Lec 5A APT for Forward and Futures Contracts Lec 5A: Arbitrage Pricing of Forwards and Futures (Hull, Ch. 5.3 and 5.4) Suppose the spot price of an asset (e.g., stock) evolves as shown below. The risk-free rate r = 4.88%/yr(c.c.). Consider a Forward contract on 1 share of stock with 1-yr to “expiration”: Spot Prices Forward Prices $140 B ? B How to determine $120 A F = ? forward prices? $100 C ? C (A: by trial and error) Suppose at B forward price = $120. Is this okay? Arbitrage Strategy: ▸ Buy forward contract, “exercise” immediately, Pay $120 ▸ Receive the stock, sell it in the cash market: CFB = -0 - 120 + 140 = $20 ➟ To preclude arbitrage the forward price must = $140 Using the same logic, the forward price at C must = ? . Lec 5A APT for Forward and Futures Contracts dfdf

Lec 5A APT for Forward and Futures Contracts Forward Price at A: Spot Price Forward Prices $140 B $140 B $120 A F = ? A $100 C $100 C Suppose F = $100, Spot Price = $120 ➟ Forward price seems low, spot high. ▸ Strategy: {Buy forward contract, buy a Bond} and {Short the Stock}. ▸ Arbitrage Portfolio: {-S, +F, +B(PV = $120)} ➟ CFA = +120-0-120 = 0 At Expiration, If mkt ↑ SB = 140, Bond matures for 120e0.0488 = $126. Use $100 and the forward contract to buy the stock and cover the short. ➟ CF = +126-100 = +$26 If mkt ↓ S C =100, Bond matures for 120(e0.0488) = $126. Use $100 plus the forward to buy stock and cover the short. CF=126-100=+$26 Lec 5A APT for Forward and Futures Contracts dfdf

Lec 5A APT for Forward and Futures Contracts This must be an ARB OPPORTUNITY. Strategy doesn’t cost anything and yet it pays $26 whether the market goes ↑ or ↓ . To preclude the arbitrage, at A Forward price must = $126 = 120(e0.0488)= er S0 Question: How do you create a synthetic forward contract? Answer: Recall the arbitrage strategy 1. Buy the stock now and sell it forward. 2. Sell a bond also to finance the stock purchase. {+S, -F, -B } ➟ Net CF0 = $0. Therefore, {+F}={+S, -B} ➟ Long Forward = Long stock financed with a bond also, {+S} = {+F, +B} ➟ Long Stock = Long Forward + Long a Bond {+S, -F} = {+B} ➟ Buy stock and sell it forward = a risk-free bond Lec 5A APT for Forward and Futures Contracts dfdf

Lec 5A APT for Forward and Futures Contracts Another Example: Suppose the spot price evolves as shown below. (Assume 1 year to expiration, r = 0.0976/yr (c.c.), hence 1 period = 6 months and r = 4.88%/period (c.c.). ) SP500 Price tree Forward Price tree B 140 D B 140 D A 120 A 126 S0 =100 100 E F0 = ? 100 E 80 84 C 60 F C 60 F t = 0 1 2 t = 0 1 2 At expiration, the forward price must = Spot (or cash) price We already know: At B , Forward price must = $126 = SB (er) At C , Forward PriceC = $84 = Spot PriceC (e0.0488) What is Arb free price at A ? Lec 5A APT for Forward and Futures Contracts dfdf

Lec 5A APT for Forward and Futures Contracts SP500 Price tree Forward Price tree B 140 D B 140 D A 120 A 126 S0 =100 100 E F0 = ? 100 E 80 84 C 60 F C 60 F t = 0 1 2 t = 0 1 2 At A , does this price make sense: F = 90.25? It seems low (relative to the spot price) ➟ Buy Forward, Short Stock {+F contract, -S, +B(PV=$100) }. ➟ CF A = -0 + 100 - 100 = 0 At Expiration (2 periods later), if spot price ↑ SD = 140, Bond matures for 100(e2*0.0488) = 110.25. Settle forward contract: Pay 90.25, receive one share, then cover short. CF = +110.25-90.25= $20 Lec 5A APT for Forward and Futures Contracts dfdf

