Valuation of Equities and Currencies Using the Black-Scholes Model

Some underlying instruments have carry benefits. These benefits include dividends for stock options, foreign interest rates for currency options, and coupon payments for bond options.

The BSM model should be adjusted to incorporate carry benefits in the option value. Let the carry benefit be a continuous yield, \(\gamma\). The carry adjusted BSM model is expressed as:

$$ \begin{align*} \text{European call}: c_0 & =S_0e^{-\gamma T}N\left(d_1\right)-e^{-rT}KN\left(d_2\right) \\ \text{European put}: p_0 & =e^{-rT}KN\left({-d}_2\right)-S_0e^{-\gamma T}N\left(-d_1\right) \end{align*} $$

Where:

$$ d_1=\frac{\ln{\left(\frac{S_0}{K}\right)}+\left(r-\gamma+\frac{\sigma^2}{2}\right)T}{\sigma\sqrt T } $$

and

$$ d_2=d_1-\sigma\sqrt T $$

It is worth noting that carry benefits lower the expected future value of the underlying. Further, an increase in carry benefits lowers the value of a call option and raises the put option’s value.

BSM Valuation for Equities

Assume that the underlying equity has a continuously compounded dividend yield \(\gamma=\delta\). The BSM model can be adjusted for dividends as follows:

$$ \begin{align*} \text{European call}: c_0 & =S_0e^{-\delta T}N\left(d_1\right)-e^{-rT}KN\left(d_2\right) \\ \text{European put}: p_0 &=e^{-rT}KN\left({-d}_2\right)-S_0e^{-\delta T}N\left(-d_1\right) \end{align*} $$

Where:

$$ d_1=\frac{\ln{\left(\frac{S_0}{K}\right)}+\left(r-\delta+\frac{\sigma^2}{2}\right)T}{\sigma\sqrt T } $$

and

$$ d_2=d_1-\sigma\sqrt T $$

The arbitrageur of a dividend-paying stock receives dividend payments when they long the stock and pays dividends when they short the stock. Dividends reduce the number of shares to buy for calls and the number of shares to short-sell for puts. The higher the dividends, the lower the value of \(d_1\) and hence the lower the value of \(N\left(d_1\right)\).

Example: Valuing Stock Options Using the BSM Model

Consider a stock that is trading on the London Stock Exchange at £50. A trader believes that the stock price will rise in the next month and decides to buy one-month call options with an exercise price of £53. The risk-free annual rate of interest is 2%, and the yield on the stock is £0.35%. The volatility of the stock is 20%.

The BSM model inputs are as follows:

• The spot price of the underlying = £50
• Exercise price = £53
• Expiration = 1 month
• Risk-free rate = 2%
• Dividend yield = 0.35%.
• Volatility = 0.20 

BSM Valuation of Currencies

The BSM model can also be used to value foreign exchange options. The carry benefit for a foreign exchange option is the continuously compounded foreign risk-free interest rate.

The values of European call and put options are determined using the following formulas:

$$ \begin{align*} \text{European call}: c_0 &=S_0e^{-r^fT}N\left(d_1\right)-e^{-rT}KN\left(d_2\right) \\ \text{European put}: p_0 &=e^{-rT}KN\left(-d_2\right)-S_0e^{-r^fT}N\left(-d_1\right) \end{align*} $$

Where:

$$ d_1=\ln{\left(\frac{S_0}{K}\right)}+\frac{\left(r-r^f+\frac{\sigma^2}{2}\right)T}{\sigma\sqrt T} $$

and

$$ d_2=d_1- \sigma\sqrt T $$

Note that:

\(r\) = Domestic risk-free rate.

\(r^f\) = Foreign risk-free rate.

Example: BSM Model Applied to Value Options on Currency

A swiss exporter will receive Euros for his watches. The exporter purchases a three-month put option with an exercise price K = 1.07CHF/EUR to protect themselves against a decrease in the EUR exchange rate. The current exchange rate is 1.08CHF/EUR.

The BSM model inputs for this currency option are as follows:

• The underlying, \(S_0\) (the value of the domestic currency per unit of the foreign currency) = 1.08CHF/EUR
• \(r\) = The annualized swiss risk-free rate
• \(r^f\) = The EUR rate (Carry rate)
• Time to expiration = 0.25 years (three months)

Question

An Australian importer has to pay fixed pound (£) amounts every three months for goods. The spot price of the currency pair is 1.78A$/£. If the exchange rate rises to, say 1.80 A$/£, then the Aussie will have weakened as it will take more Aussie dollars to buy one pound. The importer believes that the Australian dollar will depreciate in the following months. The importer, therefore, decides to buy an at-the-money spot pound call option to protect herself against this depreciation. The risk-free Australian rate is 3.00%, and the British risk-free rate is 2.00%.

The underlying price, the risk-free rate, and the carry rate to use in the BSM model to get the pound call option value is most likely:

1. 1.78, 2.00%, 3.00%.
2. 0.56, 2.00%, 3.00%.
3. 1.78, 3.00%, 2.00%.

Solution

The correct answer is C.

The underlying, \(S_0\) (the value of the domestic currency per unit of the foreign currency) = 1.78A$ /£. The risk-free rate is the Australian rate, 3.00%, and the carry rate is the British rate of 2.00%

Reading 34: Valuation of Contingent Claims

LOS 34 (h) Describe how the Black–Scholes–Merton model is used to value European options on equities and currencies.