Binomial Option Pricing Model Finance (Derivative Securities) 312 Tuesday, 3 October 2006 Readings:...
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Transcript of Binomial Option Pricing Model Finance (Derivative Securities) 312 Tuesday, 3 October 2006 Readings:...
Binomial Option Binomial Option Pricing ModelPricing ModelFinance (Derivative Securities) 312
Tuesday, 3 October 2006
Readings: Chapter 11 & 16
Simple Binomial ModelSimple Binomial Model
Suppose that:• Stock price is currently $20• In three months it will be either $22 or $18• 3-month call option has strike price of 21
Stock Price = $22Option Price = $1
Stock Price = $18Option Price = $0
Stock price = $20Option Price = ?
Option PricingOption Pricing
Consider a portfolio:• long shares, short 1 call option
Portfolio is riskless when 22– 1 = 18 = 0.25
22– 1
18
Option PricingOption Pricing
Riskless portfolio:• long 0.25 shares, short 1 call option
Value of the portfolio in three months:• 22 x0.25 – 1 = 4.50
Value of portfolio today (r = 12%):• 4.5e–0.120.25) = 4.3670
Value of shares: 0.25 20 = 5Value of option: 5 – 4.367 = 0.633
GeneralisationGeneralisation
A derivative lasts for time T and is dependent on a stock
Su ƒu
Sd ƒd
Sƒ
GeneralisationGeneralisation
Consider the portfolio that is long shares
and short 1 derivative
The portfolio is riskless when Su – ƒu = Sd – ƒd or
Su– ƒu
Sd– ƒd
ƒu df
Su Sd
GeneralisationGeneralisation
Value of portfolio at time T:• Su– ƒu
Value of portfolio today:• (Su – ƒu)e–rT
Cost of portfolio today: • S– f
Hence ƒ = S– (Su– ƒu )e–rT
GeneralisationGeneralisation
Substituting for we obtain
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
wherep
e d
u d
rT
Risk-Neutral ValuationRisk-Neutral Valuation
Variables p and (1 – p) can be interpreted as the risk-neutral probabilities of up and down movements
Value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate
Expected stock price: pS0 u + (1 – p)S0 d
• Substitute for p, gives S0erT
Risk-Neutral ValuationRisk-Neutral Valuation
Since p is a risk-neutral probability
20e0.12(0.25) = 22p + 18(1 – p) p = 0.6523Alternatively, using the formula:
Su = 22 ƒu = 1
Sd = 18 ƒd = 0
S ƒ
p
(1– p )
6523.09.01.1
9.00.250.12
e
du
dep
rT
Risk-Neutral ValuationRisk-Neutral Valuation
Su = 22 ƒu = 1
Sd = 18 ƒd = 0
Sƒ
0.6523
0.3477
Value of option:• e–0.12(0.25) (0.6523 x 1 + 0.3477 x 0) = 0.633
Two-Step TreeTwo-Step Tree
Value at node B • e–0.12(0.25)(0.6523 x 3.2 + 0.3477 x0) = 2.0257
Value at node A • e–0.12(0.25)(0.6523 x2.0257 + 0.3477 x0) =
1.2823
201.2823
22
18
24.23.2
19.80.0
16.20.0
2.0257
0.0
A
B
C
D
E
F
Valuing a Put OptionValuing a Put Option
504.1923
60
40
720
484
3220
1.4147
9.4636
A
B
C
D
E
F
Valuing American Valuing American OptionsOptions
505.0894
60
40
720
484
3220
1.4147
12.0
A
B
C
D
E
F
DeltaDelta
Delta () is the ratio of the change in the price of a stock option to the change in the price of the underlying stock
The value of varies from node to node
Determining Determining uu and and dd
Determined from stock volatility t
t
eud
eu
1
contract futures afor 1
rate free-risk
foreign theis herecurrency w afor
index on the yield
dividend theis eindex wherstock afor
stock paying dnondividen afor
)(
)(
a
rea
qea
ea
du
dap
ftrr
tqr
tr
f
Tree ParametersTree Parameters
Conditions:
ert = pu + (1 – p)d
2t = pu2 + (1 – p)d 2 – [pu + (1 – p)d ]2
u = 1/ d Where t is small:
tr
t
t
ea
du
dap
ed
eu
Complete TreeComplete Tree
S0
S0u
S0d S0 S0
S0u2
S0d 2
S0u3
S0u
S0d
S0d 3
Example: Put OptionExample: Put Option
Parameters• S0 = 50; K = 50; r = 10%; = 40%;
• T = 5 months = 0.4167; t = 1 month = 0.0833
Implying that:
u = 1.1224; d = 0.8909;
a = 1.0084; p = 0.5076
Example: Put OptionExample: Put Option89.070.00
79.350.00
70.70 70.700.00 0.00
62.99 62.990.64 0.00
56.12 56.12 56.122.16 1.30 0.00
50.00 50.00 50.004.49 3.77 2.66
44.55 44.55 44.556.96 6.38 5.45
39.69 39.6910.36 10.31
35.36 35.3614.64 14.64
31.5018.50
28.0721.93
Effect of DividendsEffect of Dividends
For known dividend yield:• All nodes ex-dividend for stocks multiplied by
(1 – δ), where δ is dividend yield
For known dollar dividend:• Deduct PV of dividend from initial node• Construct tree• Add PV of dividend to each node before ex-
dividend date