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Forward contracts, delivery price, and value

What you will be able to do

Section titled “What you will be able to do”

After this lesson, you should be able to:

  • read the long and short obligations of a forward;
  • distinguish delivery price, current forward price, and contract value;
  • calculate a fair delivery price with deterministic income;
  • calculate the signed value of an existing forward.

An obligation, not an option

Section titled “An obligation, not an option”

The exchange occurs at the forward-delivery-time Tfwd\explain{forward-delivery-time}{T_{\mathrm{fwd}}}. Under the physical-settlement convention, the long pays the forward-delivery-price Kfwd\explain{forward-delivery-price}{K_{\mathrm{fwd}}} and receives one unit of the asset. If the derivative-underlying-value at delivery is STfwd\explain{derivative-underlying-value}{S}_{\explain{forward-delivery-time}{T_{\mathrm{fwd}}}}, the local long-forward-expiry-payoff per unit is the asset value minus the delivery price:

ΠTfwdlong=STfwd−Kfwd\explain{long-forward-expiry-payoff}{\Pi_{\explain{forward-delivery-time}{T}_{\mathrm{fwd}}}^{\mathrm{long}}} =\explain{derivative-underlying-value}{S}_{\explain{forward-delivery-time}{T_{\mathrm{fwd}}}}-\explain{forward-delivery-price}{K_{\mathrm{fwd}}}
(4.1.1)

The payoff of the short is the negative of the payoff of the long. Hull distinguishes this bilateral obligation from an option holder’s exercise right and gives the corresponding long and short payoff directions[1].

Fair delivery price at inception

Section titled “Fair delivery price at inception”

The forward-price F0,Tfwd\explain{forward-price}{F}_{0,\explain{forward-delivery-time}{T_{\mathrm{fwd}}}} is the delivery price that would make a new contract worth zero at valuation-time 00. Suppose the underlying is worth S0\explain{derivative-underlying-value}{S}_0 and the underlying-income-present-value paid before delivery is I0\explain{underlying-income-present-value}{I_0}. The long does not own the asset before delivery and does not receive the income. So the prepaid value of the asset delivered is S0−I0\explain{derivative-underlying-value}{S}_0-\explain{underlying-income-present-value}{I_0}.

Under the stated cash-and-carry assumptions, with the delivery-time discount-factor D(0,Tfwd)\explain{discount-factor}{D}(0,\explain{forward-delivery-time}{T_{\mathrm{fwd}}}), the law of one price gives the forward price:

F0,Tfwd=S0−I0D(0,Tfwd)\explain{forward-price}{F}_{0,\explain{forward-delivery-time}{T_{\mathrm{fwd}}}} =\frac{\explain{derivative-underlying-value}{S}_0-\explain{underlying-income-present-value}{I_0}}{\explain{discount-factor}{D}(0,\explain{forward-delivery-time}{T_{\mathrm{fwd}}})}
(4.1.2)

Hull derives the no-income and known-income forward prices by matching a spot purchase and financing strategy to the forward exchange [2].

For S0=100\explain{derivative-underlying-value}{S}_0=100, I0=4\explain{underlying-income-present-value}{I_0}=4, and D(0,Tfwd)=0.95\explain{discount-factor}{D}(0,\explain{forward-delivery-time}{T_{\mathrm{fwd}}})=0.95, the forward price is:

F0,Tfwd=100−40.95=101.0526316\explain{forward-price}{F}_{0,\explain{forward-delivery-time}{T_{\mathrm{fwd}}}}=\frac{100-4}{0.95}=101.0526316
(4.1.3)

The delivery price is not a premium paid at inception. A new contract with a fair delivery price has signed value zero, because its delivery price equals the current forward price.

An existing forward can gain or lose value

Section titled “An existing forward can gain or lose value”

After inception, Kfwd\explain{forward-delivery-price}{K_{\mathrm{fwd}}} stays fixed, but the current forward price can change. For the local forward-contract-quantity N\explain{forward-contract-quantity}{N}, the forward-contract-value to the long at time zero is the discounted difference between the current forward price and the delivery price:

V0fwd=ND(0,Tfwd)(F0,Tfwd−Kfwd)\explain{forward-contract-value}{V}_0^{\mathrm{fwd}} =\explain{forward-contract-quantity}{N} \explain{discount-factor}{D}(0,\explain{forward-delivery-time}{T_{\mathrm{fwd}}}) \left(\explain{forward-price}{F}_{0,\explain{forward-delivery-time}{T_{\mathrm{fwd}}}}-\explain{forward-delivery-price}{K_{\mathrm{fwd}}}\right)
(4.1.4)

If N=10\explain{forward-contract-quantity}{N}=10, the current forward price is 98, the contract delivery price is 102, and the discount factor is 0.94, the value to the long is:

V0fwd=10(0.94)(98−102)=−37.60\explain{forward-contract-value}{V}_0^{\mathrm{fwd}}=10(0.94)(98-102)=-37.60
(4.1.5)

The short’s value is USD +37.60 under the same inputs. Hull derives this discounted forward-price difference for an existing contract [3].

