Computing Expected Value

Question

Consider an investor trading on a stock market. At any given moment he can:

1. Buy a stock (if doesn't have one)
2. Sell a stock (if he does have one)
3. Do nothing

At any given time $T$ (where $T=0,1,...,100$) he either buys/sells a stock for price $P$ with probability ($Pr=.5$) or does nothing ($Pr=.5$). Sample table looks like:

Time         Price
==================
0            100
1            90
2            120
3            115

The investor chooses randomly. How do I calculate the expected value of his total profits? Any suggestions? I guess I need to use decision trees but don't know how to start.

Answer 1

If you read this as saying that

• at any one time the person either holds $1$ stock or holds $0$ stock, each with probability $\frac12$, and
• at that time the stock is worth the price at that time,

then, using linearity of expectation, the expected value of the profits is half the change in the stock price from the initial price to the current price,

This is the same expected value as buying $\frac12$ stock at the beginning and holding it through the whole process: random buying and selling simply increases the uncertainty

Answer 2

Forget for a while about the prices, and look weather the investor has stock or not. This is a 0/1 variable (call it $X_t$), whose state depends on the state at the previous time. Namely:

• if $X_t=0$, then $X_{t+1} = 1$ (buy) with probability $\frac 1 2$ and $X_{t+1} = 0$ (do nothing) with probability $\frac 1 2$.

• if $X_t = 1$, then $X_{t+1} = 0$ (sell) with probability $\frac 1 2$ and $X_{t+1} = 1$ (do nothing) with probability $\frac 1 2$.

This, you can put in a decision tree. Then compute the value of the stock by multiplying by the price of the stock at time t: $V_t=P_t \cdot X_t$, and take the average.

As the problem is totally symmetric, my guess is that $x_t$ will average $\frac 1 2$ on the long run. So, if $P_t$ and $X_t$ are independent (contrary to any real case situation) $V_t$ will average to $E(V_t) = E(X_t) \cdot E(P_t) = \frac 1 2 E(P_t)$.