1. 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 = = 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 = = -10%
S. Mann, 2010

2. 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

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

4. 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

5. 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

6. 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
V0 = D S0 - C0
Vd = D Sd - Cd = D
Choose D (delta) so that Vu= Vd
S. Mann, 2010

7. Finding the Hedge ratio
Vu = D
V0 = D S0 - C0
Vd = D
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

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

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

10. Risk-neutral probability: Using “Trick” to value call
K = 50 ; U = 1.1 , D = 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 = ,
R(h) - D = =
U - D = = 0.18
so p = /0.18 =
Then C0 = B(0,T) x [pCu + (1-p)Cd = 0.95[ (0.737) 5 + (1-.737) 0 ]
= [ x $5.00]
= [$3.68]
= $3.50
S. Mann, 2010

11. 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