Concepts Description
Pricing and Valuation of Forward Commitments
Forward contracts Long forward position (Long) : Party to the forward contract agrees to buy the financial / physical asset
Short forward position (Short) : Party to the forward contract agrees to sell the financial / physical asset
No‐arbitrage principle Price of the forward contract : contract prce of the underlying asset of the forward contract.
No‐arbitrage pinciple : forward price that make the value of both long and short position = 0 at initiation
Assumptions:
‐ Transaction costs = 0
‐ No restriction on short sales or the use of short sales proceeds
‐ Both borrowing and lending can be done in unlimited amount at risk‐free rate of interest
Simple calculation :
1
T = forward contract term (Years)
Equity forward contracts with Stock, stock portfolio, Equity index : have expected dividend payments over the life of the contract → Must adjust the spot price for PV of those expected dividends
Discrete Dividends
1
T : 365 days basis After the initiation, value of stock may change → Long / Short posi on in a forward contract has value
To calculate the value of long position in a forward contract : make adjustment for PV of remaining expected dividends @ time t
1
Equity index forward contract with Calculate the future price of an equity index forward contract : assume dividend are paid continuously
continuous dividends
In which:
T : 365 days basis
1
1
Forward contracts on fixed income Calculation : similar with equity forward contracts with dividends
securities
1
T : 365 days basis Value of forward contract after initiation, but prior to expiration : same with equity forward contracts with dividends
1
Bond futures contracts Bond futures contracts : allow the short an option to deliver any of several bonds, which satisfy the delivery terms of the contract
At settlement, buyer pays full price
T : 365 days
Calculation of bond futures price :
1
Quoted future price = Future price ÷ Conversion factor
Conversion factor : used to adjust the long payment at delivery, so that more valuable bonds received larger payment
LIBOR‐based loans and forward Eurodollar deposit : USD deposits in banks outside of US
rate agreements Applied ending rate : LIBOR or Euribor
Forward price in an Forward Rate Agrement (FRA) : is a forward interest rate
T : 360 days basis ‐ LIBOR rates : add‐on rates and quoted on a 30/360 day basis in annual terms
‐ Long position in FRA is long the rate, and wins when the rate increases
‐ Interest on underlying loan won't be paid until the end of the loan. Payoff will only occur at the expiraation of the FRA (Payoff is the PV of interest savings on the loan)
E.g.: 1x4 FRA (90‐day loan, 30 days from now) ‐ Interest is paid at the end of the loan (day 120); payoff occurs at the expiration of the FRA (day 30)
Price of FRA : (AxB FRA)
1
Value of FRA at Maturity : 1
1
1
Value of FRA before Maturity :
Currency forward contracts Currency forward price : risk‐free rate return in home country = Bought 1 unit of foreign currency @ Spot rate + Invest @ foreign risk‐free rate + Exchange the proceeds of the
investment at maturity @ Forward price
T : 365 days basis 1
1
Forward price and Spot price are quoted as : Price currency/Base currency
Value of Currency forward contracts after initiation :
1
, Future contracts Future contracts : are marked to market by clearinghouse
Mark to market : Adjusting the margin balance in a future account each day for the change in the value of the contract from the previous trading day, based on the settlement price
Value of future contract = Current future price ‐ Previous market‐to‐market price
Interest rate swap Interest rate swap : swap fixed rate to floating rate
Initiation of the swap : choose fixed rate → PV of floa ng payments = PV of fixed‐rate payments
T : 360 days basis After initiation : floating rate changes → value of swap of 1 party is posi ve, value of swap of the other party is nega ve
Computing swap fixed rate:
‐ Step 1 : Computing discount factor (Z)
1
1
360
‐ Step 2 : Computing periodic swap fixed rate
1
∑
‐ Step 3 : Annual swap fixed rate = Periodic swap fixed rate × Number of settlement periods per year
Computing MV of interest rate swap:
360
Currency swap Currency swap : swap fixed rate loan in Currency A with fixed rate loan in Currency B
Calculating interest rate of currency swap : similar to interest rate swap
T : 360 days basis Value of currency swap after initiation = PV of CF expected to receive ‐ PV of CF obligated to pay
Equity swap Equity swap :
(1) swap fixed rate payments with return from equity index, or
T : 360 days basis (2) swap equity return with equity return
Calculating fixed interest rate of equity swap : similar to interest rate swap
Pricing and Valuation of Forward Commitments
Forward contracts Long forward position (Long) : Party to the forward contract agrees to buy the financial / physical asset
Short forward position (Short) : Party to the forward contract agrees to sell the financial / physical asset
No‐arbitrage principle Price of the forward contract : contract prce of the underlying asset of the forward contract.
