Option Hedging in Jump-Diffusion Markets by Malliavin Calculus

Subject Areas : Journal of Investment Knowledge

Minoo Bakhsh Mohammadlou ¹ , Rahman Farnoosh ²

1 - PhD student, Iran University of Science and Technology
2 - Associated Professor, Iran University of Science and Technology (Corresponding Author)

Received: 2017-09-19 Accepted : 2018-01-20 Published : 2019-09-23

Keywords: Jump diffusion market, Hedging strategy, Malliavin calculus, Clark-Ocone formula, Residual risk,

Abstract :

We obtain the hedging strategy in a jump-diffusion market by minimizing the variance of the residual risk. We calculate the residual risk by two formulas: the Ito's formula and the jump-diffusion version of the Clark-Ocone formula. The results show that Malliavin calculus can generate the hedging strategy under weaker assumptions. Thus afterward we do not require to check the strong condition  on  and the condition  with bounded derivative is sufficient.    

References:

*       Benth, F., Nunno, G.Di., Lokka, A., Oksendal, B., Proske, F. (2003), Explicit representation of the minimal variance portfolio in markets driven by Levy processes. Mathematical Finance. 13(1), 55-72.

*       Bermin, H.P. (2003), Hedging Options: The Malliavin calculus approach versus the Δ-Hedging approach, Mathematical Finance, 13(1), 73-84.

*       Black, F., Scholes, M. (1973), The pricing of options and corporate liabilities, journal of political economy, 81, 637–659.

*       Bo, L., Tang, D., Wang, Y. (2017), Optimal investment of variance-swaps in jump-diffusion market with regime-switching, Journal of Economic Dynamics and Control, 83, 175-197.

*       Cont, R., Tankov, P. (2004), Financial modeling with jump processes, Financial Mathematics Series, Chapman and Hall/CRC.

*       Follmer, H., Schweizer, M. (1991), Hedging of contingent claims under incomplete information, Applied Stochastic Analysis, Stochastic Monographs, 5, Goldon and Breach, 389-414.

*       Follmer, H., Sondermann, D. (1990), Hedging of non-redundant contingent claims, in Contributions to Mathematical Economics, North-Holland, 205-223.

*       Harrison, J.M., Pliska, S.R. (1981), Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and Their Applications, 11, 215-280.

*       Harrison, J.M., Pliska, S.R. (1983), A stochastic calculus model of continuous trading: complete markets, Stochastic Processes and Their Applications, 15, 313-316.

*       Kim, N., Lee, Y. (2018), Estimation and prediction under local volatility jump-diffusion model, Physica A: Statistical Mechanics and its Applications, 491, 729-740.

*       Liu, Y., Jiang, I. M., Hsu, W. (2017), Compound option pricing under a double exponential jump-diffusion model, The North American Journal of Economics and Finance, In press, corrected proof.

 

*       Lokka, A. (2004), Martingale representation of functional of Levy processes, Stochastic Analysis and Applications, 22(4), 867-892.

*      Nunno, G. D., Oksendal, B., Proske, F. (2009), Malliavin calculus for Levy processes with applications to Finance, Springer.‎‎‎‎‎‎