Binomial Price Evolution Introduction to Binomial Option pricing S. Mann, 2010 Binomial Price Evolution Su = S(0) U asset price one period later, if the return is positive. Su = S x U example: U = 1.1, dollar return = 1.10. Percentage return = 1.10 - 1 = 10% S(0) asset price at start (now) Sd = S(0) D asset price one period later, if the return is negative. Sd = S x D example: D = 0.9, dollar return = 0.9. Percentage return = .90 - 1 = -10% S. Mann, 2010

Evolution Example Su = 55 S(0) = 50 Sd = 46 Let U = 1.1 and D = 0.92 asset price one period later, if the return is positive. Su = S (1.1); % return is 10% S(0) = 50 Sd = 46 asset price one period later, if the return is negative. Su = S (0.92); % return is - 8% S. Mann, 2010

Binomial Call Outcomes Su S(0) Sd Cu = max(0, Su - K) C Cd = max(0, Sd - K) S. Mann, 2010

Example evolution: U = 1.1, D= 0.92 Su = S(0)U = 50(1.1) = 55 S(0) Sd = S(0)D = 50(0.92) = 46 Cu = max(0, Su - K) = 5 C Cd = max(0, Sd - K) = 0 S. Mann, 2010

Binomial Call Valuation K = 50 ; U = 1.1 , D = 0.92 Su = 55 Cu = 5 S(0)=50 C0 Sd = 46 Cd = 0 Price Call by forming riskless portfolio. A riskless portfolio must earn riskless rate (r) or arbitrage is possible. [ U > (1+r) > D] Choose portfolio so that Vu= Vd Vu = D Su - Cu V0 = D S0 - C0 Vd = D Sd - Cd S. Mann, 2010

Desired Outcome of hedge Portfolio, V: K= 50 ; U = 1.1 , D = 0.92 Cu = 5 Su = 55 C0 S(0)=50 Cd = 0 Sd = 46 Vu = D Su - Cu = D 55 - 5 V0 = D S0 - C0 Vd = D Sd - Cd = D 46 - 0 Choose D (delta) so that Vu= Vd S. Mann, 2010

Finding the Hedge ratio Vu = D 55 - 5 V0 = D S0 - C0 Vd = D 46 - 0 Find D so that Vu = Vd : Vu = Vd D Su - Cu = D Sd - Cd Cu - Cd Solve for D to find: D = = 5/9 Su - Sd S. Mann, 2010

Outcome: holding the hedge portfolio Vu = (5/9) 55 - 5 = 25.56 V0 = 5/9 S0 - C0 Vd = (5/9) 46 - 0 = 25.56 Portfolio of V = (5/9)S - C pays $25.56 risklessly. A riskless bond paying 25.56 costs B(0,T) x 25.56 Two portfolios Same Payoffs Different costs Arbitrage opportunity S. Mann, 2010

Pricing the call by absence of arbitrage Vu = (5/9) 55 - 5 = 25.56 V = 5/9 S - C Vd = (5/9) 46 - 0 = 25.56 T-Bill paying 25.56 costs B(0,T)(25.56) V = (5/9) S - C = B(0,T) 25.56 C = (5/9) S - (25.56) B(0,T) = 27.78 - (25.56)B(0,T) If B(0,T) = 0.95, B(0,T)25.56 = 24.48 C = 27.78 - 24.28 = 3.50 S. Mann, 2010

Risk-neutral probability: Using “Trick” to value call K = 50 ; U = 1.1 , D = 0.92 B(0,T) = 0.95 Cu = 5 Price Call same way as before, but use algebra trick: define R(h) = 1/B(0,T) = riskless return Then define: C0 Cd = 0 So that R(h) = 1/.95 = 1.0526, R(h) - D = 1.0526 - 0.92 = 0.1326 U - D = 1.10 - 0.92 = 0.18 so p = 0.1326/0.18 = 0.737 Then C0 = B(0,T) x [pCu + (1-p)Cd = 0.95[ (0.737) 5 + (1-.737) 0 ] = 0.95 [ 0.737 x $5.00] = 0.95 [$3.68] = $3.50 S. Mann, 2010

Binomial pricing using “risk-neutral” probabilities C = Present value of “expected” payoff = PV[ E[ CT ] ] = B(0,T)[ p CU + (1-p) CD ] (one period model) = B(0,T)[p2CUU + 2 p(1-p) CUD + (1-p)2CDD] (2 periods) Value Put the same way: P = Present value of “expected” payoff = PV[ E[ PT ] ] = B(0,T)[ p PU + (1-p) PD ] (one period model) = B(0,T)[p2PUU + 2 p(1-p)PUD + (1-p)2PDD] (2 periods) S. Mann, 2010