No‐arbitrage pinciple : forward price that make the value of both long and short position = 0 at initiation
Assumptions:
‐ Transaction costs = 0
‐ No restriction on short sales or the use of short sales proceeds
‐ Both borrowing and lending can be done in unlimited amount at risk‐free rate of interest
Simple calculation :
1
T = forward contract term (Years)
Equity forward contracts with Stock, stock portfolio, Equity index : have expected dividend payments over the life of the contract → Must adjust the spot price for PV of those expected dividends
Discrete Dividends
1
T : 365 days basis After the initiation, value of stock may change → Long / Short posi on in a forward contract has value
To calculate the value of long position in a forward contract : make adjustment for PV of remaining expected dividends @ time t
1
Equity index forward contract with Calculate the future price of an equity index forward contract : assume dividend are paid continuously
continuous dividends
In which:
T : 365 days basis
1
1
Forward contracts on fixed income Calculation : similar with equity forward contracts with dividends
securities
1
T : 365 days basis Value of forward contract after initiation, but prior to expiration : same with equity forward contracts with dividends
1
Bond futures contracts Bond futures contracts : allow the short an option to deliver any of several bonds, which satisfy the delivery terms of the contract
At settlement, buyer pays full price
T : 365 days
Calculation of bond futures price :
1
Quoted future price = Future price ÷ Conversion factor
Conversion factor : used to adjust the long payment at delivery, so that more valuable bonds received larger payment
LIBOR‐based loans and forward Eurodollar deposit : USD deposits in banks outside of US
rate agreements Applied ending rate : LIBOR or Euribor
Forward price in an Forward Rate Agrement (FRA) : is a forward interest rate
T : 360 days basis ‐ LIBOR rates : add‐on rates and quoted on a 30/360 day basis in annual terms
‐ Long position in FRA is long the rate, and wins when the rate increases
‐ Interest on underlying loan won't be paid until the end of the loan. Payoff will only occur at the expiraation of the FRA (Payoff is the PV of interest savings on the loan)
E.g.: 1x4 FRA (90‐day loan, 30 days from now) ‐ Interest is paid at the end of the loan (day 120); payoff occurs at the expiration of the FRA (day 30)
Price of FRA : (AxB FRA)
1
Value of FRA at Maturity : 1
1
1
Value of FRA before Maturity :
Currency forward contracts Currency forward price : risk‐free rate return in home country = Bought 1 unit of foreign currency @ Spot rate + Invest @ foreign risk‐free rate + Exchange the proceeds of the
investment at maturity @ Forward price
T : 365 days basis 1
1
Forward price and Spot price are quoted as : Price currency/Base currency
Value of Currency forward contracts after initiation :
1
, Future contracts Future contracts : are marked to market by clearinghouse
Mark to market : Adjusting the margin balance in a future account each day for the change in the value of the contract from the previous trading day, based on the settlement price
Value of future contract = Current future price ‐ Previous market‐to‐market price
Interest rate swap Interest rate swap : swap fixed rate to floating rate
Initiation of the swap : choose fixed rate → PV of floa ng payments = PV of fixed‐rate payments
T : 360 days basis After initiation : floating rate changes → value of swap of 1 party is posi ve, value of swap of the other party is nega ve
Computing swap fixed rate:
‐ Step 1 : Computing discount factor (Z)
1
1
360
‐ Step 2 : Computing periodic swap fixed rate
1
∑
‐ Step 3 : Annual swap fixed rate = Periodic swap fixed rate × Number of settlement periods per year
Computing MV of interest rate swap:
360
Currency swap Currency swap : swap fixed rate loan in Currency A with fixed rate loan in Currency B
Calculating interest rate of currency swap : similar to interest rate swap
T : 360 days basis Value of currency swap after initiation = PV of CF expected to receive ‐ PV of CF obligated to pay
Equity swap Equity swap :
(1) swap fixed rate payments with return from equity index, or
T : 360 days basis (2) swap equity return with equity return
Calculating fixed interest rate of equity swap : similar to interest rate swap
