Lesson Objective: To analyze the mechanics of options, including calls and puts, option pricing in a two-state world via a simplified binomial model, and option trading strategies.

In-Depth Notes:

1. Option Fundamentals:
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 (strike price) on or before a specified date (expiration date). The seller (writer) of the option has the obligation to fulfill the contract if the buyer exercises the option.

• Call Options: Give the buyer the right to buy the underlying asset. The buyer profits if the price of the underlying asset rises above the strike price (plus the premium paid). Call options are used for bullish strategies (expecting prices to rise).

• Put Options: Give the buyer the right to sell the underlying asset. The buyer profits if the price of the underlying asset falls below the strike price (minus the premium paid). Put options are used for bearish strategies (expecting prices to fall) and for portfolio protection.

• Option Premium: The price paid by the buyer to the seller for the option. The premium is the market price of the option.

• Strike Price (Exercise Price): The price at which the underlying asset can be bought (call) or sold (put).

• Expiration Date: The date on which the option expires. After expiration, the option is worthless.

• American vs. European Options:

  • American Options: Can be exercised at any time up to and including the expiration date. Most exchange-traded options (including equity options in the US) are American-style.

  • European Options: Can only be exercised on the expiration date. European options are common in European markets and are the standard for many index options.

2. Option Pricing – The Binomial Model:
The binomial model is a simple but powerful framework for pricing options. It assumes that the price of the underlying asset can only move up or down by a specified amount over a given period.

• The One-Period Binomial Model: The model calculates the expected payoff of the option at expiration and discounts it back to the present at the risk-free rate. The key steps are:

  1. Define the Up and Down Factors: Determine the factors by which the underlying price can move up (u) or down (d).

  2. Calculate the Risk-Neutral Probability: Calculate the risk-neutral probability (p) of an up move: p = (1 + r - d) / (u - d), where r is the risk-free rate.

  3. Calculate the Expected Payoff: Calculate the expected payoff of the option at expiration: Expected Payoff = p × Option Payoff (Up) + (1 - p) × Option Payoff (Down).

  4. Discount to Present Value: Discount the expected payoff back to the present value at the risk-free rate: Option Price = Expected Payoff / (1 + r).

• The Two-Period Binomial Model: Extends the one-period model to two periods, allowing for more complex price paths. The model can be extended to multiple periods to provide a more accurate approximation of option prices.

3. Option Trading Strategies:
Options can be combined to construct a wide variety of trading strategies, each with a specific risk-return profile.

• Directional Strategies (Bullish/Bearish):

  • Buy Call: Bullish strategy with limited risk (premium paid) and unlimited upside.

  • Buy Put: Bearish strategy with limited risk (premium paid) and limited downside (the underlying price cannot fall below zero).

  • Covered Call: Buy the underlying asset and sell a call option (income generation strategy). This strategy provides downside protection (by the premium received) but caps upside potential.

  • Protective Put: Buy the underlying asset and buy a put option (portfolio insurance). This strategy limits downside risk while allowing for upside participation.

• Neutral Strategies (Income Generation):

  • Sell Covered Call: Generates income (premium) from selling calls on shares you own. The risk is that the stock is called away (sold) if the stock price rises above the strike price.

  • Cash-Secured Put: Sell a put option and set aside cash to buy the stock if the put is exercised. This strategy generates income (premium) and may result in buying the stock at a lower price.

• Volatility Strategies (Straddles and Strangles):

  • Long Straddle: Buy a call and a put with the same strike price and expiration date. This strategy profits from a large move in either direction (high volatility). The risk is limited to the total premium paid.

  • Long Strangle: Buy a call and a put with different strike prices (call strike above put strike). This is a cheaper version of the straddle but requires a larger price move to be profitable.

4. The “Greeks” – Measuring Option Risk:
The “Greeks” are measures of the sensitivity of an option’s price to various factors. They are critical for risk management and for constructing and managing option strategies.

• Delta (Δ): The rate of change of the option price with respect to the price of the underlying asset. Delta measures the option’s price sensitivity to a $1 change in the underlying price.

• Gamma (Γ): The rate of change of delta with respect to the price of the underlying asset. Gamma measures the convexity of the option price.

• Theta (Θ): The rate of change of the option price with respect to time. Theta measures the time decay of the option.

• Vega (V): The rate of change of the option price with respect to volatility. Vega measures the option’s sensitivity to changes in implied volatility.

• Rho (ρ): The rate of change of the option price with respect to the risk-free interest rate.