Martingales, Stock Wealth, and the Limits of Random-Walk Assumptions
Summary
The answer uses the martingale concept to explain why, under a simplified random-walk model, an increase in portfolio wealth has the same expected value at a later time. A martingale is defined by the condition that the conditional expectation of a future value, given the information available now, equals its current value. In this framing, the expected future wealth is unchanged even though realized outcomes can differ.
The discussion cautions that this conclusion depends on the model and the probability measure. Actual stock prices need not be martingales under real-world probabilities, and expected returns may differ from zero. In derivatives pricing, the absence of arbitrage supports risk-neutral probability measures under which discounted prices are martingales, but those expectations are not the same as real-world forecasts. The answer also distinguishes the martingale idea from the Markov property, which says the current state contains all relevant information from the past for predicting the future; real markets may violate that property too.
Key ideas
- A martingale's expected future value, conditional on current information, equals its present value.
- A random-walk assumption can make portfolio wealth a martingale in expectation.
- The martingale conclusion does not imply that realized wealth stays constant.
- Risk-neutral pricing measures differ from real-world probability forecasts.
- The Markov property is a separate assumption that past information beyond the current state is irrelevant.
Tags
Full text
# Why are changes in stock market wealth considered permanent?
# Why are changes in stock market wealth considered permanent?
Assume stock prices follow a random walk. If my investments go up by 1,000 dollars on the stock market today and I keep that money invested, in expectation, how much are my investments worth 1 year from now?
I see that the answer is 1,000 dollar, i.e., I should think about my new level of wealth as permanent.
But can someone provide me with the math and the intuition for this?
## Answer by Kevin (score 3, accepted)
https://quant.stackexchange.com/a/49723
I guess the concept you're looking for are martingales. These are stochastic processes which remain on their current level (in expectation!).
Ignoring some technical conditions, a stochastic process $(X_t)$ is called a martingale if for all time points $t\geq s$, $$\mathbb{E}[X_t|\mathcal{F}_s]=X_s.$$ Here, $(\mathcal{F}_s)$ refers to a filtration, the information set available at time $s$. So, given the knowledge (information) at time $s$, your best prediction for the future value $X_t$ is the current value $X_s$.
For instance, the process $(X_t)$ could model the wealth of a portfolio. This would align with your question. However, the question is whether such a portfolio really satisfies the above property. Probably it will not. Real life stock prices are not really martingales: is the best guess (expected value) for Apple's price in one year's time really today's price? Perhaps not. If you however simplify reality and assume that stock prices are simple random walks, then stock prices are indeed martingales, see here.
Derivatives pricing is obviously built upon the concept of martingales and indeed, given the absence of arbitrage strategies, there exist artifical probability measures under which discounted stock prices are indeed martingales. But the expectation in this risk-neutral world is vastly different from the real world expectations (because real world investors are risk averse).
A related concept would be the Markov property. Here, you require that the filtration $(\mathcal{F}_s)$ is generated by the random variable $X_s$, i.e. $\mathbb{E}[X_t|\mathcal{F}_s]=\mathbb{E}[X_t|\sigma(X_s)]$ for all $t\geq s$. This means, past information does not matter at all for predicting stock prices. Again, in the real world, you will find plenty of violations of the Markov property.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.