## Forwards vs. Futures

I’ve covered Forwards and Futures in previous posts, and now that I’ve covered the basics of Stochastic Interest Rates as well, we can have a look at the difference between Forwards and Futures Contracts from a financial perspective.

As discussed before, the price of a Forward Contract is enforceable by arbitrage if the underlying is available and freely storable and there are Zero Coupon Bonds available to the Forward Contract delivery time. In this case, the forward price is

$F(t,T)&space;=&space;S(t)&space;\cdot&space;{1&space;\over&space;{\rm&space;ZCB}(t,T)}$

In this post I’m going to assume a general interest rate model, which in particular may well be stochastic. In such cases, the price of a ZCB at the present time is given by

${\rm&space;ZCB}(t,T)&space;=&space;{\mathbb&space;E}\Big[&space;\exp{\Big\{\int_t^T&space;r(t')&space;dt'\Big\}&space;}&space;\Big]$

Futures Contracts are a bit more complicated, and we need to extend our earlier description in the case that there are interest rates. The basic description was given before, but additionally in the presence of interest rates, any deposit that is in either party’s account is delivered to the OTHER party at the end of each time period. So, taking the example from the previous post, on day 4 we had $4 on account with the exchange – if rates on that day were 10% p.a., over that day the$4 balance would accrue about 10c interest, which would be paid to the other party.

Let’s say we’re at time s, and want to calculate the Futures price to time T. Our replication strategy is now as follows, following the classic proof due to Cox, Ingersall and Ross but in continuous time. Futures Contracts are free to enter into and break out of due to the margin in each account, so entering X Futures Contracts at time t and closing them at time t+dt will lead to a net receipt (or payment if negative) of $\inline&space;{\rm&space;X}\cdot\big[&space;H(t+dt,T)&space;-&space;H(t,T)\big]$. From t+dt to T, we invest (borrow) this amount at the short rate and thus recieve (need to pay)

${\rm&space;X}\cdot\big[&space;H(t+\tau,T)&space;-&space;H(t,T)\big]\cdot\prod_t^T&space;\big(&space;1&space;+&space;r(t)\tau&space;\big)$

and now moving to continuous time

${\rm&space;X}\cdot\big[&space;H(t+dt,T)&space;-&space;H(t,T)\big]\cdot\int_t^T&space;e^{&space;r(t)}\&space;dt$

We follow this strategy in continuous time, constantly opening contracts and closing them in the following time period [I’m glossing over discrete vs. continuous time here – as long as the short rate corresponds to the discrete time step involved this shouldn’t be a problem], and investing our profits and financing our losses both at the corresponding short rate. We choose a different X for each period [t,t+td] so that $\inline&space;{\rm&space;X}(t)&space;=&space;\int_s^t&space;\exp{\{r(t')\}}dt'$. We also invest an amount H(s,T) at time s at the short rate, and continually roll this over so that it is worth $\inline&space;H(s,T)\cdot&space;\int_s^T&space;\exp{\{r(t)\}}dt$ at time T

Only the final step of this strategy costs money to enter, so the net price of the portfolio and trading strategy is H(s,T). The net payoff at expiry is

$H(s,T)\cdot&space;\int_s^T&space;e^{r(t)}dt&space;+&space;\sum_s^T&space;{\rm&space;X}\cdot[H(t+dt,T)-H(t,T))]\cdot\int_t^T&space;e^{r(t)}dt$

$=&space;H(s,T)\cdot&space;\int_s^T&space;e^{r(t)}dt&space;+&space;\sum_s^T&space;\int_s^t&space;e^{r(t)}dt\cdot[H(t+dt,T)-H(t,T))]\cdot\int_t^T&space;e^{r(t)}dt$

$=&space;H(s,T)\cdot&space;\int_s^T&space;e^{r(t)}dt&space;+&space;\int_s^T&space;e^{r(t)}dt&space;\cdot&space;\sum_s^T&space;[H(t+dt,T)-H(t,T))]$

