1Lec 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 $140B ? B How to determine $120 A F = ? forward prices? $100C ? 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: CF B = = $20 ➟ To preclude arbitrage the forward price must = $140 Using the same logic, the forward price at C must = ?.

2 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)} ➟ CF A = = 0 At Expiration, If mkt ↑ S B = 140, Bond matures for 120e = $126. Use $100 and the forward contract to buy the stock and cover the short. ➟ CF = = +$26 If mkt ↓ S C =100, Bond matures for 120(e ) = $126. Use $100 plus the forward to buy stock and cover the short. CF= =+$26

3 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(e )= e r S 0 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 CF 0 = $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

4 Lec 5A APT for Forward and Futures Contracts Another Example: Suppose the spot price evolves as shown below. (Assume 1 year to expiration, r = /yr (c.c.), hence 1 period = 6 months and r = 4.88%/period (c.c.). ) SP500 Price tree Forward Price tree B140 DB140 D A 120 A 126 S 0 = E F 0 = ? 100 E 8084 C60 F t = 01 2t = 01 2 At expiration, the forward price must = Spot (or cash) price We already know: At B, Forward price must = $126 = S B (e r ) At C, Forward Price C = $84 = Spot Price C (e ) What is Arb free price at A ?

5 Lec 5A APT for Forward and Futures Contracts SP500 Price tree Forward Price tree B140 DB140 D A120 A 126 S 0 = E F 0 = ? 100 E 8084 C60 F t = 01 2t = 01 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 At Expiration (2 periods later), if spot price ↑ S D = 140, Bond matures for 100(e 2* ) = Settle forward contract. Pay 90.25, receive one share, then cover short. CF = = $20

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

7 Lec 5A APT for Forward and Futures Contracts Price Forwards that generate a known Income (Sec 5.5, 5.6) Suppose the spot price evolves as shown, starting at S 0 = $50 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→60B A 40 F A 40 F S 0 = GF 0 = ? G 33.33→18.33 C C F H t = 01 2t = 01 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: F 0

8 Lec 5A APT for Forward and Futures Contracts Stock Price tree Forward Price tree B 90 E 90 E 75→60B A 40 F A 40 S 0 = GF 0 = ? G 33.33→18.33 C C F H t = 01 2t = 01 2 Note1: PV($15 Dividend) = 15e (½) = $14. Note2: {+S} = {+F, +B(PV=$14), +B(FV=F)} = synthetic stock Strategy at point A. t=0 {+S, -B(PV=$14), -F)} ➟ CF 0 = = -$36 t= ½ year, receive div = $15, use it to pay back the bond. CF 1/2 = 0 t=1 sell the stock thru the forward and receive $ F 0 Since the strategy has no risk, the ROR must = ➟ 36e = F 0 ➟ F 0 = $41.33 The arb-free forward price at point A is F 0 = $41.33

9 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 A120 A 126 S 0 = G F 0 = G C 60 F 60 H t = 01 2t = 01 2 Suppose you go long a forward contract at time 0 Forward Price: F 0 = $ At time ½ mkt ↑ to S B = 120. The forward price at B = $126 Long wants to close the forward contract. How? Just sell a 6-months contract at F B = 126.

10 Lec 5A APT for Forward and Futures Contracts Note that at time 1, You will ➀ pay $ 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 = = +$15.75 (for sure). Q: What is the Gain at t = 1/2 ? A: Gain = PV of Gain = $15 (=15.75 e ) | ––––––––––––––|–––––––––––––––––––––––––––––| 0½ 1 Therefore, the mark-to-market value of the original contract is $15.

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