Risk measurement and Implied volatility under Minimal Entropy Martingale Measure for Levy process

Document Type: Research Paper

Authors

Department of Applied Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

10.22034/amfa.2020.674944

Abstract

This paper focuses on two main issues that are based on two important concepts: exponential Levy process and minimal entropy martingale measure. First, we intend to obtain   risk measurement such as value-at-risk (VaR) and conditional value-at-risk (CvaR) using Monte-Carlo methodunder minimal entropy martingale measure (MEMM) for exponential Levy process. This Martingale measure is used for the exponential type of the processes such as exponential Levy process. Also, it can be said MEMM is a kind of important sampling method where the probability measure with minimal relative entropy replaces the main probability. Then we are going to obtain VaR and CVaR by Monte-Carlo simulation. For this purpose, we have to calculate option price, implied volatility and returns under MEMM and then obtain risk measurement by proposed algorithm. Finally, this model is simulated for exponential variance gamma process. Next, we intend to develop two theorems for implied volatility under minimal entropy martingale measure by examining the conditions. These theorems consider the asymptotic implied volatility for the case that time to maturity tends to zero and infinity.

Keywords


[1]   Bakhshmohammadlou, M., Farnoosh, R., Numerical Solution of Multidimensional Exponential Levy Equation by Block Pulse Function, Advances in Mathematical Finance and Applications, 2020, 5(2), P. 247-259. Doi: 10.22034/amfa.2020.1873599.1260

[2]    Carr, P., Geman, H., Madan, D. B., Yor, M., The fine structure of asset returns: An empirical investigation, The Journal of Business, 2002, 75(2), P.305-332. Doi:10.1086/338705

[3]   Cont, R., Tankov, P., Financial Modeling with Jump Processes, 2004, Chapman & Hall/CRC, Boca Raton. ISBN: 9781584884132

[4]     Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M., Stricker, C., Exponential hedging and entropic penalties, Mathematical finance, 2002, 12(2), P. 99-123. Doi: 10.1111/1467-9965.02001

[5] Fujiwara, T., Miyahara, Y., The minimal entropy martingale measures for geometric Lévy processes, Finance and Stochastics, 2003, 7(4), P.509-531. Doi:10.1007/s007800200097

[6] Fujisaki, M., Zhang, D., Evaluation of the MEMM, parameter estimation and option pricing for geometric Lévy processes, Asia-Pacific Financial Markets, 2009, 16(2), P.111-139. Doi: 10.1007/s10690-009-9089-1

[7] Glasserman, P., Heidelberger, P., Shahabuddin. P., Importance sampling and stratification for value-at-risk, IBM Thomas J. Watson Research Division,1999. (www.gsb.columbia.edu/faculty/pglasserman/Other/pg-var.pdf)

 [8] Glasserman, P., Heidelberger, P., Shahabuddin, P., Variance reduction techniques for estimating value-at-risk, Management Science, 2000, 46(10), P.1349-1364. ISBN:0-7803-6582-8

[9] Glasserman, P., Heidelberger, P., Shahabuddin, P., Efficient Monte Carlo methods for value-at-risk, IBM Thomas J. Watson Research Division, 2000.

[10] Hubalek, F., Sgarra, C., Esscher transforms and the minimal entropy martingale measure for exponential Lévy models, Quantitative finance, 2006, 6(02), P.125-145. Doi: 10.1080/14697680600573099

[11] Izadikhah, M., Improving the Banks Shareholder Long Term Values by Using Data Envelopment Analysis Model, Advances in Mathematical Finance and Applications, 2018, 3(2), P.27-41.

 Doi: 10.22034/amfa.2018.540829

[12] Jeanblanc, M., Klöppel, S., Miyahara, Y., Minimal fq-martingale measures for exponential Lévy processes, The Annals of Applied Probability, 2007, 17(5/6), P.1615-1638. Doi: 10.1214/07-AAP439

[13] Jeanblanc, M., Yor, M., Chesney, M., Mathematical methods for financial markets, Springer Science & Business Media, 2009. ISBN-13: 978-1852333768

[14] Kind, A., Pricing American-style options by simulation, Financial markets and portfolio management, 2005, 19(1), P. 109-116. Doi: 10.1007/s11408-005-2300-0

[15] Miyahara Y., Moriwaki N., Volatility Smile/Smirk Properties of [GLP & MEMM] Models, Discussion Papers in Economics, Nagoya City University, 405, 2004, P.1-16.

[16] Miyahara, Y., Option pricing in incomplete markets: Modeling based on geometric Lévy processes and minimal entropy martingale measures, World Scientific, 2011, 3. ISBN-13:978-1848163478

[17] Navidi, S., Rostamy-Malkhalifeh, M., Banihashemi, S., Using MODEA and MODM with Different Risk Measures for Portfolio Optimization, Advances in Mathematical Finance and Applications, 2020, 5(1), P. 29-51. Doi: 10.22034/amfa.2019.1864620.1200

[18] Paziresh, M., Jafari, M., Feshari, M., Confidence Interval for Solutions of the Black-Scholes Model, Advances in Mathematical Finance and Applications, 2019, 4(3), P. 49-58. Doi: 10.22034/amfa.2019.1869742.1231

[19] Tankov, P., Financial modeling with Lévy processes, 2010. HAL Id: cel-00665021

[20] Tahmasebi, M., Yari, G. H., Minimal relative entropy for equivalent martingale measures by low-discrepancy sequence in Lévy process, Stochastics, 2019, P. 1-18. Doi: 10.1080/17442508.2019.1642339

[21] Tehranchi, M. R., Asymptotics of implied volatility far from maturity, Journal of Applied Probability, 2009, 46(3), P. 629-650. Doi: 10.1239/jap/1253279843

[22] Tehranchi, M. R., Implied volatility: long maturity behavior, Encyclopedia of Quantitative Finance, 2010. Doi: 10.1002/9780470061602.eqf08011

[23] Tilley, J. A., Valuing American options in a path simulation model, Insurance Mathematics and Economics, 1995, 2(16), P. 169. Doi:10.1016/0167-6687(95)91768-h