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.