Knowledge check 4.1.1 Forward contracts and value

Link to Knowledge check 4.1.1: Forward contracts and value

Model boundary and review note

Section titled “Model boundary and review note”

This lesson uses a toy cash-and-carry model for an investment asset with known income and deterministic discounting. It does not model futures-style daily settlement, repo specialness, collateral, counterparty credit, taxes, borrowing asymmetry, delivery options, or an asset that cannot be financed or shorted on the assumed terms.

The formulas are implemented in a pure tested domain module. Sources, notation, examples, code, and answer keys remain draft pending independent human review.

References

Section titled “References”
  1. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 1 §1.3, printed pp. 5-7, forward contracts and payoffs. draft ↩
  2. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 5 §§5.3-5.5, printed pp. 103-108, cash-and-carry forward pricing and known income. draft ↩
  3. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 5 §5.7, printed pp. 109-111, valuation of existing forward contracts. draft ↩
Notation used on this page (14)
NNForward contract quantitydraft

Positive number of underlying units delivered under the forward contract. Contract value and payoff scale linearly with this quantity.

N>0\explain{forward-contract-quantity}{N}>0

Units: units of underlying

ΠTfwdlong\Pi_{T_{\mathrm{fwd}}}^{\mathrm{long}}Long forward expiry payoffdraft

Signed delivery-time value received by the long after exchanging the fixed delivery payment for the underlying. Per unit it equals underlying value at delivery minus the contractual delivery price.

Units: stated currency at forward delivery time

A(0,t)A(0,t)Accumulation factordraft

Grows one current unit over a stated future horizon under the selected rate model.

Units: dimensionless currency-units per current currency-unit

mmCompounding frequencydraft

Number of equal compounding periods per year under the stated rate convention, a positive integer fixed by the model convention.

Units: compounding periods per year

StS_tDerivative underlying valuedraft

Value at model time t of one unit of the asset or claim named as the derivative's underlying; a positive quoted value, while a position in the underlying carries its own signed quantity.

Units: stated currency per unit of underlying at model time

D(0,t)D(0,t)Discount factordraft

Converts one deterministic future unit into value at valuation time.

Units: current currency-units per future currency-unit

VtfwdV_t^{\mathrm{fwd}}Forward contract valuedraft

Signed current value of an existing forward from the named counterparty's perspective; positive to the long when the current forward price exceeds the contract's fixed delivery price under the lesson model.

Units: stated currency at model time

KfwdK_{\mathrm{fwd}}Forward delivery pricedraft

Contractual currency amount per unit of underlying paid by the long at delivery and received by the short.

Units: delivery-time currency per unit of underlying

TfwdT_{\mathrm{fwd}}Forward delivery timedraft

Future model time when the forward counterparties exchange the underlying and delivery payment; a contract date shared by the long and short, not the underlying asset's maturity.

Units: model-years from the stated valuation time

Ft,TfwdF_{t,T_{\mathrm{fwd}}}Forward pricedraft

Delivery price that would give a newly struck forward for the stated delivery time zero current value; a quoted contract rate rather than a cash amount received at quotation time.

Units: delivery-time currency per unit of underlying

j(m)j^{(m)}Nominal annual ratedraft

Annualized rate quote that must be paired with its compounding frequency.

Units: decimal rate per year

rmr_mPeriodic ratedraft

Rate applied once in each compounding period under the stated convention, derived from the stated nominal annual quote in this model.

Units: decimal per compounding period

I0I_0Underlying income present valuedraft

Valuation-time value of deterministic cash income paid by the underlying after valuation and before the contract horizon, forward delivery or the matched option expiry. It is subtracted from spot value because the forward or option position does not receive that income.

Units: stated currency at valuation time per unit of underlying

00Valuation timedraft

Common origin from which later model times and present values are measured.

Units: years from the valuation date