# American Institute of Mathematical Sciences

August & September 2019, 12(4&5): 969-978. doi: 10.3934/dcdss.2019065

## A pricing option approach based on backward stochastic differential equation theory

 Information Academy of Renmin University of China, Beijing, China

* Corresponding author: Xiao-Qian Jiang

Received  August 2017 Revised  December 2017 Published  November 2018

In option pricing, backward stochastic differential equation (BSDE) has wide application and Black-Scholes model is one of the classic pricing model. However, the model needs many preconditions which causes the implementing environment of model to approach perfection, leading to large deviation in actual application. Therefore, this article study the optimization problem of option pricing model under limited conditions intensively. It means that when random volatility is given, the option pricing formula with random interest rate is proposed and corresponding revision is also provided. Then we adopt call option and put option of Standard Poor's 500 index options to perform empirical research. The results indicate the assumption of random volatility is closer to reality. Compared to tradition models, the approach proposed in this article has enough theoretical basis. It is proved to own simple modeling method and higher accuracy which also shows certain reference significance to option pricing.

Citation: Xiao-Qian Jiang, Lun-Chuan Zhang. A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-978. doi: 10.3934/dcdss.2019065
##### References:
 [1] L. Andersen and J. Andreasen, Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Review of Derivatives Research, 4 (2000), 231-262. [2] T. Björk and H. Hult, A note on Wick products and the fractional Black-Scholes model, Finance & Stochastics, 9 (2005), 197-209. doi: 10.1007/s00780-004-0144-5. [3] R. Carmona, F. Delarue and G. E. Espinosa, et al., Singular forward-backward stochastic differential equations and emissions derivatives, Annals of Applied Probability, 23 (2013), 1086–1128. doi: 10.1214/12-AAP865. [4] D. M. Chance and Ph. D. CFA, Option pricing: The black-scholes-merton model[m]// essays in derivatives: Risk-transfer tools and topics made easy, Second Edition, John Wiley & Sons, Inc., 27 (2011), 133-137. [5] F. Chen, J. Shen and H. Yu, A new spectral element method for pricing european options under the black choles and merton jump diffusion models, Journal of Scientific Computing, 52 (2012), 499-518. doi: 10.1007/s10915-011-9556-5. [6] S. Datta, R. D'Mello and I. D. Mai, Executive compensation and internal capital market efficiency, Journal of Financial Intermediation, 18 (2009), 242-258. [7] J. D. Finnerty, Modifying the Black-Scholes-Merton model to calculate the cost of employee stock options, Journal of Applied Finance, 40 (2005), 2-32. [8] G. Franke, R. C. Stapleton and M. G. Subrahmanyam, When are Options Overpriced?: The Black-Scholes model and alternative characterisation of the pricing kernel, Social Science Electronic Publishing, 3 (1999), 79-102. [9] G. Gonz lez-Rivera, T. H. Lee and S. Mishra, Forecasting volatility: A reality check based on option pricing, utility function, value-at-risk, and predictive likelihood, International Journal of Forecasting, 20 (2004), 629-645. [10] C. B. Hyndman and P. O. Ngou, A convolution method for numerical solution of backward stochastic differential equation, Methodology & Computing in Applied Probability, 19 (2017), 1-29. doi: 10.1007/s11009-015-9449-4. [11] M. Jackson, When a good standard development process fails, Interlending & Document Supply, 33 (2005), 53-55. [12] J. R. Liang, J. Wang and W. J. Zhang, et al., Option pricing of a bi-fractional Black erton choles model with the Hurst exponent, Applied Mathematics Letters, 23 (2010), 859–863. doi: 10.1016/j.aml.2010.03.022. [13] J. D. Macbeth and L. J. Merville An empirical examination of the black-scholes call option pricing model, An empirical examination of the black-scholes call option pricing model, Journal of Finance, 34 (1979), 1173-1186. [14] A. G. Malliaris, R. W. Kolb and J. A. Overdahl, The black choles option pricing model, Social Science Electronic Publishing, 22 (2008), 247-263. [15] J. I. Pea, G. Rubio and G. S. Serna, Bid-ask spread and option pricing, Social Science Electronic Publishing, 7 (2000), 351-374. [16] Y. Peng, B. Gong and H. Liu, et al. Parallel computing for option pricing based on the backward stochastic differential equation, Proceedings of High Performance Computing and Applications, Second International Conference, Shanghai, China, (2010), 325–330. [17] B. A. Shadwick and W. F. Shadwick, On the computation of option prices and sensitivities in the Black?Scholes?Merton model, Quantitative Finance, 2 (2002), 158-166. [18] J. Yong, Linear forward-backward stochastic differential equations, Applied Mathematics & Optimization, 39 (1999), 93-119. doi: 10.1007/s002459900100.

