Chapter 2: Forward and futures contracts Flashcards
DEF: Forward Contract
A forward contract is an agreement between two parties in which one party agrees
to sell a particular asset at an agreed FORWARD PRICE , on the MATURITY DATE (or
delivery date), and the other party agrees to buy that asset at that price and on that date.
party receiving asset has a LONG position and delivering has SHORT position
- forward contracts have no price added on - forward price depends on whether the asset provides income
DEF: SHORT SELLING
Short-selling is selling something which one does not own.
may own a positive or a negative amount of assets.
In this course we will assume that any tradable asset can be bought, sold and
short-sold.
to close short position buy asset back and return to owner eg margin account with stock broker
THM 3 current prices of assets and p1 and p2 certainty that their prices q1 and q2 at some point in the future
Let p1 and p2 be the current prices of two assets, and suppose that we know with
certainty that their prices q1 and q2 at some point in the future satisfy q1 ≤ q2. Then p1 ≤ p2.
In particular, if we know two assets have the same price at some point in the future, then they must have the same price at any prior time.
THM 4 forward contract (FC) on asset PROVIDING NO INCOME
Consider a forward contract on an asset which provides no income and whose
price at the present (its spot price) is S. The forward price is F = Se^(rT) where T is time to maturity and r = Y (T ), the T -year spot interest rate.
THM 5 forward contract (FC) on asset PROVIDING INCOME
Consider a forward contract on an asset that provides income, with maturity date
in T years. Let S be the spot price of the asset and let I be the present value of the income generated
by the asset until the maturity of the forward contract. Then F = (S − I)e^rT where r = Y (T ), the T -year spot interest rate.
PROOF
THM 5 forward contract (FC) on asset PROVIDING INCOME
PROOF Let S be the spot price of the asset and let I be the present value of the income generated
by the asset until the maturity of the forward contract. Then F = (S − I)e^rT
Proof: Suppose that the asset produces payments at times 0 less than t₁ less than … less than t_n less than T ( n payments, in between 0 and T exclusive)
whose present values are I₁,…,I_n. Let r₁ = Y(t₁),..,r_n =Y(t_n).
•If F is less than (S-I)e^(rT):
•Take the long position in the forward contract. Short sell the asset for S. Deposit (S-I) for T years at spot interest rate r.
For every 1 ≤ k ≤ n deposit I_k for t_k years at spot rate r_k.
For T years make the payments due from the asset, withdrawing corresponding deposits.
After T years have the asset delivered for F, using the balance of deposit (S-I)e^(rT)
Use the asset to close the short position.
•If F is larger than (S-I)e^(rT):
Take the short position in the forward contract. Borrow (S-I) at spot interest rate r. Then for every 1 ≤ k ≤ n BORROW I_k for t_k years at spot interest rate r_k. AFter t_k years the balance of the kth loan equals the kth payment from the asset. I_k = e^(-r_kt_k) *kth payment from asset.
Buy the asset for S ( total we have from k+1 loans).
Use payments from the asset to repay the corresponding loans.
After T years deliver the asset, collect F.
Loan is now (S-I)e^rT (we’ve paid back the loans corresponding to the k payments) , pay back and pocket F- (S-I)e^rT.
PROOF
THM 4 forward contract (FC) on asset PROVIDING NO INCOME
PROOF
Proof:
•If F is larger than Se^(rT):
•Take short position in the FC, borrow £S for T years at the spot interest rate r, BUY ASSET FOR S, wait T years and DELIVER the asset for agreed price F, repaying loan Se^(rT) to pocket F- Se^(rT) (larger than 0)
•If F is less than Se^(rT):
• Take the long position in the FC. Short sell the asset for S and deposit this at spot interest rate r.
Wait T years.
Balance of deposit is Se^rT, use this to pay forward contract F and receive asset (have it delivered) to CLOSE short position. Pocket Se^rT -F , larger than 0.
Foreign Exchange forward contracts
FOREIGN CURRENCY as an ASSET . example of a forward contract on an asset that PROVIDES INCOME ( via interest)
No arbitrage arguments
- used to prove theorems and correct prices
- example of two portfolios and theorem 3 for equality of current prices ie 2 portfolios at time in future is known to be identical so by theorem 3 deduce the spot values are equal.
THM 6 FORWARD PRICE: forward contract on one unit of FOREIGN CURRENCY whose SPOT RATE= S
maturity date = T years in future
yield curve in the foreign currency = Y_f (t).
F = forward rate, r = Y (T ) and r_f = Y_f (T ).
Consider now a forward contract on one unit of FOREIGN CURRENCY whose SPOT RATE= S (costs S units of domestic currency to purchase one unit of foreign currency) . Let the maturity date be T years in the future, and let the yield curve in the foreign currency be Y_f (t).
Let
F be the forward rate, i.e., the party with the short position in the forward contract will deliver in T years one unit of foreign currency and receive a payment of F units of domestic currency. Write
r = Y (T ) and r_f = Y_f (T ). We have F=Se^((r-r_f)T)
PROOF THM 6 Foreign currency, one unit, forward contract with forward price F
F=Se^((r-r_f)T)
CONSIDER 2 PORTFOLIOS:
•portfolio A: Long position in forward contract and Fe^-rT units of domestic currency earning interest rate r for T years
•portfolio B: e^-(r_fT) units of foreign currency earning interest rate r_f
At t=T
A generates F units of DOMESTIC and used to pay for one unit of foreign currency.
B has one unit of foreign currency
By theorem 3 both portfolios must have same value at any time t st 0 ≤ t ≤ T.
At t=0 F e^−rT = Se^−rf T
Futures contract
similar to a forward contract: it is also an agreement to deliver an asset at
an agreed price F, the futures price, and at an agreed date, the maturity date or delivery date
Again, one can have a short position (a commitment to deliver the asset) or a long position (a
commitment to buy the asset.)
Futures are settled daily- Marking to Market, invalidating the no-arbitrage arguments used to
produce forward prices as interest from cash flows stochastic
THM 7 The Futures-Forwards Equivalence Principle
(The Futures-Forwards Equivalence Principle)). If interest rates are deterministic
then forward price and futures price coincide.
Example: Taking a short position on May 2015 brent crude futures, futures price $24.10. changes to $24.20 and $24
Example: Taking a short position on May 2015 brent crude futures, futures price $24.10. Agree to produce 1,000 US barrels for $24.10. If at the end of the day futures price is $24.20 you lose 0.10 per barrel, pay broker $1000*0.1=$100 and futures price changed to $24.10. If next day drop of 0.20 then broker pays you $200 and futures price changes ro $24
