Currency Options

Raquel Fonseca

## Keywords

Currencies, Derivatives, Foreign Exchange Rates

Unreviewed

# Definition

Alternatively to the use of forward exchange contracts or futures contracts, the investor could use currency options to hedge his/her risk. Options have the comparative advantage of maintaining a certain degree of flexibility in the hedging, as, while protecting against a downside risk, they do not stop the investor from profiting from unexpected upward movements of the foreign exchange rates. An option [3] “is the right, but not the obligation, to buy (or sell) an asset under specified terms”. Depending on whether we obtain the right to buy or sell the underlying asset, the option is called a call or a put, respectively.

The exercise of an option is a function of its value at expiration date. The investor who buys a call option expects the asset price to increase, so that at expiration date he/she can buy the asset at the strike price $K$ and sell it at the spot price $S$. His/her gain is then the $\max(0,S-K)$, because if the spot price decreases to less than the strike price, the option is not exercised. The buyer of a put option, on the other hand, believes that the spot price will decrease below the strike price, which would allow him/her to profit by $K-S$. At expiration date, his/her gain can be translated as $\max(0,K-S)$.

Increased flexibility, however, does not come without a price attached. In order to be able to exercise his/her right, the buyer of the option must pay a premium up front, which will not be recovered under any circumstance. An option premium depends on the strike and spot prices of the underlying asset, the time to expiration of the option, and the volatility of the underlying asset. The premium reflects both the intrinsic value of the option (whether or not the strike price is less, greater or equal to the spot price) and the time value (the possibility of changes in the spot price that could potentially create a gain for the investor). The seller of the option, often called the option writer receives the premium, and this represents his/her only gain. While the losses of the option buyer are limited by the premium paid, there is no limit to the losses that may be incurred by the option writer.

Together with the premium to be paid, the contract of an option must also specify:

1. Expiration date: options may be distinguished between ‘American’ or ‘European’ options, depending on whether these may be exercised before the expiration date or not, respectively.
2. Strike price: options may be classified as ‘in-the-money’, ‘at-the-money’, or ‘out-of-the-money’, when considering the relationship between the strike price and the spot price of the underlying asset (above, equal or below, respectively).
3. Contract size: similarly to futures, one option contract refers to a certain number of stocks or units of the underlying asset, and not only one unit. The price paid for an option is therefore not related to the quantity of the underlying asset, thus the potential for leveraging gains.

# Option Pricing

Option pricing models are built on the assumption that stock prices (or prices of the underlying asset) follow a ‘geometric Brownian motion’, where:

(1)
\begin{align} dS = \mu Sdt + \sigma Sdz \end{align}

Equation 1 suggests that changes in the stock prices $S$ have two sources: a rate of return $\mu$, expressed as a percentage of the stock price and dependent on time, and a volatility $\sigma$, which adds ‘noise’ and variability to the process and is expressed as a percentage of the stock price and of a random variable that follows a Wiener process (i.e., with a mean zero and a variance one). Behind this formula is the assumption that stock prices follow a Markov process, where the price tomorrow depends only on the price today and is not influenced by the prices in the past. The price today incorporates already all the information from the past.

If stock prices behave in accordance with equation 1, then by Ito's Lemma it can be shown that, [2]:

(2)
\begin{align} d\ln S =\left (\mu - \frac{\sigma^2}{2}\right) dt + \sigma Sdz \end{align}

Prices are then said to have a lognormal distribution, which takes only non negative values and is skewed to the left with different mean, median and mode. The most widely used option pricing model, the Black-Scholes model, relies on the assumption that stock prices are lognormally distributed. Additional assumptions needed to derivate the pricing formulas are, [2]:

1. Short selling is allowed.
2. No transaction costs. Assets are perfectly divisible.
3. No dividends are distributed.
4. No riskless arbitrage opportunities.
6. Constant risk free interest rate.

