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Binomial Option Pricing Model Finance (Derivative Securities) 312 Tuesday, 3 October 2006 Readings: Chapter 11 & 16
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Simple 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 = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price = ?
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Option Pricing Consider a portfolio: long shares, short 1 call option Portfolio is riskless when 22 – 1 = 18 = 0.25 22 – 1 18
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Option Pricing Riskless portfolio: long 0.25 shares, short 1 call option Value of the portfolio in three months: 22 x 0.25 – 1 = 4.50 Value of portfolio today ( r = 12%): 4.5 e –0.12 0.25) = 4.3670 Value of shares: 0.25 20 = 5 Value of option: 5 – 4.367 = 0.633
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Generalisation A derivative lasts for time T and is dependent on a stock Su ƒ u Sd ƒ d SƒSƒ
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Generalisation Consider the portfolio that is long shares and short 1 derivative The portfolio is riskless when Su – ƒ u = Sd – ƒ d or Su – ƒ u Sd – ƒ d
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Generalisation Value of portfolio at time T: Su – ƒ u Value of portfolio today: (Su – ƒ u )e –rT Cost of portfolio today: S – f Hence ƒ = S – ( Su – ƒ u )e –rT
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Generalisation Substituting for we obtain ƒ = [ pƒ u + (1 – p)ƒ d ]e –rT where
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Risk-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: pS 0 u + (1 – p)S 0 d Substitute for p, gives S 0 e rT
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Risk-Neutral Valuation Since p is a risk-neutral probability 20 e 0.12(0.25) = 22 p + 18 (1 – p) p = 0.6523 Alternatively, using the formula: Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 S ƒS ƒ p (1 – p )
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Risk-Neutral Valuation Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 SƒSƒ 0.6523 0.3477 Value of option: e –0.12(0.25) (0.6523 x 1 + 0.3477 x 0) = 0.633
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Two-Step Tree Value at node B e –0.12(0.25) (0.6523 x 3.2 + 0.3477 x 0) = 2.0257 Value at node A e –0.12(0.25) (0.6523 x 2.0257 + 0.3477 x 0) = 1.2823 20 1.2823 22 18 24.2 3.2 19.8 0.0 16.2 0.0 2.0257 0.0 A B C D E F
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Valuing a Put Option 50 4.1923 60 40 72 0 48 4 32 20 1.4147 9.4636 A B C D E F
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Valuing American Options 50 5.0894 60 40 72 0 48 4 32 20 1.4147 12.0 A B C D E F
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Delta 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
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Determining u and d Determined from stock volatility
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Tree Parameters Conditions: e r t = pu + (1 – p)d 2 t = pu 2 + (1 – p)d 2 – [pu + (1 – p)d ] 2 u = 1/ d Where t is small:
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Complete Tree S0S0 S0uS0u S0dS0d S0S0 S0S0 S0u2S0u2 S 0 d 2 S0u3S0u3 S0uS0u S0dS0d S 0 d 3
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Example: Put Option Parameters S 0 = 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
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Example: Put Option
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Effect 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
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