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Derivatives Pricing a Forward / Futures Contract
Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles
Derivatives 01 Introduction |2
Valuing forward contracts: Key ideas
• Two different ways to own a unit of the underlying asset at maturity: – 1.Buy spot (SPOT PRICE: S0) and borrow
=> Interest and inventory costs – 2. Buy forward (AT FORWARD PRICE F0)
• VALUATION PRINCIPLE: NO ARBITRAGE • In perfect markets, no free lunch: the 2 methods should cost the same.
You can think of a derivative as a mixture of its constituent underliers, much as a cake is a mixture of eggs, flour and milk in carefully specified proportions. The derivative’s model provide a recipe for the mixture, one whose ingredients’ quantity vary with time. Emanuel Derman, Market and models, Risk July 2001
Derivatives 01 Introduction |3
Discount factors and interest rates
• Review: Present value of Ct • PV(Ct) = Ct × Discount factor
• With annual compounding:
• Discount factor = 1 / (1+r)t
• With continuous compounding: • Discount factor = 1 / ert = e-rt
Derivatives 01 Introduction |4
Forward contract valuation : No income on underlying asset
• Example: Gold (provides no income + no storage cost) • Current spot price S0 = $1,340/oz • Interest rate (with continuous compounding) r = 3% • Time until delivery (maturity of forward contract) T = 1
• Forward price F0 ?
Strategy 2: buy spot and borrow
Buy spot -1,340 + ST
Borrow +1,340 -1,381
0 ST -1,381
Strategy 1: buy forward
0 ST –F0
t = 0 t = 1
Should be
equal
Derivatives 01 Introduction |5
Forward price and value of forward contract
• Forward price:
• Remember: the forward price is the delivery price which sets the value of a forward contract equal to zero.
• Value of forward contract with delivery price K
• You can check that f = 0 for K = S0 e r T
rTeSF 00 =
rTKeSf −−= 0
Derivatives 01 Introduction |6
Arbitrage
• If F0 ≠ S0 e rT : arbitrage opportunity
• Cash and carry arbitrage if: F0 > S0 e rT
• Borrow S0, buy spot and sell forward at forward price F0
• Reverse cash and carry arbitrage if S0 e rT > F0 • Short asset, invest and buy forward at forward price F0
Derivatives 01 Introduction |7
Arbitrage: examples
• Gold – S0 = 1,750, r = 1.14%, T = 1 S0 erT = 1,770
• If forward price =1,800 • Buy spot -1,750 +S1
• Borrow +1,750 -1,770 • Sell forward 0 +1,800 – S1
• Total 0 + 30 • If forward price = 1,700
• Sell spot +1,750 -S1
• Invest -1,750 +1,770 • Buy forward 0 S1 – 1,700 • Total 0 + 70
Repurchase agreement (repo)
• An important kind of forward contract. • Sale of a security + agreement to buy it back at a fixed price.
(=reverse cash-and-carry, sale coupled with long forward) • Underlying security held as a collateral • Most repo are overnight • Counterparty risk => haircut = Value of collateral - Loan
Schematic repo transaction
TRADER MARKET REPO DEALER
buy bond at Pt
pay Pt
deliver bond
get Pt - haircut
MARKET
sell bond at PT
get PT
TRADER REPO DEALER
get bond
Time t
Time T
Pay (Pt-Haircut)(1+RepoRate * (T-t))
The repo market
February 18, 2013 10
Institutional investors
Non financial firms
Financial institution
Deposit (overnight)
Collateral Market value
+ haircut
(No checking account insured by FDIC)
Huge repo market – exact size unknown
Spot rates and the yield curve
Derivatives 02 Pricing forwards/futures |11 18/02/13
Derivatives 02 Pricing forwards/futures |12 18/02/13
Forward on a zero-coupon : example
• Consider a: • 6- month forward contract • on a 1- year zero-coupon with face value A = 100 • Current interest rates (with continuous compounding)
– 6-month spot rate: 4.00% – 1-year spot rate: 4.30%
• Step 1: Calculate current price of the 1-year zero-coupon • I use 1-year spot rate • S0 = 100 e –(0.043)(1) = 95.79
• Step 2: Forward price = future value of current price • I use 6-month spot rate • F0 = 95.79 e(0.04)(0.50) = 97.73
Derivatives 02 Pricing forwards/futures |13 18/02/13
Forward on zero coupon
• Notations (continuous rates): A ≡ face value of ZC with maturity T* r* ≡ interest rate from 0 to T* r ≡ interest rate from 0 to T R ≡ forward rate from T to T* • Value of underlying asset: S0 = A e-r* T*
• Forward price: F0 = S0 er T = A e (rT - r* T*)
= A e-R(T*-T)
t T T*
Ft
A
r
r*
R
Derivatives 02 Pricing forwards/futures |14 18/02/13
Forward interest rate
• Rate R set at time 0 for a transaction (borrowing or lending) from T to T*
• With continuous compounding, R is the solution of: A = F0 eR(T* - T)
• The forward interest rate R is the interest rate that you earn from T to T* if you buy forward the zero-coupon with face value A for a forward price F0 to be paid at time T.
