Striking a balance between mathematical rigour and market practice and written by experienced practitioner Antonio Castagna, the book shows readers how to correctly build an entire volatility surface from the market prices of the main structures. Starting with the basic conventions related to the main FX deals and the basic traded structures of FX Options, the book gradually introduces the main tools to cope with the FX volatility risk.

It then goes on to review the main concepts of option pricing theory and their application within a Black-Scholes economy and a stochastic volatility environment. The book also introduces models that can be implemented to price and manage FX options before examining the effects of volatility on the profits and losses arising from the hedging activity. All rights reserved. Versand In den Warenkorb. The situation is still worse for portfolios including many options with different strikes and different maturities.

Many articles and books are devoted to this task and we refer to them for further analysis of this issue, since we deem it is beyond the scope of this study to investigate it.

FX Options and Smile Risk

We summarize the results of this section in the following Facts 2. That is why we have to shift our attention on the volatility risk. Those are the questions we try to answer in this section. This volatility may be simply time dependent that is: deterministically dependent from time , or it can be stochastic. We do not know and we do not want to model the true nature of the process commanding the evolution of the price S. So it seems sensible to think that in some way the performance of the replicating strategy is determined by the implied volatility and its link with the realized volatility.

This intuition is confirmed by the following analysis.

We summarize the results of this section in the following: Facts 3. Nevertheless it cannot be used to estimate the risk in the real world; in fact, everyday the book is marked to the market, so to have a revaluation as near as possible to the true current value of the assets and other derivatives.

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That means that the book is revaluated at current market conditions regarding the price of the underlying asset and the implied volatility we drop for the moment the fact that also the interest rates are updated to the current levels. We summarize the results of this section in the following: Facts 4.

Flat Volatility Smile. Convex Volatility Smile. Volatility Risk: Exposures to the Volatility Matrix In the real world the entire volatility surface is stochastic; this does not mean that at any time each implied volatility, corresponding at a given strike and maturity, moves in an erratic way completely unrelated with all the other volatilities. As a stylized fact, we can identify, for a given expiry, three kind of movements of the smile. The second kind of movement is a change in the curvature of the smile; it is a symmetric movement.

Positive Sloped Volatility Smile. Negative Sloped Volatility Smile.

FX options and smile risk

The slope can be also negative; for example in Figure 6 the dashed line draws a volatility smile with a a negative change of the slope. The volatility smile for a given expiry moves according to the three basic movements we have described above. Although it is very useful to disentangle amongst them, one should never forget that in the real world the volatility smile is just the combined result of the three.

In Figure 7 the dashed line draws a volatility smile produced by a composite change summing up the three basic ones described above. This is a very realistic smile. From the very quick overview of the basic movements of the smile we may infer that we need a model which is able to cope with the features of the option markets. In what follows we introduce a possible model capable to accomplish this task: it has been designed for FX option markets, but it can be easily adapted to other markets.

The Uncertain Volatility Model. The intuition behind the UVUR model is as follows. The exchange rate process is just a BS geometric Brownian motion where the asset volatility and the domestic and foreign risk free rates are unknown, and one assumes different joint scenarios for them.

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In this model, both interest rates and volatility are stochastic in the simplest possible manner. Further details can be found in Brigo, Mercurio and Rapisarda The analytical tractability at the initial time is extended to all those derivatives which can be explicitly priced under the BS paradigm. We provide below an example of calibration to real market FX data, as of 9 May , when the spot exchange rate was 1. Domestic and foreign discount factors for the relevant maturities. The Vega contribution turns out to be several orders of magnitude smaller than the Vanna and Volga terms in all practical situations, hence one neglects it.

Since the Vanna-Volga method is a simple rule-of-thumb and not a rigorous model, there is no guarantee that this will be a priori the case. The attenuation factors are of a different from for the Vanna or the Volga of an instrument. This is because for barrier values close to the spot they behave differently: the Vanna becomes large while, on the contrary, the Volga becomes small. Hence the attenuation factors take the form:.

Both of these quantities offer the desirable property that they vanish close to a barrier.

FX options and smile risk (eBook, ) []

For example, for a single barrier option we have. Similarly, for options with two barriers the survival probability is given through the undiscounted value of a double-no-touch option. The first exit time FET is the minimum between: i the time in the future when the spot is expected to exit a barrier zone before maturity, and ii maturity, if the spot has not hit any of the barrier levels up to maturity.

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