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A put option gives its holder the right to sell an underlying asset at a specified strike price. A European put option may be exercised only at maturity, whereas an American put option permits earlier exercise.

A contingent claim whose payoff depends on the underlying price history, rather than only its terminal price. Asian options use averages, barrier options use whether a threshold has been crossed, and lookback options use running extrema. In the Black-Scholes model, the Brownian Martingale representation theorem still gives a replicating strategy; its portfolio wealth need not be a function of the current stock price alone. Adding a running integral or extremum can give a finite-dimensional state for pricing.

The monetary amount delivered by a contingent claim at its payment time, represented by a random variable measurable with respect to the information then available. For a European call option it is $(S_T-K{)}^{+}$, and for a European put option it is $(K-S_T{)}^{+}$. A payoff is the delivered amount, whereas the claim is the contract and its current price is the cost of acquiring or replicating that contract.

A constructible regular polygon is a regular polygon admitting a compass-and-straightedge construction. The Gauss-Wantzel theorem classifies its possible numbers of sides.

Set $S_2=\sqrt{15}{\beta }^{3/2}{v}_{\eta }^{2}h$ and $r=(15\beta {)}^{3/4}\eta x$. The Kolmogorov equation for structure functions and the constant-skewness turbulence closure give the displayed ordinary differential equation, with $h(0)=0$. Its small-separation balance is $h\sim x^2$; its large-separation balance is $h\sim x^{2/3}$. The first matches differentiable velocity increments, and the second recovers the assumed Kolmogorov two-thirds law.

A fee proportional to terminal portfolio wealth but reduced by accumulated portfolio quadratic variation changes the optimal risk exposure of a manager with constant relative risk aversion utility. Write $A=\sigma {\sigma }^T$, $b=\mu -r1$ and assume $A$ is invertible. The fee process has drift $r+b^T\pi -\epsilon {\pi }^{T}A\pi /2$ and volatility ${\sigma }^T\pi$. Its Hamilton-Jacobi-Bellman equation therefore maximizes $b^T\pi -(R+\epsilon ){\pi }^{T}A\pi /2$, giving ${\pi }^{\ast }=A^{-1}b/(R+\epsilon )$. The fee increases effective relative risk aversion from $R$ to $R+\epsilon$, independently of its positive scale $a$.

There is also a direct global bound. Put $p=1-R$, $v^{\ast }={\sigma }^T{\pi }^{\ast }$, and tilt probability by $\exp (p{v}^{\ast }\cdot W_T-p^2|v^{\ast }{|}^{2}T/2)$. Under this probability let $Z$ be the stochastic exponential of $(1+\epsilon )\int ({\sigma }^T\pi -v^{\ast })\cdot d{W}^{\ast }$. The terminal fee ratio satisfies $(y_T/y_T^{\ast }{)}^p=Z_T^{p/(1+\epsilon )}$. A nonnegative local martingale has expectation at most one. The Jensen inequality, applied to a concave positive power when $p>0$ and a convex negative power when $p<0$, proves that the constant exposure maximizes expected CRRA utility. Effective risk aversion one uses logarithmic utility.

Logarithmic utility on positive portfolio wealth has derivative $1/x$ and relative risk aversion one. Its utility conjugate is $V(y)=-\log y-1$, since the maximizing wealth is $1/y$. It satisfies the Inada conditions.

For zero cash interest, constant excess drift $\mu$, volatility $\sigma$, and both terminal and consumption utility $2\sqrt{x}$, set $A=\sigma {\sigma }^T$ and assume $\mu$ belongs to the image of a linear map $A$. Let $A^{+}$ be the Moore-Penrose inverse and $q={\mu }^T{A}^{+}\mu$. Then $h^{\prime }+qh+1=0$, $h(T)=1$, so $h(t)=e^{q(T-t)}+(e^{q(T-t)}-1)/q$ for $q>0$, and $h(t)=1+T-t$ for $q=0$. The Hamilton-Jacobi-Bellman equation has the displayed solution. Maximizing dollar investment and consumption are ${\theta }_t=2X_t{A}^{+}\mu$ and $C_t=X_t/h(t)$. The resulting positive portfolio wealth is a geometric Brownian motion with a deterministic time-dependent drift, and finite-horizon moment bounds justify equality in verification.