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##### References:
 [1] L. Andersen and J. Andreasen, Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Review of Derivatives Research, 4 (2000), 231-262. [2] T. Björk and H. Hult, A note on Wick products and the fractional Black-Scholes model, Finance & Stochastics, 9 (2005), 197-209. doi: 10.1007/s00780-004-0144-5. [3] R. Carmona, F. Delarue and G. E. Espinosa, et al., Singular forward-backward stochastic differential equations and emissions derivatives, Annals of Applied Probability, 23 (2013), 1086–1128. doi: 10.1214/12-AAP865. [4] D. M. Chance and Ph. D. CFA, Option pricing: The black-scholes-merton model[m]// essays in derivatives: Risk-transfer tools and topics made easy, Second Edition, John Wiley & Sons, Inc., 27 (2011), 133-137. [5] F. Chen, J. Shen and H. Yu, A new spectral element method for pricing european options under the black choles and merton jump diffusion models, Journal of Scientific Computing, 52 (2012), 499-518. doi: 10.1007/s10915-011-9556-5. [6] S. Datta, R. D'Mello and I. D. Mai, Executive compensation and internal capital market efficiency, Journal of Financial Intermediation, 18 (2009), 242-258. [7] J. D. Finnerty, Modifying the Black-Scholes-Merton model to calculate the cost of employee stock options, Journal of Applied Finance, 40 (2005), 2-32. [8] G. Franke, R. C. Stapleton and M. G. Subrahmanyam, When are Options Overpriced?: The Black-Scholes model and alternative characterisation of the pricing kernel, Social Science Electronic Publishing, 3 (1999), 79-102. [9] G. Gonz lez-Rivera, T. H. Lee and S. Mishra, Forecasting volatility: A reality check based on option pricing, utility function, value-at-risk, and predictive likelihood, International Journal of Forecasting, 20 (2004), 629-645. [10] C. B. Hyndman and P. O. Ngou, A convolution method for numerical solution of backward stochastic differential equation, Methodology & Computing in Applied Probability, 19 (2017), 1-29. doi: 10.1007/s11009-015-9449-4. [11] M. Jackson, When a good standard development process fails, Interlending & Document Supply, 33 (2005), 53-55. [12] J. R. Liang, J. Wang and W. J. Zhang, et al., Option pricing of a bi-fractional Black erton choles model with the Hurst exponent, Applied Mathematics Letters, 23 (2010), 859–863. doi: 10.1016/j.aml.2010.03.022. [13] J. D. Macbeth and L. J. Merville An empirical examination of the black-scholes call option pricing model, An empirical examination of the black-scholes call option pricing model, Journal of Finance, 34 (1979), 1173-1186. [14] A. G. Malliaris, R. W. Kolb and J. A. Overdahl, The black choles option pricing model, Social Science Electronic Publishing, 22 (2008), 247-263. [15] J. I. Pea, G. Rubio and G. S. Serna, Bid-ask spread and option pricing, Social Science Electronic Publishing, 7 (2000), 351-374. [16] Y. Peng, B. Gong and H. Liu, et al. Parallel computing for option pricing based on the backward stochastic differential equation, Proceedings of High Performance Computing and Applications, Second International Conference, Shanghai, China, (2010), 325–330. [17] B. A. Shadwick and W. F. Shadwick, On the computation of option prices and sensitivities in the Black?Scholes?Merton model, Quantitative Finance, 2 (2002), 158-166. [18] J. Yong, Linear forward-backward stochastic differential equations, Applied Mathematics & Optimization, 39 (1999), 93-119. doi: 10.1007/s002459900100.
Call option price regulation under different $\sigma$
When considering uderlying transaction behavior of the option, S & P 500 index changes in the period from NOV 5th, 2013 to Nov 3rd, 2015
Comparison of hidden volatility from BS and EBS
European call option price of three models
 BS CBS EBS $\sigma =$ 0.05 4.5062 6.1752 6.5952 $\sigma =$ 0.1 5.3283 7.8540 7.2335 $\sigma =$ 0.15 6.7893 8.3348 8.0334 $\sigma =$ 0.2 7.2439 9.4538 8.6571 $\sigma =$ 0.25 8.5466 10.6532 10.4295 $\sigma =$ 0.3 10.8991 12.5298 13.8195 $\sigma =$ 0.5 17.9923 21.0404 22.6319
 BS CBS EBS $\sigma =$ 0.05 4.5062 6.1752 6.5952 $\sigma =$ 0.1 5.3283 7.8540 7.2335 $\sigma =$ 0.15 6.7893 8.3348 8.0334 $\sigma =$ 0.2 7.2439 9.4538 8.6571 $\sigma =$ 0.25 8.5466 10.6532 10.4295 $\sigma =$ 0.3 10.8991 12.5298 13.8195 $\sigma =$ 0.5 17.9923 21.0404 22.6319
Theoretical price and risk index of three models
 BS CBS EBS Call option Put option Call option Put option Call option Put option Theoretical price 7.1618 4.8643 7.9394 4.0217 8.4352 3.9280 Delta 0.4935 -0.4205 0.4857 -0.425668 0.466891 -0.42234 Gamma 0.01823 0.0191 0.01356 0.0165 0.009885 0.0098 Theta -0.007143 -0.00814 -0.0085201 -0.008536 -0.00972 -0.009755 Vaga 0.362788 0.361785 0.375942 0.378952 0.394567 0.394756 Rho -0.071415 -0.071415 -0.071415 -0.071415 -0.071415 -0.071415
 BS CBS EBS Call option Put option Call option Put option Call option Put option Theoretical price 7.1618 4.8643 7.9394 4.0217 8.4352 3.9280 Delta 0.4935 -0.4205 0.4857 -0.425668 0.466891 -0.42234 Gamma 0.01823 0.0191 0.01356 0.0165 0.009885 0.0098 Theta -0.007143 -0.00814 -0.0085201 -0.008536 -0.00972 -0.009755 Vaga 0.362788 0.361785 0.375942 0.378952 0.394567 0.394756 Rho -0.071415 -0.071415 -0.071415 -0.071415 -0.071415 -0.071415
The significantly volatility reducing estimation of classical Black-Scholes and revised model
 Classic BS model (%) EBS model with drift (%) Max error over period 0.86 0.59 Avdrage error of each day 0.26 0.12 Average error of all prices 0.14 0.07 Std. Dev. Of error of each day 0.13 0.06 Std. Dev. Of error of all prices 0.11 0.06
 Classic BS model (%) EBS model with drift (%) Max error over period 0.86 0.59 Avdrage error of each day 0.26 0.12 Average error of all prices 0.14 0.07 Std. Dev. Of error of each day 0.13 0.06 Std. Dev. Of error of all prices 0.11 0.06
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