Forwards

I’ve mentioned Forwards many times in the blog so far, but haven’t yet given any description of what they are, and it’s about time for a summary! Forwards were probably the first ‘financial derivative’ to be traded, and they still occupy a central role in the field today.

At the most basic level, a Forward Contract is a delayed sale between two parties. The contract is signed now and the sale and the price agreed to, but the transaction only happens at a later date. We can immediately see why these contracts would be popular, as they insulate the people who have signed the contract from risk due to price variations. For example, an airline can foresee fairly accurately its fuel requirements over the next year, so might wish to enter into a Forward Contract with an oil company for a certain amount of fuel to be delivered next year, in order not to be exposed to the risk of price rises. Meanwhile, the oil company knows its costs of production, and might well also decide that as long as the contract price is profitable it is locking in business for the future and avoiding the risk of price falls harming its operations.

The first thing to note is that, unlike options, the contract is binding – both parties are obliged to enter into the transaction at the given date, there is no optionality in this contract. The agreed price has different names, but for consistency with other products quants tend to call it the Strike Price, K.

Forward Contracts are the most simple form of derivative. A derivative is a financial instrument that derives it’s price from some other (‘underlying’) instrument. In the example above, the underlying was the cost of oil, but it could equally well have been foreign exchange rates, stock prices or interest rates – indeed, all of these are different sorts of risks that companies will in general want to hedge out. What is the price of a Forward Contract and how does it depend on the underlying? If the price of oil rises after the contract above had been signed, the airline company would breath a sign of relief – it has locked in the forward price for its oil via the Forward Contract so the contract is clearly a valuable asset for it. By contrast, the oil company will be miffed, as it could have sold the oil on the market for a better price. If prices fall, situations will be reversed. How can we express this mathematically?

As a simple example, I will consider a Forward Contract for me to buy a stock from a bank in a year’s time for $100. We can calculate the price of this contract using a technique called ‘replication’. What we need to do is construct a portfolio that exactly matches the payments implicit in the Forward Contract, and according to the ‘Law of One Price’, the price of these two matching portfolios must be the same. Call now t=0 and the expiry of the contract t=T.

Portfolio 1: 1 Forward Contract on S at strike price $100 and at time T. Transactions are: at time T, I pay $100, and receive one unit of stock. This will be worth S(T), the price of the underlying at that time. There are no net payments at time t=0.

How can we match this? Well, we will need to ensure the receipt of an amount of cash S(T) at time T, and an ideal way of ensuring this is by buying a unit of the stock at t=0 and holding it, since this is guaranteed to be worth S(T) at t=T. We also need to match the payment of the strike price, $100, at t=T. An ideal product to use for this is a Zero Coupon Bond maturing at t=T (this is equivalent to borrowing money and you can see it either way. We’re making all of the usual assumptions about being able to go long and short equally easily etc.). Recall that Zero Coupon Bonds are worth $1 at time t=T, and worth

    \[{\rm ZCB}(0,T) = {\mathbb E} \Bigr[ e^{-\int_0^T r(t')dt'} \Bigl] = e^{-rT}\]

at t=0, with the last equality holding only in the case of constant interest rates. So, we can short 100 ZCBs at t=0, these will mature at t=T when we will have to pay the holder $100. The holder will pay us the present value at t=0, which is 100 \cdot ZCB(0,T). We’ve constructed a portfolio that matches all of the payments of portfolio 1 at t=T:

Portfolio 2: Long 1 underlying stock S, short 100 ZCBs; close all positions at t=T.

How much does it cost to enter into portfolio 2? As we’ve said, this will identically be the price of portfolio 1. The cost is

    \[{\rm Fwd}(0,T) = S(0) - K \cdot {\rm ZCB}(0,T)\]

    \[= {\rm ZCB}(0,T) \cdot \Bigl({S(0) \over {\rm ZCB}(0,T)} - K \Bigr)\]

We can see that this price is a linear function of K – being long a Forward Contract is better if the strike is lower! We see that its price doesn’t depend on any assumptions about the underlying growth rate (the only assumption was that we can readily trade it on the market at the two times t=0 and t=T), and depends instead on the risk-free interest rate r(t). For this reason the contract is described as ‘Model Independent’ – its price doesn’t depend on any assumptions we make about the world, we can enforce it by no arbitrage using prices of things that are available to buy right now. Compare that to the hedging strategy for a vanilla option discussed here, which relied on continuous re-hedging in the future, and is hence open to risk if our model of future volatility and interest rate isn’t perfect (and it won’t be!).

