Forward pricing
Assets with no cash flows
Assets with know discrete cash flows
Assets with continuous cash flows (index)
Finance 7523 Spring 1999
Assistant Professor Steven C. Mann
Neeley School , TCU
Forward price of a stock
S(0) = $25. Stock pays no dividends
is(6 month) = 7.12% ( T = 1/2 )
if you borrow $25 today you repay $25[1+ 0.0712(1/2)] = $25.89
f = six month forward price of stock
Consider strategy:
borrow $25
buy stock
sell 6-month forward
total
now
25.00
- 25.00
0
0
T=6 months later
-$25.89
S(T)
- [ S(T) - f ]
f - 25.89
= f - S(0)[ 1 + is T]
Arbitrage-free forward pricing:
f (0,T) = S(0)(1 + isT) = $25.89
Cash and Carry forward pricing
Forward contract with delivery date T; spot asset with no cash flows
“Cash and carry” strategy:
buy asset at cost S(0)
borrow asset cost
sell forward at f(0,T)
now
- S(0)
+ S(0)
0
at date T
S(T)
-S(0)(1+isT)
-[ S(T) -f(0,T)]
Total
0
f(0,T) - S(0)(1+isT)
f(0,T) = S(0)( 1 + isT)
Forward price is “future value”
of spot price
f(0,T) [ 1/(1+isT)] = S(0)
S(0) = f(0,T)B(0,T)
S(t) = f(t,T) B(t,T)
Spot price is “present value”
of the forward price
Forward valuation
(post-initiation and “off-market”)
Value = V[forward price, time]
at initiation, value is zero:
V[ f(0,T), 0 ] ð 0.
At maturity:
V[ f(0,T), T ] = S(T) - f(0,T)
at some time t (post initiation):
V[ f(0,T), t ] = ?
1) Valuation by offset:
At time t:
value at T:
total value at T:
value at t:
long f(0,T).
Sell f(t,T)
S(T) - f(0,T)
- [ S(T) - f(t,T) ]
f(t,T) - f(0,T)
B(t,T)[ f(t,T) - f(0,T) ]
2) Valuation by algebra:
V[f(0,T),t] = PV( V[f(0,T),T]) = PV[ S(T) - f(0,T)]
note S(t) = PV[S(T)]; and S(t) = f(t,T)B(t,T)
so
(prior page)
V(f(0,T),t] = f(t,T)B(t,T) - f(0,T)B(t,T)
=B(t,T) [ f(t,T) - f(0,T)] = PV(price difference)
Example - problem 2.3: forward pricing and valuation
Non-dividend paying asset; S(0) = $65
contract maturity is 90 days
simple interest rate is 4.50% ; daycount is actual/365.
a) find forward price
f(0,90/365) = S(0)(1+is(90/365)) = $65(1 + 0.045(90/365) ) = $65.72
value of contract is zero.
b) You are asked to value a 90-day forward on this asset with
delivery price = $60. This is an “off-market’ forward: value is nonzero.
Value of long forward with off-market delivery price:
value = PV( difference in forward prices)
= B(0,T)[ market forward price - contract forward price]
= B(0,T) [ 65.72 - 60.00]
= (1 + 0.045(90/365)) -1 [$ 5.72]
=(0.98903)($5.72) = $5.66
Example - problem 2.4: forward pricing and valuation
Prob 2.4:
S(0) = $45. Non-dividend paying asset
contract maturity is 100 days
simple interest rate is 4.75% ; daycount is actual/365.
a) find current forward price f(0,100/365)
f(0,100/365)
= $45.00(1+ 0.0475(100/365)) = $45.59
b) You are long 100 day forward to buy asset at $50.25.
If you sell a 100 day forward at current price, what is payoff at T?
At maturity:
long:
short
net:
S(T) - $50.25
-[ S(T) - $45.59]
$45.59 - $50.25 =
- $4.66.
c) what is the present value of your net position?
PV
= B(t,T)[ f(t,T) - f(0,T)]
= [1/(1+0.0475(100/365)](-4.66) = (0.9872)(-4.66) = -$4.60
Assets with known cash flows
Example:
12 month T-note
par = $1000.
Coupon=10% semi-annual
$50
0
!
