Description
Binomial Option Pricing Model
Binomial Option Pricing Model
Binomial Option Pricing Model
Cox, J. C., S. A. Ross, and M. Rubinstein. “Option Pricing: A Simplified Approach.” Journal of Financial Economics 7 (October 1979): 229 – 64.
A Simple Binomial Model Binomial Option Pricing Model
Assumptions
The stock price follows a random walk Arbitrage opportunities do not exist In the limit, as the time step becomes smaller, BOPM leads to the lognormal assumption of stock prices
A stock price is currently $20 In three months it will be either $22 or $18
Stock Price = $22 Stock price = $20 Stock Price = $18
A Call Option
A 3-month call option on the stock has a strike price of 21. Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0
Setting Up a Riskless Portfolio
Consider the Portfolio: long ? shares short 1 call option 22? – 1
18?
Portfolio is riskless when 22? – 1 = 18? or
? = 0.25
Valuing the Portfolio
(Risk-Free Rate is 12%)
Valuing the Option
The portfolio that is long 0.25 shares short 1 option is worth 4.367 The value of the shares is 5.000 (= 0.25 × 20 ) The value of the option is therefore 0.633 (= 5.000 – 4.367 )
The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22 × 0.25 – 1 = 4.50 The value of the portfolio today is 4.5e – 0.12×0.25 = 4.3670
Generalization
A derivative lasts for time T and is dependent on a stock
Generalization
(continued)
Consider the portfolio that is long ? shares and short 1 derivative S0u? – ƒu
S0 ƒ
S0u ƒu S0d ƒd
S0d? – ƒd The portfolio is riskless when S0u? – ƒu = S0d? – ƒd or
?=
ƒu ? fd S 0u ? S 0 d
Generalization
(continued)
Generalization
(continued)
Value of the portfolio at time T is S0u? – ƒu Value of the portfolio today is (S0u? – ƒu)e–rT Another expression for the portfolio value today is S0? – f Hence ƒ = S0? – (S0u? – ƒu )e–rT
Substituting for ? we obtain
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
where
p=
e rT ? d u?d
Original Example Revisited
p
p as a Probability
It is natural to interpret p and 1-p as probabilities of up and down movements The value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate
S0u = 22 ƒu = 1 S0d = 18 ƒd = 0
S0 ƒ
(1 –
p)
Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from 20e0.12 ×0.25 = 22p + 18(1 – p ) which gives p = 0.6523 Alternatively, we can use the formula
p
e rT ? d e 0.12 × 0.25 ? 0 .9 p= = = 0 .6523 u?d 1 .1 ? 0 . 9
S0 ƒ
S0u ƒu S0d ƒd
(1 –
p)
Valuing the Option Using RiskNeutral Valuation
0.65
0.34
A Two-Step Example
24.2 22 20 18 16.2 Each time step is 3 months K=21, r=12% 19.8
23
S0u = 22 ƒu = 1 S0d = 18 ƒd = 0
S0 ƒ
77
The value of the option is e–0.12×0.25 (0.6523×1 + 0.3477×0) = 0.633
Valuing a Call Option
D
22 20 A 1.2823 2.0257 18 0.0
24.2 3.2 19.8 0.0 16.2 0.0
A Generalization
The length of a time step is now ?t rather T.
B E C F
ƒ = e–r?T[ pƒu + (1 – p)ƒd ]
where
p= e r?T ? d u?d
Value at node B = e–0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257 Value at node A = e–0.12×0.25(0.6523×2.0257 + 0.3477×0) = 1.2823
ƒ = e–2r?T[ p2ƒuu + 2p(1-p)fud+(1 – p)2ƒdd]
A Put Option Example; K=52
K = 52, time step = 1yr r = 5%
60 50 A 4.1923
B E
What Happens When an Option is American
72 0 48 4 32 20
D
K = 52, time step = 1yr r = 5%
50 A 5.0894
D
60
72 0 48 4 32 20
B E
1.4147 40
C
1.4147 40
C
9.4636
F
12.0
F
At node C, the value of the option is 9.4636, whereas the payoff from early Exercise is 12
Choosing u and d
One way of matching the volatility is to set
u = e?
?t ?t
Choosing u and d: Example
American option with K = 52, time step = 1yr, r = 5%, ? = 30%, the life of the option is 2 years, and there are two time steps
d = 1 u = e ??
where ? is the volatility and ?t is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein
The Probability of an Up Move
a?d u?d
p=
a = e r?t for a nondividen d paying stock a=e
( r ? q ) ?t
Thank you
for a stock index wher e q is the dividend yield on the index
doc_522974811.pdf
Binomial Option Pricing Model
Binomial Option Pricing Model
Binomial Option Pricing Model
Cox, J. C., S. A. Ross, and M. Rubinstein. “Option Pricing: A Simplified Approach.” Journal of Financial Economics 7 (October 1979): 229 – 64.
