In order to answer this question, let’s first define the concepts of martingale, sub-martingale, and super-martingale in the context of a stochastic process.
A stochastic process {Xt} on a probability space (Ω, ℱ, P) is said to be a:
1. Martingale if, for all s < t,
E(Xt|ℱs) = Xs
2. Sub-martingale if, for all s < t,
E(Xt|ℱs) ≥ Xs
3. Super-martingale if, for all s < t,
E(Xt|ℱs) ≤ Xs
Here, ℱs represents the information available up to time s, and E(Xt|ℱs) represents the expected value of the random variable Xt given the available information.
Now let’s examine whether stock prices are martingale, sub-martingale, or super-martingale.
Let {St} denote the stock price at time t. Stock prices are often assumed to follow a geometric Brownian motion. Under the risk-neutral measure, the stock price process is given by
dSt = rStdt + σStdWt
where r is the risk-free interest rate, σ is the stock’s volatility, and Wt is a standard Brownian motion.
Under the risk-neutral measure, the expectation of the stock price at a future time, given the current price, will have the drift term that takes into account the time value of money. Therefore, we have:
E(St|ℱs) = Sser(t − s)
Since r is positive and for any s < t, er(t − s) > 1, it follows that:
E(St|ℱs) > Ss
This inequality implies that stock prices form a sub-martingale under the risk-neutral measure.
It is essential to note that the above conclusions are model-dependent, as we have assumed that stock prices follow geometric Brownian motion under the risk-neutral measure. Other models could lead to different results. In practice, stock prices tend to be more complex and may exhibit different behavior that may not fit perfectly with any of the three categories (martingale, sub-martingale, or super-martingale).