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The forward price (or sometimes
forward rate The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a ''forward rate''.. Forward rate calculation To extract the forward rate, we n ...
) is the agreed upon price of an
asset In financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value of ownership that c ...
in a
forward contract In finance, a forward contract or simply a forward is a non-standardized contract between two parties to buy or sell an asset at a specified future time at a price agreed on at the time of conclusion of the contract, making it a type of derivat ...
. Using the
rational pricing Rational pricing is the assumption in financial economics that asset prices - and hence asset pricing models - will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is use ...
assumption, for a forward contract on an underlying asset that is tradeable, the forward price can be expressed in terms of the
spot price In finance, a spot contract, spot transaction, or simply spot, is a contract of buying or selling a commodity, security or currency for immediate settlement (payment and delivery) on the spot date, which is normally two business days after the ...
and any dividends. For forwards on non-tradeables, pricing the forward may be a complex task.


Forward price formula

If the underlying asset is tradable and a dividend exists, the forward price is given by: : F = S_0 e^ - \sum_^N D_i e^ \, where :F is the forward price to be paid at time T :e^x is the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
(used for calculating continuous compounding interests) :r is the
risk-free interest rate The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free ra ...
:q is the
convenience yield A convenience yield is an implied return on holding inventories. It is an adjustment to the cost of carry in the non-arbitrage pricing formula for forward prices in markets with trading constraints. Let F_ be the forward price of an asset with in ...
:S_0 is the
spot price In finance, a spot contract, spot transaction, or simply spot, is a contract of buying or selling a commodity, security or currency for immediate settlement (payment and delivery) on the spot date, which is normally two business days after the ...
of the asset (i.e. what it would sell for at time 0) :D_i is a
dividend A dividend is a distribution of profits by a corporation to its shareholders. When a corporation earns a profit or surplus, it is able to pay a portion of the profit as a dividend to shareholders. Any amount not distributed is taken to be re-i ...
that is guaranteed to be paid at time t_i where 0< t_i < T.


Proof of the forward price formula

The two questions here are what price the short position (the seller of the asset) should offer to maximize his gain, and what price the long position (the buyer of the asset) should accept to maximize his gain? At the very least we know that both do not want to lose any money in the deal. The short position knows as much as the long position knows: the short/long positions are both aware of any schemes that they could partake on to gain a profit given some forward price. So of course they will have to settle on a fair price or else the transaction cannot occur. An economic articulation would be: :(fair price + future value of asset's dividends) − spot price of asset = cost of capital : forward price = spot price − cost of carry The future value of that asset's dividends (this could also be coupons from bonds, monthly rent from a house, fruit from a crop, etc.) is calculated using the risk-free force of interest. This is because we are in a risk-free situation (the whole point of the forward contract is to get rid of risk or to at least reduce it) so why would the owner of the asset take any chances? He would reinvest at the risk-free rate (i.e. U.S. T-bills which are considered risk-free). The spot price of the asset is simply the market value at the instant in time when the forward contract is entered into. So and his net gain can only come from the opportunity cost of keeping the asset for that time period (he could have sold it and invested the money at the risk-free rate). let :''K'' = fair price :''C'' = cost of capital :''S'' = spot price of asset :''F'' = future value of asset's dividend :''I'' = present value of ''F'' (discounted using ''r'' ) :''r'' = risk-free interest rate compounded continuously :''T'' = length of time from when the contract was entered into Solving for fair price and substituting mathematics we get: : K = C + S - F \, where: :C = S(e^ - 1) \, (since e^ = 1 + j \, where ''j'' is the effective rate of interest per time period of ''T'' ) : F = c_1 e^ + \cdots + c_n e^ where ''ci'' is the ''i''th dividend paid at time ''t i''. Doing some reduction we end up with: : K = (S - I)e^. \, Notice that implicit in the above derivation is the assumption that the underlying can be traded. This assumption does not hold for certain kinds of forwards.


Forward versus futures prices

There is a difference between forward and futures prices when interest rates are
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
. This difference disappears when interest rates are deterministic. In the language of
stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
, the forward price is a martingale under the
forward measure Forward is a relative direction, the opposite of backward. Forward may also refer to: People * Forward (surname) Sports * Forward (association football) * Forward (basketball), including: ** Point forward ** Power forward (basketball) ** ...
, whereas the futures price is a martingale under the
risk-neutral measure In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or '' equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price u ...
. The forward measure and the risk neutral measure are the same when interest rates are deterministic.


See also

*
Forward rate The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a ''forward rate''.. Forward rate calculation To extract the forward rate, we n ...
*
Forward measure Forward is a relative direction, the opposite of backward. Forward may also refer to: People * Forward (surname) Sports * Forward (association football) * Forward (basketball), including: ** Point forward ** Power forward (basketball) ** ...
*
Convenience yield A convenience yield is an implied return on holding inventories. It is an adjustment to the cost of carry in the non-arbitrage pricing formula for forward prices in markets with trading constraints. Let F_ be the forward price of an asset with in ...
*
Cost of carry The cost of carry or carrying charge is the cost of holding a security or a physical commodity over a period of time. The carrying charge includes insurance, storage and interest on the invested funds as well as other incidental costs. In intere ...
*
Backwardation Normal backwardation, also sometimes called backwardation, is the market condition where the price of a commodity's forward or futures contract is trading below the ''expected'' spot price at contract maturity. The resulting futures or forward ...
*
Contango Contango is a situation where the futures price (or forward price) of a commodity is higher than the ''expected'' spot price of the contract at maturity. In a contango situation, arbitrageurs or speculators are "willing to pay more owfor a co ...


References


Bibliography


''Binomial Models in Finance -'' van der Hoek & Elliott

''Martingale Methods in Financial Markets'' - Musiela & Rutkowski
{{Derivatives market Derivatives (finance) Financial economics