Profit and Loss

We define the profit and loss of a trading strategy.

Suppose a stock has price X_t at time t. If we buy one share of stock at time t_0 the potential profit and loss at time t > t_0, or mark-to-market, is X_t - X_{t_0} assuming we could sell the share at time t for X_t. If we actuall sell the share at time t_1 > t_0 the realized profit and loss is X_{t_1} - X_{t_0}. This simple observation corresponds to the mathematical definition of stochastic integration.

What the math does not accurately reflect is that the “price” X_t is somewhat nebulous in the absence of an actual transaction. Your brokerage account will record the exact time, price, and amount of every transaction executed. Anyone who has ever traded knows the “price” X_t they watch during trading hours is not necessarily what will ultimately show up in their account.

If \Delta_t = 1 when t_0 < t \le t_1 and is zero otherwise, then the P&L is \int_0^t \Delta_s\,dX_s = \int_0^t 1_{(t_0,t_1]}(s)\,dX_s = \begin{cases} 0 & t \le t_0 \\ X_t - X_{t_0} & t_0 < t \le t_1 \\ X_{t_1} - X_{t_0} & t > t_1 \\ \end{cases}

Instead of one share, we could buy any number of shares \Gamma_0 based on information available at time t_0 and sell \Gamma_0 shares at time t_1. \int_0^t \Gamma_0 1_{(t_0, t_1]}(s)\,dX_s = \begin{cases} 0 & t \le t_0 \\ \Gamma_0(X_t - X_{t_0}) & t_0 < t \le t_1 \\ \Gamma_0(X_{t_1} - X_{t_0}) & t > t_1 \\ \end{cases}

This corresponds to the elementry trading strategy buy \Gamma_0 at t_0 and sell \Gamma_0 at t_1.

Instead of fixed times t_j we can generalize to stopping times \tau_j.

A trading strategy is a finite number of increasing stopping times \tau_0 < \tau_1 < \cdots < \tau_n and trades \Gamma_0, \Gamma_1, \dots, \Gamma_n where \Gamma_j is a function of information available at time \tau_j and \sum_j \Gamma_j = 0. Every trading strategy must close out.

The position is the accumulation of trades {\Delta_t = \sum_{\tau_j < t} \Gamma_j}. Note the strict inequality. It takes some time for a trade to settle. We can write this as {\Gamma_t = \sum_{s<t} \Gamma_s} where {\Gamma_s = \Gamma_j 1(s = \tau_j)}. Note \Delta_t is a linear combination of elementary treades and we define the profit and loss at time t by the stochastic integral \int_0^t \Delta_s\,dX_s.

It is possible to incorporate transaction costs by replacing \Gamma_j with, e.g., (1 + \epsilon\mathop{\rm{sgn}}(\Gamma_j))\Gamma_j where \epsilon is the proportional transaction cost and \mathop{\rm{sgn}}(x) is the sign of x.