Good evening,

A few days away from the exam, I am taking a bit of time to post the main picture of some topics on the curriculum, which I think can be simply explained. This post is dedicated to **forwards**.

The rationale behind forwards is very simple. Assume there is an asset $S$ which is worth $S_0$ today. You want to enter a contract with somebody to agree to buy the asset at time $T$ at the forward price $FP$. This might seem a bit complicated at first glance for people not familiar with finance mainly because you do not know what $S_T$ (the price of $S$ at time $T$) will be. Well, the truth in the matter is that… it doesn’t matter. Indeed, there is a way you can replicate the action of buying a stock at time $T$ by doing the following:

- At time $t=0$:
- Borrow $S_0$ at the current rate $R$
- Buy the asset at $S_0$

- At time $t=T$:
- Repay what you borrowed at time 0 with interest for $S_0 \cdot (1+R)^T$
- Keep the stock at the value $S_T$.

So, the net investment at $t=0$ is 0, and at $t=T$, you have the stock and you pay $S_0 \cdot (1+R)^T$. So, this is exactly exactly the same thing as buying the stock forward. Hence, you can deduce that:

$$FP=S_0 \cdot (1+R)^T$$

The results for the *law of one price*, and we call that an arbitrage argument because if the price was any different from stated above, the you could make instant risk-free profit by doing the strategy previously stated (or its opposite).

For example, assume a stock is worth $S_0=100\$$ today, that the interest rate $R=10\%$, and that you want to buy the stock forward in 1 year. Then, $FP=100 \cdot.(1+10\%)=110$.

If somebody is willing to buy it forward for 115, then enter the forward contract as a seller, thus agreeing selling $S$ for 115 in a year. Borrow 100 today, buy the stock, hold the stock, repay your loan plus interest in a year for 110 and give the stock to the counterparty for the agreed 115. You get a free lunch of 115-110=5.

If somebody wants to sell the stock forward for 105, you should agree to enter the contract as a buyer. Sell the stock short for 100 today, invest the proceeds for the interest rate today, collect the interest invested of 110 in a year, and buy back the stock as agreed for 105. You make a free lunch of 110-105=5.

That’s it. It’s easy. The forward price is the price today invested at the interest rate. What we just did implies that the value today of a forward contract is 0, by definition. However, the value of the contract will evolve between time 0 and expiration T. Clearly at expiration, the value of the contract is given by:

$$V_T=S_T-FP$$

When time $t$ is between 0 and T, we get the following result:

$$V_t = S_t – \frac{FP}{(1+R)^{T-t}}$$

This is quite logical and you can always check that the value at $t=0$ is 0:

$$V_0=S_0 – \frac{FP}{(1+R)^{T-0}}=S_0 – \frac{S_0 \cdot (1+R)^T}{(1+R)^T}=0$$

And for $t=T$

$$V_T=S_T – \frac{FP}{(1+R)^{T-T}}=S_T – FP$$

Notice that this is the value for the **long** side, i.e. for the person agreeing on buying the asset at expiration for the forward price. Because derivatives are **zero-sum games**, *the value of the short side is the opposite of the value of the long side*.

That’s it. I’ll come back with variants later.