Option Pricing and Valuation

TL;DR

Option pricing models help us figure out a fair price for an option contract by considering several key factors. The Black-Scholes-Merton model is the most famous for European options, but other methods like binomial trees are useful for American options and provide intuitive steps. Understanding these models helps you determine if an option is over- or under-priced in the market.

1. The Mental Model

Think of an option's price like the price of insurance. How much you'd pay depends on how likely the "insured event" (stock price movement) is, how big the potential payout could be, and how long until the insurance expires.

2. The Core Material

When you're trying to price an option, you're essentially calculating its fair theoretical value. This value comes from a combination of intrinsic value (how much it's "in the money" right now) and extrinsic value (also called time value). Extrinsic value accounts for the potential for the option to become more profitable before it expires.

The key factors influencing an option's price are often called the "Greeks" but let's just focus on what they represent:

1. Current Stock Price (S): The price of the underlying asset right now.
2. Strike Price (K): The price at which you can buy or sell the underlying.
3. Time to Expiration (T): How much time is left until the option expires (usually in years). The more time, the more chance for the stock price to move favorably, so longer dated options generally cost more.
4. Volatility ($\sigma$): How much the stock's price is expected to fluctuate. Higher volatility means a greater chance of big price swings, which increases the value of options (both calls and puts).
5. Risk-Free Interest Rate (r): The return you could get on a risk-free investment (like a U.S. Treasury bond). This affects the present value of future payouts.
6. Dividends (q): If the underlying stock pays dividends, this can reduce the stock price and thus affect option value.

The Black-Scholes-Merton (BSM) Model

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The BSM model is a cornerstone for pricing European options. It assumes the stock price follows a lognormal distribution and that volatility and interest rates are constant. While the full formula looks complex, it essentially calculates the present value of the probability-weighted payoffs.

Here's a conceptual breakdown of its inputs and how they influence the price:

graph TD
    A["Current Stock Price (S)"] --> B["Option Price (C or P)"]
    C["Strike Price (K)"] --> B
    D["Time to Expiration (T)"] --> B
    E["Volatility ($\sigma$)"] --> B
    F["Risk-Free Rate (r)"] --> B
    G["Dividends (q)"] --> B
    B --> H["Call Price (C)"]
    B --> I["Put Price (P)"]
    H -- "$S \uparrow, K \downarrow, T \uparrow, \sigma \uparrow, r \uparrow, q \downarrow$" --> "Call Price $\uparrow$"
    I -- "$S \downarrow, K \uparrow, T \uparrow, \sigma \uparrow, r \downarrow, q \uparrow$" --> "Put Price $\uparrow$"

Binomial Option Pricing Model

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For American options, which can be exercised before expiration, the BSM model isn't directly applicable because it doesn't account for early exercise. The Binomial Option Pricing Model is a discrete-time model that's more flexible. It works by building a "tree" of possible stock prices over time, from today until expiration.

At each step, the stock price can either move up by a certain factor (u) or down by another factor (d). You then work backward from the expiration date, calculating the option's value at each node. At each node for an American option, you compare the intrinsic value (if exercised) with the value if held until the next period, and choose the higher of the two.

Put-Call Parity

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This is a fundamental relationship between the prices of European calls and puts with the same strike price and expiration date. It states that:

Call Price (C) + Present Value of Strike Price (Ke^-rT) = Stock Price (S) + Put Price (P)

This relationship holds true in an efficient market; if it doesn't, an arbitrage opportunity exists. It helps you value one option if you know the price of the other, the stock price, the strike, and the risk-free rate.

3. Worked Example

Let's use a simplified scenario to illustrate the binomial model's logic for a European call option. We'll build a one-step binomial tree.

Scenario:
* Current Stock Price (S) = $100
* Strike Price (K) = $100
* Time to Expiration (T) = 1 year
* Risk-Free Rate (r) = 5% per year
* In one year, the stock price can either go up to $120 or down to $90.

Steps:

1. Calculate the option's payoff at expiration in each state:

   • If stock goes up to $120: Call Payoff = Max($120 - $100, 0) = $20
   • If stock goes down to $90: Call Payoff = Max($90 - $100, 0) = $0

2. Determine the probability of an upward movement (p) using risk-neutral valuation.
   This formula helps us pretend investors are indifferent to risk, and all assets earn the risk-free rate.
   The up factor (u) = $120/$100 = 1.2
   The down factor (d) = $90/$100 = 0.9

   p = (e^(rT) - d) / (u - d)
p = (e^(0.05 * 1) - 0.9) / (1.2 - 0.9)
p = (1.05127 - 0.9) / 0.3
p = 0.15127 / 0.3
p = 0.50423

   The probability of a downward movement is 1 - p = 1 - 0.50423 = 0.49577.

3. Calculate the expected payoff at expiration and discount it back to today:

   Call Price = e^(-rT) * [p * (Up Payoff) + (1 - p) * (Down Payoff)]
Call Price = e^(-0.05 * 1) * [0.50423 * $20 + 0.49577 * $0]
Call Price = 0.951229 * [$10.0846 + $0]
Call Price = 0.951229 * $10.0846
Call Price = $9.593

So, based on this one-step binomial model, the fair value of this call option is approximately $9.59.

4. Key Takeaways

• Option pricing accounts for both intrinsic value (how "in-the-money" it is) and extrinsic value (time and volatility potential).
• The Black-Scholes-Merton model is the standard for European option valuation, assuming constant volatility and risk-free rates.
• The Binomial Option Pricing Model is more flexible, especially for American options, by building a tree of possible price movements.
• Key factors influencing option prices include the underlying stock price, strike price, time to expiration, volatility, and interest rates.
• Put-Call Parity provides a theoretical link between European call and put prices with the same strike and expiration.

Common Mistakes to Avoid:
- Don't confuse intrinsic value with the option's total premium; the premium includes time value.
- Forgetting that the BSM model is designed for European options and has limitations for American options.
- Underestimating the impact of volatility; higher volatility significantly increases option premiums.
- Ignoring the time value decay (theta); options lose value as they approach expiration.
- Not understanding that option models provide theoretical values, and market prices can deviate due to supply and demand.

5. Now Try It

Using the same scenario from the worked example (S=$100, K=$100, T=1 year, r=5%, Up=$120, Down=$90), calculate the fair price of a European put option using the one-step binomial model.

What to do:
1. Calculate the put option's payoff at expiration for both the "up" ($120) and "down" ($90) stock price scenarios.
2. Use the same risk-neutral probability p calculated in the example (since it's for the underlying stock's movement, not the option itself).
3. Calculate the discounted expected payoff to find the put option's price today.

What success looks like: You should arrive at a put price of approximately $4.50.