• Replacing A and F0 by their values:
TTrTTrR
−
−= *
**
In previous example: %60.450.01
50.0%41%30.4=
−
×−×=R
Derivatives 02 Pricing forwards/futures |15 18/02/13
Forward rate: Example
• Current term structure of interest rates (continuous): – 6 months: 4.00% ⇒ d = exp(-0.040 × 0.50) = 0.9802 – 12 months: 4.30% ⇒ d*
= exp(-0.043 × 1) = 0.9579 • Consider a 12-month zero coupon with A = 100 • The spot price is S0 = 100 x 0.9579 = 95.79 • The forward price for a 6-month contract would be:
– F0 = 97.79 × exp(0.04 × 0.50) = 97.73 • The continuous forward rate is the solution of: • 97.73 e0.50 R = 100
⇒ R = 4.60%
Forward rate - illustration
Derivatives 02 Pricing forwards/futures |16 18/02/13
Derivatives 02 Pricing forwards/futures |17 18/02/13
Forward borrowing
• View forward borrowing as a forward contract on a ZC You plan to borrow M for τ years from T to T*
Define: τ = T* - T The simple interest rate set today is RS
You will repay M(1+RS τ) at maturity • In fact, you sell forward a ZC
The face value is M (1+RS τ) The maturity is is T* The delivery price set today is M
• The interest will set the value of this contract to zero
Derivatives 02 Pricing forwards/futures |18 18/02/13
Forward borrowing: Gain/loss
• At time T* : • Difference between the interest paid RS and the interest on a loan made at
the spot interest rate at time T : rs M [ rs- Rs ] τ
• At time T: • ΠT = [M ( rs- Rs ) τ ] / (1+rSτ)
Derivatives 02 Pricing forwards/futures |19 18/02/13
FRA (Forward rate agreement)
• Example: 3/9 FRA • Buyer pays fixed interest rate Rfra 5% • Seller pays variable interest rate rs 6-m LIBOR • on notional amount M $ 100 m • for a given time period (contract period) τ 6 months • at a future date (settlement date or reference date) T ( in 3 months, end of
accrual period)
• Cash flow for buyer (long) at time T: • Inflow (100 x LIBOR x 6/12)/(1 + LIBOR x 6/12) • Outflow (100 x 5% x 6/12)/(1 + LIBOR x 6/12)
• Cash settlement of the difference between present values
Derivatives 02 Pricing forwards/futures |20 18/02/13
FRA: Cash flows 3/9 FRA (buyer)
• General formula:CF = M[(rS - Rfra)τ]/(1+rS τ) • Same as for forward borrowing - long FRA equivalent to cash settlement of
result on forward borrowing
(100 rS 0.50)/(1+rS 0.50)
T=0,25 T*=0,75
(100 x 5% x 0.50)/(1+rS 0.50)
Derivatives |21
Review: Long forward (futures)
0 0rTF S e=
TS
0 T
0S
0 0f =t
0 0( )rT rTt t tf S F e F F e− −= − = −
0S
Synthetic long forward: borrow S0 and buy spot
Pay forward price
Receive underlying asset
Value of forward contract
Derivatives |22
Review Long forward on zero-coupon
0 0rTF S e=
TS
0 T
0S
0 0f =t
0 0( )rT rTt t tf S F e F F e− −= − = −
* *
0r TS Ae−=
*T
A
τ
0RF e τ=
(1) Face value of zero-coupon given
(2) Calculate forward price which set the initial value of the contract equal to 0.
