Options et Marchés Spéculatifs
Class note 2- 1
OPTIONS ET MARCHES SPECULATIFS
Professor André Farber
Class note 2
Pricing Forward and Futures
INTRODUCTION
Notations
Key idea today
Discount factors and interest rates
2
2
2
3
VALUING A FORWARD CONTRACT
5
Case 1: no income on underlying asset
5
Valuation
Forward price :
Arbitrage 1: Cash and carry
Arbitrage 2: Reverse cash and carry
Basis
Example: forward on zero-coupon (= term deposit)
Forward rate
Case 2: known cash income
Case 3: known dividend yield
Case 4: Consumption assets
VALUING A FUTURES CONTRACT
Forward price & expected future price
5
5
6
6
9
10
11
12
13
14
15
19
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 2
INTRODUCTION
Notations
F0 :
Forward price set at time 0
f :
Value of forward contract at time 0
K :
Delivery price
T :
Maturiy
(Reminder: When contract initiated : K = F0  f = 0)
Key idea today
1. DECOMPOSITION OF A FORWARD CONTRACT:
Two different ways to own a unit of the underlying asset at
maturity:
1.Buy spot (spot price: S0) and borrow
2. Buy forward (AT FORWARD PRICE Ft)
2. VALUATION PRINCIPLE:
NO ARBITRAGE : in perfect markets, no free lunch.
The 2 methods should cost the same.
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Options et Marchés Spéculatifs
Class note 2- 3
Discount factors and interest rates
d(T) discount factor = present value at time 0 of 1 unit of currency
at time T
Note: 1/d(T) is the future value at time T of 1 unit of currency invested at time t
rs(T) simple interest rate over period 0,T
d (T ) 
1
1  rs (T )  T
ra(T) annually compounded interest rate over period 0,T
d (T ) 
1
(1  ra (T ))T
rn(T) interest rate with compounding n times per annum
d (T ) 
1
r (T ) nT
(1  n
)
n
r(T) continuously compounded interest rate over period 0,T
d (t , T ) 
1
 e  r (T ) T
e r (T )T
 exp  r (T )T 
To shift from continuous compounding (rate r) to compounding n times per annum
(rate rn) , use the following formulas
er  (1 
rn n
)
n
Hence:
rn n
r
)  n  ln( 1  n )
n
n
rn  n(e r / n  1)
r  ln( 1 
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Options et Marchés Spéculatifs
Class note 2- 4
Numerical illustrations:
Present and future value calculations
One discount factor, several underlying interest rate...
Maturity
Discount factor
Simple (6m)
Annual
Continuous
6 months (0.5 year)
0.9804
4%
4.04%
3.96%
From Simple
To
Simple
0.9804 = 1/[1+(4%)(0.5)]
0.9804 = 1/[1+(4.05%)0.5]
0.9804 = 1/[1+(4%)(0.5)]
Annual
4%
Continuous
4%=
.5
Annual
4.04% =
[(1.0404) -1]
{exp[(3.96%)(.5)]-1}-1
4.04%
4.04% =
[1+(.04)(.5)]²-1
Continuous
4%=
exp(3.96%)-1
3.96%=
3.96%=
ln[1+(.04)(.5)]/(0.5)
ln(1.0404)
3.96%
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 5
VALUING A FORWARD CONTRACT
Case 1: no income on underlying asset
Valuation
No arbitrage opportunity
Consequence : in a perfect capital market, V
value of forward contract = value of synthetic forward contract
f =S0 - PV(K) = S0 – K d(T)
With continuously compounded interest rate:
f = S0 - K e-rT
Forward price :
Delivery price such that f = 0
F0 = S0/d(T) = FV(S0)
With continuously compounded interest rate:
F0 = S0 erT
NOTE :
f  ( F0  K )e  rT
f>0