$=&space;H(s,T)\cdot&space;\int_s^T&space;e^{r(t)}dt&space;+&space;\int_s^T&space;e^{r(t)}dt&space;\cdot&space;[H(T,T)-H(s,T))]$$=&space;H(T,T)&space;\cdot&space;\int_s^T&space;e^{r(t)}dt$

And H(T,T) is S(T), so the net payoff of a portfolio costing H(s,T) is

$=&space;S(T)&space;\cdot&space;\int_s^T&space;e^{r(t)}dt$

How does this differ from a portfolio costing the Forward price? Remembering that in Risk-Neutral Valuation, the present value of an asset is equal to the expectation of its future value discounted by a numeraire. In the risk-neutral measure, this numeraire is a unit of cash B continually re-invested at the short rate, which is worth $\inline&space;B(t,T)&space;=&space;e^{\int_t^T&space;r(t')dt'&space;}$, so we see that the Futures Price is a martingale in the risk-neutral measure (sometimes called the ‘cash measure’ because of its numeraire). So the current value of a Futures Contract on some underlying should be

$H(t,T)&space;=&space;{\mathbb&space;E}^{\rm&space;RN}\big[&space;S(T)&space;|&space;{\cal&space;F}_t&space;\big]$

ie. the undiscounted expectation of the future spot in the risk-neutral measure. The Forward Price is instead the expected price in the T-forward measure whose numeraire is a ZCB expiring at time T

$F(t,T)&space;=&space;{\mathbb&space;E}^{\rm&space;T}\big[&space;S(T)&space;|&space;{\cal&space;F}_t&space;\big]$

We can express these in terms of each other remembering F(T,T) = S(T) = H(T,T) and using a change of numeraire (post on this soon!). I also use the expression for two correlated lognormal, which I derived at the bottom of this post

\begin{align*}&space;F(t,T)&space;&=&space;{\mathbb&space;E}^{T}\big[&space;F(T,T)&space;|&space;{\cal&space;F}_t&space;\big]&space;\\&space;&=&space;{\mathbb&space;E}^{T}\big[&space;S(T)&space;|&space;{\cal&space;F}_t&space;\big]&space;\\&space;&=&space;{\mathbb&space;E}^{T}\big[&space;H(T,T)&space;|&space;{\cal&space;F}_t&space;\big]&space;\\&space;&=&space;{\mathbb&space;E}^{RN}\big[&space;H(T,T)&space;{B(t)\over&space;B(T)}&space;{{\rm&space;ZCB}(t,T)\over&space;{\rm&space;ZCB(T,T)}}|&space;{\cal&space;F}_t&space;\big]&space;\\&space;&=&space;{\rm&space;ZCB}(t,T){\mathbb&space;E}^{RN}\big[&space;H(T,T)&space;{1\over&space;B(T)}|&space;{\cal&space;F}_t&space;\big]&space;\\&space;&=&space;{\rm&space;ZCB}(t,T){\mathbb&space;E}^{RN}\big[&space;H(T,T)\big]&space;{\mathbb&space;E}^{RN}\big[&space;e^{-\int_t^T&space;r(t')dt'}&space;\big]&space;e^{\sigma_H&space;\sigma_B&space;\rho}&space;\\&space;&=&space;H(t,T)&space;\cdot&space;e^{\sigma_H&space;\sigma_B&space;\rho}&space;\\&space;\end{align*}

where $\inline&space;\sigma_H$ is the volatility of the Futures price, and $\inline&space;\sigma_B$ is the volatility of a ZCB – in general the algebra will be rather messy!

As a concrete example, let’s consider the following model for asset prices, with S driven by a geometric brownian motion and rates driven by the Vasicek model discussed before

${dS&space;\over&space;S}&space;=&space;r(t)&space;dt&space;+&space;\sigma_S&space;dW_t$

$dr&space;=&space;a&space;\big[&space;\theta&space;-&space;r(t)\big]&space;dt&space;+&space;\sigma_r&space;\widetilde{dW_t}$

And (critically) assuming that the two brownian processes are correlated according to rho

$dW_t&space;\cdot&space;\widetilde{dW_t}&space;=&space;\rho&space;dt$

In this case, the volatility $\inline&space;\sigma_B$ is the volatility of $\inline&space;{\mathbb&space;E}\big[&space;e^{-\int_t^T&space;r(t')dt'}\big]$, and as I discussed in the post on stochastic rates, this is tractable and lognormally distributed in this model.

