Option Pricing and Valuation

TL;DR

Option pricing is about figuring out an option's fair value, balancing its potential payoff against various risk factors. Mathematical models help estimate this value by considering market inputs and the option's characteristics. Understanding these models lets you make informed decisions about buying or selling options.

1. The Mental Model

Think of an option's price like the price of insurance. You pay a premium upfront for the right, but not the obligation, to take an action in the future. The price of that insurance depends on how likely it is you'll need to use it, and how much it might be worth if you do.

2. The Core Material

When you price an option, you're trying to determine its fair value. This isn't just about what the option could be worth at expiry, but also what it's worth right now given all the uncertainties. Options have two main components to their price: intrinsic value and extrinsic value (or time value).

Intrinsic Value

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This is the immediate profit you'd make if you exercised the option right now.
* For a call option, intrinsic value is Max(0, (Underlying Price - Strike Price)).
* For a put option, intrinsic value is Max(0, (Strike Price - Underlying Price)).
If an option has intrinsic value, it's considered in-the-money (ITM). If it has no intrinsic value, it's out-of-the-money (OTM) or at-the-money (ATM).

Extrinsic Value (Time Value)

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This is the amount of the option's premium that isn't intrinsic value. It represents the value investors place on the chance that the option will become profitable (or more profitable) before it expires.
Extrinsic Value = Option Price - Intrinsic Value
Several factors influence extrinsic value:

• Time to Expiration: More time means more opportunity for the underlying asset's price to move favorably, so extrinsic value is higher. It decays over time, especially accelerating closer to expiry (this is called theta decay).
• Volatility: Higher expected volatility in the underlying asset's price means a greater chance of large price swings, which increases the probability of the option ending up in-the-money. So, higher volatility generally leads to higher extrinsic value for both calls and puts.
• Interest Rates: Affects the cost of holding the underlying asset (for calls) or the proceeds from selling it (for puts). Higher interest rates generally increase call prices and decrease put prices.
• Dividends: Expected dividends reduce the underlying stock price, making calls less valuable and puts more valuable.

Option Pricing Models

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While simple formulas give you intrinsic value, calculating fair extrinsic value requires more complex models. The most famous is the Black-Scholes-Merton (BSM) model.

The BSM model helps estimate the theoretical price of European-style options (which can only be exercised at expiry). It assumes:
1. No dividends during the option's life.
2. Efficient markets with no transaction costs.
3. Risk-free rate and volatility are constant.
4. Stock price movements follow a log-normal distribution.

The model uses these inputs:
* S: Current stock price
* K: Option strike price
* T: Time to expiration (in years)
* r: Risk-free interest rate (annualized)
* σ: Volatility of the stock (annualized standard deviation of returns)

Here's how these factors influence option prices:

graph TD
    A["Increase in Stock Price (S)"] --> B["Increase Call Price"]
    A --> C["Decrease Put Price"]

    D["Increase in Strike Price (K)"] --> E["Decrease Call Price"]
    D --> F["Increase Put Price"]

    G["Increase in Time to Expiration (T)"] --> H["Increase Call Price"]
    G --> I["Increase Put Price"]

    J["Increase in Volatility (σ)"] --> K["Increase Call Price"]
    J --> L["Increase Put Price"]

    M["Increase in Risk-Free Rate (r)"] --> N["Increase Call Price"]
    M --> O["Decrease Put Price"]

    P["Increase in Dividends"] --> Q["Decrease Call Price"]
    P --> R["Increase Put Price"]

Other models exist, like binomial tree models, which are more flexible for American options (exercisable any time up to expiry) and options with dividends. These models break the time to expiration into smaller steps, allowing for various price paths for the underlying asset.

Implied Volatility

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Instead of predicting future volatility, market participants often use option prices to infer what volatility the market expects. This is called implied volatility. If an option's market price is higher than what BSM would predict using historical volatility, it suggests the market is pricing in higher implied volatility. This is a crucial concept because it tells you what the market collectively believes about future price swings.

3. Worked Example

Let's say you're looking at a call option for a stock:

• Stock Price (S): \$100
• Strike Price (K): \$95
• Time to Expiration (T): 0.5 years (6 months)
• Risk-Free Rate (r): 2% (0.02)
• Volatility (σ): 20% (0.20)

First, let's find the intrinsic value:
For a call, Max(0, (S - K)) = Max(0, ($100 - $95)) = Max(0, $5) = $5.

Now, imagine the market price for this call option is \$8. This means the total premium is \$8.

The extrinsic value (time value) would be:
Extrinsic Value = Market Price - Intrinsic Value
Extrinsic Value = $8 - $5 = $3

This \$3 represents the market's expectation of further price movement and the value of having 6 months until expiration. If this option was OTM, say with a strike of \$105, its intrinsic value would be \$0, and its entire market price (e.g., \$2) would be extrinsic value.

To get the \$8 market price, a sophisticated model like Black-Scholes would be used. If you plugged in the S, K, T, r, and a volatility (implied volatility), it would output that \$8. If BSM with our 20% historical volatility output, say, \$7.50, it means the market is pricing in a slightly higher implied volatility than 20% to reach the \