Options payoff and Greeks

~9 min read

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Why it matters

If you cannot draw a payoff diagram and name the first-order Greeks, you will not survive a derivatives interview. Options are nonlinear bets: limited loss if you buy, unbounded or large loss if you sell. Greeks turn that nonlinearity into a risk report — delta for direction, gamma for curvature, vega for vol, theta for time. Pricing models (Black–Scholes, trees, Monte Carlo) exist to turn these payoffs into a present value; the payoff itself is model-free.

Core concepts

Call vs. put, long vs. short. A European call pays \(\max(S_T - K, 0)\) at \(T\); a put pays \(\max(K - S_T, 0)\). Long option: you paid premium, payoff is floored at zero. Short option: you received premium and have the opposite payoff — that is where blow-ups live.
Intrinsic vs. time value. Intrinsic is the immediate-exercise value \(\max(\pm(S-K), 0)\). Time value is extra you pay for optionality (vol, remaining time). At expiry, time value is zero and price = payoff.
Moneyness. ITM / ATM / OTM is about \(S\) vs \(K\) (or forward vs \(K\)). A 1-delta call is deep ITM; a 25-delta put is OTM. Market makers quote by delta as often as by strike.
Put-call parity (European, same \(K,T\)): \(C - P = DF(T)\,(F - K) = S_0 e^{-qT} - K e^{-rT}\). Equivalently \(C + Ke^{-rT} = P + S_0\) (no dividends). If it fails, you replicate the cheap side and short the rich side — model-free arb (see Forwards / futures for \(F\)).
American vs. European. American can exercise early. For a non-dividend call, early exercise is suboptimal (better to sell than exercise). Puts, and calls on dividend-paying underlyings, can rationally exercise early.
The Greeks (sensitivities of value \(V\)):
- Delta \(\Delta = \partial V/\partial S\): share-equivalent. Call delta in \((0,1)\), put in \((-1,0)\).
- Gamma \(\Gamma = \partial^2 V/\partial S^2 = \partial\Delta/\partial S\): how fast delta moves. Long vanilla options have positive gamma.
- Vega \(\partial V/\partial\sigma\): long options like vol. (Not a Greek letter; still called vega.)
- Theta \(\partial V/\partial t\): usually negative for long options (time decay).
- Rho \(\partial V/\partial r\): smaller, more relevant for long-dated.
Gamma–theta tradeoff. Long gamma / long vega typically pays theta. You make money from big moves and bleed if the underlying sits still. Short options collect theta and get hurt by jumps and vol spikes.

Mental model

  Long call payoff at T (ignore premium):

  P&L |
      |           /
      |         /
      |       /
      |_____/___________  S_T
            K

  Long put: mirror, pays when S_T < K.

  Greeks as a dashboard, not a personality:
    delta  -> "how many shares am I synthetically long?"
    gamma  -> "how fast will that share-equivalent change?"
    vega   -> "what if implied vol moves 1 point?"
    theta  -> "what does one day of sitting still cost me?"

An option price is the discounted risk-neutral expected payoff. The payoff diagram is the contract; the Greeks are the local Taylor expansion of the pricing function.

Interview questions

Draw the P&L at expiry of a long call you bought for premium \(c\). Where is breakeven? Answer: Hockey stick: zero below \(K\), then \(S_T - K\). After premium, P&L is \(\max(S_T-K,0) - c\). Breakeven is \(S_T = K + c\). Below \(K\) you lose the whole premium, not more.
State put-call parity and one arbitrage if \(C - P > S - Ke^{-rT}\) (no dividends). Answer: Parity: \(C + Ke^{-rT} = P + S\). If call minus put is too expensive vs. forward, short call, long put, long stock, short the bond (i.e. borrow \(Ke^{-rT}\)). Terminal payoffs cancel; you keep the initial mispricing.
Why is a vanilla call’s delta between 0 and 1, and why is ATM delta not exactly 0.5 in Black–Scholes? Answer: A call is a slice of the stock (never more than 1 share, never a short). In BS, \(\Delta_{\text{call}} = e^{-qT} N(d_1)\), which is \(N(d_1)\) with \(q=0\). ATM-spot (\(S=K\)) still has \(d_1 = (r + \sigma^2/2)\sqrt{T}/\sigma > 0\), so delta \(> 0.5\). ATM-forward is closer to 0.5, still not exactly because of the \(\sigma^2/2\) term.
You are long a 1-month ATM straddle. What are you long/short in Greek space, and when do you make money? Answer: Long gamma, long vega, short theta, delta roughly flat. You make money if realized vol is high (spot moves a lot) or implied vol rises; you lose if the spot sits still (theta) or implied vol falls.
A trader says “I’m short gamma.” What does that actually mean for P&L over a day? Answer: Their delta-hedged P&L is roughly \(\tfrac12 \Gamma (\Delta S)^2 + \Theta \Delta t + \text{vega}\, \Delta\sigma\). Short gamma means they lose on a large \(|\Delta S|\) and typically collect theta if nothing happens. Jumps are lethal for short-gamma books.

Watch

Call payoff diagram — Khan Academy. Hockey-stick payoff for the long call.
Put-call parity — Khan Academy. Stock + put vs. call + bond, same terminal payoff.
FRM: Stock option Greeks — Bionic Turtle. Delta, gamma, vega, theta, rho as sensitivities, in eight minutes.