Forward pricing

1
Forward pricing
Assets with no cash flows Assets with known
discrete cash flows Assets with continuous cash
flows (index)
Finance 70420 Spring 2004 Associate Professor
Steven C. Mann The Neeley School of Business at
TCU
2
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 251 0.0712(1/2) 25.89 f
six month forward price of stock Consider
strategy now T6 months later borrow 25
25.00 -25.89 buy stock -
25.00 S(T) sell 6-month forward 0 -
S(T) - f total 0 f - 25.89 f
- S(0) 1 is T
Arbitrage-free forward pricing f (0,T)
S(0)(1 isT) 25.89
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Cash and Carry forward pricing
Forward contract with delivery date T spot
asset with no cash flows
Cash and carry strategy now at date
T buy asset at cost S(0) - S(0) S(T) borrow
asset cost S(0) -S(0)(1isT) sell forward at
f(0,T) 0 - S(T)
-f(0,T) Total 0 f(0,T) - S(0)(1isT)
Forward price is future value of spot price
f(0,T) S(0)( 1 isT) f(0,T) 1/(1isT)
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
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Forward valuation (post-initiation and
off-market)
Value Vforward price, time at initiation,
value is zero V f(0,T), 0 ? 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 long
f(0,T). Sell f(t,T) value at T S(T) -
f(0,T) - S(T) - f(t,T) total value at T
f(t,T) - f(0,T) value at t B(t,T) f(t,T)
- f(0,T) 2) Valuation by
algebra Vf(0,T),t PV( Vf(0,T),T) PV
S(T) - f(0,T) note S(t) PVS(T) and
S(t) f(t,T)B(t,T) (prior page) so 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)
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Example 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)(1is(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
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Example forward pricing and valuation
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 S(T)
- 50.25 short - S(T) - 45.59 net 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/(10.0475(100/365)(-4.66)
(0.9872)(-4.66) -4.60
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Assets with known cash flows
Example 12 month T-note par 1000. Coupon10
semi-annual
Zero-coupon yield curve (simple
interest) month T is(T) B(0,T)
6 1/2 7.18 0.9653 9 9/12 7.66 0.9456 12
1 7.90 0.9267
50 1050
0 .5 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 T6
months
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Forward pricing assets with known cash flows
Strategy 1 cost now t1 6 months at T9
months a) buy bond 1021.39
50.00 Bc(9,12) b) borrow PV(coupon) - 50.00
B(0,6) - 50.00 total 1021.39 - 48.27
0 Bc(9,12) S(0) - d(t1)B(0,t1)
Strategy 2 cost now t1 6 months at T9
months a) enter long forward 0
0 Bc(9,12) - f (0,9) b) lend PV( f (0,9)) f
(0,9) B(0,9) 0 f (0,9)
(buy bill ) total f (0,9) B(0,9)
0 Bc(9,12)
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(t1)B(0,t1)
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General forward pricing for assets with known
cash flows
Strategy 1 cost now at t1 at T a) buy
asset S(0) d(t1) S(T) b) borrow PV(d(t1)) -
d(t1) B(0,t1) - d(t1) total S(0) -
d(t1)B(0,t1) 0 S(T)
Strategy 2 cost now at t1 at T a)
enter long forward 0 0 S(T) - f
(0,T) b) lend PV( f (0,T)) f (0,T) B(0,T)
0 f (0,T) (buy bill
) total f (0,T) B(0,T) 0 S(T)
Each strategy has same payoff must have same
cost to avoid arbitrage
in general f (0,T)B(0,T) S(0) -
d(t1)B(0,t1) for N known flows f (0,T)B(0,T)
S(0) - S d(ti)B(0,ti)
N

i1
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Example forward pricing - asset pays dividends
S(0) 63.375 Bill prices stock pays
dividends 1.50 in 1 month 1
month 0.9967 2.00 in 7 months 7 month
0.9741 12 month 0.9512 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) S(0) - d(1)B(0,1) -
d(7)B(0,7) f (0,12) (0.9512) 63.375 -
1.50(0.9967) - 2.00(0.9741) f (0,12)
59.93/(0.9512) f (0,12) 63.01
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Assets with continuous payouts (index, currency)
for N known flows f (0,T)B(0,T) S(0) - S
d(ti)B(0,ti) S(0) - PV(cash payout to
time T) if asset pays continuous yield then
PV(dividends to time T) S(0) 1 - exp(-dyT)
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)
N
i1
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Example Index forward pricing
SP500 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) S(0)exp(-dyT) f
(0,95) 495.00 (0.993514) / (0.98426)
499.37. 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
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Commodity Forwards Storage cost
Storage Define G cost of storing asset
for (0,T) (per unit) paid time 0. Cost of
carry strategy cost now value at date
T buy asset at cost S(0), pay storage S(0)
G S(T) borrow asset cost and storage cost
-S(0) G -S(0)G(1isT) sell forward at
f(0,T) 0 - S(T)
-f(0,T) Total 0 f(0,T) - S(0)
G(1isT) 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(10.03875(180/365))
2.25 377.28 / oz.
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Commodity Forwards Convenience yield
Convenience yield Define Y(0,T) present
value (time 0) of benefits provided by holding
asset. Cost of carry strategy cost
now value at date T buy asset at cost S(0), pay
storage S(0) G S(T) borrow asset cost
and storage cost -S(0) G
-S(0)G(1isT) receive convenience yield 0
Y(0,T)(1isT) sell forward at f(0,T) 0
- S(T) -f(0,T) Total 0 f(0,T)
S(0) G -Y(0,T)(1isT) 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 exp0.01356
368(1.01365) 373.02
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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)(1isT). If iI gt is
market price gt model price. iI lt
is market price lt model price. If
iI gt is arbitrage strategy cost
now value at date T buy asset at cost S(0)
S(0) S(T) borrow asset cost -S(0) -S(0)(1isT)
sell forward at f(0,T) 0 -S(T) -f(0,T)
Total 0 f(0,T) - S(0)(1isT)
S(0)(1iIT) - S(0) )(1isT) S(0)(iI -
is)T gt 0
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is(1) is(1,2)
Forwards compared to futures
h1 h12
is(2)
h2
0 1 2
Forward price f(0,2) f(1,2)
S(2) forward cash flow 0 0 S(2) -
f(0,2) forward contract total time 2 cash flow
S(2) - f(0,2) futures price F(0,2) F(1,2)
S(2) futures cash flow 0 F(1,2) -
F(0,2) S(2) - F(1,2) futures contract
total time 2 cash flow S(2) - F(1,2)
F(1,2)-F(0,2)(1is(1,2)h12) S(2) - F(1,2)
F(1,2) - F(0,2) (F(1,2)
-F(0,2))(1is(1,2)h12) S(2) - F(0,2)
F(1,2)-F(0,2)(1is(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)(1is(1,2)h12) if f(0,2)
F(0,2) then expected margin account earnings are
zero.