Forward Prices, Spot Prices, and Expected Future Stock Prices

# Forward price vs. expected future stock price


# Forward price vs. expected future stock price












The forward price of a maturity-$T$ forward on a stock paying no income or dividend is $F=e^{r(T-t)}S_t$.

In discussing this, Stephen Blyth says:

> Note that the forward price depends only on the current stock price $S_t$ , the interest rate $r$ and the time to maturity $T – t$. It often surprises those encountering forwards for the first time that the determination of the forward price does not depend on the growth rate or standard deviation of the stock, or indeed any distributional assumptions about $S_T$. The forward price says nothing further about predicting where the stock will be at time $T$ than the spot price does.

But, Hull says (p. 124):

> Suppose that $k$ is an investor’s required return for this investment. The present value of this investment is $-Fe^{-r(T-t)}+E[S_T]e^{-k(T-t)}$. We can assume that all investments in securities markets are priced so that they have zero net present value. This means that $\mathbf{F=e^{(r-k)(T-t)}E[S_T].}$

I have several questions:

- Doesn't the latter imply that the forward price does in fact depend on the expected future stock price, contrary to what Blyth said?

- Hull goes on to say that the discount rate $k$ should be $=r, >r, and <r$ (respectively) if the returns from the asset are uncorrelated/positively correlated/negatively correlated with the market. Why is this so?

## Answer by Mats Lind (score 4)

https://quant.stackexchange.com/a/81436

Starting with question 1, Blyth does not contradict that "the forward price does in fact depend on the expected future stock price". What he actually says is that given the spot price, predictions says nothing further. So the answer is, the forward depends on expectations, but that does not stand agaist what Blyth says. Hull's and Blyth's statements are indeed in perfect agreement as you see if you arrange them like:

A. a relation between the spot and expectations: $S_t=e^{(-k)(T-t)}E[S_T]$

B. a relation between forward and spot: $ {F=e^{r(T-t)}S_t}$

A. and B. combines into Hull's formulation: ${F=e^{(r-k)(T-t)}E[S_T].}$

On question 2, Blyth apperently sees "the market" as the single source of risk (CAPM that is?). Then you need a discount relative to future expectations, produced in Hull's formula from a negative $r-k$, to compensate for the risk born by the asset through beeing positively correlated to the single, unhedgeable, source of risk. Zero correlation, no non-diversifiable costly risk, hence no return other than the risk free, and $k=r$ making the forward price equal to expectations. Positive $r-k$ is the premium paid for the hedge to market risk that the negative correlation gives.

## Answer by Rylan (score 2)

https://quant.stackexchange.com/a/81424

For question 1:

The first argument you cite is based on saying that if the forward price $F(t, T)$ was anything besides $S_te^{r(T-t)}$ then we could find an arbitrage by trading in the stock and the money market account. In most of quantitative finance, arguments of this form are what's used for pricing.

In the edition of Hull I have (11th), he begins the chapter with an argument similar to what I described above, and he introduces the argument you posted nearer the end. My interpretation is that if you're willing to make a number of additional assumptions (that are motivated by economic theory) then you can also have that the futures price bears that relationship with the expected spot price. But you don't have to make those assumptions to get the price from the first argument.