Currency 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:
 Current asset price
 Exercise price
Time to expiry of option
 Volatility
 Risk-free money rate
 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-Kohl Hagen 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 risk less 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 recognized 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 recognized 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 realize 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.

What is currency options delta? The most obvious risk an option trader faces is the movement in the underlying asset (spot exchange rates). The effect this has on an option position depends on the direction and the extent of this movement. Generally, if the book is long of options, any loss experienced is limited. However, if the book is short of options, then losses are potentially unlimited.
  
Traders must therefore be aware of the sensitivity of the value of each option in their book to movements in the spot exchange rate and this is reflected in the delta of the option.
  
 Delta = Change in option value / Change in underlying asset
  
 A helpful interpretation of "Delta" is that it reflects the probability of exercise; although this is not exactly true, as the delta of some exotic options can be significantly greater than one. In the case of standard options, however, it is a useful interpretation. The gamma effect means that position deltas move as the asset price moves and predictions of revaluation profit and loss based on position deltas are therefore not accurate, except for small moves.
  
Gamma (G) measures the response of an option's Delta to changes in rates.
Gamma = Change in Delta / Change in Underlying Asset
  
Bought options have positive gamma while sold options have negative gamma. A portfolio's gamma will be the weighted sum of its option's gammas and the resulting gamma will be determined by the dominant options in the portfolio. In this regard, options close to the money with short time to expiry have a dominant influence on the portfolio's gamma. The Gamma of an option increases as the option matures and decreases with volatility.
  
A portfolio with a positive gamma gets longer as the market goes up and shorter as the market goes down, which is ideal. A portfolio with negative gamma gets shorter as the market goes up and longer as the market goes down.
  
A portfolio with a positive gamma is more attractive than a negative gamma portfolio with time decay being the mitigating factor. A negative gamma means the rate of losses increases as losses are sustained and the rate of profit falls as profits are experienced.
  
With delta neutral positions, the sign of Gamma is useful. If Gamma is negative, the portfolio profits so long as the spot rate remains stable. If Gamma is positive, the portfolio will only profit from large movements in spot rates in either direction.
  
To adjust the Gamma of a portfolio a trader must buy or sell additional option contracts as the Gamma of a cash position is zero. Options are more expensive the longer the time to maturity. Hence all options will become less valuable the longer they are held. This decay of time value (theta) will thus work in favor of short positions in options and against long positions (whether calls or puts).
  
Bought options have negative thetas, sold options have positive thetas. Generally a portfolio with a positive theta will gain in value as time passes and nothing else changes, whereas a portfolio with a negative theta will decay over time if nothing changes. In general, as the expiry date comes closer the value of the option falls, all other things being equal. This is intuitively correct because there is less time in which a price movement can occur. This rate of decay tends to accelerate as the option approaches its maturity date and decelerates as the option moves out of-the-money.
  
Options close to the money will have more influence on the position theta than options far from the money. This is due to an option having its maximum time value (ie. price less intrinsic value) when it is at the money. Most time decay occurs near to expiry. It is at this time that a positive theta will be most profitable, assuming that the strategy can he kept delta hedged. Risk reversals refer to the difference in pricing for a Put and a Call for a 25 Delta option. Supply and demand in the options market often means that these Puts and Calls, which theoretically should trade at the same volatility level, have differing prices.
  
The name 'Risk Reversal' comes from the fact that traders actually make markets in the price difference between the 25 delta puts and calls. The affect of buying or selling the risk reversal is to change the risk to being:
  
Long Calls / Short Puts    or    Long Puts / Short Calls Volatility is the expected degree of fluctuation in the spot rate. The most serious losses which occur in option portfolios are often the result of jumps in volatility.  The derivative of the option price with respect to volatility is called the vega. A positive vega position will result in profits from an increase in volatility. To create a positive vega, a trader needs to dominate the portfolio with bought options, bearing in mind that the vega will be dominated by those options that are close to the money and have significant time to expiry remaining.
  
VEGA = Change in Premium
Change in Volatility as all options rise in value with increases in volatility, a long position wig benefit farm an increase in volatility whilst a short position will lose money (Again this is riot always true in the case of exotic options).
  
The volatility is normally annualized and expressed as a percentage. This volatility can be estimated from a sample of historical data for the asset price- The problem with this is to decide how many days to go back, and there is no fixed answer to this. The historical estimate assumes, of course, that the past is a good guide to the future.
  
In an exchange traded market, one can readily deduce the market's view of volatility by taking the market value of an option and all the other inputs except volatility, and 'backing out' the volatility which would be needed in the formula to make the market price equal the fair value. This is called the implied volatility. The problem with this can be that options at different strikes with the same maturity may have different implied volatilities. This implied volatility is basically what the market trades in fact, and many markets are even quoted in volatility terms. Volatility becomes a traded entity in its own right and is the essence of modern option markets.