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**Optimal price for the hedge of a European-type contingent claim.**
*(Ukrainian, English)*
Zbl 1195.91164

Teor. Jmovirn. Mat. Stat. 77, 133-140 (2007); translation in Theory Probab. Math. Stat. 77, 147-154 (2008).

The author deals with the \((B, S)\) financial market described by the system of stochastic equations:
\[
dS_t = a(t, S_t)S_t dt + \sigma_tS_t dW_t, dB_t = r_tB_t dt, B_0 = 1,t\in [0, T],
\]
where \(W_t\) is a standard Brownian motion, \(S_0\) is a given random variable independent of \(W_t\), and \(r_t\) is a nonnegative progressively measurable stochastic process considered as the interest rate. It is assumed that the volatility \(\sigma_t=\rho(\tilde{\sigma}_t)\), where \(\rho\in C^2_b (R)\), is an increasing function and that \(\tilde{\sigma}_t\) is a stochastic process determined by the equation
\[
d\tilde{\sigma}_t=\tilde{\alpha}(t,\tilde{\sigma}_t)dt+\tilde{\beta}(t,\tilde{\sigma}_t)dB^H_t+\varepsilon \tilde{\beta}(t,\tilde{\sigma}_t )dV_t.
\]
Here \(\varepsilon\) is a positive real number, \(B^H_t\) is fractional Brownian motion with Hurst parameter \(H \in (3/4, 1)\) independent of the process \(W_t\), the process \(V_t\) is a standard Brownian motion independent of the processes \(W_t\) and \(B^H_t\). The classical model with standard Brownian motions is considered, for example, by G. Kallianpur and R. L. Karandikar [Introduction to option pricing theory, Boston: Birkhäuser (2000; Zbl 0969.91003)]. S. V. Posashkov [Visn., Ser. Fiz.-Mat. Nauky, Kyïv. Univ. Im. Tarasa Shevchenka 2005, No. 2, 56–61 (2005; Zbl 1089.91503)] proved that the market is incomplete if the volatility is governed by fractional Brownian motion. However, an equation for the optimal hedging price of a contingent claim that locally minimizes the risk is not obtained in the mentioned article.

The main feature of the model considered in this paper is the long range dependence of the volatility described by a mixture of standard Brownian motion and fractional Brownian motion with Hurst parameter in the interval \((3/4, 1)\). The optimal hedging price for a European contingent claim minimizing a certain risk is obtained. A partial differential equation for the optimal price of a given contingent claim is derived.

The main feature of the model considered in this paper is the long range dependence of the volatility described by a mixture of standard Brownian motion and fractional Brownian motion with Hurst parameter in the interval \((3/4, 1)\). The optimal hedging price for a European contingent claim minimizing a certain risk is obtained. A partial differential equation for the optimal price of a given contingent claim is derived.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

### MSC:

91G20 | Derivative securities (option pricing, hedging, etc.) |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

60J35 | Transition functions, generators and resolvents |

60G22 | Fractional processes, including fractional Brownian motion |