Calibration of the foreign exchange (FX) local volatility model is critical in calculating the value and risk sensitivities of FX structured products, most notably power reverse dual currency (PRDC) notes, which are the most traded of all exotic FX structured products.
The problem of calibration is to determine the parameters of a specific stochastic process that best match the prices of a set of input instruments (also known as calibration instruments). To calibrate the FX local volatility, the calibration instruments will be a set of European-exercise FX options with as wide a range of maturities and strikes as possible. Long dated products are needed as the nominal lifetime of PRDC notes can be in the order of 30 years (although their callable and knockout features considerably reduce their duration).
The calibration is done by a sequence of techniques. This approach was chosen due to its balance between scope and tractability; it offers an essentially closed-form solution to the problem of calibrating to the FX volatility surface, while keeping a realistic form for the FX local volatility surface.
In its most general form, the no-arbitrage process followed by the FX rate is given by
and represent the domestic and foreign short rates, both are assumed to follow a mean-reverting Hull-White processes:
and = the domestic and foreign mean-reversion constant (which are assumed constant in time),
and = the domestic and foreign short rate volatilities (which are also assumed constant in time).
The extra term in the foreign interest rate process arises from the fact that the whole instrument is priced in a single measure, one that is risk-neutral from the point of view of a domestic investor.
In order to perform the calibration, one must choose a specific form for the local volatility function. Following, we choose a functional form for known as the "constant elasticity of variance" or CEV parameterization, where
The name CEV derives from the stochastic process introduced in reference. There are two time-dependent parametric functions to determine, and . The time-dependent "level" is fixed at the outset. We have chosen the functional form for convenience in the calibration process.
The problem of calibration then becomes one of finding the two time dependent parameters and that best fit the volatility structure of the FX options used as calibration instruments.
In order to develop a realistic term structure for the FX process, we rewrite the process (Equation 1) into a process describing the forward FX rate
which results from the expectation of the process (Equation 1). The resulting process can be written in the deceptively simple form
= a Brownian motion constructed from the three underlying processes and
= the time-dependent local volatility for the forward FX rate, which is related to the local volatility of the spot FX rate.
As a consequence of writing the process in terms of the forward FX rate, the input to the local volatility function always appears in the combination where is defined as the ratio of prices of domestic to foreign zero coupon bonds:
Therefore there are two stochastic factors "hidden" in the process (Equation 6). The bulk of the computational effort in the paper is to replace the process (Equation 6) with a different process that contains one single stochastic factor, namely the forward FX rate, and whose marginal distribution agrees with (Equation 6) – this is the so-called "Markovian projection" technique. Enforcing this condition guarantees that the prices of European options are identical in the two processes. This allows for a method of calibration that is essentially "instantaneous". One can calibrate this "secondary" process to a series of forward FX options with minimal computational effort (i.e., without PDE’s or Monte Carlo methods), and obtain a parameterization of the local volatility function (Equation 3).
The FX calibration is performed in two separate stages. In the first stage, the FX process (Equation 5) is projected onto a displaced diffusion
where and are related to the functions and . The parameters and are chosen such that the Black option prices resulting from the process (Equation 6) best match a set of FX options with a given maturity across a set of strikes. This is done for each maturity in the set of calibration instruments.
The second stage of the calibration uses this information to choose a set of and related to the time varying CEV parameters via the representations
These formulae represent piecewise constant "indicator functions", in which and when , and when et cetera.
Inverting the functional form of and from Equation 1 (and assuming that the Hull-White parameters are not time varying), one can obtain a closed form solution for the piecewise constant parameter and quarterly for dates between the valuation date and the furthest option expiry.
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