The Value of a Forward Contract

forward-price

The value of a forward contract and the forward price of an asset are something we will use again and again in various places.  Of equal importance are the techniques we can use to arrive at the equations for these.  We begin with some definitions and then I will take you though my thought process when constructing the proof/argument.

Forward Contract Definitions

It’s very easy to get mixed up with the terminology related to forwards so we start by making some clear definitions and then work our way up from there.

Definition (Forward Contract)
A long forward contract is an agreement to buy a specified amount of a specified asset at a specified price and on a specified future date.  The agreement is non-standard and over-the-counter (OTC).

At a high level a forward contract agrees a price today for future delivery of some asset.  If the price increases between now and delivery then you make a profit and if it decreases you make a loss.

Now a forward contract is very different from a futures contract.  Futures are exchange traded and standardised contracts where forwards are customised OTC contracts.  There are many other differences such as the risks involved, size and term of contracts, how easy it is to close out a position and the liquidity of contracts.  In this post we deal with forward contracts only.

We use the following notation:

  • V_t^F for the value of the forward contract at time t \leq T;
  • T for the expiry time (agreed future purchase date);
  • K for the strike price (agreed price at which we will purchase the asset);

The underlying asset (or just underlying) we purchase under the contract will vary.  For now we will consider a single stock which pays a continuously compounded dividend in stock (a scrip dividend).  Such an underlying has value S_t at time t.  Also worth mentioning here is that we assume any cash earns a continuously compounded interest rate of r and that this is constant (a short or continuously compounded rate).

From the definition of a forward contract we get the payoff at expiry as

    \begin{displaymath} f_T = S_T - K \end{displaymath}

and the payoff diagram is shown below.

forwardpayoff

From the payoff diagram we can see we make a profit if S_T > K, a loss if S_T < K and break even when S_T = K.  Our losses are capped at K.  This is an important feature of forwards which differs from options – there is no initial premium but we can make a loss on the contract and so be in a worse position than not holding the contract at all at time T.

Definition (Forward Price)
The Forward Price F_t of an asset is the value of K such that V_t^F = 0.

Our goal in the remainder of this post will be first to construct arguments to obtain the value of a forward contract on our underlying asset and then to obtain the forward price from this.

The Value of a Forward Contract

A common question/task (for example in exams or interviews) is: “derive the forward price of …. “.  Whenever I encounter such problems I always take a similar approach:

  1. I know I’m going to use a “No Arbitrage Principle” argument1We will be making the usual assumptions involved in such arguments such as ignoring transaction costs, taxes, infinite divisibility of assets, availability of a short risk free rate etc … ;
  2. Under this argument we construct two portfolios at time t such that they have the same value at T.  This value will be the payoff of the forward contract and the strike that sets this to zero the forward price;
  3. One of these portfolios A will be a long forward contract: a contract to purchase one underlying asset at time T for strike price K and with payoff f_T = S_T - K;
  4. The other portfolio B we construct from the forward contract payoff by working backwards, considering each of the components of the payoff and how they change over time;
  5. Once we have our porfolios we actually construct our no-arbitrage argument to arrive at our results;

So in short our key goal in solving the problem is to construct a portfolio B such that V_T^B = S_T - K from tradable assets.  For the stock component of this I start by asking myself: “how does the value of a portfolio which contains only the underlying asset change over time?”.  In our particular case in this post we have

    \begin{displaymath} N_t S_t \longrightarrow N_T S_T \end{displaymath}

where N_t is the number of assets held in the portfolio at time t.  It’s not usual to see the number of assets and value of an asset separated out in such arguments but I think it makes understanding my thought process easier so I have included it.  If we have N_t = 1 then N_T = \e^{q \tau} where \tau = T - t.  This is because our stock pays out a dividend continuously in stock (our scrip dividend) so the number of stock held in our portfolio increases.  The price/value of the stock may change independently of the number we hold and is represented by S_t.  From this portfolio we can write

