Worst-Case Optimal Investment with a Random Number of Crashes

Abstract

We study a portfolio optimization problem in a market which is under the threat of crashes. At random times, the investor receives a warning that a crash in the risky asset might occur. We construct a strategy which renders the investor indifferent about an immediate crash of maximum size and no crash at all. We then verify that this strategy outperforms every other trading strategy using a direct comparison approach. We conclude with numerical examples and calculating the costs of hedging against crashes.

Worst-Case Portfolio Optimization in a Market with Bubbles

“…This shows that a random number of total crashes introduces an additional longterm effect on the optimal strategies (see also Belak et al [4]). …”

“…In addition, it is assumed that the maximum crash size is fixed a priori and does not change with time. In the present paper we extend the model in Belak et al [4] to allow for changing maximum crash sizes and changing market coefficients by proposing a regime switching model. That is, we assume that the market is in one of d + 1 states, where in the first state no bubble is present and hence the investor does not have to fear a crash.…”

“…Note that the optimization problem is the same as in Korn and Wilmott [22], however, observe that the crucial difference is that we allow for a random number of crashes (where Korn and Wilmott [22] assume to know an upper bound on the total number of crashes). This leads to optimal portfolio strategies where the crash risk has not only a short-term impact (as in Korn and Wilmott [22]) but also a long-term impact (see Belak et al [4]). …”

“…In a previous paper of Belak et al [4], the warnings occur at the jump times of an independent Poisson process, i.e. the times at which warnings occur are exponentially distributed.…”

“…where the family (f i ) i∈E solves the coupled system of ordinary differential equations ∂ ∂t f i (t) = − 1 2 4) and where Q = (q i,j ) i,j∈E denotes the transition rate matrix of the underlying Markov chain. Similarly, in the case of logarithmic utility, the value function can be written as V RS (t, x, i) = log(x) + f i (t), (1.5) where the family (f i ) i∈E solves A model closely related to our model was recently introduced in Capponi and Figueroa-López [8].…”