Discrete-Time Portfolio Strategies Approximate Continuous Trading

Summary

This explanation clarifies the notation used when approximating a continuous-time trading strategy. The portfolio holding, denoted by phi, is a number of shares that may vary over time. The superscript n in phi^(n) labels an approximation built using n time steps across the interval from the start to the horizon; it is neither a derivative nor a power.

On each interval, the approximate strategy holds the value of the continuous-time strategy observed at the interval’s starting point. Its cumulative gain is represented by a sum of holdings multiplied by successive changes in the underlying Brownian price process, or equivalently by a stochastic integral. Increasing the number of steps is described as improving the approximation, and the notation commonly leads to a limit as n grows without bound. The brief answer is conceptual: it does not state formal convergence conditions or discuss whether a particular strategy is admissible or self-financing.

Key ideas

• The superscript n indexes a discrete approximation and does not indicate a derivative or exponent.
• The time horizon is divided into n intervals, with the strategy held constant within each interval.
• Each interval’s holding is taken from the continuous-time strategy at the interval’s start.
• The approximate gain can be expressed as a sum of holdings times price increments or as a stochastic integral.
• Convergence as the time-step count grows requires conditions not detailed in the explanation.

Tags

# What does $\phi^{(n)}(t)$ mean for a portfolio?


# What does $\phi^{(n)}(t)$ mean for a portfolio?












I am currently in the process of deciphering the notes written by an instructor. I missed the class.

He writes:

> Continuous time trading: $S(T) = W(T)$, "Bachelier Model". $\phi(t)$ denotes number of shares held at time t. Approximation is $\phi^{(n)}(t) = \phi(t_i), t_i < t < t_{i+1}, t_i = \frac{i}{n}T$. Gain from strategy $\phi^{(n)}$ is $\sum_{i=1}^n \phi(t_{i-1})[W_{t_i} > - W_{t_i -1}] = \int_0^T \phi^{(n)}(s) dW(s).$

I don't understand what $n$ or $\phi^{(n)}$ is. Is it a derivative?

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/31455

n is the number of time steps used in the discrete time solution $\phi^{(n)}$ to approximate the continuous time solution $\phi$ (the time period $T$ is being broken up into n steps). The higher the n the better the approximation. In the expression $\phi^{(n)}$ n is a super-script, not an exponent nor a derivative.

I have a feeling that later on you are going to be told that $n \rightarrow \infty $ ...

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