Options Pricing Overview

While option traders do not need to understand the actual mathematics of the option pricing model in order to trade currency options successfully, many traders find it useful to have some understanding of how the various components of the pricing model affect an option’s premium.

A good starting point from which to understand the pricing of a currency option is to break the option premium down into its two component parts - its intrinsic value and its time value. The premium of any option will be equal to the simple sum of its intrinsic value and its time value, although while the intrinsic value is relatively easy to calculate, the time value component of an option’s price requires fairly advanced mathematical modelling techniques to determine it with any degree of accuracy.

Historically, the most famous option pricing theory work was done by Black and Scholes in the early 1970’s. Their seminal work permitted the options markets to start trading on a much better theoretical basis than just supply and demand, and their Black-Scholes option pricing model that was originally used to determine the “fair value” of stock options has now been modified for valuing options in a number of different markets. Specifically, the forex market typically uses the Garman-Kohlhagen model for pricing currency options, while the Black model is used for evaluating options on futures, and the Binomial model is used to determine theoretical pricing for American-style options.

The primary inputs into the Garman-Kohlhagen currency option pricing model are:

  1. The current market price of the underlying spot, forward or futures forex contract.
  2. The call currency and the put currency.
  3. The strike price.
  4. The expiration date of the option.
  5. The delivery date of the option.
  6. The delivery date of the premium.
  7. The risk-free interest rates for both currencies.
  8. The option style (European-style, American-style, etc.)
  9. The market implied volatility corresponding to the expiration date, delta and strike.

The basic idea incorporated into these option pricing models is to assume that a dynamic hedge should be able to be created for the option with the underlying instrument, and that the price of the underlying instrument follows some sort of stochastic process in which upward movements in the underlying are as likely as downward movements. Then, the idea that the resulting riskless portfolio should earn the risk-free interest rate is used in order to solve for the theoretical option price.

The solution to the equation based on the given assumptions and the known values of the option at expiration for varying values of the price of the underlying instrument, provides the theoretical or “fair” value of the option at any point in time. The solution also determines both the hedging mechanism and the quantity of the underlying instrument or other options that are required to create a riskless portfolio.

In fact, it is now widely recognised that these pricing models have reasonable validity in the case of equities, currencies and commodities. The principal difficulties relate to the constant volatility and constant interest rate assumptions of these models, and this issue becomes especially significant for longer-dated options. Furthermore, many interest-rate market participants find these models insufficient for pricing interest rate options, and  especially so for those with longer expirations. While the pricing model issues encountered with the other financial instruments are significant, a substantial conceptual problem arises when assuming a constant risk-free interest rate for the life of the option, while at the same time using a stochastic process for the interest-rate-related instrument on which the option is written!

In addition, one needs to realise that the fair value of an option calculated according to a Black-Scholes-type pricing model only makes sense in the context of the riskless hedge argument. As any seasoned trader can tell you, it is possible to buy options under the pricing model’s fair value and lose money; or to buy an option above the model’s fair value and make money, and vice versa. Basically, the only way the fair value can be locked in is by maintaining the dynamic hedge, either through making transactions in the underlying instrument itself, or by means of using other suitable option trades combined in a portfolio approach. Even then, usually some degree of sensitivity exists to assumptions about the stochastic process of the underlying instrument’s price and its volatility.

 

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