by Cornelis W. Oosterlee and Lech A. Grzelak
When the financial sector is in crisis, stocks go down and investors escape from the market to reduce their losses. Global banks then decrease interest rates in order to increase cash flow: this may lead to an increase in stock values, since it becomes less attractive for investors to keep their money in bank accounts. It is clear, therefore, that movements in the interest rate market can influence the behaviour of stock prices. This is taken into account in the so-called hybrid models currently being developed by researchers Grzelak (Delft University of Technology) and Oosterlee (CWI) from the Netherlands.
Looking in a dictionary, we see that hybrids are, for example, creatures combining body parts of more than one real species (such as mermaids or centaurs). The hybrid contracts from the financial industry are based on products from different asset classes, like stock, interest rate and commodities. Often these products have different expected returns and risk levels. Proper construction of a new hybrid product may give reduced risk and an expected return greater than that of the least risky asset. A simple example is a portfolio containing a stock with a high risk and return and a bond with a low risk and return. If one introduces an equity component into a pure bond portfolio the expected return will increase. If the percentage of the equity in the portfolio is increased, it eventually starts to dominate the structure and the risk may increase with a higher impact for a low or negative correlation.
Advanced hybrid models can be expressed by a system of stochastic differential equations (SDEs), for example for stock, volatility and interest rate, with a full correlation matrix. Such an SDE system typically contains many parameters that should be determined by calibration with financial market data. This task is challenging: European options need to be priced repeatedly within the calibration procedure, which should therefore be done extremely fast and efficiently.
At a major financial institution like a bank, one can distinguish a number of tasks that must be performed in order to price a new financial derivative product. First, the new product is defined as the market asks for it. If this is a derivative product, then there is an underlying, modelled by stochastic differential equations (SDEs). Each asset class has different characteristics, leading to different types of SDEs. To achieve a reasonable model that is related to the present market, one calibrates the SDEs. These products also form the basis for the hedge strategies used by the banks to reduce the risk associated with selling the new product.
Once the asset price model is determined, the new derivative product is modelled accordingly. The product of interest is then priced by means of a Monte Carlo simulation for the integral version of the problem, or by the numerical approximation of a partial differential equation. The choice of numerical pricing method is thus based on whether one is aiming for the model calibration, in which speed of a pricing method is essential, or for the pricing of the new contract, for which robustness of the numerical method is of highest importance.
Fourier-based option pricing methods are computationally fast, and can be used relatively easily for the hybrid models mentioned above. They are particularly well suited for the calibration process, as their efficiency is obvious for pricing basic option products. They work whenever the so-called characteristic function of the asset price process, ie the Fourier transform of the probability density function, is available.
The contribution of the Dutch group to pricing algorithms is the development of the COS option pricing method, based on Fourier cosine expansions. The basis of this method is that the Fourier-cosine expansion of a probability density function is directly related to the characteristic function. This not only allows probability density functions to be recovered but in addition, means highly efficient option pricing can be performed simultaneously for a large number of basic options. A unique property of the pricing method is that the pricing error decreases exponentially with the number of terms in the cosine expansion. Accurate pricing with the COS method therefore takes place in just a few milliseconds. Experience has been gained for a variety of stochastic processes for the modelling of the assets, like sophisticated hybrid models.
At present, the researchers are improving the applicability of their pricing method. They have already generalized it successfully to the pricing of options with early exercise opportunities, as well as to so-called swing options, which allow contract holders to buy or sell electricity at certain times in the contract. Another application, related to the recent credit crisis, is the calibration of credit default swaps, one of the problematic product classes of recent years, modelled by means of jump processes to the last year's market. A remarkable correspondence between model and market prices was achieved, and calibration could be performed within reasonable time, due to the use of the COS method.
Cornelis W. Oosterlee
CWI, The Netherlands
Tel: +31 20 592 4108