×

Risk-minimizing hedging strategies for insurance payment processes. (English) Zbl 0983.62076

Let \((\Omega,{\mathcal F},P)\) be a probability space with filtration \({\mathbf F}=(\mathcal F_t)_{0\leq t\leq T}\). Consider a financial market consisting of two assets with discounted price processes \(X=(X_t)_{0\leq t\leq T}\) and \(Y=(Y_t)_{0\leq t\leq T}\), respectively, where \(X\) is the discounted price process associated with some risky asset (a stock), and the discounted price process \(Y\) associated with the locally risk-free asset (saving account) which is assumed to be constant. A trading strategy is a process \(\varphi=(\xi,\eta)\), where \(\xi_t\) is interpreted as the amount of stocks at time \(t\), and \(\eta_t\) is the discounted deposit on the saving account at time \(t\). The pair \(\varphi_t=(\xi_t,\eta_t)\) is called the portfolio held at time \(t\). The value process associated with the strategy \(\varphi=(\xi,\eta)\) is defined by \[ V_t(\varphi)=\xi_tX_t+\eta_t,\quad 0\leq t\leq T. \] Consider an agreement between two partners, a seller (hedger) and a buyer specifying certain payments given by an \({\mathbf F}\)-adapted process \(A=(A_t)_{0\leq t\leq T}\). The simplest considered situation is where \(A\) is of the form \(A_t=-\kappa+\mathbf{1}_{\{t\geq T\}}H,\;0\leq t\leq T,\) for some constant \(\kappa\) and \(H\in L_2({\mathcal F}_T,P)\) (payments take place at time 0 and time \(T\) only). Let \(C(\varphi)=(C_t(\varphi))_{0\leq t\leq T}\) be a cost process defined by the equation \[ C_t(\varphi)=V_t(\varphi)-\int_0^t\xi_udX_u+A_t,\quad 0\leq t\leq T, \] with the initial condition \(C_0(\varphi)=V_0(\varphi)+A_0\). \(C_t(\varphi)\) comprises the hedger accumulated costs during \([0,t]\) and \(V_t(\varphi)\) may be interpreted as the value of the portfolio \(\varphi_t=(\xi_t,\eta_t)\) held at time \(t\) after the payment \(A_t\). The risk process of the strategy \(\varphi=(\xi,\eta)\) is defined by \(R_t(\varphi)= {\mathbf E}\left[(C_T(\varphi)-C_t(\varphi))^2 |{\mathcal F}_t\right]\). A strategy is said to be risk-minimizing if it minimizes \(R_t(\varphi)\) for all \(t\). Define a martingale \(V_t^{\ast}={\mathbf E}\left[A_T |{\mathcal F}_t\right],\;0\leq t\leq T\). This martingale can be uniquely decomposed by use of the Galtchouk-Kunita-Watanabe decomposition [see H. Föllmer and D. Sondermann, Contributions to Mathematical Economics, Hon. G. Debreu, 206-223 (1986; Zbl 0663.90006)] as \[ V_t^{\ast}=V_0^{\ast}+\int_0^t\xi_u^AdX_u+L_t^A, \] where \(L^A\) ia a zero-mean martingale orthogonal to \(X\) and \(\xi^A\) is a predictable process.
The author proves that there exists a unique \(0\)-admissible (\(V_T(\varphi)=0\)) risk-minimizing strategy \(\varphi=(\xi,\eta)\) for \(A\) given by \[ (\xi_t,\eta_t)=(\xi^A_t,V_t^{\ast}-A_t-\xi^A_tX_t),\quad 0\leq t\leq T. \] The presented result extends the corresponding result of Föllmer and Sondermann. The author presents some examples related to insurance. These include a general unit-linked life insurance contract driven by a Markov jump process and a claim process from non-life insurance, where the claim size distribution is affected by a traded price index.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
60G44 Martingales with continuous parameter
60G35 Signal detection and filtering (aspects of stochastic processes)

Citations:

Zbl 0663.90006
PDF BibTeX XML Cite
Full Text: DOI