Is there a more rigorous way to show the expected price of a stock?

Question

I am studying for the Society of Actuaries' Financial Math exam and I found the solution to this practice problem to be not convincing enough:

Is there a more mathematical / rigorous way of showing why $P > 105$? I am just not convinced by the author's solution above.

For instance, from another study manual, I know we can let $S_0$ be the current price and let $F_{0, 1}$ be the forward price for delivery in one year. Let $r$ be the force of interest.

Then either $F_{0, 1} = S_0e^{rT}, F_{0, T} = S_0e^{rT} - FV(\text{dividends})$, or $F_{0, 1} = S_0e^{(r - \delta)T}$, depending on whether the stock is non-dividend-paying, pays discrete dividends, or has dividend yield $\delta$, respectively.

I don't know where to go from here, but I feel like there has got to be a way to show without a doubt that the expected price of P is greater than 105.

Answer 1

Think about it this way: The buyer of the forward contract is certain to get the forward $F_{0,1}$ in one year. Therefore the present value of this future payment should be discounted by the risk-free rate, say, $r_0$. The expected value of the stock in one year, $E_{0,1}$, however, is not known, and so the present value of the expected value will be obtained by discounting the expected value by some higher rate than the risk-free rate, say, $r_1>r_0$, which reflects the fact that buying a stock is risky. As both present values must give the same current stock price, we get, assuming continuous discounting:

$$e^{-r_0}F_{0,1}=e^{-r_1}E_{0,1}$$

since $e^{-r_0}>e^{-r_1}$, we must have $E_{0,1}>F_{0,1}$.

(I did assume no dividends)

Answer 2

This question is clearly nonsense.

The pricing formula for forwards is derived from a model independent arbitrage argument. Therefore the "real world" expectation could be any value and is dependent on personal opinion.

Looking outside mathematical arguments to the real world, the fact that the violation of the given answer had a name, contango, shows that the answer doesn't follow from the information in the question.