussed the valuation of European-style options on non-dividend paying stocks. In practice, there are companies that do pay quarterly dividends to their shareholders (e.g., Apple, Microsoft, and Bank of America). Additionally, when valuing options on indices, the issue of valuing options on dividend-paying stocks becomes more pressing, especially if the index is mostly made up of dividend-paying stocks (e.g., S&P 500 Index).

To be able to value options on dividend-paying stocks, I will first assume that dividends are paid continuously (a very reasonable assumption when valuing options on indices but not when valuing options on a single stock^{[1]}). Since a dividend has the effect of decreasing the value of a stock, assuming that a stock pays a continuously compounded dividend rate of q, equation (3.1) takes the form of

(3.5)

where

S is the price of the stock.

r is the annualized continuously compounded risk-free rate. q is the annualized continuously compounded dividend yield. σ is the annualized volatility of the index return.

dz is the random variable drawn from a standard normal probability density function.

dS is the small change in the stock price over a small time interval dt.

As earlier, observing that the diffusion equation given by equation (3.5) is equivalent to assuming that In ST is assumed to be normally distributed with a mean of

and a variance of

where

is the price of the stock at time t.

is the zero rate corresponding to a maturity (T) using zero rate curve at time t.

is the zero-dividend rate corresponding to a maturity (T) using zero dividend curve at time t.

is the spot-volatility rate corresponding to a maturity (T) using spot volatility curve at time t. is the time today.

is the time when the option matures.

It readily follows that the formula for pricing the call option on a dividend-paying stock can be obtained by taking an approach similar to equation (3.2). As a consequence, one has

(3.6)

Equation (3.6) can be simplified to obtain

(3.7a)

whereandare as defined earlier.

The formula to price a put option on a dividend-paying stock can similarly be shown to be

(3.7b)

Equations (3.7a) and (3.7b) can easily be programmed on the Microsoft Excel spreadsheet as shown in Table 3.4.

TABLE 3.4 Valuing European-Style Options on Dividend-Paying Stocks

Although Table 3.4 shows the valuation of an option on dividend-paying stocks using equations (3.7a) and (3.7b), these equations can also be used to value options on indices and currencies. Since the application of the equations to value index options is straightforward, I will only discuss the adaptation of equations (3.7a) and (3.7b) to value currency options.

Like stock and index options, currency options also trade on both the exchange and OTC markets where the underlying asset could be either a spot currency or currency futures or currency forward rate. For the sake of convenience, I will assume that the underlying is a spot-currency rate. Because currency rates can be quoted in two ways (e.g., USD/EUR and EUR/USD), one has to understand the context in which terms like call and put options are used, since a call option on a currency rate that is written one way (e.g., USD/EUR) is the same as a put option on the same currency rate that is written in a reciprocal manner (e.g., EUR/USD).

Carman and Kohlhagen (1983) observed that currency rate movements are a function of both the domestic risk-free rate and foreign risk-free rate of that currency. They then went on to assume that the foreign risk-free rate can be treated as a continuously compounded dividend rate and modified equation (3.5) to arrive at

(3.8)

where

S is the currency-exchange rate (expressed as the number of domestic units divided by foreign units).

r is the annualized continuously compounded domestic risk-free rate. rf is the annualized continuously compounded foreign risk-free rate. a is the annualized volatility of spot currency rate returns. dz is the random variable drawn from a standard normal probability density function.

dS is the small change in the exchange rate over a small time interval dt.

Given the similarity of equation (3.8) to equation (3.5), one can easily use equations (3.7a) and (3.7b) to value the currency options by replacing qt,T in the formulae with rft,T. Doing this, one can arrive at the setup laid out in Table 3.5 when the underlying currency rate is the CAD/USD (Canadian dollar/U.S. dollar) pair.

TABLE 3.5 Valuing European-Style Currency Options

From Table 3.5, it can be seen that the option premiums^{[2]} (given in cells B10 and B11) are denominated in CAD/USD (or domestic/foreign). For this to be converted to a dollar amount, one would need a notional in U.S. dollars so that the premiums paid for the option are denominated in Canadian dollars.^{[3]}

[1] The reason for this stems from the fact that dividends are usually paid no more than four times a year on a given stock (i.e., discrete payment amounts at discrete times).

[2] The call (put) option in Table 3.5 refers to the call (put) on USD or put (call) on CAD in the nomenclature of the currency markets. Sometimes this can be further abbreviated as USD call (call on USD) or a USD put (put on USD) and so on.

[3] In the event the premium needs to be paid in USD, one only needs to convert the premium in CAD using the spot rate. In the event the notional is given in CAD, one would need to convert the notional to USD using the strike rate.

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