FX Barrier Options by Zareer Dadachanji

Author:Zareer Dadachanji
Language: eng
Format: epub
ISBN: 9781137462763
Publisher: Palgrave Macmillan
Published: 2015-10-15T00:00:00+00:00

where all the market data parameters (spot, volatility and interest rates) are held constant.

I have named this risk measure specifically in terms of the Black–Scholes model, because when we move beyond the Black–Scholes model (as we will do in Chapters 4 and 5), we will see that there are all sorts of different ways in which we can calculate theta.

When the option maturity is a long way off, the difference between the mathematical and practitioner theta will be small. But as we approach expiry, the difference will become much larger.

For the specific case of vanilla options, the rate of change in option value with respect to time is always negative, and is termed time decay. This is illustrated by Figures 3.33. It is no coincidence that the theta profile looks like the negative of the vega and gamma profiles: just as vega and gamma are fundamentally related to each other, so too is theta intimately connected. We can understand this intuitively if we consider the option value as a function of variance σ 2(T − t), so that an increase in volatility σ increases the variance, and a step forward in time t decreases the variance. In practice, the option value will also depend on the forward and the discount factor, which will bring the Domestic and Foreign interest rates into the relationship. A relationship between theta, vega and the rhos has been derived by Reiss and Wystup [24], and can be written in the following form:

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