Option pricing models

Idea: Price by copying the payoff

Build the option's payoff from things that already have prices. No guess about the future goes in.

• Hold $\Delta$ shares and borrow cash so the position pays what the call pays
• Any other price allows an arbitrage

Example: A one-step call

Stock $60$ ends at $80$ or $50$, cash earns $5\%$, strike $65$. Price the call.

1. Payoffs $15$ and $0$: $\Delta =\frac{15}{30}=\frac{1}{2}$
2. Loan today: $\frac{25}{1.05}=\frac{500}{21}$
3. Copy: $30-\frac{500}{21}=\frac{130}{21}$

Formula: The pricing weight

$q=\frac{S_0(1+r)-S_d}{S_u-S_d}$

The price is the payoff averaged with weight q, then discounted. The real chance never appears.

Quick check: Find q

Same tree: stock $60$, ends $80$ or $50$, cash $5\%$. What is $q$?

Idea: More steps: the binomial tree

Value end nodes by payoff, then step back: each node is the discounted $q$ average of its children.

• $n$ steps need only $n+1$ end nodes
• American options: at each node keep the larger of waiting and exercising

Formula: At-the-money shortcut

$C\approx 0.4S_0\sigma \sqrt{T}$

With r near $0$: $0.4\times 200\times 30\%\times \frac{1}{2}=12$, against about $11.96$ from Black-Scholes.

Common mistake: Pricing with your own view

Discounting the expected payoff under what you think the stock will do.

Instead A price comes from what you can hedge with. Being bullish does not make a call dearer.

Idea: Monte Carlo for path payoffs

Simulate many paths under $q$, average the payoffs, discount.

• Error falls like $\frac{1}{\sqrt{N}}$: half the error takes $10000\times 4=40000$ paths
• Early exercise is its weak spot

Quick check: The model-free check

Same tree, a put with strike $65$. Price it, then check $C-P=S_0-\frac{K}{1+r}$.