Options Pricing

While it is not necessary to understand the actual mathematics of the option pricing model, it is useful to have some understanding of how the various components of the model affect the option premium.

The major inputs to models of this type are:

  1. Current asset price
  2. Exercise price
  3. Time to expiry of option
  4. Volatility
  5. Risk-free money rate
  6. Holding benefit or dividend rate on underlying instrument

The most famous option pricing theory work was done by Black and Scholes. Their work has been subsequently modified for valuing options in a number of markets (e.g. Garman-Kohlhagen for currencies, Black for futures, the Binomial model for American options). The basic idea of these models is to specify the condition that a dynamic hedge should be able to be created between an option and the underlying instrument, and then to use the fact that the resulting riskless portfolio should earn the risk-free rate. The solution to the equation specifying the condition, given the known boundary values of the option at expiry provides the fair value of the option at any time and the hedging mechanism required. It is assumed that the price of the underlying instrument follows some sort of stochastic process.

In fact, it is now widely recognised that these 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, and are especially significant for longer dated options. It is equally widely recognised that the models become increasingly shaky for interest rate options, especially long-dated ones. Certainly, the problems encountered for the other instruments are no less, but there is also a massive conceptual problem in assuming a constant money 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.

It is important to realise that the fair value of an option calculated according to a Black and Scholes type model only makes sense in the context of the riskless hedge argument. It is certainly possible to buy options under fair value, as determined by you, and to lose money; or to sell option above your fair value and lose money The only way the fair value can be locked in is by maintaining the dynamic hedge, either through the underlying instrument itself or by means of other suitable options in a portfolio approach. Even then, there is usually some degree of sensitivity to assumptions about the underlying stochastic process and its volatility.

A good starting point from which to understand the pricing of a currency option is to break the option premium down into two parts - its intrinsic value and its time value For any option, the premium will be equal to the sum of its intrinsic value and its time value.

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