    \begin{displaymath} S_t \e^{-q \tau} \longrightarrow S_T. \end{displaymath}

So our portfolio B will contain \e^{-q\tau} of asset S_t at time t which will have value S_t \e^{-q\tau}.  The payoff for the forward contract also contains a short position in cash of value K.  In a similar way to the above we can construct a portfolio containing only cash and ask how this changes over time.  We have

    \begin{displaymath} -K \e^{-r \tau} \longrightarrow -K. \end{displaymath}

Hence we can construct our portfolio B as

  • Long in \e^{-q\tau} of the underlying asset, which has value S_t \e^{-q\tau};
  • Short in K\e^{-r\tau} of cash;

and we can now write down the values of our two portfolio at time t as:

  • Portfolio A: V_t^A = V_t^F;
  • Portfolio B: V_t^B = S_t \e^{-q\tau} - K\e^{-r\tau};

Then by considering the values of these two portfolios at time T we have

    \begin{displaymath} V_T^A = V_T^B = S_T - K. \end{displaymath}

This is exactly what we want and is a result of how we constructed portfolio B.  So far so good! Next, by the NAP we must have V_t^A = V_t^B otherwise we could construct an arbitrage portfolio2I use my usual technique for constructing a NAP portfolio here: construct a portfolio C such that V_t^C = 0 but V_T^C > 0.  This is more of a special case of the general definition of an arbitrage portfolio that is easier to get your head around so I use it where possible!.  For example if V_t^A < V_t^B we construct a portfolio with zero value at time t that has a non-zero value at time T as

    \begin{displaymath} V_t^C = \underbrace{V_t^A - V_t^B}_{\text{assets}} + \underbrace{\varepsilon_t}_{\text{cash}} = 0 \end{displaymath}

Where \varepsilon_t > 0.  Then

    \begin{displaymath} V_T^C = \underbrace{V_T^A - V_T^B}_{=0} + \varepsilon_T \geq 0 \end{displaymath}

Where \varepsilon_T \geq 0 is the value of cash \varepsilon_t which has earned interest between t and T.   We can do something similar for V_t^A > V_t^B and so come to the conclusion that V_t^A = V_t^B and hence

    \begin{displaymath} V_t^F = \e^{-r\tau}\left\{ S_t\e^{(r-q)\tau} - K \right\} \end{displaymath}

Which is exactly what we are aiming for.

The Forward Price

Now that we have the value of the forward contract we can derive the forward price from this as the value of K such that V_t^F = 0.  By looking at the term in brackets in our forward price formula we get

    \begin{displaymath} \boxed{ F_t = S_t\e^{(r-q)\tau} } \end{displaymath}

This is exactly what we were aiming for!

Other Underlying Assets and Cost of Carry

Using similar arguments as above we can construct the forward price for various other underlying assets.  Examples include assets which have a storage cost, assets with discrete dividends, coupon paying bonds and even forward swap rates.  I won’t go into the details in this post and instead will maybe include some future posts for one or two of the most important of these other asset classes.

One case I will write down specific equations for is an asset which has a continuously compounded storage cost of u.  An example of such an asset might be gold bullion – we would pay to have it stored in a secure location and this payment may be in the form of giving up some of our gold.  A storage cost then works in the opposite direction to a dividend and the value of the asset changes over time slightly differently.  Our portfolio of a single asset will change over time as

    \begin{displaymath} S_t \e^{(u-q) \tau} \longrightarrow S_T. \end{displaymath}

Note that in the above we have left in both the dividend rate and the storage cost to give us the most general form of growth (and equations below).  An example of an asset which might have both could be shares in an investment fund where management charges are taken as a reduction in the number of shares held and dividends are the net income from the underlying assets backing the shares.

When we consider storage costs we can use the cost of carry c defined as

    \begin{displaymath} c = r + u - q \end{displaymath}

and our equations become

    \begin{displaymath} V_t^F = \e^{-r\tau}\left\{ S_t\e^{c\tau} - K \right\} \end{displaymath}

and

    \begin{displaymath} \boxed{ F_t = S_t\e^{c\tau} } \end{displaymath}

Footnotes   [ + ]

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