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documents/academic/options_pricing_notes.md

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