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 = = -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 = Percentage return = = 10% Introduction to Binomial Option pricing S. Mann, 2007

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%

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

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

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

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 =  V u =  S u - C u = 

Finding the Hedge ratio Find  so that V u = V d : V 0 =  S 0 - C 0 V u =  V d =  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

Outcome: holding the hedge portfolio V 0 = 5/9 S 0 - C 0 V u = (5/9) = V d = (5/9) = Portfolio of V = (5/9)S - C pays $25.56 risklessly. A riskless bond paying costs B(0,T) x Two portfolios Same Payoffs Different costs Arbitrage opportunity

Pricing the call by absence of arbitrage V = 5/9 S - C V u = (5/9) = V d = (5/9) = T-Bill paying costs B(0,T)(25.56) V = (5/9) S - C = B(0,T) C = (5/9) S - (25.56) B(0,T) = (25.56)B(0,T) If B(0,T) = 0.95, B(0,T)25.56 = C = = 3.50

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 = , R(h) - D = = U - D = =0.18 so  = /0.18 = Then C 0 = B(0,T) x [  C u + (1-  )C d = 0.95[ (0.737) 5 + (1-.737) 0 ] = 0.95 [ x $5.00] = 0.95 [$3.68] = $3.50

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)