Lec 12B: BOPM for Puts (Hull, Ch.12) Single-period BOPM ▸ Replicating Portfolio ▸ Risk-Neutral Probabilities How to Exploit Arbitrage Opportunities Multi-period BOPM ▸ Two-period Risk-Neutral Probabilities ▸ Two-period Replicating Portfolios ▸ Four Period European Puts, with R-N probabilities ▸ Four Period American Puts, with R-N probabilities Value of early exercise Lec 12B BOPM for Puts dfdf

Single Period BOPM for Puts (Replication) (p.2) Example: Consider a P0E(K = $50, T=1yr). Our job is to find a “fair” price for this Put. Stock, Put, and bond prices evolve as follows (r = 25%/yr) : t=0 T=1 t=0 T=1 t=0 T=1 100 (SU ) 0 ( PU ) 1.25 S0 = 50 P0 = ? B0 = 1.0 25 (SD) 25 (PD ) 1.25 Lec 12B BOPM for Puts dfdf

Construct a Stock/Bond portfolio that replicates the Put exactly. Replication Solution. Construct a Stock/Bond portfolio that replicates the Put exactly. Portfolio T=1 CF0 SD = 25 SU =100 Short 1 Share -25 -100 +50 Buy a Bond (FV=100) 100 100 -80 Sell 3 Puts -75 0 +3P0 0 0 3P0-60 = ? At time 1, CFs from this portfolio (strategy) = 0. Therefore, to avoid arbitrage: 3P0 - 30 = 0, ⇒ P0 = $10 Lec 12B BOPM for Puts dfdf

Arbitrage Opportunities (p. 2) Suppose (physical) puts sell for P0 = $15, can we make some free money? YES. 3 Actual Puts = $45 ⇒ Sell 3 Synthetic Puts = {-S, +B(FV=100)} , CF0 = +50 - 100/1.25 = -30 ⇒Buy Arbitrage Portfolio: {-1 share, +B(FV = $100), -3P} Portfolio T=1 CF0 SD = 25 SU =100 Short 1 Share -25 -100 +50 Buy a Bond (FV=100) 100 100 -80 Sell 3 Puts -75 0 +3(15) 0 0 +15 Arb Profit At time 1, CFs from this portfolio (strategy) = 0. Therefore, to avoid arbitrage: ⇒ P0 = $10 Lec 12B BOPM for Puts dfdf

How to create a synthetic Put? Theory of BOPM (p. 3) Given: t=0 T=1 t=0 T=1 100 (SU ) 0 (PU ) S0 = 50 P0 = ? 25 (SD) 25 (PD ) How to create a synthetic Put? Let, Δ = Shares of Stock, B = $ amount of Bonds. If the stock ↑ Δ100+(1.25)B = 0 If the stock ↓ Δ25 +(1.25)B = 25 Solve for two unknowns. Δ*= (0-25)/(100-25)= -1/3 i.e., short 1/3 shares and B* = -[100(-1/3)]/1.25 = +$26.67 (i.e. +B(PV=$26.67) 1 Physical Put = { -(1/3) sh of Stock, +B(PV=26.67, T=1yr) } At t=0 P0 = -1/3(50) + 26.67 = $10 (same as before) Lec 12B BOPM for Puts dfdf

“Risk-Neutral Probabilities” (p. 4) Assume a risk-neutral economy. Then, all assets are priced to yield the risk-free rate of return. Why? For the stock: S0 = [100p + 25(1 - p)]/(1.25) = $50 ➟ solve for p: p =½ p= prob. of SU = $100, and 1 - p = ½ = prob. of SD = $25 Put price: P0 = [0(1/2) + 25(1/2)]/(1.25) = $10 (same as before). Does Put-Call Parity hold? +S +P = +C +B +50 +10 =? +20 + 50/1.25 Yes. In general, Put price: P0 = [p PU + (1-p) PD ]/(1+R) , p =[(1+r)-d]/(u-d) Lec 12B BOPM for Puts dfdf

