Maximizing the Chance of Reaching a Wealth Target Under Geometric Brownian Motion

Summary

The document poses a finite-horizon allocation problem: an investor starts with a given amount, allocates a fraction of wealth to a stock following geometric Brownian motion, and seeks to maximize the probability of finishing above a target. It defines wealth dynamics when the invested fraction can depend on current wealth and time remaining.

For a simpler constant-fraction strategy, it derives a lognormal distribution for terminal wealth and proposes maximizing the probability of exceeding the target by differentiating that probability with respect to the allocation fraction. This provides a tractable starting point, but it does not solve the original state- and time-dependent control problem. It also gives no numerical parameters, optimized allocation, or discussion of constraints such as borrowing, transaction costs, or model error.

Key ideas

• The stock price is modeled with geometric Brownian motion.
• The allocation fraction may depend on current wealth and time remaining.
• With a constant allocation fraction, terminal wealth is lognormally distributed.
• The constant-fraction case can be optimized by differentiating the probability of clearing the terminal wealth target.

Tags

# Finding the optimal strategy of a stock trading game


# Finding the optimal strategy of a stock trading game












Assume there is a stock, its price at time $t$ is $S(t)$. Its price is changing according to a geometric brownian motion, that is $dS=S(\mu dt+\sigma dW)$. You start with $\\\$1$ and you are allowed to choose a fraction $f$ amout of your net worth to put into the stock at any given time. After a period of time $T$, if your net worth is greater than $\\\$e$, you win, otherwise you lose. What strategy should you choose to maximise the probability of winning? (A strategy is a function that inputs the current wealth, the remaining time until $T$, and outputs the fraction $f$).

Let the wealth at time $t$ be $N(t)$, for a given strategy $f$, the change of wealth $dN(t)=f(N(t),T-t)N(t)(\mu dt+\sigma dW)$

A simpler version of this question would be asking for a fixed $f$ for all time (instead of a function of time). This will gives $dN(t)=fN(t)(\mu dt+\sigma dW)$, which means $N(T)$~$lognormal(f\mu T-\frac{(f\sigma)^2T}{2},(f\sigma)^2T)$. Then calculate the cdf and take the derivative with respect to $f$ to find the maximum.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.