37. Pricing and Valuation of Forward Commitments Flashcards
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.
Arbitrageur’s 2 Rules
Rule #1 Do not use your own money.
Rule #2 Do not take any price risk.
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.
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.
Value Additivity Principle
The value of a portfolio is simply the sum of the values of each instrument held in the portfolio.
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.
Convergence
The property of forward and futures contracts in which the derivative price becomes the spot price at expiration of the derivative.
The market value of a long position in a forward contract value is
VT( T) = ST – F0( T)
The market value of a short position in a forward contract value is
VT( T) = F0( T) – ST
The market value of a long position in a futures contract value before marking to market is
vt( T) = ft( T) – ft–( T)
The market value of a short position in a futures contract value before marking to market is
vt( T) = ft–( T) – ft( T)
The futures contract value after daily settlement is
vt( T) = 0
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
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
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.
Advanced Set
The reference interest rate is set at beginning of the settlement period.
Advanced Settled
An arrangement in which the settlement is made at the beginning of the settlement period.
Settled in Arrears
An arrangement in which the interest payment is made at the end of the settlement period.
Cash Flow Table for Deposit and Lending Strategy with FRA
Cash Flows for Financed Position in the Underlying Instrument Combined with a Forward Contract
Cash Flows for Financed Position in the Underlying Instrument
Cash Flows for Financed Position in the Underlying with Forward
Cash Flows for FRA Valuation
Cash Flows for the Valuation of a Long Forward Position
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 = PVCI0,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 γT = FVCI0,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
BT( T + Y) – F0( T).
or,
(ST – AIT) – F0( T)
Fixed-income forward or futures price including the conversion factor aka “adjusted price”
B0( T + Y) + AI0 – PVCI0, 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
CFi = focus is on cash flows
APi = the accrual period,
rFLT, i denotes the observed floating rate appropriate for Time i
NADi denotes the number of accrued days during the payment period,
NTDi denotes the total number of days during the year applicable to cash flow i
rFIX 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
Receive-floating, pay-fixed net cash flow
Receive-fixed, pay-floating net cash flow
Cash Flows for Receive-Fixed Swap Hedge with Bonds
Value of Swap - receive fixed, pay floating
The value of buying a fixed-rate bond and issuing a floating-rate bond.
Fixed Bond Rate eq
Swap Pricing eq
Floating Leg Cash Flow
Fixed Leg Cash Flow
Value of a fixed rate swap at some future point in Time t
The sum of the present value of the difference in fixed swap rates times the stated notional amount (denoted NA)
Value of a fixed-rate bond in Currency k
Cash Flows for Currency Swap Hedged with Bonds
Currency Swap Valuation Equation
Cash flows for the equity leg of an equity swap
Equity swap cash flows
Cash flows for the fixed interest rate leg of an equity swap
Cash Flows for Receive-Fixed Equity Swap Hedged with Equity and Bond
Value of the notional amount of equity
