Options: Basics and Put–Call Parity

Options are the most widely known class of derivatives.
Unlike forwards and futures — which obligate both parties — an option grants rights but not obligations.
This asymmetry makes option pricing and hedging central to modern quantitative finance.

1. What Is an Option?

An option is a contract giving its holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price (the strike, $K$) before or at a specified date (the maturity, $T$).

Type	Action	Right to	Payoff at Maturity
Call	Buy	$S_T$	$\max(S_T - K, 0)$
Put	Sell	$S_T$	$\max(K - S_T, 0)$

Key parameters:

• $S_T$: price of the underlying at expiry
• $K$: strike price
• $T$: time to expiration
• $r$: risk-free rate
• $q$: continuous yield or dividend rate

2. Economic Meaning

Options introduce nonlinear exposure:

• A call provides upside participation with limited downside.
• A put provides downside protection.

These features make options valuable tools for both hedging and speculation.

Hedging Example

An investor holding a stock can buy a put to protect against large losses — creating a protective put position that behaves like insurance.

Speculative Example

A trader expecting volatility can buy both a call and a put (a straddle).
If the market moves significantly in either direction, the combined position profits.

3. Payoff and Profit

The option payoff is the cash flow at expiry, while profit accounts for the premium paid ($C_0$ or $P_0$).

Option	Payoff	Profit
Call	$\max(S_T - K, 0)$	$\max(S_T - K, 0) - C_0$
Put	$\max(K - S_T, 0)$	$\max(K - S_T, 0) - P_0$

The point where profit crosses zero is the break-even price:

• Call: $S_T = K + C_0$
• Put: $S_T = K - P_0$

4. Moneyness and Intuition

The relationship between $S_T$ and $K$ determines the moneyness of an option:

Condition	Call Option	Put Option
$S_T > K$	In the Money (ITM)	Out of the Money (OTM)
$S_T = K$	At the Money (ATM)	At the Money (ATM)
$S_T < K$	Out of the Money (OTM)	In the Money (ITM)

Moneyness affects both the option’s intrinsic value and time value: \(\text{Option Premium} = \text{Intrinsic Value} + \text{Time Value.}\)

5. European vs. American Exercise

• European options can be exercised only at expiration ($T$).
• American options can be exercised anytime up to $T$.

For non-dividend-paying assets, an American call is never optimal to exercise early because holding preserves time value.
However, for puts or dividend-paying assets, early exercise can be rational.

6. Put–Call Parity

A cornerstone relation linking calls, puts, and forwards is Put–Call Parity.
It arises from the equality of two portfolios with identical payoffs.

Portfolios

Portfolio	Composition	Payoff at $T$
A	Long Call + Short Put	$S_T - K$
B	Long Forward (deliver at $K$)	$S_T - K$

Since both have the same payoff, their current values must be equal:

\[C_0 - P_0 = S_0 e^{-qT} - K e^{-rT}.\]

If this equality fails, arbitrage exists:

• If LHS > RHS → short (Call–Put combo), long (synthetic forward).
• If LHS < RHS → opposite trade.

7. Practical Applications

1. Synthetic Positions:
   Build one asset’s payoff using others:
   • Long Call + Short Put = Synthetic Forward
   • Long Stock + Long Put = Protective Put
   • Short Stock + Long Call = Synthetic Put
2. Hedging and Risk Management:
   • Portfolio insurance using index puts.
   • Option collars for downside protection with capped upside.
3. Volatility Trading:
   • Long straddle or strangle for volatility exposure.
   • Short variance swaps or options for income generation.

8. “Price” vs “Intrinsic Value”

• Price is the market value of the option, determined by supply and demand.
  • Also called premium.
  • We could see it as the how much the “right” is worth today, the “right” including exercise value and the time value.
• Intrinsic Value is the immediate exercise value:
  • Call: $\max(S_T - K, 0)$
  • Put: $\max(K - S_T, 0)$
  • This is immediate exercise value, ignoring time value.
• Time Value is the extra amount paid over intrinsic value, reflecting the probability of future favorable movements.

9. Summary

• Options give asymmetric rights, not symmetric obligations.
• Payoff nonlinearity introduces convex exposure to price and volatility.
• Put–Call parity links option prices to forward values and ensures arbitrage-free pricing.
• Understanding payoffs and parity prepares you for the next step: the Binomial Model, where we build discrete-time option pricing intuition.

Next up: Options Pricing: Binomial Model