Pricing future and forward contracts

• Mechanism of margin accounts for future contracts;
• Martingale approach and risk-neutral measure;
• Pricing models using generalized risk-free interest rate.

Hey all!

Derivative contracts can be used for hedging a determined position as well as to speculate the future movement of an underlying index or asset. One can use, for instance, the several types of options in order to bet on a direction of the prices or index. An option will give the right, but not the obligation, for the holder to buy or sell an underlying for a predetermined delivery price. In this context, an investor holding a long position in an option has a loss limited to the acquisition price of the derivative. The profit is unlimited. Conversely, for an investor holding a short position, the situation inverts, that is, the profit is limited whereas the losses are unlimited. 

Unlike options, both parties holding futures or forward contracts assume an obligation regarding the delivery price of the contract. Therefore, both long and short positions on futures or forward contracts might present unlimited losses and profits. The basic mechanism of a future and a forward contract is similar. In principle, two participants agree to trade a determined asset or index for a determined value in the end of a period in the future. When the contract is on interest rates, a common practice of the market involves two important dates, the setting and the payment date. In the setting date, the interest rate is settled while the payment is done after the tenor period (related to interest rate of the contract). 

Margin account

As mentioned above forward and a future contracts present a similar concept regarding the expectation and the obligations assumed. Nevertheless, there are several differences between these contracts, such as the fact that future contracts are regulated whereas forward contracts are traded in the OTC market. Other differences reflect significantly in the pricing procedure and will be discussed in the following session.

Managing default risk: margin account

Since both contracts (future and forward) involve an obligation with unlimited loss, there is a high risk of default of the counterpart. Being a regulated contract, the participants of a future contract are required to maintain funds in a margin account with the corresponding broker. The broker is also required to maintain an account in some clearinghouse that diary manages the loss and profits among the different contracts held by the brokers. So, what is purpose of the margin account? In the end of each business day, the value of the future contract is updated according to the movements of the future index (or asset) in that corresponding day. The differences in the value of the future contract set the profits and losses and then are transferred from a margin account to another. When the margin account is below a certain level, the investor is required to provide additional funds, if not, the position of the corresponding investor is automatically closed (with some additional cost for the investor).
In general, this marked-to-market system of the future contracts is not applied in forward contracts, which ends to bring difference in the pricing process as will be demonstrated in the next sections. Notice however, that this is not a restrict rule and some forward contracts may be subject to a system similar to margin accounts.

Risk-neutral measure and Martingale processes

Before going further in the pricing models, I decided to make some brief comments on the pricing method that I will use for both contracts.
Let´s start with the concept of risk-neutral measure. "Measure" hereafter will denote the distribution of probabilities which follows determined assumptions. In the risk-neutral measure, it is assumed that the average return of any portfolio equals the one from the money market account. Put in other words, no matter the risk of the investment, the investors "accept" a return equal to the one obtained with a risk-free interest rate (which is considerably lower than the real return of the asset). The return of the money market account is given by:

$d \beta = r \beta dt$

resulting in:

$\beta (t)= exp \{\int_{0}^{t} r(u) du \}$

The value $\beta (t)$ denotes the value of $ \$1$ deposited in the money market account after a period $t$ subject to the time-dependent risk-free interest rate $r(t)$.
In the risk-neutral measure, the probability distribution follows the assumptions mentioned above. At a first glimpse, the assumptions of the risk-neutral measure seems very unrealistic and in fact they are. The question here is why does one can price correctly a derivative using risk-neutral measure? A technical answer for this question relies on a connection between the real measure and the risk-neutral measure using the Girsanov´s Theorem and the concept of measure equivalence (Shreve´s book volume II brings an interesting and quite technical discussion on this topic). Informally, let´s say that both real and risk-neutral measures agree about what is possible or not in terms of the movements of the market. One can think that once the probabilities between the measures agree one does not need to think in terms of the real return of the assets, and can use the risk-free return instead. The main advantage of using risk-neutral measure is the guarantee of arbitrage-free models. More on this topic will be given in a future post.

The parameter $\beta$ defined above will be used as a numeraire in the description of a Martingale process involving the value $V(.)$ of the derivatives:

$\frac{V(S(t))}{\beta (t)}=E[ \frac{V(S(T))}{\beta (T)} |F(t)]$

where the expectation value operator $E[.]$ is defined within the risk-neutral measure and $F(t)$ denotes the filtration at time $t$.

Pricing forward contracts

The payoff of this contract at maturity can be given by:

$f_w(T)=S(T)-K$

where $S(T)$ is the value of the asset at maturity $T$ and $K$ is the delivery price. In order to price the contract at the time $t$, I will use the Martingale approach using the expected value under risk-neutral measure:

$f_w(t)=\beta (t) E[ \frac{(S(T) - K)}{\beta (T)} |F(t)]$

In general, a forward contract has no initial cost, then $F_w(t_0=0) =0$. Using this, we can determine the delivery price:

$E[ \frac{(S(T)-K)}{\beta(T)} |F(t_0)]=0$

$E[ \frac{S(T)}{\beta(T)} |F(t_0)]=E[ \frac{K}{\beta(T)} |F(t_0)]$

According to the Martingale approach, the ratio $\frac {S(T)}{\beta (T)}$ is a Martingale under the risk-neutral measure. Therefore:

$\frac{S(t_0)}{\beta (t_0)} = K*E[  \frac{1}{\beta(T)} |F(t_0)]$

One can notice that the expectation value that remains in the expression above is simply the price at $t_0=0$ of a zero-coupon bond with maturity at $T$:

$P(0,T) = E[\frac{1}{\beta (T)}|F(t_0)]$

Zero-coupon bonds are bonds that do not pay intermediate coupons during the life time of the bond. Notice that this is the simplest expression for a zero-coupon bond, not including default risk of the counterpart, for instance. 

