Forward and Futures Contract
1 Forward contract An agreement to buy or sell an asset on a fixed date in the future, called the delivery time, for a price specified in advance, called the forward price.
Short forward position The party who agrees to sell the asset is said to be taking a short forward position. The other party, who agrees to buy the asset at delivery, is said to have a long forward position.
Notations The time at which forward contract is exchanged is \(0\), delivery time \(T\), and the forward price \(F(0,T)\). The time \(t\) market price of the asset be \(S(t)\). (No payment is made by any party at time \(0\), when the forward contract is exchanged). The payoff at delivery is \(S(T)-F(t,T)\) for a long forward position and \(F(t,T)-S(T)\) for a short forward position.
Forward price
Let \(r\) denote risk free rate under continuous compounding which is assumed to be constant throughout.
Theorem For a stock paying no dividends, the forward price is
\[F(0,T) = S(0) e^{rT}\]
if the contract is initiated at a time \(t<T\), then \(F(t,T) = S(t)e^{r(T-t)}\).
The proof of the above theorem can be easily done using no-arbitrage principle.
Theorem The forward price of a stock paying exactly one dividend \(d\) at time \(t\), where \(0<t<T\), is
\[F(0.T) = [S(0) - e^{-rt}d]e^{rT}\]
The proof of the above theorem can also be done using no-arbitrage principle. The above formula can be generalized to the case when dividends are paid more than once:
\[F(0,T) = [S(0) - d_0]e^{rT},\]
where \(d_0\) is the present value of all dividends due during the lifetime of the forward contract.
Dividend yield
Dividends are often paid continuously, with an interest \(r_d>0\), the interest is called (continuous) dividend yield.
Theorem The forward price for stock paying dividends continuously at a rate \(r_d\) is
\[ F(0,T) = S(0)e^{(r-r_d)T}\]
In general, if the contract is initiated at time \(t<T\), then
\[F(t,T) = S(t)e^{(r-r_d)(T-t)}\]
Value of forward contract
The value of a forward contract, at a time \(t\), can be defined as the price one has to pay currently to initiate the forward contract, started at time \(0\), with maturity \(T\) and payoff \(F(0,T)\), at time \(t\). Every forward contract has value zero, when initiated. As the price of the commodity changes, the value of the forward contract too, changes. In particular, at delivery time, the value of a long forward contract is \(S(T) - F(0,T)\). The value, of course, can be zero, positive or negative.
Theorem For any \(t\) such that \(0 \le t \le T\) the time \(t\) value of a long forward contract with forward price \(F(0,T)\) is given by
\[V(t) = [F(t,T) - F(0,T)]e^{-r(T-t)}\]
It should be noted that the above theorem works fine even if there are dividends involved! Now, for a stock paying no dividends, we see that \(V(t) = S(t) - S(0)e^t\).2
Futures
One of the the two parties of a forward contract will be losing money (unless the payoff at \(T\) is \(0\)). Futures contracts are designed to eliminate such risk. It costs nothing to initiate a futures position.
A futures contract involves an underlying asset and a specified time of delivery, a stock with prices \(S(n)\) for \(n=0,1,\cdots\) and time \(T\). In addition to the usual stock prices, the market dictates the so called futures prices \(f(n,T)\) for each step \(n=0,1,\cdots\) such that \(n\tau \le T\). A futures contract involves a random cash flow, known as marking to market. Namely, at each time set \(n=1,2,\cdots\) such that \(n\tau \le T\), the holder of a long futures position will receive the amount
\[f(n,T) - f(n-1,T)\]
if positive, or will have to pay it if negative. The opposite payments apply for a short futures account. We impose the following two conditions:
- The futures price at delivery is \(f(T,T) = S(T)\).
- At each time step \(n=0,1,\cdots\) such that \(n\tau \le T\) the value of a futures position is zero.
Working
- Initial margin: Each investor entering into a futures contract has to pay a deposit, called the initial margin, which is kept at the clear house as a collateral. Typically, a fixed percentage of current price of the stock.
- In case of a long futures position, the amount \(f(n,T)-f(n-1,T)\) is added to the deposit if positive or subtracted if negative at each time step \(n\) (the negative amount is added or subtracted in case of a short futures account).