In a Brownian complete market with constant coefficients and nonsingular $\sigma$, maximize expected CRRA utility of terminal portfolio wealth divided by a geometric stock index. Writing $p=1/R$ and $q=n^{-1}1$, the pointwise marginal condition gives $w_T^{\ast }={\lambda }^{-p}{\zeta }_T^{-p}{J}_T^{1-p}$. Its state-price budget constraint determines ${\lambda }^{-p}=w_0/\mathbb{E}[({\zeta }_T{J}_T{)}^{1-p}]$. Conditional lognormal moments and replication then give the displayed constant dollar fractions. The first term is ordinary optimal risk exposure; the second hedges the random benchmark. The bank holds the remaining fraction, which can be negative when borrowing is allowed.

For a self-financing portfolio earning both running reward $e^{-\rho t}{W}^{\gamma }$ and terminal reward $W_T^{\gamma }$, $0<\gamma <1$, with no consumption withdrawals, Hamilton-Jacobi-Bellman equation gives $J(W,t)=A(t)W^{\gamma }$ and the displayed constant risky fraction. The coefficient satisfies $A^{\prime }+\kappa A+e^{-\rho t}=0$, $A(T)=1$, where $\kappa =\gamma r+\gamma (\mu -r{)}^2/(2(1-\gamma ){\sigma }^2)$. Its positive solution is $e^{\kappa (T-t)}+{\int }_t^T{e}^{\kappa (s-t)-\rho s}ds$. Risk aversion is $1-\gamma$; increasing $\gamma$ raises risky investment and can require borrowing.

For Gaussian terminal prices with positive definite covariance $V$, set $M=(\gamma V{)}^{-1}$ and $\eta =(S_0^{T}M\mu -w_0)/(S_0^{T}M{S}_0)$, assuming $S_0\ne 0$. The normal moment-generating function turns expected exponential utility into a strictly concave mean-minus-variance objective. Its free optimum has cash $S_0^{T}M{S}_0(1+r-\eta )$. If this is positive, keep the free optimum; otherwise the cash constraint binds and the multiplier becomes $\eta$. Thus zero optimal cash is equivalent to $\eta \ge 1+r$, including equality.

With positive welfare weights $a_j$, maximize ${\sum }_j{a}_j{\beta }_j^t[-e^{-{\gamma }_j{c}_j}/{\gamma }_j]$ subject to ${\sum }_j{c}_j=d$. Interior first-order conditions give $a_j{\beta }_j^t{e}^{-{\gamma }_j{c}_j}=\nu$. Put $\Gamma =({\sum }_j{\gamma }_j^{-1}{)}^{-1}$, $\log \bar{\beta }=\Gamma {\sum }_j{\gamma }_j^{-1}\log {\beta }_j$ and $\log A=\Gamma {\sum }_j{\gamma }_j^{-1}\log a_j$. ThenIf consumption must be nonnegative, the interior formula is replaced by $c_j=\max \{0,(\log a_j+t\log {\beta }_j-\log \nu )/{\gamma }_j\}$, with $\nu$ determined by resource clearing. These are Pareto allocations; realizing them as a competitive equilibrium with one productive asset requires feasible financing. For common ${\beta }_j=\beta$ and initial ownership $\Gamma /{\gamma }_j$, constant holdings and $C_t^j=(\Gamma /{\gamma }_j)d_t$ give such an equilibrium whenever fundamental prices and the utility sums are finite.

For exponential utility $(1-e^{-ax})/a$ on unrestricted terminal portfolio wealth in the Black-Scholes model, the optimal dollar holding in the stock is the displayed deterministic amount. Set $\vartheta =(\mu -\rho )/\sigma$ and let $W^Q=W+\vartheta t$ be the Brownian motion under the risk-neutral measure. The terminal optimizer is $X^{\ast }=w_0{e}^{\rho T}+\vartheta W_T^Q/a$. Its marginal utility is proportional to the state-price density, and the concave supporting-tangent inequality proves global optimality under the state-price budget constraint. The expected utility at the optimum is $(1-\exp (-a{w}_0{e}^{\rho T}-{\vartheta }^{2}T/2))/a$. A nonnegative-wealth constraint changes this optimizer.

For unrestricted final portfolio wealth in an $n$-period binomial market, write $R$ for the bank growth factor, $a>0$ for absolute risk aversion, and $Z=dQ/dP$ for the binomial-market probability density. The displayed optimizer has the state-price budget constraint ${\mathbb{E}}_Q{H}^{\ast }=R^n{w}_0$. Its exponential utility derivative satisfies $e^{-a{H}^{\ast }}=\lambda Z$ for a constant $\lambda >0$. The concave supporting-tangent inequality bounds any competing expected utility by this value, because the expected linear error is $\lambda {\mathbb{E}}_Q(H-H^{\ast })=0$. Strict concavity proves uniqueness, and market completeness supplies replication. The formula is altered by a constraint requiring nonnegative final wealth. With physical up probability $p$ and risk-neutral probability $q$, ${\mathbb{E}}_Q\log Z=n[q\log (q/p)+(1-q)\log ((1-q)/(1-p))]$.

