37. Pricing and Valuation of Forward Commitments Flashcards by

Forward Commitment

Derivative instrument in the form of a contract that provides the ability to lock in a price or rate at which one can buy or sell the underlying instrument at some future date or exchange an agreed-upon amount of money at a series of dates.

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Arbitrageur’s 2 Rules

Rule #1 Do not use your own money.

Rule #2 Do not take any price risk.

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Law of One Price

A principle that states that if two investments have the same or equivalent future cash flows regardless of what will happen in the future, then these two investments should have the same current price.

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Carry Arbitrage

A no-arbitrage approach in which the underlying instrument is either bought or sold along with an opposite position in a forward contract.

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Value Additivity Principle

The value of a portfolio is simply the sum of the values of each instrument held in the portfolio.

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At Market

When a forward contract is established, the forward price is negotiated so that the market value of the forward contract on the initiation date is zero.

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Convergence

The property of forward and futures contracts in which the derivative price becomes the spot price at expiration of the derivative.

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The market value of a long position in a forward contract value is

VT( T) = ST – F0( T)

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The market value of a short position in a forward contract value is

VT( T) = F0( T) – ST

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The market value of a long position in a futures contract value before marking to market is

vt( T) = ft( T) – ft–( T)

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The market value of a short position in a futures contract value before marking to market is

vt( T) = ft–( T) – ft( T)

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The futures contract value after daily settlement is

vt( T) = 0

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Carry Benefits

Benefits that arise from owning certain underlyings; for example, dividends, foreign interest, and bond coupon payments. Alternatively, carry benefits decrease the burden of carrying the underlying instrument through time; hence, these benefits are subtracted in the forward pricing equation. As benefits increase, price decreases. gamma

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Carry Costs

Costs that arise from owning certain underlyings. They are generally a function of the physical characteristics of the underlying asset and also the interest forgone on the funds tied up in the asset. The financing costs that come from the rate of interest and the carry costs that are common to physical assets are equivalent concepts. Carry costs, like the rate of interest, increase the burden of carrying the underlying instrument through time; hence, these costs are added in the forward pricing equation. theta

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Fwd Rate Agreement

A forward contract calling for one party to make a fixed interest payment and the other to make an interest payment at a rate to be determined at the contract expiration.

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Advanced Set

The reference interest rate is set at beginning of the settlement period.

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Advanced Settled

An arrangement in which the settlement is made at the beginning of the settlement period.

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Settled in Arrears

An arrangement in which the interest payment is made at the end of the settlement period.

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Cash Flow Table for Deposit and Lending Strategy with FRA

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Cash Flows for Financed Position in the Underlying Instrument Combined with a Forward Contract

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Cash Flows for Financed Position in the Underlying Instrument

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Cash Flows for Financed Position in the Underlying with Forward

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Cash Flows for FRA Valuation

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Cash Flows for the Valuation of a Long Forward Position

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Cash Flows Related to Carrying the Underlying through Calendar Time

Cash Flows with Forward Contract Market Price Too High Relative to Carry Arbitrage Model

Forward Price

Forward Value

FRA Fixed Rate

FRA Notation

Future value of underlying

FV of underlying adj for carry cash flows

FV of underlying adjusted for carry

PV of difference in forward prices

Settlement amount at h for receive-fixed

FRA Timeline

Settlement amount at h for receive-floating

Value of a Forward Contract at Initiation and Expiration

Equation for Vg( 0, h, m), the value of the FRA at Time g that was initiated at Time 0, expires at Time h, and is based on m-day Libor.

Accrued Interest

Conversion Factor - Fixed Income Futures

Cheapest to Deliver - Fixed Income Futures

Futures or Forward Price

The futures or forward price is simply the future value of the underlying in which finance costs, carry costs, and carry benefits are all incorporated, or:

Fixed Income Bond Notation

Time 0 is the forward contract trade initiation date Time T is the contract expiration date T + Y = the underlying instrument’s current time to maturity Y is the time to maturity of the underlying bond at Time T, when the contract expires. B₀( T + Y) = the quoted price observed at Time 0 of a fixed-rate bond that matures at Time T + Y and pays a fixed coupon rate For bonds quoted without accrued interest, let AI₀ denote the accrued interest at Time 0. The carry benefits are the bond’s fixed coupon payments, γ0 = PVCI_0,T, meaning the present value of all coupon interest paid over the forward contract horizon from Time 0 to Time T The corresponding future value of these coupons is γ_T = FVCI_0,T Finally, there are no carry costs, and thus θ₀ = 0

Fixed Income Bond Price

S₀ = Quoted bond price + Accrued interest = B₀( T + Y) + AI₀

Total Profit or Loss on a Long Futures Position

B_T( T + Y) – F₀( T). or, (S_T – AI_T) – F₀( T)

Fixed-income forward or futures price including the conversion factor aka “adjusted price”

B₀( T + Y) + AI₀ – PVCI_{0, T}

Full spot price minus the present value of the coupons over the life of the forward or futures contract.

In equilibrium, to eliminate an arbitrage opportunity

Conversion Factor Adjusted FV of Underlying Adj for Carry

Covered Interest Rate Parity or Interest Rate Parity

The relationship among the spot exchange rate, the forward exchange rate, and the interest rates in two currencies that ensures that the return on a hedged (i.e., covered) foreign risk-free investment is the same as the return on a domestic risk-free investment. Also called interest rate parity.

Interest Rate Swap Notation

FLT - floating leg FIX - fixed leg CF_i = focus is on cash flows AP_i = the accrual period, r_{FLT, i} denotes the observed floating rate appropriate for Time i NAD_i denotes the number of accrued days during the payment period, NTD_i denotes the total number of days during the year applicable to cash flow i r_FIX denotes the fixed swap rate.

Interest Rate Parity (Annual)

Interest Rate Parity (Continuous)

Forward Pricing and Valuation Expressions - Generic

Receive Floating, Pay Fixed Swap

Equivalent to being long a floating-rate bond and short a fixed-rate bond. Assuming both bonds were purchased at par, the initial cash flows are zero and the par payments at the end offset each other.

Generic Swap Cash Flows: Receive-Floating, Pay-Fixed