$1050
Zero-coupon yield curve
(simple interest)
month T
is(T)
B(0,T)
6
9
12
1/2
9/12
1
7.18% 0.9653
7.66% 0.9456
7.90% 0.9267
1
Spot bond price Bc(0,12) =
=
=
=
$50.00 B(0,6) + $1050.00 B(0,12)
$50.00(.9653) + $1050.00(.9268)
$48.27
+ $ 973.12
$1021.39
Forward contract does not receive coupon at T=6 months
Forward pricing: assets with known cash flows
Strategy 1:
cost now
t1 =6 months
at T=9 months
a) buy bond
b) borrow PV(coupon)
1021.39
- 50.00 B(0,6)
+ 50.00
- 50.00
Bc(9,12)
total
1021.39 - 48.27
0
Bc(9,12)
[S(0) - d(t1)B(0,t1)]
Strategy 2:
cost now
t1= 6 months
at T=9 months
a) enter long forward
b) lend PV( f (0,9))
0
f (0,9) B(0,9)
0
0
Bc(9,12) - f (0,9)
f (0,9)
f (0,9) B(0,9)
0
Bc(9,12)
(buy bill )
total
Each strategy has same payoff: must have same cost to avoid arbitrage
f (0,9) B(0,9) = $1021.39 -28.27 = $973.12
f (0,9)
= $ 973.12/0.9457 = $1029.03
in general: f (0,T)B(0,T) = S(0) - d(t 1)B(0,t1)
General forward pricing for assets with known cash flows
Strategy 1:
cost now
at t1
at T
a) buy asset
b) borrow PV(d(t1))
S(0)
- d(t1) B(0,t1)
+ d(t1)
- d(t1)
S(T)
total
S(0) - d(t1)B(0,t1)
0
S(T)
Strategy 2:
cost now
a) enter long forward
b) lend PV( f (0,T))
0
f (0,T) B(0,T)
0
0
S(T) - f (0,T)
f (0,T)
f (0,T) B(0,T)
0
S(T)
at t1
at T
(buy bill )
total
Each strategy has same payoff: must have same cost to avoid arbitrage
in general: f (0,T)B(0,T) = S(0) - d(t 1)B(0,t1)
for N known flows:
N
f (0,T)B(0,T)
= S(0) -
Σ
i=1
d(ti)B(0,ti)
Example: forward pricing - asset pays dividends
Problem #2.7
S(0) = 63 * = $63.375
stock pays dividends:
$1.50 in 1 month
$2.00 in 7 months
Bill prices:
1 month:
7 month :
12 month:
Find price of one-year forward contract written on stock.
Use:
f (0,T) B(0,T) = S(0) - PV(dividends)
f (0,12) B(0,12)
f (0,12) (0.9512)
f (0,12)
f (0,12)
= S(0) - d(1)B(0,1) - d(7)B(0,7)
= $63.375 - $1.50(0.9967) - $2.00(0.9741)
= $59.93/(0.9512)
= $63.01
0.9967
0.9741
0.9512
Assets with continuous payouts (index, currency)
for N known flows:
f (0,T)B(0,T)
N
= S(0) - Σ
d(ti)B(0,ti)
i=1
= S(0) - PV(cash payout to time T)
if asset pays continuous yield
then PV(dividends to time T) = S(0)[ 1 - exp(-dyT)]
(J&T ch 2 appendix - note typo )
so that f (0,T)B(0,T) = S(0) - [ S(0) [ 1 - exp(-dyT)]]
f (0,T)B(0,T) = S(0) exp(-dyT)
write B(0,T) as continuous discount factor: B(0,T) = exp(-rT)
then
f (0,T) = S(0) exp ( (r-dy)T)
Example: Index forward pricing
Problem #2.9: S&P500 Index = 495.00
div yield = 2.50% continuous (365 day year)
a) Given 95-day discount rate =5.75% (360 day year), find f (0,95 days)
B(0,95) = 1 - 0.0575(95/360) = 0.984826
exp(-dyT) = exp(-0.025(95/365)) = 0.993514
f (0,95)B(0,95)
f (0,95)
= S(0)exp(-dyT)
= 495.00 (0.993514) / (0.98426)
b) One day later index is at 493. 94-day discount is 5.75%.
What is the value of the contract in part (a)?
B(0,94) = 1 - 0.0575(94/360) = 0.984986
exp(-dyT) = exp(-0.025(94/365)) = 0.993582
f (0,94) = 493.00(0.993582)/(0.984986) = 497.20
value of prior contract
= B(0,94) ( 497.20 - 499.37)
= 0.984986 (-2.17) = -2.04
= 499.37.