A nonzero uniform equilibrium of the conserved-field real amplitude equation obeys $\mu =R^2-\alpha R$. Fixed zero mean excludes the constant $B$ perturbation. Its homogeneous amplitude eigenvalue is $-r$. For a nonzero Fourier wavenumber $k$, the stability matrix has trace $-r-(1+\sigma )k^2$ and determinant $\sigma k^2(k^2+r-2\delta R^2)$. The displayed conditions are necessary and sufficient for strict linear stability on a period $2L$. In an infinite-domain limit the nonzero-mode condition becomes $r\ge 2\delta R^2$; a long-wave instability need not occur in a sufficiently short domain.

For $0<\delta <1$ and $\alpha \ne 0$, a broad localized plateau in the conserved-field real amplitude equation has effective parameter $\nu =\mu -\delta ⟨A^2⟩$ and stationary equation $A^{\prime \prime }+\nu A+\alpha A^2-(1-\delta )A^3=0$. Equal endpoint values of its first integral give $R=2\alpha /[3(1-\delta )]$, $\nu =-2{\alpha }^2/[9(1-\delta )]$. If the plateau occupies fraction $h$ of a long periodic interval, $⟨A^2⟩\simeq h{R}^2$ supplies the displayed relation. It applies when both the plateau and exterior are long relative to the front width; $0<h<1$ gives its leading existence interval.

For equal heat extraction, the density increase caused by brine rejection can exceed that caused by cooling near the freezing point because the thermal density response is weak. Using conservation of salt during freezing, the ratio is $R=(\partial \rho /\partial S)S_0{f}_i/[(1-f_i)|\partial \rho /\partial T|\Delta T]$, with $f_i=c_{p,w}\Delta T/L_f$. This is a parcel comparison, not a guarantee that freezing overcomes any ambient density stratification.

Let liquid mass $M$ freeze incrementally, and let each new ice increment retain salinity $rS$ with constant $0\le r\le 1$, where $S$ is the instantaneous well-mixed liquid salinity. Conservation gives $d(MS)=rSdM$, hence $dS/S=(r-1)dM/M$. Integrating yields $S/S_0=(M/M_0{)}^{r-1}$. For frozen mass fraction $f$, $\Delta S=S_0[(1-f{)}^{-(1-r)}-1]\simeq (1-r)S_{0}f$. Specifying a fixed bulk ice salinity instead produces a different, though first-order compatible, finite-mass balance.

Basic radiance is radiance divided by the square of the refractive index. For radiance $L$ at a fixed wavelength,Lossless propagation conserves reduced radiance along each light ray; $L$ itself is conserved only when the endpoint refractive indices agree. A larger irradiance at a smaller image can result from admitting more directions without increasing reduced radiance. Uniform wave absorption can lower radiant flux while leaving the occupied geometric etendue unchanged; an opaque aperture removes part of that occupied set. ^[ref]

Place two equal point masses of $1\mathrm{kg}$ each on opposite sides of a fixed origin, at equal radius $r$, and let them share a signed angular velocity $\omega =\dot{\theta }$ in a plane. Their total mass is $M=2\mathrm{kg}$ and their moment of inertia about the perpendicular axis is $I=M{r}^2$. Radial pulling forces have zero torque about that origin, soTake an initial radius $r_i=1\mathrm{m}$ and angular velocity ${\omega }_i=1rad/s$, and a final radius $r_f=0.5\mathrm{m}$. ThenandThese are endpoint states with ${\dot{r}}_i={\dot{r}}_f=0$. During the pull, the full kinetic energy also contains radial motion: $K=M{\dot{r}}^2/2+K_{\mathrm{rot}}$. Assume the pulling apparatus supplies all net work to the two point masses; the work-energy theorem then fixes its net work at $+3\mathrm{J}$, including any radial acceleration and braking. Each mass spirals inward during the pull, allowing its inward force to do positive work. Conservation of angular momentum therefore predicts the final angular velocity without requiring kinetic energy conservation. For an apparatus included in a larger system, its stored energy supplies the increase consistently with conservation of energy. OpenStax uses the same mechanism to explain a skater pulling her arms inward. ^[ref]