- Maintenance margin: If the deposit falls below a certain level, known as maintenance margin, the clearing house will issue a margin call, requesting the investor to make a payment and restore the deposit to the level of initial margin.
- The futures position can be closed at any time, in which case the deposit will be returned to the investor. In particular, if the investor fails to respond to the margin call, the clearing house will forcefully close the account.
Pricing of \(f\)
Theorem If the interest rate is constant, then \(f(0,T) = F(0,T)\).
Hedging with futures
It is possible to hedge risk associated with a stock by using the futures market, i.e., if you wishes to sell the stock at some point in the future, using a futures account, one can minimize the risk associated with this.
The difference between the spot and futures prices is called the basis:
\[ b(t,T) = S(t)- f(t,T)\]
Clearly, the Basis converges to zero as \(t\rightarrow T\). In a market with constant interest rates, it is given explicitly by
\[ b(t,T) = S(t)(1-e^{r(T-t)})\]
Similarly, if the asset pays dividends at the rate \(r_d>r\), then the basis is positive and is given by
\[ b(t,T) = S(t)(1-e^{(r-r_d)(T-t)})\]
Hedging with \(N\) futures account
We may invest in \(N\) futures account to decrease the risk. In this case, the basis is given by \(b_{N}(t,T) = S(t) - N f(t,T)\) and hence the variance is:
\[ \text{Var}(b_N(t,T)) = \sigma_s^2 + N^2\sigma_f^2 - 2 N\sigma_s \sigma_f \rho\]
and the quadratic expression has a minimum at
\[ N = \rho \frac{\sigma_s}{\sigma_f}.\]
This ratio is known as optimal hedge ratio.
- If the interest rate is constant, it can be easily shown that \(N = e^{-r(T-t)}\).
Futures on stock index
A stock exchange index is a weighted average of a selection of stock prices with weights proportional to the market capitalization of stocks. An index of this kind is approximately proportional to the market portfolio. We can treat index as a security, for the purpose of the futures market.
Beta factor
(Topic from Portfolio management)
It is important to understand how the return \(K_V\) of a given portfolio or a single security will react to trends affecting the whole market. To this end, we can plot the values of \(K_V\) for each market scenario agains those of the return \(K_M\) on the market portfolio and compute the line of best fit, also known as the regression line or the charecterestic line. The equation of the best fit will be
\[ y = \beta_V x + \alpha _ V. \]
One can show that the gradient \(\beta_V\) and the intercept \(\alpha_V\) of the line of best fit,
\[ \beta_V = \frac{\text{Cov}(K_V, K_M)}{\sigma^2_M}, \quad \alpha_V = \mu_V - \beta_V\mu_M. \]
Further, it can be shown that the expected return of a portfolio \(\mu_V\) is a linear function of the beta coefficient \(\beta_V\) of the portfolio,
\[ \mu_V = r_F + (\mu_M - r_F) \beta_V.\]
where \(r_F\) is the risk free return rate.
The expected return on a portfolio over a time step of length \(\tau\) is given by
\[ \mu_V = r_F + (\mu_M - r_F)\beta_V\]
By \(V(n)\), we shall denote the value of the portfolio at the \(n\)th time step and we assume that the index is equal to the market portfolio. Hence the futures price is given by
\[ f(n,T) = M(n)(1+r_F)^{T-n}\]
We can form a new porfolio with value \(\bar{V}(n)\) by supplementing the original portfolio with \(N\) short futures contract on the index with delivery time \(T\). The value of the new portfolio at the nth step is given by
\[ \bar{V}(n) = V(n) -n(f(n,T) - f(n-1,T)) \]
The return on the portfolio in the first step is denoted by \(K_{\bar{V}}\).
Proposition If
\[ N = (\beta_V - a)\frac{(1+r_F)V(0)}{f(0,T)} \]
Then \(\beta_{\bar{V}} = a\) for any given number \(a\).
The proof of the above proposition boils down to calculating \(\text{Cov} (K_{\bar{V}}, K_M)/\sigma_M^2\). A straightforward corollary is that when \(a=0\), \(\beta_{\bar{V}} = 0\) and hence \(\mu_{\bar{V}} = r_F\).
Notes
- In case of hedging with futures account, there can be a possibility of loss or gain because of the marking to the market payments. But this does not exist when one hedges using a forward contract and the only downside of this is the default risk.