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Forward Pricing 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: now T=6 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 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)(1+isT) sell forward at f(0,T) 0 -[ S(T) -f(0,T)] Total 0 f(0,T) - S(0)(1+isT) Forward price is “future value” f(0,T) = S(0)( 1 + isT) of spot price f(0,T) [ 1/(1+isT)] = S(0) S(0) = f(0,T)B(0,T) Spot price is “present value” of the forward price S(t) = f(t,T) B(t,T) 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: 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: 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) (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) 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: 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/(1+0.0475(100/365)](-4.66) = (0.9872)(-4.66) = -$4.60 Assets with known cash flows Example: Zero-coupon yield curve 12 month T-note (simple interest) par = $1000. month T is(T) B(0,T) Coupon=10% semi-annual 6 1/2 7.18% 0.9653 $50 $1050 9 9/12 7.66% 0.9456 12 1 7.90% 0.9267 0 ! 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 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 T=9 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) 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: N f (0,T)B(0,T) = S(0) - S d(ti)B(0,ti) i=1 Example: forward pricing - asset pays dividends Problem #2.7 S(0) = 63 * = $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 Assets with continuous payouts (index, currency) for N known flows: N f (0,T)B(0,T) = S(0) - S d(t )B(0,t ) i=1 i i = 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) = 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 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](1+isT) sell forward at f(0,T) 0 -[ 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.
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