We can see that in the case of correlated stochastic rates, these two prices are not the same – which means that Futures and Forward Contracts are fundamentally different financial products.

For two standard normal variates x and y with correlation rho, we have:

\begin{align*}&space;{\mathbb&space;E}\big[&space;e^&space;{\sigma_1&space;x}&space;\big]&&space;=&space;e^&space;{{1\over&space;2}\sigma_1^2&space;}&space;\end{align*}

and

\begin{align*}&space;{\mathbb&space;E}\big[&space;e^&space;{\sigma_1&space;x&space;+&space;\sigma_2&space;y}&space;\big]&&space;=&space;{\mathbb&space;E}\big[&space;e^&space;{\sigma_1&space;x&space;+&space;\sigma_2&space;\rho&space;x&space;+&space;\sigma_2&space;\sqrt{1-\rho^2}z}&space;\big]\\&space;&&space;=&space;{\mathbb&space;E}\big[&space;e^&space;{(\sigma_1&space;+&space;\sigma_2&space;\rho)&space;x&space;+&space;\sigma_2&space;\sqrt{1-\rho^2}z}&space;\big]\\&space;&&space;=&space;\big[&space;e^&space;{{1\over&space;2}(\sigma_1&space;+&space;\sigma_2&space;\rho)^2&space;+&space;{1\over&space;2}(\sigma_2&space;\sqrt{1-\rho^2})^2}&space;\big]\\&space;&&space;=&space;\big[&space;e^&space;{{1\over&space;2}\sigma_1^2&space;+&space;{1\over&space;2}\sigma_2^2&space;+&space;\sigma_1&space;\sigma_2&space;\rho}&space;\big]\\&space;&&space;=&space;{\mathbb&space;E}\big[&space;e^&space;{\sigma_1&space;x&space;}\big]&space;{\mathbb&space;E}&space;\big[&space;e^{\sigma_2&space;y}\big]&space;e^{&space;\sigma_1&space;\sigma_2&space;\rho}&space;\end{align*}

## Futures

I covered Forward Contracts in a post a week ago, and promised to look at the related Futures Contract soon. These contracts aim to do the same thing – provide a means of ensuring an underlying asset for future delivery at a price determined today – but their dynamics are rather different from those of the Forward Contract.

One of the main criticisms of Forward Contracts is that they are directly written between two parties. This means they are subject to credit risk – if one counterparty goes bankrupt, the contract may not be guaranteed any more. By contrast, a Futures Contract is written via an exchange. This is a central organisation that matches long and short Futures Contract counterparties, and oversees the transaction (and of course, charges a fee for its services). Futures Contracts are standardised in terms of expiry dates and contract terms, which increases liquidity in the market. The counterparty who wants to receive the underlying asset at expiry is said to be the long party, while the one providing it is the short party.

As with a Forward Contract, an underlying asset is specified. Each party must have an account with the exchange in order to enter transactions. Each day, the exchange will quote the current Futures Price, which is the price at which the transaction will happen; but to enter into the contract doesn’t cost either party (beyond the small exchange fee mentioned above). Instead, the counterparties must update their accounts daily to reflect the difference between the initial Futures Price quoted and the current Futures Price quoted. If it is above the initial price, the party who is short the contract must have this difference in their account; and if it is below the initial price, the party who is long must have this difference in their account.

