Trading Stochastic Processes

While there are many signs to consider, there is no single definitive sign to give you "the" answer. Our brains operate with a rational layer, but as illustrated in The Enigma of Reason, this does not necessarily mean we are inherently logical. Gut feelings may arise from something even more primal than reason, often leaving us uncertain about their true meaning. Embracing randomness and learning how to apply available tools to navigate uncertainty seems to be the most effective (educated guess) approach for those who aren't clairvoyant.

In this context, I will explore one trading strategies that is well known among market participants. Following the approach outlined in Bruder and Gaussel’s Risk-Return Analysis of Dynamic Investment Strategies, I will focus on Strategies where returns are proportional to the return of the risky asset. In other words, it allocates a percentage of the portfolio in the risky asset.

To establish a framework for simulating and evaluating trading strategies, it is necessary to derive certain equations involving derivatives and integrals of stochastic processes. While some readers may focus solely on the final results, others may be interested in the step-by-step derivation. Therefore, I will include key intermediate steps leading to the concluding equations.

Explicit results are remarkable because they highlight the impact of leveraging when following a particular solution path. However, taking a slightly different approach can reveal a crystal-clear parameter for entering a trade (figuratively speaking) and determining the right side to take!

Risky Asset Price Process and Trading Strategy

Assume that the price process of interest (i.e. the price of the risky asset), S(t), follows a GBM:

The trading strategy X(t) can be defined as:

The trading strategy holds a fraction f(S(t)) of wealth in the asset S.

So the trading strategy somehow defines a bounded weigthing scheme on the return of the risky asset price process. In other words, it could be, at certain point in time, +3% and -5% in other etc.

Different Solution Paths

We want to solve for X(t), the wealth. Start by rewriting the strategy equation in a more convenient way and taking into account quadratic variation of dW:

• The first term in the first integral represents the cumulative drift contribution to the portfolio and depends, when drift is constant, only on initial and final prices and is called option profile.
• The second term in the first integral is the It? correction term due to the quadratic variation.
• The term in the second integral represents the cumulative stochastic (volatility) contribution to the portfolio.

This solution provides a general framework for analyzing the portfolio dynamics under the given strategy and GBM price process. The specific behavior of X(t) depends on the choice of function f, μ and σ.

As a sanity check, if one assumes that f(S) is a constant "k" then:

And if drift and volatility are also constant, we find the result:

Which is equivalent to:

I think this is a remarkable result because it directly links the strategy's outcome to the time is held (a.k.a. holding period), the base asset and its fixed proportion within the portfolio. Additionally, it is evident that leveraging (k > 1) can have devastating consequences if the risky asset performs poorly (S(t) << S(0)) and loses value with increasing holding period (because k(1-k) < 0).

An alternative path in (4), is to assume f(S) is piecewise constant, then the solution could be written as:

And approximating first two integrals by the average values of integrands so that (8) can be simplified to:

Assuming segments are independent, to maximize log-return per segment, allocation is:

The side of the trade (long or short) is given by the sign of the term inside parentheses in (10). Assuming that "Z" is equal or below 3 (probability of 0.997 for a Normal distribution) and time duration of at least 3 weeks:

So a certain trend would be eligible to trade if risky asset's short term ratio between drift and volatility average values is, in absolute terms, at least 3/4; its side is long if the the ratio is positive and vice-versa. In practice, given existing trading costs, ratio value has to be higher.

Concluding Remarks

My goal was to provide readers with a framework for defining trading strategies in explicit mathematical terms. By leveraging analytical tools, we can express the expected impact of a strategy on each possible trajectory of a stochastic price process. Once these expectations are formulated, we can simulate multiple paths, apply the strategy to each, and analyze the average performance across all simulations to draw meaningful conclusions.

There is a clear distinction between the stylized price process that governs asset dynamics and the trading strategy itself. By combining these two elements, we derive an equation that captures the strategy’s behavior under the assumed price evolution. Typically, this equation includes at least one integral term, explicitly reflecting the path dependency of the strategy—highlighting how the final outcome depends not just on the terminal price but also on the fluctuations along the way (trading impact).

Additionally, the equation often contains a separate term that depends only on the initial and final states (option profile). In some cases, simplifying the expression requires workarounds to improve readability and interpretability, ensuring that the core takeaways remain clear while maintaining mathematical tractability.

In this context, insights do not refer to subjective intuition or gut feelings but rather to model-driven takeaways derived from the mathematical formulation. These insights emerge from the structure of the governing equations, revealing how the trading strategy interacts with the stochastic price process and what can be expected in terms of outcomes.

Two major findings were highlighted: leverage and holding period impacts, asgiven by (7), and optimum proportion to maximize log-return, when allocation is piecewise constant and dependent on risky asset's drift to volatility ratio, as given by (11).

As always, maintaining a humble approach — recognizing that models are merely lower-dimensional, simplified representations of reality — helps mitigate risks and prevent losses. Keeping this perspective ensures that model-driven decisions remain grounded in their inherent limitations.

Finally, this development was intended for illustration purposes only, as real-world processes tend to be more complex than stylized ones. The normality assumption in Geometric Brownian Motion (GBM) justified using a three-standard-deviation threshold to represent a 99.7% likelihood. A fat-tailed distribution, however, would require a much higher threshold. Nevertheless, the same steps can be applied when facing similar problems.

Enjoy!