Call & Put Options
beginnerThe building blocks of all option strategies. Understand how calls and puts work, visualize their payoffs, and explore a live options chain.
Parameters
P&L at Expiry
Scenario Analysis
How Options Work
What is an option?
An option is a contract that gives the buyer the right, but not the obligation, to buy or sell an underlying asset at a specified price (the strike) before or at a specified date (expiry). The seller (writer) has the obligation to fulfill if the buyer exercises.
Right to buy the underlying at the strike price. You profit when the stock goes up. Buying a call is a bullish bet.
Right to sell the underlying at the strike price. You profit when the stock goes down. Buying a put is a bearish bet or a hedge.
Long vs. Short
| Long (Buy) | Short (Sell) | |
|---|---|---|
| You pay/receive | Pay premium | Receive premium |
| Max profit | Unlimited (call) / Strike - Premium (put) | Premium received |
| Max loss | Premium paid | Unlimited (call) / Strike - Premium (put) |
| Risk profile | Limited risk, unlimited reward | Limited reward, unlimited risk |
Key terminology
When to use each strategy
The Black-Scholes Model
Option Pricing Formulas
The Black-Scholes model gives us a closed-form solution for European option prices. Published by Fischer Black, Myron Scholes, and Robert Merton in 1973, it remains the foundation of modern options pricing.
Where
Variables
The Greeks
The Greeks measure how an option's price changes with respect to each input variable. They are partial derivatives of the Black-Scholes formula.
Rate of change of option price with respect to the underlying price. A delta of 0.50 means the option moves $0.50 for every $1 move in the stock.
Rate of change of delta. Highest for ATM options near expiry. Tells you how quickly your delta exposure changes.
Time decay — how much value the option loses per day. Almost always negative for long options. Accelerates near expiry.
Sensitivity to volatility. How much the option price changes for a 1% change in implied volatility. Highest for ATM options with longer expiry.
Put-Call Parity
A fundamental relationship linking European call and put prices for the same strike and expiry. If this parity breaks, arbitrage is possible.
Key Assumptions
- European exercise only (no early exercise)
- No dividends paid during the option's life
- Constant risk-free rate and volatility
- Log-normal distribution of stock returns
- No transaction costs or taxes
- Continuous trading is possible
In practice, these assumptions are violated — which is why implied volatility varies by strike (the "volatility smile") and options trade at prices that differ from Black-Scholes theoretical values.
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