Numerical comparison of local risk-minimisation and mean-variance hedging (Q2771115)

Pricing and hedging of derivatives in incomplete markets still presents a problem. For this reason the authors develop a general Markovian two factor model. The model describes the dynamics of a single asset \((X_t)\) and its volatility \((Y_t)\) through coupled stochastic differential equations. The no arbitrage condition is used to assure the existence of local equivalent martingale measures. Incompleteness of the market then means the non uniqueness of these measures, so that a choice of an appropriate candidate arises. In this model a hedging strategy is then a process \((\vartheta_t, \eta_t)\) where \((\vartheta_t)\) gives the units of the risky asset \((X_t)\) and \((\eta_t)\) the amount of secure assets in the portfolio. The value and cost process are then given by NEWLINE\[NEWLINE V_T =\vartheta_t X_t +\eta_t \quad \text{and} \quad {\mathbb{C}}_t = V_t-\int_0^t \vartheta_s dX_s. NEWLINE\]NEWLINE This model is then applied to hedging of European style contingent claims \(H\) at time \(T\). The optimization criteria for the hedging in this framework are local risk minimization and mean variance hedging. The first method aims at minimizing the conditional second moment of cost increments at each moment \(t\). An appropriate choice of the minimal local martingale measure then leads to a partial differential equation for the pricing function. In the mean variance hedging with the contingent claim \(H\), the expression NEWLINE\[NEWLINE (V_0, \vartheta) \to E \Biggl(\biggl(H-V_0 -\int_0^T \vartheta_s dX_s\biggr)^2\Biggr) NEWLINE\]NEWLINE is minimized over an appropriate choice of initial values \(V_0\) and hedging strategies \(\vartheta\). For this case too the pricing function will satisfy a partial differential equation for European contingent claims. It is shown that mean variance hedging results in lower squared costs than local risk minimization. This general approach is then applied to four specific stochastic volatility models in order to describe the differences of these hedging strategies qualitatively and quantitatively. Finally the costs, prices and hedge ratios are computed numerically for all these models and various sets of parameters. The main mathematical techniques in the general part are the Girsanov transformation and the decomposition of the contingent claim \(H\) (see e.g. [\textit{M. Schweizer}, Stochastic Anal. Appl. 13, 573--599 (1995; Zbl 0873.60042)]). The necessary mathematical requirements and conditions are left rather vague in many instances unfortunately.NEWLINENEWLINEFor the entire collection see [Zbl 0967.91001].