Lec 5A APT for Forward and Futures Contracts if SE = 100, Bond matures for 100(e2*0.0488) = 110.25. Buy stock thru forward contract. Pay 90.25, receive one share, then cover short. CF = +110.25-90.25= $20 if SF = 60, Settle forward contract. Pay 90.25 and receive one share, The payoff is always +$20. To preclude this arbitrage, at point A Forward price must = 90.25+20 = $110.25 = 100 (e2*0.0488) = Stock PriceA (e2*(r/2)) In general, at t=0, Forward Price F0 must be such that the CF from {+F, -S, +B(FV=F0)} = 0 ➟ CF = -0 +S0 - F0 (e-r T) = 0 ➟ F0 = (er T)S0 done! Lec 5A APT for Forward and Futures Contracts dfdf

Lec 5A APT for Forward and Futures Contracts Price Forwards that generate a known Income (Sec 5.5, 5.6) Suppose the spot price now is S0 = $50, and evolves as shown below. Assume 1 year to expiration, r = 13.8%/yr (c.c.) Stock will pay a $15 Dividend in period 1. Assume the stock price will ↓ by full amount of dividend. Stock Price tree Forward Price tree B 90 E 90 E 75→60 B A 40 F A 40 F S0 =50 27.5 G F0 = ? 27.50 G 33.33→18.33 C C 12.22 H 12.22 H t = 0 1 2 t = 0 1 2 Stock price at B and C is ex-Dividend ($15 lower) Our goal is to find the arb-free forward price at point A: F0 Lec 5A APT for Forward and Futures Contracts dfdf

Lec 5A APT for Forward and Futures Contracts Stock Price tree Forward Price tree B 90 E 90 E 75→60 B A 40 F A 40 F S0 =50 27.5 G F0 = ? 27.50 G 33.33→18.33 C C 12.22 H 12.22 H t = 0 1 2 t = 0 1 2 Note1: PV($15 Dividend) = 15e-0.138(½) = $14. Note2: {+S} = {+F, +B(PV=$14), +B(FV = F0 )} <== synthetic stock Strategy at point A. t=0 {+S, -B(PV=$14), -F)} ➟ CF0 = -50 +14 +0 = -$36 t= ½ year, receive div = $15, use it to pay back the bond. CF1/2 = 0 t=1 sell the stock thru the forward and receive $F0 Since the strategy has no risk, the ROR must = 0.138 ➟ 36e0.138 = F0 ➟ F0 = $41.33 The arb-free forward price at point A is F0 = $41.33 Lec 5A APT for Forward and Futures Contracts dfdf

Lec 5A APT for Forward and Futures Contracts Mark to Market of Forward Contracts SP500 Price tree Forward Price tree B 140 E 140 E A 120 A 126 S0 =100 100 G F0 = 110.25 100 G 80 84 C 60 F 60 H t = 0 1 2 t = 0 1 2 Suppose you go long a forward contract at time 0 Forward Price: F0 = $110.25 At time ½ mkt ↑ to SB = 120. The forward price at B = $126 Long wants to close the forward contract. How? Just sell a 6-months contract at FB = 126. Lec 5A APT for Forward and Futures Contracts dfdf

Lec 5A APT for Forward and Futures Contracts Note that at time 1, You will ➀ pay $110.25 and buy 1 share of stock (to satisfy the first contract), and ➁ sell 1 share of stock and receive $126 (from the second contract) Net CF = -110.25 + 126 = +$15.75 (for sure). Question: What is the Gain at t = 1/2 ? Answer: Gain = PV of 15.75 Gain = $15 (=15.75 e-0.0488) +15.75 | ––––––––––––––|–––––––––––––––––––––––––––––| 0 ½ 1 Therefore, the mark-to-market value of the original contract is $15. Lec 5A APT for Forward and Futures Contracts dfdf

Lec 5A APT for Forward and Futures Contracts Thank You! (a Favara) Lec 5A APT for Forward and Futures Contracts dfdf