Deriving a Forward Price from the Spot Price and Discount Factor

# forward rate/zero coupon


# forward rate/zero coupon












We have an asset with the price process $S_t$, $0\leq t\leq T$ Further we have a zero coupon bond, and the price of it at time $t$ is denoted by $P(t,T)$ (with payoff 1 at time T). Let $F_{(t, T )}[S]$ denote the forward price, I need to argue that the following relationship holds:

$F_{(t, T )}[S] =\frac{S_t}{P(t,T)}$

Could someone please help me with the argumentation?

## Answer by blizzard16 (score 1, accepted)

https://quant.stackexchange.com/a/75243

I think there are two things to get straight for the correct intuition. Understand what forward price (1) and $P(t, T)$ with pay-off 1 at T (2) means. Let's begin with the forward price:

$F_{(t,T)}[S]$ means what would be the price of the stock $S$ at time $t$ given that you agree to make a deal prior of the asset's transfer at time $T$. At least I would assume this for clarity, correct if I am wrong.

Now the 2nd building block: zero-coupon bond that pays $FV=1$ at $T$. Since we are so lucky that we have $FV=1$, we can treat the $P(t, T)$ as a discount factor. If the $P(t, T)$ evaluates, say to $.8$. That means that 1 dollar at time $T$ is worth 0.8 dollars at time $t$.

Let's now put everything together. Usually forward prices for assets are quoted followingly: $F_0=S_0e^{rT}$, where $F_0$ is the forward price of a stock given its price $S_0$. Then the $S_0$ is multiplied the inverse of the $F_0$'s discount factor or by continuous compounding w.r.t rate of return $r$ of $S_0$ and time $T$ to arrive at the forward contracts value. See if you can connect the points.

So in your case the $P(t, T)=e^{rT}$ and essentially $P(t, T)<1$ making it useful to interpret $F_{(t,T)}[S]=S_tP(t,T)^{(-1)}$. You can verify this by putting some numbers for the $P(t,T)$ since it works here as inverse of the discount factor as it's essentially compunding the value of the stock $S$ at time $t$ to its value at $T$ that is the price of the forward contract for the stock.

Disclaimer: the reason why I am not putting any more formulas for the variables is that it's not necessary and there's no clear definition's for them as the same properties hold for multiple different types of functions that reflect asset prices and discount factors.

E: didn't really answer your question: given the context I gave you, could an arbitrage opportunity arise if the equality didn't hold? If such does, how would you make money out of it?

I hope this helps you.