Two-periods BOPM. Use R-N probabilities (p. 3) Let S0 = $50, need to find the Put price for P(K=$50, T=1 year). 1 pd = 6 months. r = 22.22%/year, or 11.11%/6 months; 1/(1+r)=0.9 Stock Price tree Put Price tree D 100 A D 0 A 70 0 S0 =50 50 B P0 =$3.71 0 B 35 10 E 25 C E 25 C t =0 1 2 t =0 1 2 at D, Put is worthless (Why?) at E, R-N probability: S0 =[50p + 25(1 - p)]/(1.1111) = 35 ➟ p=5/9 ➟ Put price: P0 = [0(5/9) + 25(4/9)]/(1.1111) = $10 at F (right now) u = 70/50 = 7/5; d = 35/50 = 7/10 p =[(1+r)-d] / (u-d) = [(1.1111-(7/10)] / (7/5-7/10) =0.5873 ➟ P0 ={0.5873(0)+0.4127(10)} /1.1111=$3.71 N.B. in this example u & d change from one period to the next. In general, it is much easier to keep u, d constant. Lec 12B BOPM for Puts dfdf

Stock Price tree Put Price tree D 100 A D 0 A 70 0 Two-periods BOPM. (p. 4) Replication: Trade a portfolio of stocks and bonds to replicate the cash flows from the Put Stock Price tree Put Price tree D 100 A D 0 A 70 0 S0 =50 50 B P0 =$3.71 0 B 35 10 E 25 C E 25 C t =0 1 2 t =0 1 2 at D, Put is worthless. Thus, (Δ=0, B=0} no stock, no bond at E: if S ↑ (E → B): Δ(50) + B(1.1111) = 0 if S ↓ (E → C): Δ( 25 ) + B(1.1111) = 25 ➟ Δ = -1, and B = 45 What is the math telling us? Short 1 share at $35, buy a bond, PV=$45, (FV=45(1.1111)=$50), CFE = +35-45= -10 ➟ Portfolio ValueE = $10 = Value of PutE Lec 12B BOPM for Puts dfdf

Stock Price tree Put Price tree D 100 A D 0 A 70 0 Two-periods BOPM. Replication: Trade a portfolio of stocks and bonds to replicate the cash flows from the Put Stock Price tree Put Price tree D 100 A D 0 A 70 0 S0 =50 50 B P0 =$3.71 0 B F 35 10 E 25 C E 25 C t =0 1 2 t =0 1 2 at F (now), (F → D): Δ(70) + B(1.1111) = 0 (F → E): Δ( 35 ) + B(1.1111) =10 ➟ Δ = -10/35, B = +18 Short 0.2857 shares at $50, buy a bond, PV=$18, CFF = +0.2857-18= -3.71 ➟ Portfolio ValueF = $3.71 = Put ValueE Lec 12B BOPM for Puts dfdf

European vs. American Puts (p. 5) First, consider a European Put(K= $35); r = 25%/period Stock Price tree Put Price tree 168.75 G 0 G B 112.50 D 0 D 75 75 H 0.096 0 H S0 =50 50 E P0 =$0.44 0.40 E 33.33 33.33I 1.611 1.67 I C 22.22F E 5.78 F 14.82J 20.18 J At G, H put is out of the money ➟ P = 0 at I put is in the money, therefore exercise. Put Value = $1.667, same at J: Put =20.18 At A thru F, use R-N approach: u = 1.5, d = 2/3; p = 0.7 For example, at F: PF ={1.67(0.7)+20.18(0.3)} /1.25=$5.78 Lec 12B BOPM for Puts dfdf

European vs. American Puts (p. 5) Now, consider the American Put(K= $35) Stock Price tree Put Price tree 168.75 G 0 G B 112.50 D 0D 75 75 H 0.096 0 H S0 =50 50 E P0 =$0.84 0.40 E 33.33 33.33 I 1.611 C 1.611 I C 22.22F 3.29 5.87 F 14.82 J 12.78 20.18 J Notes: G thru J are the same as for the European Put. At F the European price ($5.78) may be too low. Check for Arb. Opp. {+P, +S, and exercise immediately} ➟ CF = -5.78-22.22+35 =+$7 To preclude this arbitrage at F we must have PF = 12.78 At all other points (A thru F) use R-N approach: u = 1.5, d = 2/3; p = 0.7 For example, at C: PC ={0.40(0.7)+12.78(0.3)} /1.25=$3.29 Lec 12B BOPM for Puts dfdf

Thank You (a Favara) Lec 12B BOPM for Puts dfdf