---
type: medical-form
category: academic
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:

$$C = S_0 + P - \frac{X}{(1+r)^t} = \$16.36 > 15$$