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Minimum variance hedging in a model with jumps at Poisson random times. (Ukrainian, English) Zbl 1224.91162

Teor. Jmovirn. Mat. Stat. 78, 159-174 (2008); translation in Theory Probab. Math. Stat. 78, 175-190 (2009).
The author considers the problem of finding a hedging self-financing strategy for a European type call option with exercise time \(T>0\) for a one dimensional model with continuous time. The discounted price of a stock \((X_t)_{0\leq t\leq T}\) is driven by the equation \[ X_t=X_0+\int_0^tX_{s-}(ads+\sigma dW_s)+\sum_{j:0\leq\tau_j\leq t}X_{\tau_j-}U_j,\quad 0\leq t\leq T. \] Here, \((W_t)_{0\leq t\leq T}\) is a Wiener process, \((U_j)_{j\geq1}\) are independent identically distributed random variables with values in the interval \((-1,+\infty)\), \(\tau_0=0\), \(\tau_j,\;j\geq1,\) is the increasing sequence of moments where the homogeneous Poisson process \((N_t)_{0\leq t\leq T}\) jumps. For a given contingent claim \(H\), the problem is to find a minimum of the mean square deviation \(\operatorname{E}\left[H-V_0-\int_0^T\nu_sdX_s\right]^2\) over all strategies \(\nu\) and \(V_0\in\mathbb R\). If \(\nu^H\) and \(V_0^H\) are solutions of the above problem, then \(\nu^H\) is called the minimum variance hedging strategy and \(V_0^H\) is called the price of the contingent claim \(H\). The hedging strategy is found as a solution of a stochastic equation whose elements are found using the Föllmer-Schweizer decomposition [M. Schweizer, Ann. Probab. 22, No. 3, 1536–1575 (1994; Zbl 0814.60041)].

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60H05 Stochastic integrals
62P05 Applications of statistics to actuarial sciences and financial mathematics

Citations:

Zbl 0814.60041
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