The Model of Lines for Option Pricing with Jumps (original) (raw)

Abstract

This article reviews a pricing model, suitable for variance-gamma jump processes, based on the method of lines. The method accuracy is studied using European style calls as a benchmark. Implementation details for continuously and discretely monitored barrier options, and American and Bermudan options are given. function for the variance-gamma pure jump process. Consequently, the model of lines provides a method for pricing derivative claims in the variance gamma model when the model parameters reduce the arrival distribution to an Erlang distribution. As will be demonstrate later on, the general case can often be recovered with high precision by interpolation or extrapolation methods. The differential equations that arise in the model of lines are similar to the Black-Scholes equations, with the following important distinction: time derivatives are replaced by finite differences, while derivatives with respect to stock price remain intact. The pricing functions along each line are found to satisfy a system of inhomogeneous ordinary differential-difference equations, which admit simple analytic solutions for most options. Although the situation is similar to Carr's equations for American style options with randomized maturity [6], there are important differences. The key difference being that calendar time in Carr's solution must be reinterpreted as financial time in the model of lines. Because trading occurs in calendar time, not in financial time, the re-interpretation breaks risk neutrality. To restore risk neutrality, the stock price must be scaled and the option price discounted from one time-step to the next. Under this adjustment, the model of lines reproduces the exactup to negligible roundoff errors -prices of European style options in which the underlying follows a variance-gamma process. The solution scheme for European style puts and calls can easily be modified to exactly price barrier and Bermudan options contingent on information on the lines only. Path-dependent options requiring continuous monitoring, such as American options and barrier options can also be priced efficiently. However, in these cases, the model of lines produces approximate prices, as the exercise boundaries are assumed to be piecewise constant between lines. The model of lines enjoys the same calibration efficiencies and empirical explanatory power of the variance-gamma model. Nonetheless, it is still interesting to compare it with the better known stochastic volatility models. Diffusion models where volatility is stochastic have been considered by a number of authors, including Hull and White [17], Wiggins [29], Scott [27], Melino and Turnbull [22], Heston [15], [16]. Stochastic volatility models based on GARCH, such as in Duan [13], have the added advantage that the postulated process for the underlying asset is well justified by historical time series. As in jump models, the effect of stochastic volatility can formally be interpreted as inducing a random time change. Furthermore, these

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (29)

  1. C Albanese, S Jaimungal, and D. H. Rubisov. Jumping in line. to appear in Risk Magazine, February, 2001.
  2. G. Bakshi, C. Cao, and Z. Chen. Empirical performance of alternative option pricing models. Journal of Finance, 52:2003-2049, 1997.
  3. D.S. Bates. The crash of 87: Was it expected? the evidence from options markets. Journal of Finance, 46:1009-1044, 1991.
  4. D.S. Bates. Jumps and stochastic volatility: exchange rate processes implicit in deutschemark options. Review of Financial Studies, 9:69-108, 1996.
  5. F. Black and M. Scholes. The pricing of options and corporate liabilities. The Journal of Political Economy, 81:637-659, 1973.
  6. P. Carr. Randomization of the american put. Review of Financial Studies, 11:597-626, 1998.
  7. P. Carr and D. Faguet. Fast accurate valuation of american options. Working Paper, 1994.
  8. P. Carr, H. Geman, D.B. Madan, and M. Yor. The fine structure of asset returns: An empirical investigation. Working Paper, 2000.
  9. P. Carr and D. Madan. Towards a theory of volatility trading. Morgan Stanley working paper.
  10. J. Cox and A. Ross. The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3:145-166, 1976.
  11. E. Derman and I. Kani. Riding on a smile. Risk, 7:32-39, 1994.
  12. E. Derman and I. Kani. Stochastic implied trees: Arbitrage pricing with stochastic term and strike structure of volatility. International Journal of Theoretical and Applied Finance, 3:7-22, 1998.
  13. J.C. Duan. Cracking the smile. Risk, 9:55-59, 1996.
  14. H. Geman and T. Ane. Stochastic subordination. Risk, September, 1996.
  15. S. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6:327-343, 1993.
  16. S. Heston. Invisible parameters in options prices. Journal of Finance, 48:993-947, 1993.
  17. J. Hull and A. White. The pricing of options on assets with stochastic volatility. Journal of Finance, 42:281-300, 1987.
  18. J.C. Jackwerth and M. Rubinstein. Recovering probability distributions from option prices. Journal of Finance, pages 1611-1631, 1996.
  19. D. Madan and E. Seneta. The variance-gamma model for share market returns. Journal of Business, 63:511-524, 1990.
  20. D.B. Madan, P. Carr, and E.C. Chang. The variance gamma process and option pricing. European Financial Review, 2:79-105, 1998.
  21. D.B. Madan and F. Milne. Option pricing with vg martingale components. Mathematical Finance, 1:39-55, 1991.
  22. A. Melino and S. Turnbull. Pricing foreign currency options with stochastic volatility. Journal of Econometrics, 45:239- 265, 1990.
  23. R. Merton. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3:125-144, 1976.
  24. G.H. Meyer and J. Van Der Hoek. The evaluation of american options with the method of lines. Advances In Futures and Options Research, 9:265-285, 1997.
  25. M. Rubinstein. Implied binomial trees. Journal of Finance, 49:455-480, 1978.
  26. M. Rubinstein. Non-parametric tests of alternative option pricing models. Journal of Finance, 40:455-480, 1985.
  27. L. Scott. Option pricing when the variance changes randomly, theory, estimation and an application. Journal of Financial and Quantitative Analysis, 22:419-438, 1987.
  28. M. Stutzer. A simple non-parameteric approach to derivative security valuation. Journal of Finance, 51:1633-1652, 1996.
  29. J.B. Wiggins. Option values under stochastic volatility: theory and empirical estimates. Journal of Financial Economics, 19:351-377, 1987.