F0>K
f=0

F0=K
f<0

F0<K
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 6
Arbitrage 1: Cash and carry
If forward price quoted on the market (K) is greater than its
theoretical value (F0), the “true value” of the contract is negative.
(As: f = (F0-K)  d(T), f < 0)
But the market price for the contract is 0.
Hence, the contract is overvalued by the market.
 cash-and-carry arbitrage :
Sell overvalued forward:
sell forward
Buy synthetic forward:
buy spot and borrow
Arbitrage 2: Reverse cash and carry
If forward price quoted on the market (K) is less than its
theoretical value (F0), the “true value” of the contract is positive.
(As: f = (F0-K)  d(T), f > 0)
But the market price for the contract is 0.
Hence, the contract is undervalued by the market.
 reverse cash-and-carry arbitrage :
Buy undervalued forward (futures):
Buy forward
Sell synthetic forward (futures):
Short asset and borrow
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 7
Numerical illustration
Cash and carry arbitrage
Underlying asset : Gold
No income
Spot price : 250 $/oz
Maturity: 6 months
Interest rate (simple) : 4% (or 3.96% with cont.comp)
Equilibrium forward price : 250 [1+(.04)(.5)] = 255
Quoted forward (futures) price : 260
Arbitrage table
Buy spot
Borrow
Sell forward
@ 260
Total
Current date
- 250
+ 250
0
0
Delivery date
+ST
- 255 =
250[1+(.04)(.5)]
260 – ST
+5
(=260-255)
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 8
Numerical illustration
Reverse cash and carry arbitrage
Underlying asset : Gold
No income
Spot price : 250 $/oz
Maturity: 6 months
Interest rate (simple) : 4%
Equilibrium forward price : 250 [1+(.04)(.5)] = 255
Quoted forward (futures) price : 250
Arbitrage table
Short
Current date
+ 250
Delivery date
-ST
- 250
+ 255 =
250[1+(.04)(.5)]
ST –250
(borrow + sell spot)
Invest
Buy forward
@ 250
Total
0
0
+5
=255 – 250
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 9
Basis
DEFINITION : SPOT PRICE - FUTURES PRICE
bt = St - Ft
Futures price
Spot price
F =S
T
T
T
time
Depends on:
- level of interest rate
- time to maturity ( as maturity )
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Options et Marchés Spéculatifs
Class note 2- 10
Example: forward on zero-coupon (= term deposit)
A
0
T
T*
F0
Spot interest rates - notations:
r(T) = r
r(T*) = r*
Value of underlying asset :
S0  A  e
 r *T *
Forward price
F0  S0  e rT
 A  e rT r T
* *
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 11
Forward rate
Rate R set at time 0 for a transaction from T to T*
r*
0
T*
T
r
R
e
r *T *
 e e
rT
R (T * T )
=> Continuously compounded forward rate
r *T *  rT
R
T* T
Forward price of zero coupon:
F0  A  e
 R (T *  T )
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Options et Marchés Spéculatifs
Class note 2- 12
Case 2: known cash income
Ex: forward contract to purchase a coupon-bearing bond
C
0
t
T
Let I = Present value of C = PV(C)
Valuation:
f = (S0-I) - PV(K)
f  S 0  I  Ke T
Forward price : f = 0
F0  ( S 0  I )e rT
Note : as before
f  ( F0  K )e  rT
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Options et Marchés Spéculatifs
Class note 2- 13
Case 3: known dividend yield
q : dividend yield p.a. paid continuously
Examples:
Forward contract on a Stock Index
q = dividend yield
Foreign exchange forward contract:
q = foreign interest rate (continuously compounded)
Valuation:
f  S 0 e  qT  Ke  rT
Forward price:
F0  S0e
( r  q )T
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 14
Case 4: Consumption assets
1. No income
Take cost of storage into account
I = - PV of storage cost (negative income)
q = - storage cost u 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:
F0  S0e( r  u )T
The convenience yield on a consumption asset y defined so that:
F0  S 0e (c  y )T
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 15
VALUING A FUTURES CONTRACT
If the interest rate is non stochastic, futures prices and forward
prices are identical
NOT INTUITIVELY OBVIOUS:
Total gain or loss equal for forward and futures
but timing is different
Forward : at maturity
Futures : daily
PROOF:
S
F
G
r
spot price
futures price
forward price
daily interest rate
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
At time T:
Class note 2- 16
ST = GT = FT
AT T-1:
LONG 1 FWD
SHORT 1 FUTURE
TOTAL
T-1
0
0
T
ST - GT-1
-(FT - FT-1)
FT-1 - GT-1
 FT-1 = GT-1
AT T-2:
LONG (1+r) FWD
SHORT 1 FUTURE
TOTAL
T-2
0
0
T-1
(1+r)[GT-1 -GT-2]/(1+r)
-(FT-1 - FT-2)
FT-2 - GT-2
 FT-2 = GT-2
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Options et Marchés Spéculatifs
Class note 2- 17
NUMERICAL EXAMPLE
Initial spot price : 100.00
Interest rate (simple compounding) : 10%
Number of days per year : 360
FORWARD CONTRACT
cf 2 = new forward contract each month
days
year
180
150
120
90
60
30
0
0.50
0.42
0.33
0.25
0.17
0.08
0
spot
price
100.00
106.88
109.22
108.88
111.31
105.45
114.76
forward
price
105.00 (1)
111.33
112.86
111.60
113.17
106.33
114.76
cash
flow
0
0
0
0
0
0
9.76
cf 2
6.08 (2)
1.48
-1.23
1.54
-6.78
8.43
(1) Ft = St (1+r )
105.00 = 100.00 * (1+0.10* 0.50)
(2) Sell forward at 111.33
Profit in 0.42 years : 111.33 - 105 = 6.33
Present value
6.33 / (1+0.10*0.42) = 6.08
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 18
FUTURES CONTRACTS
Let us reproduce the roll over strategy with futures :
NOTE : FOR SIMPLICITY, MONTHLY MARKING TO
MARKET
days year
180
150
120
90
60
30
0
0.50
0.42
0.33
0.25
0.17
0.08
0
spot
price
100.00
106.88
109.22
108.88
111.31
105.45
114.76
futures
price
105.00
111.33
112.86
111.60
113.17
106.33
114.76
cash
flow
0
6.33
1.53
-1.23
1.54
-6.78
8.43
nb
0.960
0.968
0.976
0.984
0.992
1.000
net
c.flow
6.08
1.48
-1.23
1.54
-6.78
8.43
OMS 2000-2001 9 March, 2016
Options et Marchés Spéculatifs
Class note 2- 19
Forward price & expected future price
Is F an unbiased estimate of E(ST) ?
F < E(ST)
Normal backwardation
F > E(ST)
Contango
To understant the relation between F and E(ST), consider the
following strategy :
t
- F e-r(T-t)
0
- F e-r(T-t)
Invest
Long forward
Total
T
+F
ST - F
ST
PV = - F e-r(T-t) + E(ST) e-k(T-t) = 0
F = E(ST) e(r-k) (T-t)
If k = r
F = E(ST)
If k > r
F < E(ST)
If k < r
F > E(ST)
OMS 2000-2001 9 March, 2016