5.6 Forwards and Futures
鄭凱允
5.6.1 Forward Contracts
• Let S(t), 0  t  T , be an asset price process, and let
R(t), 0  t  T , be an interest rate process. We consider
will mature or expire at or before time T of all bonds
and derivative securities. As usual, we define the
discount process D( t )  e  R( u )du . According to the riskneutral pricing formula (5.2.30), the price at time t of a
zero-coupon bond paying 1 at time T is
t
0
1
B(t,T)=
E[D(T)| F(t)],
D(t)
0t T
(5.6.1)
Definition 5.6.1
• A forward contract is an agreement to pay a specified delivery
price K at a delivery date T, where 0  T  T for the asset
whose price at time t is S(t). The T-forward price ForS (t , T ) of
this asset at time t where 0  t  T  T is the value of K that
makes the forward contract have no-arbitrage price zero at
time t.
Theorem 5.6.2.
• Assume that zero-coupon bonds of all maturities can
be traded. Then
ForS (t , T ) 
S (t )
B (t , T )
0  t  T  T (5.6.2)
Remark 5.6.3
• The proof of Theorem 5.6.2 does not use the notion of
risk-neutral pricing. It shows that the forward price
must be given by (5.6.2) in order to preclude arbitrage.
Indeed, using (5.2.30), (5.6.1), and the fact that the
discounted asset price is a martingale under P
, we
compute the price at time t of the forward contract to
be
1
E[ D(T )( S (T )  K ) | F (t )]
D(t )
1
K

E[ D(T ) S (T ) | F (t )] 
E[ D(T ) | F (t )]
D(t )
D(t )
 S (t )  KB(t , T ).
In order for this to be zero, K must be given by (5.6.2)
5.6.2 Futures Contracts
• Consider a time interval [0,T], which we divide into
subintervals using the partition points 0= t  t  ...  t  T
We shall refer to each subinterval [ tk , tk 1) as a “day.”
• suppose the interest rate is constant within each day.
Then the discount process is given by D(0)=1 and, for
k=0,1,…,n-1,
0
tk 1
K
0
j 0
1
n
D(tk 1 )  exp{ R(u)du}  exp{ R(t j )(t j 1  t j )},
which is
F (tk )-measurable.
According to the risk-neutral pricing formula
(5.6.1), the zero-coupon bond paying 1 at
maturity T has time- t k price
1
B (t k , T ) 
E[ D (T ) | F (t k )]
D (tk )
An asset whose price at time t is S(t) has
time- t k forward price
S (tk )
Fors (tk , T ) 
,
B (tk , T )
an F (t ) -measurable quantity.
k
(5.6.3)
• Suppose we take a long position in the forward
contract at time t k . The value of this position at time
t j  tk is
Vk , j
S (tk )
1

E[ D (T )( S (T ) 
) | F (t j )]
D(t j )
B (tk , T )
S (tk )
1
1

E[ D(T ) S (T ) | F (t j )] 
E[ D (T ) | F (t j )]
D(t j )
B (tk , T ) D(t j )
 S (t j )  S (tk ) 
B (t j , T )
B(tk , T )
• If t j  tk, this is zero, as it should be. However, for
it is generally different from zero. For example, if the
interest rate is a constant r so that B(t,T)  e  r (T t ) ,
Vk , j  S (t j )  e
r ( t j t k )
S (tk )
• To alleviate to problem of default risk, the forward
contract purchaser could generate the cash flow
Vk , j  S (t j )  e
r ( t j tk )
S (tk )
V0,1
B(t1 , T )
B(t1 , T )
 S (t1 )  S (t0 ) 
 S (t1 )  S (t0 ) 
,
B(t0 , T )
B(0, T )
V1,2
B(t2 , T )
 S (t2 )  S (t1 ) 
B(t1 , T )
Vn 1, n
B (tn , T )
S (tn 1 )
 S (tn )  S (tn 1 ) 
 S (T ) 
.
B (tn 1 , T )
B (tn 1 , T )
• A better idea than daily repurchase of forward
contracts is to create a futures price FutS (t, T ) .
• The sum of payments received by an agent
who purchases a futures contract at time zero
and holds it until delivery date T is
( FutS (t1 , T )  FutS (t0 , T ))  ( FutS (t2 , T )  FutS (t1, T )) 
 ( FutS (tn , T )  FutS (tn1 , T ))  ( FutS (T , T )  FutS (0, T ))
 S (T )  FutS (0, T ).
• The condition that the value at time t k of the
payment to be received at time tk 1 be zero
may be written as
D(tk 1 )

{E[( FutS (tk 1 , T ) | F (tk )]  FutS (tk , T )},
D(tk )
• Where we have used the fact that D(t ) is F (t ) measurable to take D(t ) out of the conditional
expectation. From the equation above, we see
that
k 1
k
k 1
E[( FutS (tk 1, T ) | F (tk )]  FutS (tk , T ), k  0,1, , n  1.
(5, 6, 4)
• This shows thatFutS (tk , T )must be a discretetime martingale under P . But we also require
that Fut (T , T )  S (T ),from which we conclude that
the futures prices must be given by the
formula
S
FutS (tk , T )  E[S (T ) | F (tk )],
k  0,1,
, n.
(5, 6,5)
Indeed, under the condition that Fut (T , T )  S (T ) ,
equations (5,6,4) and (5,6,5) are equivalent.
S
• We note finally that with FutS (t, T ) given by
(5.6.5), the value at time t k of the payment to
be received at time t j is zero for every
j  k  1 . Indeed, using the F (t ) j 1
measurability of D(t ) and the martingale
property for FutS (t, T ) , we have
j
1
E[ D(t j )( FutS (t j , T )  Fut S (t j 1 , T )) | F (tk )]
D(tk )

