Every options strategy on this site rests on the same foundational math: how options are priced (Black-Scholes and beyond), what the Greeks measure (delta, theta, gamma, vega, rho), and how put-call parity connects calls and puts at the same strike. Once you understand these, every strategy becomes a deliberate combination of inputs and exposures. This page is the entry point for the foundations series.
What this page covers. Put-call parity → the Greeks (delta, theta, gamma, vega, rho) → how Greeks evolve across a trade → a worked parity example.
Put-Call Parity
Put-call parity is the relationship that makes options markets coherent. For European-style options on a non-dividend stock at the same strike and expiration:
Call Price − Put Price = Stock Price − Strike × e^(−rT)
In plain English: the price difference between a call and a put at the same strike equals the present value of the difference between the stock price and the strike. If this relationship breaks down — say a call is priced higher than the formula allows — arbitrageurs step in and restore the balance.
Put-call parity is the foundation of synthetic positions. A synthetic long stock = long call + short put (same strike, plus the discounted strike in cash). A synthetic short stock = long put + short call. These synthetics explain why bull call spreads and bear put spreads are equivalent under parity (up to a cash adjustment), why covered calls and cash-secured puts are mirror images, and why calendar spreads can be analyzed as combinations of synthetics at different expiries.
Read the full Put-Call Parity guide →
The Greeks
The Greeks measure how an option's price responds to changes in the underlying inputs. There are five first-order Greeks:
- Delta (Δ): how much the option price moves per $1 move in the underlying. Ranges 0–1 for calls, −1 to 0 for puts. ATM options have ~0.5 delta; deep ITM options have ~1 (call) or −1 (put).
- Theta (Θ): how much the option loses per day, holding all else equal. Negative for long options. Accelerates in the final 30 days.
- Gamma (Γ): the rate of change of delta per $1 move in the underlying. Largest ATM and near expiration.
- Vega (ν): how much the option price moves per 1% change in implied volatility. Positive for long options.
- Rho (ρ): how much the option price moves per 1% change in interest rates. Usually minor for short-dated options, meaningful for long-dated LEAPS.
For multi-leg strategies, the position Greeks are the sum of the leg Greeks. A bull call spread is long delta, negative theta, positive vega — net exposures that change as the underlying moves and time passes.
How Greeks Evolve Across a Trade
Greeks are not static — they change as the underlying moves and as time passes. The broad pattern across a trade's lifecycle (desk heuristic):
- Days 0–30: theta is moderate, vega is large. Position is sensitive to IV changes; directional exposure is stable.
- Days 30–14: theta accelerates, vega decays, gamma starts to matter near the strikes.
- Days 14–0: theta is large, gamma is enormous near the strikes, vega is small. Position behaves like a binary bet.
This "Greeks across the lifecycle" pattern is what drives most exit discipline: you exit not because of a calendar date, but because the position's risk profile has changed into something you didn't sign up for.
Worked Example: Pricing a Put-Call Pair with Put-Call Parity
Consider a stock at $100, a $100 strike, 90 days to expiry, and a 5% risk-free rate. By put-call parity, the call price minus the put price must equal the stock price minus the discounted strike: C − P = 100 − 100·e^(−0.05·0.25) = 100 − 98.76 = 1.24. If the listed call is trading at $3.20, the put must be trading at $1.96. If the put is offered at $1.50, an arbitrageur could sell the call at $3.20, buy the put at $1.50, buy the stock for $100, and borrow $98.76 (the discounted strike) — a net initial credit of $0.46 (3.20 − 1.50 − 100 + 98.76). At expiry the combined position nets to zero regardless of where the stock settles, so the $0.46 is locked in at inception. (Figures rounded for illustration.)
In practice this arbitrage rarely exists on the underlying exchange because market makers keep the spread tight. But the relationship still governs everything: it explains why deep ITM calls trade almost like the stock (delta ≈ 1, theta ≈ 0, vega ≈ 0); why deep OTM puts trade cheap unless the market expects a crash (delta → 0, gamma → 0); and why calendar spreads can be priced as combinations of synthetic longs and shorts at different expiries.
Two practical implications for traders: first, when implied volatility moves, both calls and puts move in the same direction but at different magnitudes depending on moneyness — the long vega trade benefits from any IV expansion; the short vega trade benefits from any IV contraction. Second, dividend-paying stocks break the basic parity formula, requiring an additional discounted dividend term; this is why options on dividend payers (large-cap utilities, REITs, energy majors) trade differently in the days leading up to ex-div dates.
Related Reading
- Understanding the VIX — vol regime context for everything above
- Volatility — skew, IV crush, gamma scalping
- Put-Call Parity: The Full Guide — the deep-dive on this page's core equation
For informational and educational purposes only. Not investment advice. Options trading involves substantial risk of loss. Past performance does not guarantee future results.