Session 9

Call Options

• European Call Option:
    - Definition: Right to buy an underlying asset on exercise date T at a price (known as strike or exercise price K) determined today.

• American Call Option:
    - Definition: Right to buy an underlying asset on or before exercise date T at the strike price K set today.

• Option buyer pays a price known as the option premium to the option seller upon purchase of the option.

• Exercise Conditions for Call Options:
    - European Call Option: On exercise date T:
      - If $S_T > K$, call option buyer exercises the option.
      - Otherwise, the option is not exercised.
    - Payoff:
      - Call buyer (LONG) at T: ${Payoff}={Max}(S_{T}-K,0)$
      - Call seller (SHORT) at T: ${Payoff}=-{Max}(S_{T}-K,0)$
      - Here, $K$ is the strike price and $S_T$ is the spot price of the asset at time T.

• American Call Option Exercise Decision:
    - At any time $t ext{ with } t ext{ satisfying } t ext{ is less than or equal to } T$ , the call option buyer must decide whether to exercise the option or not.

European Call Option Detailed Definition

• Call Option Definition:
    - Right to purchase the underlying asset at strike price K on exercise date T.

• Participants:
    - Option Buyer: Also termed as the call option holder.
    - Option Seller: Also termed as the call option writer.

• Exercise Outcomes:
    - If $S_T < K$:     - Call option buyer does not buy the asset at price K.       - Payoff: $0$       - Payoff for the seller: $0$   - If $S_T > K$:
      - Call option buyer exercises the right to purchase the asset at price K.
        - Payoff for the buyer: $S_T - K$
        - Payoff for the seller: $- (S_T - K)$

• Profit Calculation:
    - Call Option Buyer Profit:
      - $ext{Profit} = ext{Payoff} - ext{Premium Paid}$
    - Call Option Seller Profit:
      - $ext{Profit} = ext{Payoff} + ext{Premium Received}$

European Call Option Payoff and Profit at Exercise Date

• Call Option Buyer:
    - Profit on exercise date = Payoff on exercise date - Option price paid at time 0 (Ignoring time value of money).

• Call Option Seller:
    - Profit on exercise date = Payoff on exercise date + Option price received at time 0 (Ignoring time value of money).

Put Options

• European Put Option:
    - Definition: Right to sell an underlying asset on the exercise date T at the strike price K fixed today.

• American Put Option:
    - Definition: Right to sell an underlying asset on or before exercise date T at a price (strike or exercise price K) determined today.

• Option buyer pays option premium to option seller at the time of acquiring the option.

• Exercise Conditions for Put Options:
    - European Put Option:
      - If $S_T < K$, put option buyer exercises the put.
      - Otherwise, the option is not exercised.
    - Payoff:
      - Put buyer (LONG) at T: $ext{Payoff} = ext{Max}(K - S_T, 0)$
      - Put seller (SHORT) at T: $ext{Payoff} = - ext{Max}(K - S_T, 0)$

• American Put Option Exercise Decision:
    - At each moment $t$ within the interval $[0, T]$, the put option buyer must choose whether to exercise the option or not.

European Put Option Detailed Definition

• Put Option Definition:
    - Right to sell the underlying asset at strike price K on exercise date T.

• Participants:
    - Option Buyer: Also referred to as the put option holder.
    - Option Seller: Also referred to as the put option writer.

• Exercise Outcomes:
    - If $S_T < K$:     - Put option buyer exercises the right to sell, selling the asset at price K.       - Payoff: $K - S_T$       - Payoff for the seller: $- (K - S_T)$   - If $S_T > K$:
      - Put option buyer will not exercise the right to sell at price K.
        - Payoff: $0$
        - Payoff for the seller: $0$

• Profit Calculation:
    - Put Option Buyer Profit:
      - $ext{Profit} = ext{Payoff} - ext{Premium Paid}$
    - Put Option Seller Profit:
      - $ext{Profit} = ext{Payoff} + ext{Premium Received}$

Call and Put Options Payoffs on Exercise Date Summary

• Summary of Payoffs on Exercise Dates:
    - Long Call Option: Payoff based on whether market price exceeds strike price.
    - Short Call Option: Payoff is conversely as long call.
    - Long Put Option: Payoff if market price is lesser than strike price.
    - Short Put Option: Inverse of long put payoff structure.

Trading Strategies

• Protective Put:
    - Definition: Long stock position combined with a long put option.
    - Usage: Employed when stock is owned and the investor wants to limit potential losses.
    - Payoff at Maturity:
      - For $S_T < K$:       - Long Stock: Payoff = $S_T$       - Long Put: Payoff = $K - S_T$       - Total: $K$     - For $S_T > K$:
        - Long Stock: Payoff = $S_T$
        - Long Put: Payoff = $0$
        - Total = $S_T$
    - Advantage: Limits potential losses, serves as portfolio insurance, efficient risk management tool.

• Straddle:
    - Definition: Combination of a long call and long put with the same strike price and expiration date.
    - Usage: Applied when large price movements are anticipated, without prediction of the direction.
    - Payoff at Maturity:
      - For $S_T < K$:       - Long Call: Payoff = $0$       - Long Put: Payoff = $K - S_T$       - Total: $K - S_T$     - For $S_T > K$:
        - Long Call: Payoff = $S_T - K$
        - Long Put: Payoff = $0$
        - Total: $S_T - K$
      - Worst-case Scenario: Lack of price movement results in both options expiring worthless.

• Bull Spread:
    - Definition: Combination of a long call at a lower strike price (K1) and short call at a higher strike price (K2), where $K1 < K2$.   - Usage: Predicted price increase up to but not exceeding K2.   - Payoff at Maturity:     - For $S_T < K1$: Total Payoff = $0$     - For $K1 < S_T < K2$: Total Payoff = $S_T - K1$     - For $S_T > K2$: Total Payoff = $K2 - K1$
    - Financial Structure: Pay premium on call K1, receive premium on call K2, resulting in a net premium of $c1 - c2$.