Leveraged Lévy processes as models for stock prices

Abstract

Adopting a constant elasticity of variance formulation in the context of a general Lévy process as the driving uncertainty we show that the presence of the leverage effect† in this form has the implication that asset price processes satisfy a scaling hypothesis. We develop forward partial integro-differential equations under a general Markovian setup, and show in two examples (both continuous and pure-jump Lévy) how to use them for option pricing when stock prices follow our leveraged Lévy processes. Using calibrated models we then show an example of simulation-based pricing and report on the adequacy of using leveraged Lévy models to value equity structured products.

†One explanation of the documented negative relation between market volatilities and the level of asset prices (the ‘smile’ or ‘skew’), we term the ‘leverage effect’, argues that this negative relation reflects greater risk taking by the management, induced by a fall in the asset price, with a view of maximizing the option value of equity shareholders.

Keywords:

• Arbitrage pricing
• Asset allocation
• Asset pricing
• Computational finance

Acknowledgements

We are indebted to Marc Yor for discussing these precise linkages between leveraged Lévy models and Lamperti processes. We thank Gurdip Bakshi, Steve Heston and Pete Kyle for constructive discussions at the Finance Seminar of the University of Maryland.

Notes

†One explanation of the documented negative relation between market volatilities and the level of asset prices (the ‘smile’ or ‘skew’), we term the ‘leverage effect’, argues that this negative relation reflects greater risk taking by the management, induced by a fall in the asset price, with a view of maximizing the option value of equity shareholders.

†The detailed steps are available upon request.

†The Lamperti representation as in (3) relating the BESQ process and Brownian motion is exp2(W(t) + νt) = BESQ ^(δ) ×, where δ = 2ν + 2 (see, e.g., Williams 1974).

‡This transformation will benefit the discretization of the system. Because strike prices collected from real markets are often sparsely distributed in a large range, this brings difficulty to the discretization of the system when we always require the spacing of K be small enough to achieve convergence. By transforming from strike space to log strike space, we can achieve a small spacing without making the system huge.

†Details of the numerical analysis such as discretization, evaluation of double tail functions f ₁ and f ₂ using FFT, and a new scheme for fast computation as well as analytical integration results are omitted intentionally due to limited space. Interested readers are referred to my dissertation (Xiao 2005, chapter 6) and are welcome to contact the authors for notes and discussion.

‡On solving the PDE numerically, we also implement a convergence test as ΔT and Δk approach zero to demonstrate numerical stability. The results are not shown here for sake of space.

†We first simulate the CGMY random variable Z(t) at time t. This can be done in many different ways such as direct compound Poisson simulation, time-changed Brownian motion (Madan and Yor 2005), etc. We use a direct compound Poisson simulation and show in figure 2 a chi-squared fit test for 10 000 simulations using calibrated parameters C = 3.75, G = 9.76, M = 23.48 and Y = 0.5 for t = 0.01. The benchmark of the test is from simulated random variables by inverse transform of the cumulative distribution function, which is obtained from Fourier transform of the characteristic function. The test accepts the null hypothesis H0 at a significance level of 0.05.