## 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*}