Payoff-Modifying Options

Tailoring the Payout Structure

These exotic options alter the standard call or put option payoff structure in unique ways, offering specific risk-reward profiles.

1. Gap Options:

   • What it is: A gap option is a call or put option where the strike price that determines whether the option is in the money differs from the strike price that determines the payoff amount.

   • Key Features:

     • Two Strike Prices: A trigger strike (K1) that determines whether the option is in the money and a payout strike (K2) that determines the payout amount.

     • Leveraged Payoff: Can create a leveraged payoff profile, where the option's value changes more rapidly than a standard option.

   • Types:

     • Gap Call Option: Pays (S - K2) if S > K1, otherwise pays 0.
     • Gap Put Option: Pays (K2 - S) if S < K1, otherwise pays 0.

     Where S is the spot price.

   • Purpose:

     • Enhanced Returns: Designed to amplify returns when the underlying asset moves in the desired direction.
     • Specific Market Views: Allows traders to express very specific views on the magnitude of price movements.

   • Example:

     • A gap call option has a trigger strike of $100 and a payout strike of $105. If the stock price is above $100 at expiration, the option pays (S - $105). If the stock price is below $100, the option pays nothing.

   • Payoff Formulas:

     • Gap Call: max(0, S - K2) if S > K1, 0 otherwise
     • Gap Put: max(0, K2 - S) if S < K1, 0 otherwise

2. Binary Options (Digital Options):

   • What it is: A binary option (also known as a digital option) pays a fixed amount if the underlying asset's price is above or below a specified strike price at expiration; there are two distinct outcomes.

   • Key Features:

     • Fixed Payoff: Pays a fixed amount if the option is in the money at expiration.
     • All-or-Nothing: Either receives the full fixed payout or nothing at all.

   • Types:

     • Cash-or-Nothing Call: Pays a fixed amount if S > K at expiration.
     • Asset-or-Nothing Call: Pays the spot price S if S > K at expiration
     • Cash-or-Nothing Put: Pays a fixed amount if S < K at expiration.
     • Asset-or-Nothing Put: Pays the spot price S if S < K at expiration

     Where K is the strike price.

   • Purpose:

     • Simple Bets: Allow traders to make simple directional bets on the underlying asset.
     • Hedging Specific Outcomes: Can be used to hedge against specific outcomes (e.g., a project being successful only if a certain price level is reached).

   • Example:

     • A cash-or-nothing call option pays $100 if the stock price is above $50 at expiration. If the stock price is $50 or below, the option pays nothing.

   • Payoff Formulas:

     • Cash-or-Nothing Call: Fixed Amount if S > K, 0 otherwise
     • Asset-or-Nothing Call: S if S > K, 0 otherwise
     • Cash-or-Nothing Put: Fixed Amount if S < K, 0 otherwise
     • Asset-or-Nothing Put: S if S < K, 0 otherwise

3. Shout Options:

   • What it is: A shout option allows the holder to "shout" (or lock in) a minimum payoff during the option's life.

   • Key Features:

     • Shout Feature: The holder can "shout" at any time before expiration, locking in the intrinsic value of the option at that moment.

     • Minimum Payoff: At expiration, the holder receives either the option's standard payoff or the locked-in shouted value, whichever is greater.

   • Purpose:

     • Downside Protection: Provides downside protection while still allowing for upside potential.
     • Locking in Gains: Allows holders to lock in gains if the option becomes profitable before expiration.

   • Example:

     • A shout call option has a strike price of $50. The holder can "shout" when the stock price is $60, locking in a minimum payoff of $10. If, at expiration, the stock price is above $60, the holder receives the standard call option payoff (S - $50). If the stock price is below $60, the holder receives the shouted payoff of $10.

   • Payoff Formulas:

     • Call Option: max(S - K, Shout Value, 0)
     • Put Option: max(K - S, Shout Value, 0)

Comparison Table: