mathematics in finance binomial model of options pricing
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Mathematics in Finance
Binomial model of options pricing.
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Derivatives - OptionsGive the holder the right to buy or sell
the underlying at a certain date for a certain price. (European options)
• Right to buy call option• Right to sell put option• Payoff function• Cash settlement• Exchanges: AMEX, CBOT, Eurex, LIFFE, EOE, ...
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IV Derivatives - OptionsExample 1:
Long Call on stock S
with strike K=32,
maturity T,
price P=10.
Payoff function:
f(S) = max(0,S(T) – K)-4
-2
0
2
4
6
8
10
12
14
16
1 5 9 13
17
21
25
29
33
37
41
45
underlying at T
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strike
underlying maturity
volatility
Interest rate
Option value
dividends
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Derivatives - Options
Call Putstrike up down upunderlying up up downmaturity approaching down downvolatiliy up up upinterest rates up down downdividends are paid down up
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Problem: How can options be priced?
– Modelling– Black-Scholes– Solving partial differential equations– Monte-Carlo simulation– ...
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Replicating portfolio
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Binomial one period method
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Binomial one period method
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Binomial one period method
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Binomial n-period method
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Binomial n-period method
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Binomial n-period method
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Binomial n-period methodAlgorithm for binomial method
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Example
150
120
96
187.5
120
76.8
234.38
96
150
61.44
0
164.38
80
26
120.09
52.92
13.59
85.37
33.46
58.91
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Numerical implementation
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Some versions of binomial model
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Extensions of binomial model
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Black-Scholes formula
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Conclusions