dc66e42ec4
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76 lines
962 B
Markdown
76 lines
962 B
Markdown
---
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type: medical-form
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category: academic
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person:
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date: 2017-04-29
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source: options_pricing_notes.jpg
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---
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# Q1 - Options Pricing (Binomial Model)
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## (a)
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Stock price tree:
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- S₀ = 100
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- Up: 130
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- Down: 80
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Payoff tree (put option, strike = $20 implied):
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- Up: 0
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- Down: $20
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System of equations (replicating portfolio):
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$$
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\begin{cases}
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130\alpha + 1.1\beta = 0 \\
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80\alpha + 1.1\beta = 20
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\end{cases}
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$$
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Solutions:
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$$\alpha = -0.4$$
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$$\beta = 47.27$$
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Put price:
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$$P = -100 \times 0.4 + 47.27 = \boxed{\$7.27}$$
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## (b)
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Payoff tree (call option):
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- Up: 30
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- Down: 0
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System of equations (replicating portfolio):
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$$
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\begin{cases}
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120\alpha + 1.1\beta = 30 \\
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80\alpha + 1.1\beta = 0
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\end{cases}
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$$
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Solutions:
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$$\alpha = 0.6$$
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$$\beta = -43.64$$
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Call price:
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$$C = 60 - 43.64 = \boxed{\$16.36} > 15 \text{ Under priced.}$$
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Buy call and sell/short synthetic put.
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---
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Or, put-call parity:
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$$C = S_0 + P - \frac{X}{(1+r)^t} = \$16.36 > 15$$
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