Discrete-Time Portfolio Insurance with a Terminal Loss Constraint

# Portfolio insurance strategy with path dependence


# Portfolio insurance strategy with path dependence












I have the following problem.

Let us assume that $S_t$, the stock price follows, geometric Brownian moation with parameters $(\mu,\sigma^2)$. We are given an amount of money $M$ and at each point in time $t$ we invest $U_t$ in the stock and put the rest into the bank account with $0$ interest rate. We will trade at $n$ points in time $t_0 < \cdots < t_n = T$. After having done this we have a final value $V_T$ (stock gains and losses and the money on the account). We want to choose $(U_{t_i})_{i=0}^{n-1}$ such that in the end at time $T$ we have $$ P[V_T < K] \le \alpha, $$ for some protection level $K$ (e.g. $0.95 * M$) and a probability $\alpha$ (e.g. $0.01$).

How is such a problem called in the literature (in continuous time, is it "quantile" hedging?). Can you point to a paper where the solution is described? I assume that this is a rather common problem. Thanks!

EDIT: I have been discussing this problem with colleagues and we had the feeling that the usual approach of estimating $\sigma$ and finding a solution at point $0$ that is updated either after a fixed period of time or at an event (a sudden drop, or a rise in vol) is not satisfying.

Can't we incorporate in the solution at time time $0$ that if we lose in the first period then we will reduce risk for the next? Therefore we can take some more risk now. Something like this.

Or is it really the best solution to take as much risk at time $0$ to stay above $K$ with probabilty $\alpha$ and then just solve for the remaining period after we have seen the outcome at $t_1$? Can we do better? As a first step I mean in the model world.