Hence:

$K=\frac{S(t_0)}{P(0,T)}$ 

Returning to the expression of $f_w(t)$:

$f_w(t)=\beta(t) \{\frac{S(T)}{\beta(T)} - K E[ \frac{1}{\beta(T)} |F(t)] \}$

$f_w(t)=S(t) - K E[ \frac{\beta(t)}{\beta(T)} |F(t)]$

$F_w(t) = S(t) - K P(t,T)$

Replacing the $K$ in the expression for the forward contract:

$f_w(t) = S(t) - S_0 \frac{P(t,T)}{P(0,T)}$

The expression above allows one to calculate the value of the forward contract from the spot and the initial price of the asset. Additionally, the value $P(t,T)$ requires to be calculated. In principle, this calculation is not trivial, requiring the solution of stochastic differential models, such as  the Hull-White model. This calculation will be handled in a another post. For sake of simplicity, one can assume a constant risk-free interest rate $r$ resulting in:

$f_w(t)=S(t) - S_0 e^{r(T-t)}$

Pricing future contracts

The profits or losses of a future contract are settled each business day based on the current expectation of the market about the value of an underlying at maturity. Therefore, a non-arbitrage situation implies that no matter at which point (in time) an investor assumes a position in a future contract the expected profit (or loss) should be zero. As any other model, the one I will describe here must guarantee an arbitrage-free price. 

The expectation of the daily settlement within the risk neutral measure is then:

$\beta (t) E[ \sum_k \frac{ f_c(t_{k+1}, T) - f_c(t_k, T)}{\beta (t_{k+1})} | F(t)]=0$

where $T=t_N$ is the maturity. By the properties of the expectation value operator we can rewrite the expression above as:

$ E[ \beta (t_k) \frac{f_c(t_{k+1}, T) - f_c(t_k, T)}{\beta (t_{k+1})} | F(t_k)] =0$

A useful condition for the future contract is that at maturity the future value $f_c(T,T)=S(T)$ equals the spot price of the underlying. I will insert this condition in the expression above, considering to obtain the price one day before maturity:

$ E[ \beta (t_{N-1}) \frac{f_c(T, T) - f_c(t_{N-1}, T)}{\beta (T)} | F(t_{N-1})] =0$

$ E[ \beta (t_{N-1}) \frac{S(T)}{\beta (T)} | F(t_{N-1})] =E[ \beta (t_{N-1}) \frac{f_c(t_{N-1}, T)}{\beta (T)} | F(t_{N-1})]$

$E[ \beta (t_{N-1}) \frac{S(T)}{\beta (T)} | F(t_{N-1})] =f_c(t_{N-1}, T) P(t_{N-1}, T)$
Using the same process above we can calculate the price two days before maturity, obtaining:
$E[ \beta (t_{N-2}) \frac{S(T)}{\beta (t_{N-1})} | F(t_{N-2})] =f_c(t_{N-2}, T) P(t_{N-2}, t_{N-1})$
and the general case will be:
$E[ \beta (t_{k}) \frac{S(T)}{\beta (t_{k+1})} | F(t_{k})] =f_c(t_{k}, T) P(t_{k}, t_{k+1})$

$f_c(t_{k}, T)  = \frac{E[ S(T) \frac{\beta (t_{k})}{\beta (t_{k+1})} | F(t_{k})]}{E[\frac{\beta (t_{k})}{\beta (t_{k+1})} | F(t_{k})]}$

An approximation usually done in the expression above is to consider $\beta (t_{k+1})$ as $F(t_k)-$measurable, as done in Shreve´s book volume II, resulting in:

$f_c(t,T)=E[S(T) | F(t)]$

where the expectation value is taken under the risk-neutral measure.
It is worth mentioning here that assuming a constant risk-free interest rate in addition to a geometric Brownian motion model for the underlying asset, one obtains:
$f_c(t,T)=S_0 e^{r(T-t)}$
which is the same obtained for the delivery price in the forward contract.

Final remarks

Both future and forward contracts are based on an statement (or a bet) regarding the expectation of the price (or index) at a maturity date. Differences in the procedure of calculation of the profits and losses result in distinct pricing models. The models described here consider time-dependent risk-free interest rate, which brings considerable complexity for one to numerically calculate the prices. In fact, there is a whole area of research focused in interest rate models applied to this kind of situation. I will provide some briefing on this topic in a future post. At last, it is worth noticing that counterpart risk is also not included, mainly in the forward contract which is traded in the OTC market. Also this topic is quite complex and additional comments will be present in a future post.

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Live long and prosper!

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Diogo de Moura Pedroso

LinkedIn: www.linkedin.com/in/diogomourapedroso

E-mail: diogo.mourapedroso@gmail.com