Worst-case optimal investment with a random number of crashes (Q2453936)

The authors consider a financial market consisting of one risk-free asset \(B_{t}\) and one risky asset with price evolutions as in the Black-Scholes model. It is assumed that \(B_0=1\), \(dB_{t}=0\). Let \(W\) be a standard Brownian motion and let \((T_{k})_{k\in \mathbb{N}_0}\) denote the jump times of an independent Poisson process, \(T_0=0\). Let us denote the augmented filtration generated by \(W\) and the Poisson process by \((F_{t})_{t\geq0}\). The sequence \((T_{k})_{k\in \mathbb{N}_0}\) models the time points, at which the investor receives a warning about a potential crash in the market. The crash times are given by the sequence \((\tau_{k})_{k\in \mathbb{N}_0}\) of \((T_{k}\wedge T,T]\cup\{+\infty\}\)-valued stopping times with respect to \((F_{t})_{t\geq0}\), where \(T\) denotes the investment horizon. It is assumed that whenever \(T_{k}<\tau_{k}<T_{k+1}\), a crash occurs at time \(\tau_{k}\). This condition means that there is at most one crash between every two warnings. At each crash time \(\tau_{k}\), the stock price drops by relative amount \(0\leq \kappa_{k}\leq\kappa^{*}\), where \(\kappa_{k}\) is a \(F_{\tau_{k}}\)-measurable random variable and \(\kappa^{*}\in(0,1)\) is the maximum deterministic crash height. The investor does not know the crash scenarios \((\tau_{k},\kappa_{k})_{k\in \mathbb{N}_0}\) a priori, but can observe each crash whenever it occurs. The investor knows \(\kappa^{*}\) and observes \((T_{k})_{k\in \mathbb{N}_0}\). On an interval \((\tau_{k},T_{k+1})\) the investor does not have to fear a crash and trade according to the strategy \((\pi_{t}^0)_{t\in[0,T]}\), at all other times the investor uses the strategy \((\pi_{t}^1)_{t\in[0,T]}\). Given a crash scenario \(\theta=(\tau_{k},\kappa_{k})_{k\in \mathbb{N}_0}\) and a trading strategy \(\pi= (\pi_{t}^0,\pi_{t}^1)_{t\in[0,T]}\), the investor's wealth process \(X=X^{\pi,\theta}\) is given by \(X_0=x\), \(dX_{t}=\alpha\pi_{t}^1 X_{t}dt+\sigma\pi_{t}^1 X_{t}dW_{t}\), on \([T_{k},\tau_{k})\) for each \(k\), \(dX_{t}=\alpha\pi_{t}^0 X_{t}dt+\sigma\pi_{t}^0 X_{t}dW_{t}\), on \([\tau_{k},T_{k+1})\) for each \(k\), \(X_{\tau_{k}}=(1-\pi^1_{\tau_{k}}\kappa_{k})X_{\tau_{k}-}\) on \(\{\tau_{k}<T_{k}\}\) for each \(k\). Using the log utility function, the investor optimizes her expected utility under the worst possible crash scenario. The authors construct a strategy which renders the investor indifferent about an immediate crash of maximum size and no crash at all. Numerical examples, estimating the asymptotic behaviour and computing the costs of hedging against crashes, are presented.