(3) Calculate forward interest rate
Derivatives |23
Review Forward investment=buy forward zero-coupon
0F M=
TS
0 T
0S
0 0f =
(1 ) ( )1 1
S S sT T
s s
M R M R rf S M Mr r
τ ττ τ
+ −= − = − =
+ +
* *
0r TS Ae−=
*T
(1 )SA M R τ= +
τ
0RF e τ=
(1) Forward price given
(2) Calculate face value of zero-coupon which set the value of the contract equal to 0
Derivatives |24
Review Forward borrowing=sell forward zero-coupon
0F M=
TS
0 T
0S
0 0f =
(1 ) ( )1 1
S s ST T
s s
M R M r Rf M S Mr r
τ ττ τ
+ −= − = − =
+ +
* *
0r TS Ae−=
*T
(1 )SA M R τ= +
τ
0RF e τ=
Derivatives |25
Review Long Forward Rate Agreement
* *
0r TS Ae−=
1s
s
Mrrττ+
0 T
0S
0 0f =
( )1s fra
Ts
M r Rf
rτ
τ
−=
+
*Tτ
1fra
s
MRrτ
τ+
Derivatives |26
To lock in future interest rate
Borrow short & invest long
Buy forward zero coupon
Invest forward at current forward interest rate
Sell FRA and invest at future (unknown) spot interest rate
Derivatives |27
Valuing a FRA
1fra
s
MRrτ
τ+ 1 s
Mrτ
++
1s
s
Mrrττ+
0 T
*( ) (1 ) ( )t fraf M d T t M R d T tτ= × − − + × −
*Tτ
1 s
M Mrτ
+ =+
(1 )fraM R τ⇔ +
t
Derivatives 02 Pricing forwards/futures |28 18/02/13
Basis: definition
• DEFINITION : SPOT PRICE - FUTURES PRICE
• bt = St - Ft
• Depends on: • - level of interest rate • - time to maturity (↓ as maturity ↓)
Spot price
Futures price
timeT
FT
= ST
Derivatives 02 Pricing forwards/futures |29 18/02/13
Extension : Known cash income
• Ex: forward contract to purchase a coupon-bearing bond
F0 = (S0 – I )erT where I is the present value of the income
0 0.25 T =0.50
S0 = 110.76
r = 5%
C = 6
I = 6 e –(0.05)(0.25) = 5.85
F0 = (110.76 – 5.85) e(0.05)(0.50) = 107.57
Derivatives 02 Pricing forwards/futures |30 18/02/13
Known dividend yield
• q : dividend yield p.a. paid continuously
• Examples: • Forward contract on a Stock Index
r = interest rate q = dividend yield
• Foreign exchange forward contract: r = domestic interest rate (continuously compounded) q = foreign interest rate (continuously compounded)
F0 = [ e-qT S0 ] erT = S0 e(r-q)T
Derivatives |31
0€SSpot exchange rate €/$
$1
Time 0 Time T
$$ r Te
Interest rate parity
€0€r TS e
Underlying asset: one unit of foreign currency
€0.70
r$: foreign interest rate (2%)
2% 0.5$ 1.0101e × =
r€: domestic interest rate (4%)
4% 0.50€0.70 0.7141e ×× =
$0€r TF e
F0 forward exchange rate €/$
$ €0 0r T r TF e S e=
€ $( )0 0
r r TF S e −=
Derivatives 02 Pricing forwards/futures |32 18/02/13
Commodities
• I = - PV of storage cost (negative income)
• q = - storage cost per annum as a proportion of commodity price
• The cost of carry: • Interest costs + Storage cost – income earned c=r-q • For consumption assets, short sales problematic. So:
• The convenience yield on a consumption asset y defined so that:
TureSF )(00
+≤
TyceSF )(00
−=
Derivatives 02 Pricing forwards/futures |33 18/02/13
Summary
rTSeF =
Value of forward contract Forward price
No income
Known income I =PV(Income)
F=(S – I)erT
Known yield q f =S e-qT – K e-rT F = S e(r-q)T
Commodities f=Se(u-y)T- Ke-rT F=Se(r+u-y)T
rTKeSf −−=rTKeISf −−−= )(
Derivatives 02 Pricing forwards/futures |34 18/02/13
Valuation of futures contracts
• If the interest rate is non stochastic, futures prices and forward prices are identical
• NOT INTUITIVELY OBVIOUS: – èTotal gain or loss equal for forward and futures – èbut timing is different
• Forward : at maturity • Futures : daily
Derivatives 02 Pricing forwards/futures |35 18/02/13
Forward price & expected future price
• Is F0 an unbiased estimate of E(ST) ? • F < E(ST) Normal backwardation • F > E(ST) Contango
F = E(ST) e(r-k) T
• If k = r F = E(ST) • If k > r F < E(ST) • If k < r F > E(ST)
r = risk-free rate k = expected return = risk-free + risk premium CAPM:
k = r +β(kMarket − r)
Does the forward price predict the future price?
Derivatives 03 Hedging with Futures |36 18/02/13
Consider a non dividend paying stock.
S0 =E0 (S1)1+E(R)
⇔ E0 (S1) = S0 (1+E(R))
F0 = S0 (1+ rF )Forward price:
E0 (S1)−F0 = S0 E(R)− rF[ ]
Forward price bias = Risk premium on underlying asset
This is also the expected cash flow on the derivative
Derivatives 02 Pricing forwards/futures |37 18/02/13
E(S1 −F0 ) = E(S1)−F0= S0 (1+ k)− S0 (1+ r)= S0 (k − r)
Derivatives 02 Pricing forwards/futures |38 18/02/13
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