## A predictor-corrector scheme based on the ADI method for pricing american puts with stochastic volatility.(English)Zbl 1228.91077

Summary: We introduce a new numerical scheme, based on the ADI (alternating direction implicit) method, to price American put options with a stochastic volatility model. Upon applying a front-fixing transformation to transform the unknown free boundary into a known and fixed boundary in the transformed space, a predictor-corrector finite difference scheme is then developed to solve for the optimal exercise price and the option values simultaneously. Based on the local von Neumann stability analysis, a stability requirement is theoretically obtained first and then tested numerically. It is shown that the instability introduced by the predictor can be damped, to some extent, by the ADI method that is used in the corrector. The results of various numerical experiments show that this new approach is fast and accurate, and can be easily extended to other types of financial derivatives with an American-style exercise. Another key contribution of this paper is the proposition of a set of appropriate boundary conditions, particularly in the volatility direction, upon realizing that appropriate boundary conditions in the volatility direction for stochastic volatility models appear to be controversial in the literature. A sound justification is also provided for the proposed boundary conditions mathematically as well as financially.

### MSC:

 91G60 Numerical methods (including Monte Carlo methods) 91G20 Derivative securities (option pricing, hedging, etc.) 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Full Text:

### References:

 [1] Adrian, A.D.; Victor, M.Y., Probability distribution of returns in the Heston model with stochastic volatility, Quantitative finance, 2, 443-453, (2002) · Zbl 1405.91734 [2] Ball, C.A.; Roma, A., Stochastic volatility option pricing, Journal of financial and quantitative analysis, 29, 589-687, (1994) [3] Fouque, J.-P.; Papanicolaou, G.; Sircar, K.R., Derivatives in financial markets with stochastic volatility, (2000), Cambridge University Press Cambridge · Zbl 0954.91025 [4] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, The reviews of financial studies, 6, 327-343, (1993) · Zbl 1384.35131 [5] Hull, J.C.; White, A., The pricing of options on assets with stochastic volatilities, The journal of finance, 42, 281-300, (1987) [6] Feller, W., Two singular diffusion problems, Annals of mathematics, 54, 173-182, (1951) · Zbl 0045.04901 [7] Bakshi, G.; Cao, C.; Chen, Z., Empirical performance of alternative option pricing models, The journal of finance, 52, 5, 2003-2049, (1997) [8] Pan, J., The jump-risk premia implicit in options: evidence from an integrated time-series study, The journal of financial economics, 63, 3-50, (2002) [9] Tompkins, R.G., Stock index futures markets: stochastic volatility and smiles, Journal of futures markets, 21, 43-78, (2001) [10] Wu, L.; Kwok, Y.K., A front-fixing finite difference method for the valuation of American options, The journal of financial engineering, 6, 2, 83-97, (1997) [11] Zhu, S.-P.; Zhang, J., A new predictor – corrector scheme for valuating American puts, Applied mathematics and computation, 217, 9, 4439-4452, (2011) · Zbl 1237.91236 [12] Allegretto, W.; Lin, Y.-P.; Yang, H., A fast and highly accurate numerical methods for the valuation of American options, Discrete and continuous dynamical systems. series B. applications and algorithm, 8, 127-136, (2002) [13] Cox, J.; Ross, S.; Rubinstein, M., Option pricing—a simplified approach, Journal of financial economics, 7, 229-263, (1979) · Zbl 1131.91333 [14] Longstaff, F.; Schwartz, E., Valuing American options by simulation: a simple least-squares approach, The reviews of financial studies, 14, 113-147, (2001) [15] Geske, R.; Johnson, H., The American put option valued analytically, The journal of finance, 39, 1511-1524, (1984) [16] Zhu, S.-P., A new analytical-approximation formula for the optimal exercise boundary of American put options, International journal of theoretical and applied finance, 9, 7, 1141-1177, (2006) · Zbl 1140.91415 [17] Bunch, D.S.; Johnson, H., The American put option and its critical stock price, The journal of finance, 5, 2333-2356, (2000) [18] Zhu, S.