Options on Options

Second-Order Derivatives

These exotic options have, as their underlying asset, another option contract. They provide leverage and flexibility in expressing views on future volatility and price movements.

1. Compound Options (Options on Options):

   • What it is: A compound option is an option that gives the holder the right, but not the obligation, to buy or sell another option.

   • Key Features:

     • Two Strike Prices and Two Expiration Dates: One strike price and expiration date for the compound option itself, and another strike price and expiration date for the underlying option.
     • Leverage: Offers a leveraged way to bet on the price of the underlying asset.
     • Sensitivity to Volatility: Highly sensitive to changes in volatility.

   • Types:

     • Call on a Call (CoC): Gives the holder the right to buy a call option.
     • Call on a Put (CoP): Gives the holder the right to buy a put option.
     • Put on a Call (PoC): Gives the holder the right to sell a call option.
     • Put on a Put (PoP): Gives the holder the right to sell a put option.

   • Purpose:

     • Speculating on Future Volatility: Allows traders to bet on the direction of volatility.
     • Reduced Premium: Can be cheaper than buying the underlying option directly.

   • Example:

     • A call on a call option gives the holder the right to buy a call option on XYZ stock with a strike price of $50 and an expiration date of six months. The compound option itself expires in three months and has a strike price of $5.

   • Payoff (at Expiration of Compound Option):

     • CoC: Max(0, Value of Call - Strike Price of Compound Option)
     • CoP: Max(0, Value of Put - Strike Price of Compound Option)
     • PoC: Max(0, Strike Price of Compound Option - Value of Call)
     • PoP: Max(0, Strike Price of Compound Option - Value of Put)

2. Chooser Options (As You Like It Options):

   • What it is: A chooser option gives the holder the right to choose whether the option will be a call or a put at a specified future date (the choice date).

   • Key Features:

     • Choice Date: The date on which the holder must decide whether the option will be a call or a put.
     • Flexibility: Offers flexibility in adapting to changing market conditions.

   • Purpose:

     • Uncertain Market Direction: Useful when the investor is unsure of the future direction of the underlying asset but expects a significant move in either direction.

   • Example:

     • A chooser option gives the holder the right to choose, in three months, whether the option will be a call or a put on XYZ stock with a strike price of $50 and an expiration date of six months.

   • Valuation (at the Choice Date):

     • Value at Choice Date = Max(Call Value, Put Value)

   • Payoff (at Expiration, depending on the choice made):

     • If Call is Chosen: Max(0, S - K)
     • If Put is Chosen: Max(0, K - S)

Comparison Table: