Option Pricing with Stochastic Calculus
Introduction to Options
What Is an Option?
Imagine you want to buy a house. You find one you like, but you're not ready to buy it today. You could pay the owner a small fee to get the right to buy that house for a fixed price anytime in the next three months. If the housing market booms, you can buy the house at the agreed-upon lower price. If the market crashes, you can walk away, losing only the small fee. You had the choice, but not the commitment.
That's the basic idea behind a financial option. It's a contract that gives the buyer the right, but not the obligation, to buy or sell an underlying asset at a specific price on or before a certain date.
Think of options as an agreement, a contract giving you the right, but not the obligation, to buy or sell an underlying asset – typically shares listed on the NSE or BSE – at a predetermined price (called the strike price) on or before a specific date (the expiration date).
Let's break down the key terms in that agreement.
Strike Price
noun
The predetermined price at which the underlying asset can be bought or sold. This is the price you lock in.
Expiration Date
noun
The date by which the option must be exercised. After this date, the contract is worthless.
Premium
noun
The price you pay to buy the option contract. It's the fee for having the choice.
The Two Flavors of Options
Options come in two main types, depending on whether you want the right to buy or the right to sell.
A call option gives you the right to buy an asset. Think of it as "calling" the asset away from someone.
A put option gives you the right to sell an asset. Think of it as "putting" the asset to someone.
Let's see how they work. Imagine shares of XYZ Corp. are currently trading at $100 per share.
If you're optimistic and think the price will go up, you might buy a call option. You could buy a call with a strike price of $110 that expires in one month. You might pay a $2 premium per share for this right. If XYZ stock shoots up to $120, you can exercise your option to buy at $110, then immediately sell at the market price of $120 for a profit. Your bet paid off.
If you're pessimistic and think the price will fall, you might buy a put option. You could buy a put with a strike price of $90 that expires in one month. If XYZ stock drops to $80, you can exercise your option to sell at $90, securing a higher price than the market offers.
| Option Type | Your Expectation | Action | When It's Profitable (Generally) |
|---|---|---|---|
| Call | Price will rise | Right to BUY | Market Price > Strike Price |
| Put | Price will fall | Right to SELL | Market Price < Strike Price |
When Can You Use Your Option?
Besides the type of action (buy or sell), options are also categorized by when you can exercise your right. This distinction creates two styles: American and European.
American options are flexible. You can exercise your right to buy or sell at any point up to and including the expiration date. It's like having a concert ticket that's good for any show during the band's week-long residency.
European options are more rigid. You can only exercise them on the expiration date itself, not before. It's like having a ticket for a specific, one-night-only performance.
Most options on individual stocks traded in the U.S. are American style, while options on stock indexes are often European style.
This flexibility means American options are generally more valuable and carry a slightly higher premium than their European counterparts, all else being equal.
Anatomy of an Option Contract
When you trade options, you're dealing with standardized contracts. This ensures everyone is trading the same thing, which makes the market fair and efficient.
Typically, one equity option contract represents 100 shares of the underlying stock. This is the contract size. If you see an option premium quoted at $3.50, the total cost for one contract would be $3.50 per share x 100 shares, or $350.
So, every option contract clearly specifies:
- The Underlying Asset: What stock, index, or commodity the contract is for (e.g., Apple Inc. stock).
- The Type: Call or Put.
- The Strike Price: The price for the transaction (e.g., $150).
- The Expiration Date: The final day of the contract (e.g., July 19, 2024).
- The Style: American or European.
- The Size: How many shares are in one contract (usually 100 for stocks).
Understanding these basic components is the first step in exploring the world of options.