If $C$ is the price for an European call option, and $P$ the price of an European put option, the Black-Scholes pricing equations are defined as:

(3)
\begin{eqnarray} C & = & SN(d_1) - Ke^{-rT}N(d_2)\\ P & = & Ke^{-rT}N(-d_2) - SN(-d_1) \end{eqnarray}

where $S$ and $K$ are the spot and the strike price respectively, $r$ is the discount interest rate, $T$ is the time to expiration, and $N(x)$ is the standard cumulative normal probability distribution. The arguments of $N(x)$ are: (where $\sigma$ is the standard deviation of the underlying asset)

(4)
\begin{eqnarray} d_1 & = & \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}\\ d_2 & = & d_1 - \sigma\sqrt{T} \end{eqnarray}

# Currency options

Currencies are one of the possible underlying assets for options. Similarly to forwards, currency options are mainly traded on over-the-counter markets, where their specifications can be tailored to the agents own needs, but at the expense of an increased default risk (due to not benefiting from the exchange house protection). The great advantage of using options instead of forwards, is that if the foreign exchange rate moves favourably to the investor, he can still profit from that movement by not exercising the option. While the forward contract locks the foreign exchange rate to be settled in the future, the option provides an insurance, by limiting the value of a downward movement in the exchange rate.
As before, we define $S$ as the spot exchange rate, expressed in units of the base currency per unit of the foreign currency. The strike price $K$ of a currency option is the foreign exchange rate to be valid at the maturity date. Assume an US investor is expecting to receive a certain amount in GBP at some point in the future. In order to hedge against the currency risk, he would buy a put option on the GBP, therefore having the right to sell GBP at the strike price $K$ defined (or more precisely, at a certain value of the foreign exchange rate) at the maturity date. This foreign exchange rate is not binding: if the spot exchange rate in the future increases above the strike price, the option will not be exercised. (The investor could instead buy a call option on USD against GBP — note how currency options are symmetrical by definition.)

This strategy is called a ‘covered put’. As the investor holds a long position both on the asset and on a put option, his gains are unlimited, namely they are dependent on the future spot price, and his losses are limited by the strike price of the put option.

# The Garman-Kohlhagen pricing model

The pricing model for currency options is very similar to the original Black-Scholes model, and it uses the fact that holding a foreign currency is equivalent to holding a stock with a known dividend yield. Holding foreign currency provides an interest at the risk-free rate prevailing in the foreign country. The pricing model for currency options takes into account the risk-free rate of the foreign country, $r_f$, and of the home country, $r$, and similarly to the Black-Scholes model, assumes both are constant. All other assumptions remain valid. The extension of the Black-Scholes model to the pricing of currency options is owed to Mark Garman and Steven Kohlhagen [1]. If $C$ is price of a call option and $P$ of a put option, we have:

(5)
\begin{eqnarray} C & = & Se^{-r_fT}N(d_1) - Ke^{-rT}N(d_2) \\ P & = & Ke^{-rT}N(-d_2) - Se^{-r_fT}N(-d_1) \end{eqnarray}

where

(6)
\begin{eqnarray} d_1 & = & \frac{\ln(S/K) + (r - r_f + \sigma^2/2) T}{\sigma\sqrt{T}}\\ d_2 & = & d_1 - \sigma\sqrt{T} \end{eqnarray}

Equations 5 may be simplified if we assume we have a forward contract with rate $F$ with the same maturity data as the currency option. We have that $F = Se^{(r-r_f)T}$, which can be used to simplify:

(7)
\begin{eqnarray} C & = & e^{-rT}(F N(d_1) - K N(d_2))\\ P & = & e^{-rT}(X N(-d_2) - F N(-d_1)) \end{eqnarray}

where

(8)
\begin{eqnarray} d_1 & = & \frac{\ln(F/K) + (\sigma^2/2) T}{\sigma\sqrt{T}}\\ d_2 & = & d_1 - \sigma\sqrt{T} \end{eqnarray}