1. Binomial Price Evolution S(0) asset price at start (now) S d = S(0) D asset price one period later, if the return is negative. S d = S x D example: D = 0.9, dollar return = 0.9. Percentage return =.90 - 1 = -10% S u = S(0) U asset price one period later, if the return is positive. S u = S x U example: U = 1.1, dollar return = 1.10. Percentage return = 1.10 - 1 = 10% Introduction to Binomial Option pricing S. Mann, 2007

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

3. Binomial Call Outcomes S(0) SuSu SdSd C C u = max(0, S u - K) C d = max(0, S d - K)

4. Example evolution: U = 1.1, D= 0.92 S(0) S u = S(0)U = 50(1.1) = 55 C C u = max(0, S u - K) = 5 C d = max(0, S d - K) = 0 S d = S(0)D = 50(0.92) = 46

5. Binomial Call Valuation K = 50 ; U = 1.1, D = 0.92 S(0)=50 S u = 55 C0C0 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 V u = V d V 0 =  S 0 - C 0 S d = 46 C u = 5 C d = 0 V d =  S d - C d V u =  S u - C u

6. Desired Outcome of hedge Portfolio, V: K= 50 ; U = 1.1, D = 0.92 S(0)=50 S u = 55 C0C0 Choose  (delta) so that V u = V d V 0 =  S 0 - C 0 S d = 46 C u = 5 C d = 0 V d =  S d - C d =  46 - 0 V u =  S u - C u =  55 - 5

7. Finding the Hedge ratio Find  so that V u = V d : V 0 =  S 0 - C 0 V u =  55 - 5 V d =  46 - 0 V u = V d  S u - C u =  S d - C d Solve for  to find:  ==5/9 C u - C d S u - S d

8. Outcome: holding the hedge portfolio V 0 = 5/9 S 0 - C 0 V u = (5/9) 55 - 5 = 25.56 V d = (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

9. Pricing the call by absence of arbitrage V = 5/9 S - C V u = (5/9) 55 - 5 = 25.56 V d = (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

10. Risk-neutral probability: Using “Trick” to value call K = 50 ; U = 1.1, D = 0.92 B(0,T) = 0.95 C0C0 Price Call same way as before, but use algebra trick: define R(h) = 1/B(0,T) = riskless return Then define: C u = 5 C d = 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  = 0.1326/0.18 = 0.737 Then C 0 = B(0,T) x [  C u + (1-  )C d = 0.95[ (0.737) 5 + (1-.737) 0 ] = 0.95 [ 0.737 x $5.00] = 0.95 [$3.68] = $3.50

11. Binomial pricing using “risk-neutral” probabilities C = Present value of “expected” payoff =PV[ E[ C T ] ] =B(0,T)[  C U + (1-  ) C D ] (one period model) =B(0,T)[  2 C UU + 2  (1-  ) C UD + (1-  ) 2 C DD ] (2 periods) Price Put the same way: P = Present value of “expected” payoff =PV[ E[ P T ] ] =B(0,T)[  P U + (1-  ) P D ] (one period model) =B(0,T)[  2 P UU + 2  (1-  )P UD + (1-  ) 2 P DD ] (2 periods)