QuantMax

Options: fundamentals and arbitrage

5 lessons · 6 equations. Each lesson below gives its formulas and key rules; open the lesson for the full explanation.

Payoffs, moneyness and the bounds

Payoff at expiry

CT=max⁡(ST−K, 0),PT=max⁡(K−ST, 0)C_T = \max(S_T - K,\ 0), \qquad P_T = \max(K - S_T,\ 0)

The floor at zero is the entire point: you walk away rather than exercise into a loss.

Remember

  • Payoff is floored at zero; profit subtracts the premium and can be negative.

Put–call parity

The relationship

C−P=S0−Ke−rTC - P = S_0 - Ke^{-rT}

For European options on a non-dividend-paying asset, with the same strike and expiry.

Parity for options on futures

C−P=(F0−K) e−rTC - P = (F_0 - K)\,e^{-rT}

A futures contract costs nothing to enter, so there is no spot to finance and no dividend to strip out: the forward is the futures price itself. It is the form used for index and commodity options quoted against futures.

Remember

  • C−P=S0−Ke−rTC - P = S_0 - Ke^{-rT} for European options on a non-dividend payer.

Option strategies: what each one is actually a bet on

Long straddle breakevens

ST=K±(C+P)S_T=K\pm(C+P)

A call and put at the same strike cost C+PC+P; the underlying must finish beyond either breakeven to cover that premium.

Remember

  • Spreads cap a directional view and lower the breakeven; straddles trade movement.

Static arbitrage: the constraints every quote must respect

The density hidden in the prices

f(K)=erT ∂2C∂K2f(K) = e^{rT}\,\frac{\partial^2 C}{\partial K^2}

The second derivative of call prices with respect to strike is the discounted risk-neutral density. Non-negative butterflies are exactly the statement that this density is non-negative, and a finite-difference butterfly on three quoted strikes estimates it.

Remember

  • Calls fall with strike, and the fall is bounded by the discounted strike difference.

Borrow, dividends and what happens at expiry

Parity, with the real-world terms

C−P=Se−qT−Ke−rTC - P = S e^{-qT} - K e^{-rT}

The spot term is discounted by the dividend yield qq — and a stock borrow cost enters in exactly the same place, because paying to borrow is economically identical to the stock paying a dividend you do not receive.

Remember

  • Borrow cost enters parity exactly like a dividend yield.

Detailed formula cards