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A futures contract fixes a maturity and underlying quantity, but its quoted delivery price is marked to market: gains and losses from quote changes are settled through the margin account, typically daily. A newly settled position has zero contract value; the quoted futures price is not an upfront purchase price for that position. At maturity the quote equals the spot price. In the continuous-settlement idealization, discounted futures gains must be local martingales under the money-market risk-neutral measure. Thus the quote has zero pricing-measure drift. Assuming the relevant integrability makes it a true martingale, $\boxed{F_{tT}={\mathbb{E}}_Q[S_T\mid {\mathcal{F}}_t]}$.

This is a futures pricing formula. In contrast, an unsettled forward contract has delivery quote ${\mathbb{E}}_Q[D_{tT}{S}_T\mid {\mathcal{F}}_t]/{\mathbb{E}}_Q[D_{tT}\mid {\mathcal{F}}_t]$, where $D_{tT}=e^{-{\int }_t^T{r}_{u}du}$. Stochastic interest rates can make the two quotes differ. Contango means the futures quote is above spot for the maturity considered, while backwardation means it is below spot; futures curves are correspondingly described as rising or falling when comparing maturities.

For the Multivariate Ornstein-Uhlenbeck process, multiplication by the matrix exponential $e^{At}$ gives $d(e^{At}{X}_t)=e^{At}d{B}_t$, henceFor deterministic $X_0=x_0$, the Gaussian distribution has mean $e^{-At}{x}_0$ and covarianceEquivalently $\dot{C}=I-AC-C{A}^{\mathsf{T}}$, $C_0=0$. No symmetry or invertibility of $A$ is required. This covariance can also be evaluated from a single block matrix exponential: if the upper-right block of $\exp (t\left(\begin{array}{cc}-A& I\\ 0& A^{\mathsf{T}}\end{array}\right))$ is $H_t$, then $C_t=H_t{e}^{-A^{\mathsf{T}}t}$. If $X_0$ is random, these assertions hold conditionally on $X_0$; the unconditional law need not be Gaussian. A stationary Gaussian law exists when the eigenvalues of $A$ have positive real parts, but stability is unnecessary for the finite-time formulas.

For $\tau =T-t$, the conditional mean of $X_T$ is $e^{-A\tau }{X}_t$ and its conditional covariance is $C_{\tau }$. The Gaussian exponential-moment formula therefore gives the explicit quoteIt tends to $e^{b\cdot X_t}=S_t$ as $T\downarrow t$, as required.

A prepaid forward pays for delivery of an asset at a future time entirely in advance. In the Black-Scholes model with deterministic continuous dividend yield $q$, one share delivered at $T$ costs the displayed amount at $t$. Reinvesting the dividends on $e^{-q(T-t)}$ shares replicates one terminal share. An ordinary forward contract instead has no initial payment and a delivery payment fixed in advance; at constant interest rate $r$ its fair delivery price is $S_t{e}^{(r-q)(T-t)}$.

Put-call parity fixes the price difference between a European call option and a European put option on the same underlying stock, with the same strike price $K$ and expiry $T$. Assume no dividends or other payments from the stock before or at expiry, a known constant continuously compounded interest rate $r$, and a frictionless market. Borrowing and lending are available at that same interest rate; short selling of the stock, European call option, European put option, and zero-coupon bond is permitted. Settlement is certain, and the buy-and-hold portfolios below may be held until $T$ without interim funding restrictions or forced liquidation. These assumptions exclude transaction costs, taxes, borrowing fees, and execution or settlement risks from the arbitrage argument.

At expiry, the two positive parts cancel their corners:Thus a European call option and a short European put option together have the same terminal financial payoff as one stock share with a debt of $K$. A particularly readable claim replication uses two portfolios with nonnegative terminal amounts: portfolio A holds one European call option and a risk-free zero-coupon bond paying $K$ at $T$; portfolio B holds one European put option and one stock share. The zero-coupon bond costs $K{e}^{-rT}$ today, since the discount factor is $e^{-rT}$. There are no intermediate cash flows. For every terminal stock price,For the strike price $K=100$, even the following very different outcomes leave the two portfolios equal:

The law of one price forces $A_0=B_0$ in this setting: otherwise buy the cheaper portfolio and sell the more expensive portfolio using short selling. Their terminal obligations cancel in every scenario, while the initial price difference can grow in the bank account. Hence absence of arbitrage givesWith a constant per-period interest rate $i$ over $N$ discrete periods, the same reasoning gives $C_0-P_0=S_0-K(1+i{)}^{-N}$. This is a different compounding convention; $i$ is not the continuously compounded interest rate $r$.

For an illustrative stock price $S_0=100$, strike price $K=100$, expiry one year away, interest rate $r=0.05$ per year, and given European call option price $C_0=10$, put-call parity requiresBoth portfolios therefore cost approximately $105.12294245$. The inputs specify no stock price dynamics: $C_0=10$ is supplied for illustration, not calculated from the Black-Scholes model. Put-call parity determines the European put option price conditional on that European call option price; it does not determine both prices by itself. At zero interest rate, the same $S_0=K=100$ and $C_0=10$ give the exact result $P_0=10$.

To compute both directions of arbitrage, keep all the preceding inputs except the quoted European put option price. If it is $6$, portfolio B costs $106$ while portfolio A costs $105.12294245$. Buy A, sell B, and place the initial surplus $6-(10+100e^{-0.05}-100)\approx 0.87705755$ in the bank account. At expiry, A and the obligations from selling B cancel exactly, leaving $96e^{0.05}-100\approx 0.92202525$ in every scenario. If the European put option instead costs $4$, buy B for $104$ and sell A for $105.12294245$; the initial surplus is approximately $1.12294245$, growing to $100-94e^{0.05}\approx 1.18051694$. Both constructions start with zero external capital. They are model calculations under the stated frictionless market assumptions.

The beauty of put-call parity is that uncertain individual financial payoffs produce an exact relation between today's prices. No forecast of the terminal stock price and no probability distribution are needed. It provides a consistency check and lets claim replication exchange one contract for a combination of others. Dividends introduce extra cash flows, and American put options can be exercised early, so the displayed European equality cannot simply be transferred to those contracts. K. Foster's ^[ref] explain the matching portfolios; Jun Pan's ^[ref] discuss put-call parity and its dividend adjustment.

An original modern research source is Hans R. Stoll's ^[ref], The Journal of Finance 24(5), 801-824 (December 1969). Stoll studied the relation between contract prices and tested it against market data; this was a relation between prices rather than a formula fixing each price individually. Robert C. Merton's ^[ref], 28(1), 183-184, highlighted the importance of excluding premature exercise for an exact arbitrage equality. This distinction explains why the exercise assumption belongs in the statement of put-call parity.

The instantaneous continuously compounded interest rate. The continuous-time bank account grows at this rate; it equals the instantaneous forward rate at current maturity.

The maturity-indexed curve of interest rates inferred from prices of zero-coupon bonds. Continuously compounded yields $y(t,T)=-\log P(t,T)/(T-t)$ average the instantaneous forward rate over $[t,T]$. Modelling a curve requires both dependence across maturities and consistent evolution through observation time; specifying unrelated one-dimensional rate distributions is insufficient.