Commodity Forwards: Storage cost
Storage:
Define: G = cost of storing asset for (0,T) (per unit); paid time 0.
“Cost of carry” strategy:
buy asset at cost S(0), pay storage
borrow asset cost and storage cost
sell forward at f(0,T)
cost now
S(0) + G
-[S(0) + G]
0
value at date T
S(T)
-[S(0)+G](1+isT)
-[ S(T) -f(0,T)]
Total
0
f(0,T) - [S(0) +G](1+isT)
f (0,T)B(0,T) = S(0) + G
Example: 180-day Gold forward (100 troy oz.) Spot Gold S(0) = $368 / oz.
cost of storage for 180 days = 2.25 / oz., paid at time 0.
180 days simple interest rate = 3.875% annualized (actual/actual).
f (0,180)
=
(368 + 2.25)(1 + 0.03875(180/365)) = 370.25(1.01911) =
$377.33 / oz.
If storage cost G is defined to be paid at T, then f(0,T)B(0,T) = S(0) + G B(0,T)
so f (0,180) = 368(1+0.03875(180/365)) + 2.25 = $377.28 / oz.
Commodity Forwards: Convenience yield
Convenience yield:
Define: Y(0,T) = present value (time 0) of benefits provided by holding asset.
“Cost of carry” strategy:
buy asset at cost S(0), pay storage
borrow asset cost and storage cost
receive convenience yield
sell forward at f(0,T)
cost now
S(0) + G
-[S(0) + G]
0
0
value at date T
S(T)
-[S(0)+G](1+isT)
Y(0,T)(1+isT)
-[ S(T) -f(0,T)]
Total
0
f(0,T) + [S(0) + G -Y(0,T)](1+isT)
f (0,T)B(0,T) = S(0) + G - Y(0,T).
Define yn = net convenience yield = Y-G ; where Y and G are continuously compounded rates
f (0,T) = S(0) exp[(r-yn)T]
Example: 180-day Gold forward (100 troy oz.) Spot Gold S(0) = $368 / oz.
cost of storage is 0.25% as continuous annual rate.
Gold lease rate (convenience yield) is 1.50% annual continuously compounded.
Yn = 0.0150 - 0.0025 = 0.0125 ( 125 basis points)
180 day continuously compounded rate is 4.0%
f (0,180) = 368 exp [ ( 0.04 - (0.0125))(180/365)] = $368 exp[0.01356] = 368(1.01365) = $373.02
Implied Repo rates
Define: iI as implied repo rate (simple interest).
iI is defined by:
f (0,T) = S(0)(1+ iI T) ; f (0,T) and S(0) are current market prices.
Example: asset with no dividend, storage cost, or convenience yield (example: T-bill)
Given simple interest rate is, the theoretical forward price (model price) is:
f (0,T) = S(0)(1+isT).
If
iI > i s
iI < is
market price > model price.
market price < model price.
If iI > is : arbitrage strategy :
buy asset at cost S(0)
borrow asset cost
sell forward at f(0,T)
cost now value at date T
S(0)
S(T)
-S(0)
-S(0)(1+isT)
0
-[S(T) -f(0,T)] =
Total
0
f(0,T) - S(0)(1+isT)
= S(0)(1+ iIT) - S(0) )(1+isT)
= S(0)( iI - is)T > 0
is(1)
Forwards compared to futures
is(1,2)
h1
h12
is(2)
h2
0
1
2
Forward price
forward cash flow
forward contract:
f(0,2)
f(1,2)
0
0
total time 2 cash flow = S(2) - f(0,2)
S(2)
futures price
futures cash flow
F(0,2)
0
S(2)
S(2) - F(1,2)
F(1,2)
F(1,2) - F(0,2)
S(2) - f(0,2)
futures contract: total time 2 cash flow:
= S(2) - F(1,2) + [F(1,2)-F(0,2)](1+is(1,2)h12)
= S(2) - F(1,2) + F(1,2) - F(0,2) + (F(1,2) -F(0,2))(1+is(1,2)h12)
= S(2) - F(0,2)
+ [ F(1,2)-F(0,2)](1+is(1,2)h12)
Position:
buy futures, sell forward cost today = 0: total time 2 cash flow:
= f(0,2) - F(0,2) + [F(1,2) - F(0,2)](1+is(1,2)h12)
if f(0,2) = F(0,2) then expected margin account earnings are zero.