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Lecture11_PricingOptionII

# Lecture11_PricingOptionII - 1 NBA 6730 Derivative...

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Unformatted text preview: 1 NBA 6730: Derivative Securities Lecture 11: Option Pricing II 09/30/2010 George Gao NBA6730-Derivative Securities I 2 Agenda We will see how to extend the Binomial pricing model beyond plain-vanilla European calls and puts, and how the Binomial model is related to the well-known BSM model. More on Binomial model Build the Binomial tree Black-Scholes-Merton option pricing model 2 NBA6730-Derivative Securities I 3 (1) Two-Period Binomial Model The stock price tree: The price tree for a European call: ܵ ݑܵ ݀ܵ ݑݑܵ ݑ݀ܵ ݀݀ܵ ܿ ܿ ௨ ܿ ௗ ܿ ௨௨ ൌ maxሺݑݑܵ െ ܭ, 0ሻ ܿ ௨ௗ ൌ ܿ ௗ௨ ൌ maxሺݑ݀ܵ െ ܭ, 0ሻ ܿ ௗௗ ൌ maxሺ݀݀ܵ െ ܭ, 0ሻ NBA6730-Derivative Securities I 4 (1) Two-Period Binomial Model How do we get the call price today ݐ : Dynamic programming (i.e., start at the end, work backwards) From the one-period Binomial model: ௖ ೠ ୀ ࢗ ࢉ ࢛࢛ శ ሺ૚షࢗሻ ࢉ ࢛ࢊ ૚శ࢘ ௖ ೏ ୀ ࢗ ࢉ ࢊ࢛ శ ሺ૚షࢗሻ ࢉ ࢊࢊ ૚శ࢘ Recall ݍ is the risk-neutral probability of price moving up: ࢗ ൌ ૚ା࢘ିࢊ ࢛ିࢊ Once we know ܿ ௨ and ܿ ௗ , we have a one-period model and calculate the call price: ࢉ ൌ ࢗ ࢉ ࢛ ା ሺ૚ିࢗሻ ࢉ ࢊ ૚ା࢘ ൌ ࢗ ૛ ࢉ ࢛࢛ ା ૛ࢗ ૚ିࢗ ࢉ ࢛ࢊ ା ሺ૚ିࢗሻ ૛ ࢉ ࢊࢊ ሺ૚ା࢘ሻ ૛ 3 NBA6730-Derivative Securities I 5 (2) IBM Example: Non Dividend IBM is currently traded at \$100 and the simple quarterly interest rate is 3.33%. Every quarter, the price of IBM either rises by 30% or falls by 10%. What’s the value of a European call with strike price K=\$110 and 6 months to maturity? The price tree for a European call looks like: 100 130 90 169 117 81 ܿ ܿ ௨ ܿ ௗ ሺ169 െ 110ሻ ା ൌ 59 ሺ117 െ 110ሻ ା ൌ 7 ሺ81 െ 110ሻ ା ൌ 0 NBA6730-Derivative Securities I 6 (2) IBM Example: Non Dividend Risk-neutral pricing on the stock implies ݍ ൌ ଵ.଴ଷଷଷି଴.ଽ ଵ.ଷି଴.ଽ ൌ ଵ ଷ So we get the value of call at the end of first quarter: ܿ ௨ ൌ ଴.ଷଷൈହଽା଴.଺଻ൈ଻ ଵ.଴ଷଷଷ ൌ 23.55 , ܿ ௗ ൌ ଴.ଷଷൈ଻ା଴.଺଻ൈ଴ ଵ.଴ଷଷଷ ൌ 2.56 The call tree now looks like: Finally, the call price today is: ܿ ൌ ଴.ଷଷൈଶଷ.ହହା଴.଺଻ൈଶ.ହ଺ ଵ.଴ଷଷଷ ൌ 9.05 Question: how to price a call if IBM distributes a 5% dividend at the end of first quarter? ܿ 23.55 2.56 59 7 4 NBA6730-Derivative Securities I 7 (3) IBM Example: Dividend The new stock price tree looks like: The price tree for a European call looks like 100 130 → 123.5 90 → 85.5 160.55 111.15 76.95 ܿ ܿ ௨ ܿ ௗ ሺ160.55 െ 110ሻ ା ൌ 50.55 ሺ111.15 െ 110ሻ ା ൌ 1.15 ሺ76.95 െ 110ሻ ା ൌ 0 NBA6730-Derivative Securities I 8 (3) IBM Example: Dividend Risk-neutral probability is unchanged: ݍ ൌ ଵ.଴ଷଷଷି଴.ଽଵ....
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Lecture11_PricingOptionII - 1 NBA 6730 Derivative...

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