-P.; He, Z.-W., Calculating the early exercise boundary of American put options with an approximation, International journal of theoretical and applied finance, 10, 7, 1203-1227, (2007) · Zbl 1153.91581 [19] Huang, J.; Subrahmanyam, M.; Yu, G., Pricing and hedging American options: a recursive integration method, The reviews of financial studies, 9, 277-300, (1996) [20] Ševčovič, D., Analysis of the free boundary for the pricing of an American call option, European journal of applied mathematics, 12, 25-37, (2001) · Zbl 1113.91315 [21] Ševčovič, D., An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear black – scholes equation, Canadian applied mathematics quarterly, 15, 1, 77-97, (2007) · Zbl 1145.35321 [22] Stamicar, R.; Ševčovič, D.; Chadam, J., The early exercise boundary for the American put near expiry: numerical approximation, Canadian applied mathematics quarterly, 7, 4, 427-444, (1999) · Zbl 0979.91031 [23] Zhu, S.-P., An exact and explicit solution for the valuation of American put options, Quantitative finance, 6, 3, 229-242, (2006) · Zbl 1136.91468 [24] Clarke, N.; Parrott, K., Multigrid for American option pricing with stochastic volatility, Applied mathematical finance, 6, 3, 177-195, (1999) · Zbl 1009.91034 [25] Ikonen, S.; Toivanen, J., Effcient numerical methods for pricing American options under stochastic volatility, Numerical methods for partial differential equations, 24, 1, 104-126, (2008) · Zbl 1152.91516 [26] Zvan, R.; Forsyth, P.A.; Vetzal, K.R., Penalty methods for American options with stochastic volatility, Journal of computational and applied mathematics, 91, 199-218, (1998) · Zbl 0945.65005 [27] Landau, H.G., Heat conduction in a melting solid, Quarterly of applied mathematics, 8, 81, (1950) · Zbl 0036.13902 [28] Strikwerda, C.J., Finite difference schemes and partial differential equations, (1989), Chapman & Hall New York · Zbl 0681.65064 [29] Wilmott, P.; Dewynne, J.; Howison, S., Option pricing, (1993), Oxford Financial Press · Zbl 0797.60051 [30] S.-S. Lin, Finite difference schemes for Heston’s model, A Graduate Project, 2008. [31] Zhe, Z.; Lim, K.-G., A non-lattice pricing model of American options under stochastic volatility, Journal of futures markets, 26, 5, 417-448, (2006) [32] Ito, K.; Toivanen, J., Lagrange multiplier approach with optimized finite difference stencils for pricing American options under stochastic volatility, SIAM journal on scientific computing, 31, 2646-2664, (2009) · Zbl 1201.35024 [33] Fichera, G., Existential analysis of the solutions of mixed boundary value problems, related to second order elliptic equation and systems of equations, selfadjoint, Annali Della scuola normale superiore, III, 1, 1-4, 75-100, (1949) [34] Hull, J.C., Options, futures, and other derivatives, (1997), Prentice Hall Press · Zbl 1087.91025 [35] CBOE’s volatility index, 2009. Available at: http://www.cboe.com/micro/vix/volatility-qrg.pdf. [36] Douglas, J.; Rachford, H.H., On the numerical solution of heat conduction problems in two and three space variables, Transactions of the American mathematical society, 82, 421-439, (1956) · Zbl 0070.35401 [37] Craig, I.J.D.; Sneyd, A.D., An alternating-direction implicit scheme for parabolic equations with mixed derivatives, Computers and mathematics with applications, 16, 4, 341-350, (1988) · Zbl 0654.65072 [38] Hundsdorfer, W., Accuracy and stability of splitting with stabilizing corrections, Applied numerical mathematics, 42, 213-233, (2002) · Zbl 1004.65095 [39] in’t Hout, K.J.; Welfert, B.D., Stability of ADI schemes applied to convection – diffusion equations with mixed derivative terms, Applied numerical mathematics, 57, 19-35, (2006) · Zbl 1175.65104 [40] Medvedev, A.; Scaillet, O., Pricing American options under stochastic volatility and stochastic interest rates, Journal of financial economics, 98, 1, 145-159, (2010) [41] Oosterlee, C.W., On multigrid for linear complementarity probelms with application to American-style options, Electronic transactions on numerical analysis, 15, 165-185, (2003) · Zbl 1031.65072 [42] Evans, J.D.; Kuske, R.; Keller, J.B., American options on assets with dividends near expiry, Mathematical finance, 12, 3, 219-237, (2002) · Zbl 1031.91047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.