FX Option Valuation
A currency option or FX option is a contract that gives the buyer the right, but not the obligation, to buy or sell a certain currency at a specified exchange rate on or before a specified date. Currency options are one of the most common ways for corporations, individuals or financial institutions to hedge against adverse movements in exchange rates.
A currency option, also known as FX Option, is a derivative contract that grants the buyer the right but not the obligation to exchange money denominated in one currency into another currency at a pre-agreed exchange rate on a specified future date. The FX options market is the deepest, largest and most liquid market for options of any kind. Most FX derivatives trading is over the counter (OTC) and is lightly regulated.
There are call options and put options. Also a currency option could be European style or American style. Call options provide the holder the right but not the obligation to purchase an underlying currency at a specified FX rate on a future date, while Put options give the holder the right to sell an underlying currency at a specified FX rate on a future date. A European option can be exercised only at the expiration date of the option, whereas an American option can be exercised anytime during its life.
FX options contain Vanilla FX option and Listed FX option. Although both have similar characteristics, they differ mainly in two respects: First, Vanilla options are traded OTC while Listed options are exchanged-traded. Secondly, the underlying of Vanilla options is FX spot while the one of Listed options is FX future. Nevertheless, their analytics are very similar.
Currency options are one of the most common ways for corporations, individuals or financial institutions to hedge against adverse movements in exchange rates. Corporations primarily use FX options to hedge uncertain future cash flows in a foreign currency. The general rule is to hedge certain foreign currency cash flows with forwards, and uncertain foreign cash flows with options.
Options give market participants many opportunities to limit risk and increase profit. Investors buy calls when they think the FX rate will rise or sell a call if they think it will fall. Selling an option is also referred to as ‘‘writing’’ an option. On the other hand, they buy puts if they think the FX rate will fall, or sell one if they think it will rise.
One of the most common reasons for using FX options is for short-term hedges of spot FX or foreign market positions. Unlike a forward contract that locks in the FX rate for a future transaction, FX options allow the investors to benefit from favorable FX rate movements. Currency market fluctuations can have a lasting impact on cash flow whether it is buying a property, paying salaries, making an investment or settling invoices. By utilizing FX Options, business can protect themselves against adverse movements in exchange rates.
FX Options are also useful tools which can be easily combined with Spot and Forward contracts to create bespoke hedging strategies. There are many bullish, bearish and even neutral strategies that can be implemented with options contracts. Spread strategies that are used in equity options can also be used with FX options, including vertical spreads, straddles, condors and butterflies. This presentation gives an overview of FX option definition and pricing model.
The payoff of a European call option
Payoff=N_b*max(X-K,0) where Nb the notional of the base currency X the spot FX rate K the strike.
The payoff diagram of a European call option
The payoff of a European put option
Payoff=N_b*max(K-X,0) where Nb – the notional of the base currency X – the spot FX rate K – the strike
Payoff diagram of a European put option
The present value of a European call option is given by
PV(t)=[X_0 e^(-r_q (T_d-T_s ) ) Φ(d_1 )-Ke^(-r_b (T_d-T_s ) ) Φ(d_2)] where d_1,2=[ln(X_0⁄K)+r_b (T_d-T_s )-r_q (T_d-T_s)±σ^2 (T_e-t)]⁄(σ√(T_e-t)) t the valuation date X_0 the spot FX rate quoted as base/quote K the strike using the same quotation as the spot rate T_s the spot date (several days after valuation date) T_d the option delivery date T_e the option expiry date r_b the base currency interest rate for period (T_s, T_d) r_q the quote currency interest rate for period (T_s, T_d) σ the volatility corresponding to K and (T_e-t) Φ the cumulative standard normal distribution function
The present value of a European put option is given by
PV(t)=[Ke^(-r_b (T_d-T_s ) ) Φ(-d_2 )-X_0 e^(-r_q (T_d-T_s ) ) Φ(-d_1 )]
where all notations are the same as above
American FX Option FinPricing is using the Odd-Even Cox Ross Binomial model to compute prices and risk sensitivities for American-style FX OTC options.
The Odd-Even Binomial model is an extension of the Cox Ross Binomial model. Compared to Cox Ross Binomial model, the Odd-Even Binomial model provides better price accuracy. It was implemented by creating two tress, one with even number n of iterations and another with n+1 number of iterations, then taking the average of the two results. This method also helps to reduce the number of oscillations that can be observed with traditional Binomial models, where the option value depends quite heavily on the number of periods.
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