1
E[ E[ D(t j )( Fut S (t j , T )  Fut S (t j 1 , T )) | F (t j 1 )] | F (t k )]
D(tk )

1
E[ D(t j ) E[( Fut S (t j , T ) | F (t j 1 )]  D(t j ) Fut S (t j 1 , T )) | F (t k )]
D(tk )

1
E[ D(t j )( FutS (t j 1 , T )  D(t j ) Fut S (t j 1 , T )) | F (tk )]  0
D(tk )
Definition 5.6.4
The futures price of an asset whose value at
time T is S(T) is given by the formula
FutS (t , T )  E[S (T ) | F (t )], 0  t  T .
(5.6.6)
Theorem 5.6.5
the futures price is a martingale under the riskneutral measure P , it satisfiesFutS (T , T )  S (T ), and
the value of a long (or a short) futures position
to be held over an interval of time is always
zero.
• If the filtration F(t), 0  t  T , is generated by a
Brownian motion W(t), 0  t  T , then Corollary
5.3.2 of the Martingale Representation
Theorem implies that
t
FutS (t , T )  FutS (0, T )   (u)dW (u),
0t T
0
for some adapted integrand process  (i.e.,
dFutS (t , T )  (t )dW (t ) ). Let 0  t0  t1  T be given and
consider an agent who at times t between times
t 0 and t1 holds  (t ) futures contracts.
The interest rate is R(t) and the agent’s profit
X(t) from this trading satisfies
dX (t )  (t )dFutS (t , T )  R(t ) X (t )dt  (t )(t )dW (t )  R(t ) X (t )dt ,
And thus
d ( D(t ) X (t ))  D(t )(t )(t )dW (t ).
assume that at time t0 the agent’s profit is X (t0 )  0
At time t1 , the agent’s profit X (t1 ) will satisfy
t1
D(t ) X (t )   D (u ) (u )(u )dW (u ).
t0
(5.6.7)
Because Ito integrals are martingales , we have
E[ D(t1 ) X (t1 ) | F (t0 )]
t1
t0
0
0
 E[  D(u )(u )(u )dW (u )   D(u ) (u )(u )dW (u ) | F (t0 )]
t1
t0
0
0
 E[  D(u )(u )(u )dW (u ) | F (t0 )]   D(u ) (u )(u ) dW (u )
(5.6.8)
0
If the filtration F(t), 0  t  T , is not generated by a
Brownian motion, so that we cannot use Corollary
5.3.2, then we must write (5.6.7) as
t1
D(t1 ) X (t1 )   D(u ) (u )dFutS (t , T ).
(5.6.9)
t0
This integral can be defined and it will be a martingale.
We will again have
E[ D(t1 ) X (t1 ) | F (t0 )]  0
Remark 5.6.6 (Risk-neutral valuation of a cash flow).
suppose an asset generates a cash flow so that between
times 0 and u a total of C(u) is paid, where C(u) is F(u)measurable. Then a portfolio that begins with one share of
this asset at time t and holds this asset between times t and
T, investing or borrowing at the interest rate r as necessary,
satisfies
dX (u )  dC (u )  R(u ) X (u )du ,
or equivalently
d ( D(u ) X (u ))  D(u )dC (u ).
Suppose X(t)=0. Then integration shows that
T
D(T ) X (T )   D(u)dC (u).
t
the risk-neutral value at time t of X(T), which is
the risk-neutral value at time t of the cash flow
received between times t and T, is thus
T
1
1
E[ D(T ) X (T )] 
E[  D(u )dC (u ) | F (t0 )], 0  t  T .
t
D(t )
D(t )
(5.6.10)
In (5.6.10), the process C(u) can represent a
succession of lump sum payments A , A , , A
at times t  t   t , where each Ai is an F (t ) measurable random variable. The formula for
this is
1
1
2
2
n
i
n
n
C (u )   Ai I[ o ,u ] (ti ).
i 1
In this case,

T
t
n
D(u )dC (u )   D(ti ) Ai I (t ,T ] (ti ).
i 1
only payments made strictly later than time t
appear in this sum. Equation (5.6.10) says that
the value at time t of the string of payments to
be made strictly later than time t is
n
n
1
1
E[ D(ti ) Ai I (t ,T ] (ti ) | F (t )]   I (t ,T ] (ti )
E[ D(ti ) Ai | F (t )],
D(t ) i 1
D(t )
i 1
5.6.3 Forward-Futures Spread
 r (T  t )
e
If the interest rate is a constant r , then B(t,T)=
and
ForS (t , T )  e r (T t ) S (t ),
FutS (t , T )  e rT E[e  rT S (T ) | F (t )]  e rT e  rt S (t )  e r (T t ) S (t ).
we compare ForS (0, T ) and FutS (0, T ) in the case of a
random interest rate. In this case, B(0,T)= ED(T ), and
the so-called forward-futures spread is
S (0)
ForS (0, T )  FutS (0, T ) 
 ES (T )
ED(T )
1

{E[ D(T ) S (T )]  ED(T )  ES (T )}
ED(T )
1
(5.6.11)

Cov( D (T ), S (T )),
B (0, T )
If the interest rate is nonrandom, this covariance is
zero and the futures price agrees with the forward
price.