The ‘Forward Price’ is defined as the strike which makes this contract valueless for both parties – the ‘fair price’ – looking at the equation above we can see that this must be

    \[{S(0)\over {\rm ZCB}(0,T)} = S(0)\cdot{\mathbb E}\Bigl[ e^{\int^T_0 r(t') dt'}\Bigr] = S(0)\cdot e^{rT}\]

with the second equality again only valid for constant rates. The Forward Price is exactly the expectation of the underlying stock S in the risk-neutral measure, it appears across quantitative finance and often simplifies algebra, consider for example the following two ways of expressing the BS equation for vanilla option prices:

    \[C_{\rm call}(K,T,\sigma,r,\delta) = S(0)\cdot \Phi(d_1) - \delta(0,T) K \cdot \Phi(d_2)\]

    \[C_{\rm call}(K,T,\sigma,r,\delta) = \delta(0,T) \Bigl( {\rm Fwd}(0,T)\cdot \Phi(d_1) - K \cdot \Phi(d_2) \Bigr)\]

with the various terms as defined in previous posts (nb. the discount factor here from t=0 to t=T is exactly the same as a ZCB expiring at T). I certainly prefer the second form – it is expressed in terms of quantities depending on the same time T, and all of the discounting is handled outside the brackets, so it is much more easily modularised when coding it up.

Compare the payoff of a forward at expiry with that of a call and a put option. How are they related? It turns out that

    \[C_{\rm call}(K,T,\sigma,r,\delta) - C_{\rm put}(K,T,\sigma,r,\delta) = \delta(0,T)\Big({\rm Fwd}(K,T,r,\delta) - K\Big)\]

This result is called put-call parity. It comes about because if we’re long a call and short a put, we will exercise the call only if the price goes up and the counterparty will exercise the put only if the price falls – we’ve effectively locked in a sale at the strike price K at time T – such a strategy is called a synthetic forward. Note this is actually quite a deep result – although the price of both a call and a put option will depend on the model that we assume for the underlying, the difference in the two prices is model-independent and enforceable by arbitrage! If we assume an increased level of volatility over the option lifetimes, it will increase both prices by the same amount.

Payoffs at expiry for a Forward Contract compared to vanilla calls and puts.
Payoffs at expiry for a Forward Contract compared to vanilla calls and puts. Since a Forward Contract obliges to enter a transaction at expiry at strike K, the payoff can now be negative. If strike is $100 as shown above, but spot has fallen to $70, then the contract is worth -$30, as we could have bought the underlying from the market at $70 but are instead obliged to pay $100 to honour our Forward Contract. Similarly, if the price rises, our contract payoff will be positive. By comparing to the vanilla payoffs at expiry, we can see that a long call and a short put will have the same payoff as a Forward Contract, a result called put-call parity.

Up to this point, I’ve made the implicit assumptions that the underlying asset doesn’t pay any income, it’s free to hold it, and we’re able to go long or short with equal ease. Of course, in general, none of these are true. If the asset pays income like dividends, or if there are costs of storage, it is fairly easy to adjust our replication argument above to find a new price for the Forward Contract that takes into account the extra cash flows associated with holding the underlying over the period 0 \textless t \textless T. However, we can’t always go short the underlying asset – a particular case of this will be commodities, where the asset in question may not exist at the current time (ie. if you haven’t dug it out of the ground or transported it to the delivery point at t=0!), in which case there is no reason that the forward price should be related to the spot price in the deterministic way described above – this is because we have violated our assumption about being able to trade the underlying at the two times t=0 and t=T, which was implicitly vital! However, the price of the Forward Price will still converge towards the expected future spot price as we approach t=T even in this case, since a Forward Contract for transaction at the current time is simply a spot transaction!

One final word about Forward Contracts – they are ‘Over The Counter’, which means they are signed between two counterparties directly and thus carry Credit Risk – there is a chance that your counterparty might go bankrupt before the contract expiry, in which case the transaction won’t be completed. If your contract was out-of-the-money, you will still be expected to settle your position with them, while if it was in-the-money they may well not be able to pay you your full dues, so credit risk will decrease the actual value of the contract. There are advanced ways of accounting for this that I deal with in a later post (CVA – Credit Valuation Adjustment); alternatively there is a similar product called the Futures Contract, which I will deal with very soon in another post, which is superficially similar to the Forward Contract (leading to MUCH confusion in the industry!) but also attempts to deal with this credit risk.

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