1 July 2014 CFA Institute Journal Review

# Lower Bound Approximation to Basket Option Values for Local Volatility Jump-Diffusion Models (Digest Summary)

1. Butt Man-Kit, CFA

It is generally difficult to value basket options because of the lack of analytic characterization of the distribution of the underlying basket asset price process. The authors create an easily computed approximation to European basket call prices for a local volatility jump-diffusion model. Numerical tests indicate that the suggested approximation is fast and accurate.

## Whatâ€™s Inside?

The authors apply the asymptotic method of expansion, in which one value approaches another without quite meeting. For this method, they identify several approximate solutions and then combine them to find the approximate value of the lower bound of European basket call prices. If the local volatility function is time independent, then there is a closed-form expression for the approximation. The computation of the lower bound approximation (LBA) is easy and fast, which is a key advantage over other methods, such as simulation.

## How Is This Research Useful to Practitioners?

Exotic options are usually difficult to value because of the lack of analytical form expression, and using Monte Carlo simulation is time consuming. The determination of the value of basket options involves the same challenges. Previous research has mainly focused on the development of fast and accurate approximation methods as well as the discovery of tight lower and upper bounds for option values.

The authors offer an easily implemented algorithm to compute the LBA. If the local volatility function is time independent, then there is a closed-form expression for the approximation. In addition to having a closed-form expression, there are two extra advantages. First, LBA is more flexible than other methods because it can handle general local volatility functions. Second, it can deal with different jump sizes of common jumps.

Numerical tests indicate that the LBA is fast and accurate in most cases compared with Monte Carlo simulation as well as with other existing approximation methods. The LBA method performs well with overall relative errors less than 1%.

## How Did the Authors Conduct This Research?

The authors first formulate the local basket volatility jump-diffusion model and explain some known methods for pricing European basket call options. Then they apply the second-order asymptotic expansion to derive an easily computed approximation to the lower bound. Finally, they compare the numerical performance of the LBA with other methods.

The authors extend the common Black–Scholes model to the jump-diffusion model with local volatility functions. The jump component is a Poisson-driven process in which both the intensity and the size of jumps are parameterized.

Next, they use the asymptotic expansion method to expand the parameterized asset price processes to the second order. Then the authors directly apply the conditional expectation results of multiple Wiener–Itô integrals to approximate the lower bound for the basket call option values in a jump-diffusion model.

To test the performance of the LBA, they conduct some numerical tests for European basket calls with underlying asset price processes satisfying the specified stochastic differential equation. In the first test, they perform numerical tests for three constant jump sizes. In the second test, they compare the results of Monte Carlo simulation and LBA with different levels of moneyness, from deep in the money to deep out of the money. In the third test, they show the numerical results with different maturities and different local volatility functions but with the same jump size. Finally, the fourth test resembles the third test except that the jump sizes for assets are different.

The results of the tests indicate that the LBA performs well compared with Monte Carlo simulation and other approximation methods. The LBA method is also more flexible in that it does not require the Black–Scholes setting. In addition, it does not require the same jump size for all assets, which is required by other approximation methods.

## Abstractorâ€™s Viewpoint

In the jump-diffusion model, the stock price changes are modeled by the mixture of continuous (Wiener–Itô) process and the rare-event jump (Poisson) process. The model helps in approximating the occasional shocks of price changes. Despite this advantage, the jump-diffusion model is less popular than the basic Black–Scholes model because of its complexity and lack of closed-form expressions in valuing assets. The findings of this study could help practitioners apply the LBA model to approximating European basket option prices easily and accurately in seconds.