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If the claim holder receives a cash-flow rate $k(S_t,t)$ before maturity, the self-financing hedge matches the claim's cum-dividend gain $dp+kdt$. Its stock holding is the option delta $p_x$ and its bank-account value is $p-x{p}_x$. Comparing Itô formula with portfolio gains gives the displayed equation, with terminal value equal to the terminal payoff. The positive source term $k$ represents dividends paid by the claim; dividends on the underlying stock would instead alter the stock's pricing drift. Risk-neutral pricing includes both the discounted terminal payoff and the integral of discounted interim payments.

In the Black-Scholes model, suppose a stock pays continuous dividend yield $q$, a claim pays cash flow $f(t)S_{t}dt$, and its terminal financial payoff is $c{S}_T$. If the discounted cash-flow integral is finite, its risk-neutral pricing value is the displayed expression. To derive it, substitute $V=S\varphi (t)$ into the Black-Scholes equation with claim dividends, including stock drift $(r-q)S$. This gives ${\varphi }^{\prime }-q\varphi =-f$, $\varphi (T)=c$, which an integrating factor solves. The interest rate cancels because all payments are proportional to the same stock.

In the Black-Scholes model the terminal stock price conditional on current price $x$ is $xM$ for a positive lognormal multiplier $M$. Differentiation under the risk-neutral pricing expectation gives the displayed option gamma. Its sign follows that of $f^{\prime \prime }$, so convexity and concavity of the terminal payoff are preserved. An analogous first-derivative identity preserves monotonicity. With appropriate growth conditions, Gaussian integration by parts also gives $V_{\sigma }=\sigma (T-t)x^2{V}_{xx}$, fixing the sign of option vega for these payoff classes.

A convex terminal payoff has a convex pricing extension in the binomial market, because risk-neutral pricing is a positive weighted sum of rescaled copies of the payoff. Let the stock factors be $u,d$, the riskless gross factor be $R$, and $q=(R-d)/(u-d)$. The replicating portfolio in a binomial market satisfiesThe weights sum to one. Convexity orders the secant slopes on the adjacent intervals $[d^{2}s,uds]$ and $[uds,u^{2}s]$, proving the displayed inequalities. The inequalities need not be strict: an affine payoff has constant stock holding.

In the zero-interest local volatility model $d{S}_t=\sqrt{S_t}d{W}_t^Q$, the risk-neutral pricing value of the European contingent claim $e^{v{S}_T}$, for $v>0$ and $v(T-t)<2$, isThe compound Poisson transition law of a driftless square-root diffusion proves finiteness and this conditional-expectation formula. Alternatively, substituting $V(t,S)=e^{a(t)S+b(t)}$ into the pricing equation for a local volatility model gives $a^{\prime }+a^2/2=0$ and $b^{\prime }=0$, with terminal values $a(T)=v,b(T)=0$. Hence $a(t)=v/[1-v(T-t)/2]$ and $b(t)=0$. The conditional-expectation calculation verifies that this formal solution is the price of the unbounded claim. The delta hedge holds $a(t)V$ stocks and $(1-a(t)S_t)V/B_t$ bank units.

A forward start option whose call option strike is fixed at time $T_1$ as $\lambda S_{T_1}$, with expiry $T_2>T_1$ and $\lambda >0$. Its terminal financial payoff depends on the intermediate stock price as well as the final price. Two paths ending at the same price can therefore have different payoffs, so a recombining stock tree must retain enough history to value the contract.

In a homogeneous dividend-free Black-Scholes model, let $c(1,\lambda ,T_2-T_1)$ be the European call option price with current stock one and strike $\lambda$ over the remaining duration. Homogeneity gives value $S_{T_1}c(1,\lambda ,T_2-T_1)$ when the strike is fixed. The discounted stock is a martingale, so risk-neutral pricing before that date givesThe same argument holds in a homogeneous binomial options pricing model with the corresponding remaining-step call value. It depends on the normalized future pricing rule being independent of the stock level at the fixing date.

Under a risk-neutral measure $Q$, let the short rate be $r_t=g(t)+\sigma W_t^Q$, with deterministic locally integrable $g$ and constant $\sigma >0$. For a zero-coupon bond paying one at $T$, the risk-neutral pricing value isIndeed, conditional on ${\mathcal{F}}_t$, the random part of the integrated future short rate is $\sigma {\int }_t^T(T-u)d{W}_u^Q$, which is Gaussian with mean zero and variance ${\sigma }^2(T-t{)}^3/3$. Its exponential moment gives the formula. At points where $g$ is continuous, the instantaneous forward rate isFor merely locally integrable $g$, the same identity holds almost everywhere. The model allows negative short rates and gives explicit bond prices and forward curves.

John C. Cox developed mathematical methods for option pricing and risk-neutral pricing. With Stephen A. Ross and Mark Rubinstein, he introduced the Cox--Ross--Rubinstein model, whose binomial price tree gives a particularly transparent example of a replicating portfolio.^[ref]

At each node, the two possible next-step payments determine the holdings in a stock and a riskless asset. Repeating this calculation by backward induction values the final payment through the tree. For an American option, the calculation also compares continuation value with the value of immediate exercise. This turns the distinction between European and American exercise rules into a concrete computational decision.^[ref]

Institutional profile: mitsloan.mit.edu/faculty/directory/john-c-cox

Under the Black-Scholes model with interest rate $\rho$, volatility $\sigma$ and remaining time $\Delta$, risk-neutral pricing gives $V(S)=S[\log S+(\rho +{\sigma }^2/2)\Delta ]$. This follows by differentiating the lognormal exponential moment. Its option delta is $\log S+1+(\rho +{\sigma }^2/2)\Delta$ and its second stock derivative is $1/S$.

