Finding the Probability That a Stochastic Trading Strategy Ends Positive

Summary

The document poses a probability question for a continuous-time trading strategy with fixed initial capital and holdings defined as a function of a Wiener process. It expresses the terminal portfolio value as a cubic function of the Wiener process at the end of the period, then asks how to calculate the chance that this value is positive.

The problem reduces to evaluating the probability that a standard normal random variable falls in the region where the cubic polynomial is positive. The document itself does not solve that probability or show the root analysis needed to identify the relevant intervals. Its displayed stochastic integral result also appears inconsistent with Itô's formula for the stated holdings, so the terminal-value expression should be checked before computing a probability. No numerical estimate or empirical trading evidence is provided.

Key ideas

• The strategy's terminal value is intended to be expressed as a function of the Wiener process at the horizon.
• Because the horizon Wiener value is normally distributed, the probability question reduces to a normal probability over polynomial-defined intervals.
• The excerpt does not calculate the probability.
• The stated stochastic integral should be verified against Itô's formula before using the expression.

Tags

# Value of trading strategy


# Value of trading strategy












A trading strategy is defined as follows: starting capital $v_0 = 5$ and 1 risky asset holdings $\varphi_t = 3W_t^2-3t$ where $W$ is a Wiener process. The problem is to find the probability of the value of the strategy at the end of period 1 is greater than zero, $\mathbb{P}(V_1>0)$

Now, $V_1=v_0+\int_0^13W_s^2-3t\,dW_s=W_1^3-3W_1+5$

But how do I calculate $\mathbb{P}(W_1^3-3W_1+5>0)$ or have I completely gone off track?

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