TOC
Forward and Futures Price
Forward Price
Definition: The forward price for a contract is the delivery price that would be applicable to the contract if were negotiated today.
- It is the delivery price that would make the contract worth zero today
- It may be different for contracts of different maturities
Notation
- : the delivery price for a contract that was negotiated some time ago
- : the delivery date from now
- : the T-year risk-free rate
- : the price of the underlying asset today
- : the forward price today
Assumptions
- No transaction costs
- Same tax rate on all trading profits
- Possible to borrow rate and lend money at the same risk-free rate
- Take advantage of arbitrage opportunities as they appear
Non-dividend paying stock
Using one-price-policy (or No arbitrage)
- In the absence of arbitrage Portgolio A and Portfolio B have equal cash flows at exporation
- the euqation is valid because , thus tow portfolios have the same cash flow PV
thus,
- If this relationship does not hold then there are arbitrage opportunities in the market
- For continuously compounded interest rates:
- (risk neutral price)
- is the (continuously compounded) storage cost plus the interest costs less the income earned
- Concepts underlying model value,
- Replication and reproduction
- No arbitrage (must be risk free)
- Risk neutrality: so two portfolios should have the same future cash flows
is a synthetic forward price, assume is the market observed forward price
If , the rule of arbitrage is to buy low and sell high
Example:
When is too high
When is too low
Short Selling
- it involves selling securities (or assets in general) that you do not own
- In the case of shares, you must pay dividends and other benefits that the owner of the securities receives and is entiltled to
- Reasons why a short position in a stock is closed out
- The broker is no longer able to borrow shares from other clients
- The investor with the short position chooses to close out the position
- The investor does not maintain margins required on his/her margin account
- Not the reason: The lender of the shares issues instructions to close out the position. Because the lender of shares cannot issue such instructions
Consumption vs. investment assets
- Investment assets: assets held by significant numbers of people purely for investment purposes (like gold, silver, stocks, bonds). The benefit of holding investment asset often represented by the income or dividend yield.
- Consumption assets: assets held primarily for consumption (like copper, oil). The benefit of holding consumption asset is called convenience yield.
Other Types of Cost-of-Carry
Types of cost-of-carry:
In addition to risk free rates, there are other kinds of cost of carry.
Discrete intermediate income
where is the present value of the income
- The forward price is reduced by because the person who holds the forward contract is not entitled to dividends
- Dividend and income received while holding the asset bought on spot help to reduce the cost of carrying in the replication
Dividend yield
- Calculate the present value of the dividend income
thus,
- Use , then
If is the continuously compounded dividend yield during the life of the contract (T), then
Consumption assets & storage costs
- is the continuously compounded storage cost per unit expressed as a percentage of the asset value
- is the present value of the storage cost
- is the future vlaue of the sotrage cost
Convenience yield
For some assets (typocally consumption asset and typically not investment asset), there are other benefit for having the asset, then:
- is called the convenient yield (of owning the asset )
- is the carrying cost which includes interest rate, sotrage, wastage, etc
Currency Contracts
A foreign currency is analogous to a security providing a continuous dividend yield equal to the foreign risk-free interest rate.
- is the risk free interest rate on the local currency
- is the foreign risk-free interest rate
As an example, if the local currency is $ and the foreign currency is οΏ₯. Then is the $ interest rate, is the οΏ₯ interest rate, and are expressed as $ per οΏ₯
Forward exchange rate parity
Example
- Dollar is the local currency,
Higer interest rate leads to depreciation in forward. Take $ and οΏ₯ for example. Given and are direct quotes form the US personβs perspective:
Note:
- For currency forwards rates: some forward prices are quoted as the number of U.S. dollars per unit of the foreign currency and some are quoted the other way round.
- For currency spot rates: some are quoted as the number of U.S. dollars perunit of the foreign currency and some are quoted the other way round.
- For currency future rates: are quoted as the number of U.S. dollars per unit of the foreign currency.
Forward Price Summary
- Fair forward price:
- Investment assets:
- No income:
- discrete income:
- yield income:
- Consumption assets:
- No storage costs & no conv. yield:
- discrete storage costs:
- discrete storage costs & convenience yield:
- % storage costs:
- % storage costs & convenience yield:
Future Price of Investment & Consumption asset
Futures price of investment asset can be calculated from the spot price and other observable variables while the futures price of consumption asset can not. Take gold (investment asset) and copper (consumption asset) for example:
- Gold is an investment asset
- It means that significant numbers of investors hold gold purely for investment purposes
- If the futures price is too high, investors will find it profitable to increase their holding of gold and short futures contracts
- If the futures price is too low, investors will find it profitable to decrease their holding of gold and go long in the futures market
- Because of above two realiable behavior, gold market is βefficientβ and futures price of gold can be calculated from the spot price and other observable variables
- Copper is a consumption asset
- It means that investors do not in general hold it (because of holding cost etc.)
- If the futures price is too high, investors can buy copper (for manufactures) and short futures
- If the futures price is too low, investors cannot sell copper buy futures because there are no buyers
- Therefore, there is an upper bound of the future price (i.e. price cannot be too high) while no lower bound of the future price (because there is no mechanism to correct the price when itβs too low)
Forward Contract Valuation
The value of a forward contract at intermediate dates
Notation: Only ONE forward contract
- K: the delivery price of a forward contract that was negotiated at time 0 for delivery at T
- T: maturity of the forward contract
- : the value of contract at time 0, at time t
- : the time 0 forward price for delivery at T.
- : the time t forward price for delivery at T
Valuing a forward contract
- The value of a long T-maturity forward contract at time ,
- Buy at K, unwind at time t by selling the same asset at . Both delivery at T
- The value of a short T-maturity forward contract at time οΌ
- Sell at K, unwind at time t by buying the same asset at . Both delivery at T
Concept of a Fair Strike
A fair strike (or a fair delivery price) is the strike that is fair to both buyer and seller such that the forward contract is worth zero to both parties at the time when the contract is initiated
- A fair strike for the first forward contract initiated at time 0
- A fair strike for the second forward contract initiated at time t
- But the value of the first contract with strike price is not zero at time t except in very rare occassion that
Valuing a forward contract: Long position
- a long forward (enter at time 0) with a delivery price K: the value of this first contract is at time t
- a short forward (enter at time t) with a delivery price : the value of this second contract is zero at time t
Total cash flow at time T is (portfolioβs value at time T)
Total cash flow at time t is (portfolioβs value at time t)
Therefore:
Example
Calculation of future valuation is the same as forward
- Long position:
- Short position:
Forward Prices v.s. Future Prices
Example
Forward contract cash flows
Future contract cash flows
- The net position of a futures contract holder at time is which is exactly the same as that for the forward contract
- Assumption: the interest rate is constant
When
- long futures make immediate gain (daily settlement)
- gains tend to be invested at a higher than average rate (since )
- long futures make immediate loss (daily settlement)
- loss tends to be financed at a lower than average rate (since )
Long futures contract is more attractive than long forward contract and thus long future prices tend to be higher than long forward prices. (short futures short forward)
When
- long futures make immediate gain (daily settlement)
- gains tend to be invested at a lower than average rate (since )
- long futures makes immediate loss (daily settlement)
- loss tends to be financed at a higher than average rate (since )
Long futures contract is less attractive than long forward contract and thus long futures price tend to be lower than long forward prices. (short futures short forward)
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