For $\alpha >0$, $\sigma >0$, this positive stochastic process is the exponential of an Ornstein-Uhlenbeck process. The Itô formula givesThe mean level of the logarithm differs from $\mu$ by the quadratic variation correction. In a stock model, specifying this under a physical probability measure does not by itself specify risk-neutral pricing. A non-dividend-paying traded stock has drift $rS$ under a risk-neutral measure; a different drift under that measure would require a corresponding dividend or carry convention.

Mathematical finance uses probability theory, linear algebra and mathematical optimization to study financial markets, portfolio decisions and contingent claim prices. One question asks which portfolio best serves a specified objective; another asks what price a contingent claim must have to exclude arbitrage. The latter can have an exact answer even when future stock prices are uncertain.

The central idea is claim replication: construct a self-financing portfolio delivering the same financial payoff as a contract in every possible state. The law of one price then identifies the contract's price with the portfolio's initial cost. The binomial market makes this beautiful connection between uncertainty and linear algebra visible using just two equations. Risk-neutral pricing expresses the same cost as a discounted expectation.

The one-step example below fixes today's stock price at 100, terminal prices at 120 or 80, and the bank account's gross growth factor at 1.05. A European call option with strike price 100 pays 20 or zero. Its replicating portfolio in a binomial market costs $250/21$, independently of the physical up probability, provided both states remain possible. These are conclusions within the specified financial market model.

Determining the value of a European option or American option from a specified financial market model and its permitted exercise rules. Risk-neutral pricing, replicating strategies and optimal stopping address different parts of this problem.

Invest each cash payment $q_v{S}_{v}dv$ received at time $v$ in the bank account until $T$. Its accumulated terminal amount is $q_v{S}_v{e}^{r(T-v)}dv$, so the entire future stream is equivalent, path by path, to the terminal payment ${\int }_t^T{q}_v{S}_v{e}^{r(T-v)}dv$.

The risk-neutral pricing formula, including this reinvested payoff, iswhere $d{\tilde{S}}_v=r{\tilde{S}}_{v}dv+\sigma {\tilde{S}}_{v}d{W}_v^{\mathbb{Q}}$. Equivalently, discount each cash flow directly to time $t$. This is the discounted-gains martingale principle underlying the PDE in part (a).

At maturity, one stock share is worth $(X_T-m{)}^{+}/n$. For a nonnegative strike $K$, the digital call option therefore pays one exactly onUnder the risk-neutral measure, $\log (X_T/x_0)$ is normal with mean $(r-{\sigma }^2/2)T$ and variance ${\sigma }^{2}T$. The risk-neutral pricing consequently givesThe effective company-value strike is $m+nK$, because both the outstanding debt and the number of shares matter. At $K=0$, the strict positive-share event remains $X_T>m$.

At $T_1$, with underlying spot $s$, the call option on a put option paysLet $D=K_2{e}^{-r(T_2-T_1)}$. The underlying put option value is strictly decreasing and convex in $s>0$, with limits $D$ at zero and zero at infinity. If $0<K_1<D$, there is a unique $s_{\ast }$ at which that put value equals $K_1$. Hence the compound option payoff is a decreasing convex curve from $D-K_1$ to zero on $[0,s_{\ast }]$, followed by a horizontal zero segment. There is an upward kink at $s_{\ast }$. If $K_1\ge D$, the payoff is identically zero.

For $K_1=0$, the holder obtains the underlying put at no cost, so the payoff is its full value at $T_1$. The discounted put price is a martingale under the risk-neutral measure. Using the tower property in the risk-neutral pricing,Explicitly, this is $K_2{e}^{-r(T_2-t)}\Phi (-d_2)-s\Phi (-d_1)$ with $d_1,d_2$ evaluated over $T_2-t$. The intermediate choice time disappears when its exercise cost is zero.

The exchange option has terminal payoffs determined by the two successive up/down moves:The two stocks span both one-step states, so this finite market is complete. The risk-neutral pricing uses the independent transition probabilities $q=1/3$ and $1-q=2/3$. Since the synthetic risk-free rate is zero,The price is approximately 7.56 in the currency units of the stock prices.

The terminal stock prices and European put option payoffs areLet $q$ be the risk-neutral up probability. Risk-neutral pricing and the supplied discounted-price condition giveso $9q^2-27q+14=0$. Its roots are $2/3$ and $7/3$; only the first is a probability. ThereforeThe current put price is $V_0=8/1.1^2=800/121$ pounds. This interest rate is specific to part (c), and differs from the zero rate in part (b).

The risk-neutral probability obeys $R=qu+(1-q)d$. ThusSubstitution into the risk-neutral pricing recursion gives

For the forward-start call option, the strike is fixed by the intermediate stock price. The four path payoffs areThe same final price eight has different payoffs on the two middle paths, so one must retain the history at expiry. At time one, risk-neutral pricing gives $C_1^u=(3/10)(2\cdot 4)=12/5$ and $C_1^d=(3/10)(2\cdot 2)=6/5$. ThusEquivalently, its value at the strike-fixing date is $S_1/5$ by homogeneity, so the martingale property of discounted stock prices gives the initial value $S_0/5$.

For a terminal price distinct from $K$, differentiating the put's financial payoff gives ${\partial }_K(K-S_T{)}^{+}=1_{\{S_T<K\}}$. The difference quotients are bounded by one, and the continuous lognormal distribution has zero probability at the strike. The dominated convergence theorem therefore permits differentiation inside its discounted expectation. The put-price strike derivative is exactly the price of the digital put option:Equivalently, risk-neutral pricing discounts the probability $\mathbb{Q}(S_T<K\mid S_t=S)=\Phi (-d_{-})$.