This is probably best demonstrated by a simple example. Let’s say that I want to enter a Futures Contract on Day 0 for delivery of a barrel of oil on Day 5 (realistically this would be a much longer period of months or years, but in order to keep the number of steps in the example down I’m keeping it short here), the current Futures Price quoted for a contract expiring in 5 days is $100. I’m the long party as I want to receive the oil, my counterparty is short as he will be delivering. Day 1: Futures Price quoted for expiry on Day 5 increases to$101.50. The short party has to deposit $1.50 in his account to cover this increase. Day 2: Futures Price increases to$103.50. The short party has to deposit another $2 in his account to cover this increase (so it’s now$3.50 total).

Day 3: Futures Price falls to $101. The short party can withdraw$2.50 from his account to leave only $1, which covers the change from day 0. Day 4: Futures Price for delivery on Day 5 (ie. tomorrow!) crashes to$96. The short party can take all of the money out of his account, and I need to deposit $4 to cover the difference between the current Futures Price and its initial value. Day 5: Futures Price recovers to$97. I can take $1 out of my account to leave$3 total. Since this is the day the contract expires, we make the transaction today at the initially agreed Futures Price of $100, and the money in my account is returned to me (if the money was in the short party’s account, it would have been returned to him in the same way). Note also that the Futures Price on day 5 for expiry on day 5 must be equal to the spot rate at that time – so this contract obliges me to trade at a price above the current spot rate, just as a Forward Contract at these prices would have done. There are several points to make. Firstly, I’ve described a Physically Settled Futures Contract here, since I actually took delivery of a barrel of oil. The standardised contracts will specify a grade of oil and a specific delivery location – many exchanges have physical warehouses for these transactions to happen. However, the large majority of Futures Contracts today are Cash Settled: in this case, instead of taking a delivery, I exchange money with the short party to achieve the same financial payoff. In the example above, since we were transacting at$100 while the market price was only $97, I would have to pay$3 to the short party (as we shall see, it’s no coincidence that this was the amount in my account!).

This contract more-or-less eliminates credit risk due to a counterparty defaulting. Consider, for example, that the oil company had defaulted on Day 2. This might seem like bad news for me – at that point, I was expecting to pay $100 for a commodity worth$103.50. The critical point is the account with the exchange. This amount is forfeited if the counterparty wants to leave the transaction or fails to keep topping it up as required, and delivered to the other party. In this case, the short party’s account of $3.50 would have been given to me, and the contract terminated – since the current price was$103.50, I could enter the same contract with a different counterparty without taking any financial loss.

On the other hand, if I’d decided on Day 4 that I no longer wanted to be part of the contract, as it looked like I’d be paying $100 while the contract is only worth$96, I might abscond. However, the exchange still has my $4 on account which it can take and give to the short counterparty, so once again they aren’t hurt by my reneging. The only risk to either party is that there is a large shift in price in a very short time, so that the counterparty is unable to credit a sufficient amount to his account before defaulting. Although it more-or-less deals with credit risk, this contract does introduce liquidity risk. Considering again Day 2, the company had had to deposit money in their account to cover the position. Even though the contract eventually ends up profitably for the short party, it is conceivable that cash-flow concerns would make them unable to honour their intermediate commitments, in which case they would have to break the contract at that stage and take a loss. For a Forward Contract, as there is no payment until the expiry date this couldn’t happen. Aside from these issues, this contract seems to be essentially the same as a Forward Contract, since it guaranteed a transaction on Day 5 at$100. However, so far we have ignored interest rates, which turn out to make a subtle difference to the final payoff (and which make the maths interesting!).

I’m going to to another post soon comparing these two contracts in the presence of interest rates, but I’ll tell you the answer right now to whet your appetite: in the presence of deterministic interest rates, the two payoffs are the same. This isn’t surprising – if we know what will happen in the future then we can hedge by replication both of the contracts (Forward Contracts in the way described in the previous post on them; Futures Contracts as I will describe next time – but have a go and try to construct a replication strategy yourself) so that their values are fixed today. However, this is actually quite unrealistic and most real models of interest rates include stochastic interest rates of some type – in this case, the Futures Contract becomes a path-dependent derivative and we can’t construct a static hedge for it at the current time, since we don’t know what the interest rate will look like at each stage! More to come soon.