Understanding Option Pricing Using Binomial Option Pricing Model

Lec 12A: BOPM for Calls (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 ▸ Pricing European Calls (Stock pays Dividends) ▸ Pricing American Calls (Stock with Low vs. High Dividends). Lec 12A BOPM for Calls

Price Calls by Replication(p.2) Example: Consider a C0E(K = $50, T=1yr). Our job is to find a “fair” price for this call. Stock, Call, and bond prices evolve as follows (r = 25%/yr) : t=0 T=1 t=0 T=1 t=0 T=1 100 (SU ) 50 (CU ) 1.25 S0 = 50 C0 = ? B0 = 1.0 25 (SD) 0 (CD ) 1.25 Lec 12A BOPM for Calls

Replication Solution. Construct a Stock/Bond portfolio that replicates the Call exactly. Portfolio T=1 CF0 SD = 25 SU =100 Buy 2 Shares +50 +200 -100 Sell Bond (FV=50) -50 -50 +40 Sell 3 Calls 0 -150 3C0 0 0 3C0-60 = ? At time 1, CFs from this portfolio (strategy) = 0. Therefore, to avoid arbitrage: 3C0 - 60 = 0, ⇒ C0 = $20 Lec 12A BOPM for Calls

Arbitrage Opportunities(p. 2) Suppose C0 = $25, can we make some free money? YES. 3 Synthetic calls = {+2 shares of Stock, -B(FV=50)} = +100 - 50/1.25 = $60 (Cheap) ⇒ Buy 3 Real calls = $75 ⇒ Sell Arbitrage Strategy: Sell 3 Real Calls, and buy 3 Synthetic Calls Arbitrage Portfolio: {+2 shares, -B(FV = $50), -3C} Portfolio T = 1 SD = 25 SU =100 CF0 Buy 2 shares $50 200 -100 Sell Bond (FV=50) -50 -50 +40 Sell 3 Calls 0 -150 +75 0 0 Arb. = +15 Lec 12A BOPM for Calls

Theory of BOPM(p. 3) Given: t=0 T=1 t=0 T=1 100 (SU ) 50 (CU ) S0 = 50 C0 = ? 25 (SD) 0 (CD ) How to create a synthetic Call Option? Let, Δ = Shares of Stock, B = $ amount of Bonds. If the stock ↑ Δ100+(1.25)B = 50 If the stock ↓ Δ25 +(1.25)B = 0 Solve Two equations for two unknowns: Δ*= (50-0)/(100-25)= 2/3 shares long, i.e., buy 2/3 shares and B* = -[25(2/3)]/1.25 = -$13.33 ( -B(PV=$13.33) 1 Physical Call = { +(2/3) shares of Stock, -B(PV=13.33, T=1yr) } At t=0 C0 = ⅔(50) - 13.33 = $20 (same as before) Lec 12A BOPM for Calls

Observations: (p. 4) a) Call Price = f(S0, r, K, T) b) Equally important, pricing does not depend on: 1) Probability of stock price ↑ or ↓ 2) Expected ROR on stock, and 3) Appropriate risk-adjusted discount rate for the stock or for the option! This is the fundamental contribution of Arbitrage Pricing Theory. Lec 12A BOPM for Calls

“Risk-Neutral Probabilities”(p. 4) Recall traditional approach to discounting (from the CAPM): 1. Find Expected Cash flow 2. Determine Risk-Adjusted Discount Rate (RRADR). Then, 3. Price = E(CF)/(1+RRADR) Let, p = prob(S = 100), 1-p = prob(S = 25) For the stock: S0 = [100p + 25(1 - p)] /(1+RRADR) For the call: C0 = [50p + 0(1-p)] /(1+RRADR) In a risk-averse economy, finding p and the discount rate for the Stock and the Call is not easy Lec 12A BOPM for Calls

Alternative: 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 Call price: C0 = [50(1/2) + 0(1/2)]/(1.25) = $20 (same as before). In general, Call price: C0 = [p CU + (1-p) CD ]/(1+R) p =[(1+r)-d] / (u-d) . For this example, u=100/50=2, d=25/50=1/2 p =[(1+r)-d] / (u-d) =[(1.25)-0.5] / (2-0.5) = 0.5 Lec 12A BOPM for Calls

Two-periods BOPM. Use R-N probabilities(p. 5) Let S0 = $50, need to find the call price for C(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 Call Price tree D 100 A D 50 A 70 25 S0 =50 50 B C0 =$13.20 0 B F 35 F 0 E 25 CE 0 C t =0 1 2 t =0 1 2 at D, R-N probability: S0 =[100p + 50(1 - p)]/(1.1111) = 70 ➟ p = 5/9 ➟ Call price: C0 = [50(5/9) + 0(4/9)]/(1.1111) = $25 at E, Call is worthless (Why?) 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 ➟ C0 ={0.5873(25)+0} /1.1111=$13.20 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 12A BOPM for Calls

Two-periods BOPM. (p. 6) Replication: Trade a portfolio of stocks and bonds to replicate the cash flows from the call Stock Price tree Call Price tree D 100 A D 50 A 70 25 S0 =50 50 B C0 =$13.20 0 B 35 0 E 25 CE 0 C t =0 1 2 t =0 1 2 at D, structure {Δ shares of stock, plus a bond} to replicate Call CFs if S ↑ (D → A): Δ(100) + B(1.1111) = 50 if S ↓ (D → B): Δ( 50 ) + B(1.1111) = 0 ➟ Sol’n: Δ = 1, and B = -$45 What is the math telling us? at D: Sell a bond, PV=$45, (FV=45(1.1111)=$50), Go long 1 share at $70 ➟ Portfolio ValueD = 1(70) -45=$25 = CallD Lec 12A BOPM for Calls

Two-periods BOPM. (p. 6) Stock Price tree Call Price tree D 100 A D 50 A 70 25 S0 =50 50 B C0 =$13.20 0 B 35 0 E 25 CE 0 C t =0 1 2 t =0 1 2 at E: structure {Δ shares of stock, plus a bond} to replicate Call CFs if S ↑ (E → B): Δ(50) + B(1.1111) = 0 if S ↓ (E → C): Δ( 25 ) + B(1.1111) = 0 ➟ Sol’n: Δ = 0, and B = 0 What is the math telling us? Buy 0 Shares and buy/Sell a bond, PV=$0, (FV=0(1.1111)=$0) ➟ Portfolio ValueE = 0 (= Value of CallE ) Lec 12A BOPM for Calls

Two-periods BOPM. (p. 6) Stock Price tree Call Price tree D 100 A D 50 A 70 25 S0 =50 50 B C0 =$13.20 0 B F 35 F 0 E 25 CE 0 C t =0 1 2 t =0 1 2 at F (t=0): find {Δ shares of stock, plus a bond} to replicate Call CFs if S ↑ (F → D): Δ(70) + B(1.1111) = 25 if S ↓ (F → E): Δ( 35 ) + B(1.1111) = 0 ➟ Sol’n: Δ = 5/7 = 0.714 shares, and B = -$22.50 What is the math telling us? Buy 0.714 Shares and Sell a bond PV=$22.50, FV=22.50(1.1111)=$25 ➟ Portfolio ValueF = 50(5/7)-22.50 = $13.20 (= Value of CallF ) Lec 12A BOPM for Calls

Thank You! (a Favara) Lec 12A BOPM for Calls

Understanding Option Pricing Using Binomial Option Pricing Model