A Simple Binomial Model Binomial Option Pricing Model
Assumptions
The stock price follows a random walk Arbitrage opportunities do not exist In the limit, as the time step becomes smaller, BOPM leads to the lognormal assumption of stock prices
A stock price is currently $20 In three months it will be either $22 or $18
Stock Price = $22 Stock price = $20 Stock Price = $18
A Call Option
A 3-month call option on the stock has a strike price of 21. Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0
Setting Up a Riskless Portfolio
Consider the Portfolio: long ? shares short 1 call option 22? – 1
18?
Portfolio is riskless when 22? – 1 = 18? or
? = 0.25
Valuing the Portfolio
(Risk-Free Rate is 12%)
Valuing the Option
The portfolio that is long 0.25 shares short 1 option is worth 4.367 The value of the shares is 5.000 (= 0.25 × 20 ) The value of the option is therefore 0.633 (= 5.000 – 4.367 )
The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22 × 0.25 – 1 = 4.50 The value of the portfolio today is 4.5e – 0.12×0.25 = 4.3670
Generalization
A derivative lasts for time T and is dependent on a stock
Generalization
(continued)
Consider the portfolio that is long ? shares and short 1 derivative S0u? – ƒu
S0 ƒ
S0u ƒu S0d ƒd
S0d? – ƒd The portfolio is riskless when S0u? – ƒu = S0d? – ƒd or
?=
ƒu ? fd S 0u ? S 0 d
Generalization
(continued)
Generalization
(continued)
Value of the portfolio at time T is S0u? – ƒu Value of the portfolio today is (S0u? – ƒu)e–rT Another expression for the portfolio value today is S0? – f Hence ƒ = S0? – (S0u? – ƒu )e–rT
Substituting for ? we obtain
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
where
p=
e rT ? d u?d
Original Example Revisited
p
p as a Probability
It is natural to interpret p and 1-p as probabilities of up and down movements The value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate
S0u = 22 ƒu = 1 S0d = 18 ƒd = 0
S0 ƒ
(1 –
p)
Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from 20e0.12 ×0.25 = 22p + 18(1 – p ) which gives p = 0.6523 Alternatively, we can use the formula
p
e rT ? d e 0.12 × 0.25 ? 0 .9 p= = = 0 .6523 u?d 1 .1 ? 0 . 9
S0 ƒ
S0u ƒu S0d ƒd
(1 –
p)
Valuing the Option Using RiskNeutral Valuation
0.65
0.34
A Two-Step Example
24.2 22 20 18 16.2 Each time step is 3 months K=21, r=12% 19.8
23
S0u = 22 ƒu = 1 S0d = 18 ƒd = 0
S0 ƒ
77
The value of the option is e–0.12×0.25 (0.6523×1 + 0.3477×0) = 0.633
Valuing a Call Option
D
22 20 A 1.2823 2.0257 18 0.0
24.2 3.2 19.8 0.0 16.2 0.0
A Generalization
The length of a time step is now ?t rather T.
B E C F
ƒ = e–r?T[ pƒu + (1 – p)ƒd ]
where
p= e r?T ? d u?d
Value at node B = e–0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257 Value at node A = e–0.12×0.25(0.6523×2.0257 + 0.3477×0) = 1.2823
ƒ = e–2r?T[ p2ƒuu + 2p(1-p)fud+(1 – p)2ƒdd]
A Put Option Example; K=52
K = 52, time step = 1yr r = 5%
60 50 A 4.1923
B E
What Happens When an Option is American
72 0 48 4 32 20
D
K = 52, time step = 1yr r = 5%
50 A 5.0894
D
60
72 0 48 4 32 20
B E
1.4147 40
C
1.4147 40
C
9.4636
F
12.0
F
At node C, the value of the option is 9.4636, whereas the payoff from early Exercise is 12
Choosing u and d
One way of matching the volatility is to set
u = e?
?t ?t
Choosing u and d: Example
American option with K = 52, time step = 1yr, r = 5%, ? = 30%, the life of the option is 2 years, and there are two time steps
d = 1 u = e ??
where ? is the volatility and ?t is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein
The Probability of an Up Move
a?d u?d
p=
a = e r?t for a nondividen d paying stock a=e
( r ? q ) ?t
Thank you
for a stock index wher e q is the dividend yield on the index
doc_522974811.pdf