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VaR calculations often require the valuation of complex instruments over a large
set of scenarios. As complex derivatives use computationally expensive methods
for pricing purposes, full valuation of these instruments on every scenario is not
a viable solution. In this paper, we describe a method to approximate expensive
pricing functions that allows for fast and accurate VaR calculations.
Introduction
The valuation of complex derivatives can be a computationally intensive task. It is often the case that closed form solutions do not exist, forcing the use of more expensive numerical methods such as Monte Carlo simulation and finite difference schemes. Such methods are most useful in applications where only a few price valuations are required (as in trading). For risk management purposes, and particularly for VaR, we need hundreds of valuations to obtain the P&L distribution of a complex instrument.
For example, if we wanted to calculate 95% VaR using 1,000 scenarios, we would price each instrument 1,000 times and compute VaR as the 50th largest loss scenario. This VaR estimate will have simulation error since it depends on a finite sample of scenarios. As the sample size increases, the simulation error will decrease. Therefore, there is a clear tradeoff between accuracy and computation time: on one hand, we want to run as many simulations as possible to reduce the simulation error, but on the other hand, we do not want to incur the extra cost of running additional simulations. Alternatively, we could reduce the computational expense by using a simplified form for the pricing function and hence incurring pricing error. If the pricing error on the tails of the distribution is of the same magnitude or smaller than the simulation error, we could gain a lot of speed without losing much accuracy by using a simpler, though not as precise, pricing function.
There are two commonly used simplifications for complicated pricing functions. The first, usually called the delta method, fits the pricing function with a line (or a plane if we have a function of multiple variables), taking only into account the local linear sensitivities to changes in the underlying factors. A second method, which improves upon the delta approach, fits a quadratic function to the pricing function. The basic idea behind quadratic approximations is to be able to account for the curvature – or convexity – in the pricing function. These quadratic approximations usually fit the local first and second order sensitivities of the pricing function, thus providing a good fit only on a narrow interval around the current price. Another approach that has been explored by Studer (1999) is to fit a quadratic function to the true pricing function on the range of interest without using the local sensitivities. This paper describes the use of quadratic functions to approximate the true, and possibly very expensive, pricing functions to speed up VaR calculations.
In the next section, we will reproduce the basic results from Studer's quadratic approximation method and explore the numerical efficiency of the algorithm. Section 3 presents a practical example that describes the computational gains achieved with this method as well as the accuracy tradeoff. Section 4 concludes.
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