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For a fixed twice-differentiable terminal payoff with sufficient growth control, differentiate its lognormal distribution pricing integral. Gaussian integration by parts gives the displayed identities. Convexity implies $p_{xx}\ge 0$, so increasing spot volatility cannot lower the price. The interest rate sensitivity depends on the sign of the cash position $p-x{p}_x$. Calendar-time sensitivity is $p_t=\rho (p-x{p}_x)-\frac{1}{2}{\sigma }^2{x}^2{p}_{xx}$. Thus a convex payoff with a negative cash position has a nonincreasing price at fixed spot when $\rho \ge 0$. Negative rates invalidate that conclusion: $f(x)=x-K$ gives $p=x-K{e}^{-\rho (T-t)}$ and $p_t=-\rho K{e}^{-\rho (T-t)}>0$ when $\rho <0$.

When the market price of risk is constant and nonzero spot volatility gives access to the driving Brownian motion, changes in volatility only rescale the dollar holding needed for fixed Brownian portfolio exposures. With unrestricted holdings and constant interest and discount rates, the CRRA utility value is consequently independent of the factor state. Zero volatility requires separate treatment.

In a one-factor market whose filtration is the usual augmentation of the natural Brownian filtration, with nonzero spot volatility and local martingale deflator $Y$, the Brownian martingale representation theorem constructs a nonnegative replicating strategy for a bounded nonnegative contingent claim. The minimal initial cost among nonnegative self-financing portfolios is $\mathbb{E}[Y_T{\xi }_T]$.

In a one-factor diffusion market, the market price of risk is the excess drift per unit spot volatility. It is the coefficient in the Brownian motion part of a local martingale deflator, $d{Y}_t=-Y_t(r_{t}dt+{\lambda }_{t}d{W}_t)$.

The vega of an European option is the derivative of its price with respect to spot volatility, holding its other parameters fixed. A dividend-free European call option under the Black-Scholes formula has $\nu =S_0\sqrt{T}\varphi (d_1)>0$. Differentiation proves this because $S_0\varphi (d_1)=K{e}^{-rT}\varphi (d_2)$ and ${\partial }_{\sigma }(d_1-d_2)=\sqrt{T}$. Positivity proves uniqueness of Black-Scholes implied volatility for prices strictly between the limiting call-price bounds.

Keep the maturity $T$ fixed and write $F(t,T)=\exp L(t,x_t)$, whereAs calendar time increases, ${\partial }_{t}a=\alpha a$, so $L_x=a$, $L_{xx}=0$ and $L_t=\alpha a(x-\tilde{\mu })-{\sigma }^2{a}^2/2$. Applying the Itô formula to $L$ and then to its exponential givesThe Itô formula's exponential correction cancels the displayed drift of $L$. The zero drift also follows from the tower property of conditional expectation: for $t\le u\le T$, ${\mathbb{E}}_t[F(u,T)]={\mathbb{E}}_t[S_T]=F(t,T)$. In particular $F(T,T)=S_T$, while the quote's instantaneous spot volatility approaches $\sigma$ as expiry approaches.

This is a martingale under the same measure used in the stipulated conditional expectation. To interpret it as futures pricing, that measure must be a suitable pricing measure. For a non-dividend-paying traded stock at a deterministic interest rate, the absence of arbitrage instead gives a forward contract delivery price $S_t{e}^{r(T-t)}$. A physical-measure expectation under the given mean-reverting drift is not automatically that price; the question's expectation convention is sufficient for the calculation above, but not for this additional identification.

Let ${\varphi }_T(y)=(2\pi T{)}^{-1/2}{e}^{-y^2/(2T)}$ be the normal distribution density. The Brownian reflection principle gives the killed endpoint densityIndeed, reflection after the first hit of $a$ maps paths ending below $a$ that have hit the barrier to paths ending at $2a-y$; symmetry gives ${\varphi }_T(2a-y)={\varphi }_T(y-2a)$.

By the Girsanov theorem, weighting the path law by $\exp (\nu W_T-{\nu }^{2}T/2)$ makes the coordinate process a Brownian motion with drift $\nu$. Completing the two squares yieldsIntegration up to $x\le a$ proves the joint endpoint and maximum law for drifted Brownian motion:Here $\Phi$ is the standard normal distribution function. Setting $x=a$ also gives a continuous barrier distribution, so equality at the barrier has zero probability.

For the up-and-in option, use the risk-neutral measure of the Black-Scholes model with interest rate $\rho$ and spot volatility $\sigma >0$. ThenAbove $a$, the barrier has certainly been reached. Below $a$, subtracting the killed density from the unrestricted endpoint density leaves $e^{2a\nu }{\varphi }_T(y-2a-\nu T)$. Thus the discounted expected value of its payoff isIn the second integral put $z=y-2a$, and defineThen $S_0{e}^{\sigma y}=S_0{e}^{\sigma z}/\kappa$, $z\le -a$ means $S_0{e}^{\sigma z}\le \kappa b$, and $e^{2a\nu }={\kappa }^{-\nu /\sigma }$. Both integrals now use the unrestricted terminal stock distribution. This proves that the reflected European payoff for an up-and-in option isTheir initial risk-neutral pricing values agree for payoffs with finite absolute expectations in these integrals. The identity concerns prices; the two terminal payoffs need not agree along individual paths.

Let $T=t_0$, $\tau =T-t$, and let $N$ have the standard normal distribution. For a twice differentiable payoff with enough growth control for the following expectations and derivatives, risk-neutral pricing in the dividend-free Black-Scholes model givesThe conditional discounted payoff is a martingale under the risk-neutral measure. Applying the Itô formula to $e^{-\rho s}V(S_s,s)$ and setting its drift to zero therefore gives the Black-Scholes equationThe delta hedge holds $\Delta =V_x$ shares; the remaining bond value is $\beta =V-x\Delta$, or $h=\beta /e^{-\rho (T-t)}$ units of the zero-coupon bond. Substitution in the Itô formula and the displayed equation gives $dV=\Delta dS+hdB$, proving replication and self-financing.

Differentiating the pricing expectation in the current stock price gives its option delta and option gamma:Thus a nondecreasing payoff has a nondecreasing price; a nonincreasing payoff has a nonincreasing price. A convex payoff has $\Gamma \ge 0$, and a concave payoff has $\Gamma \le 0$. This is convexity preservation in Black-Scholes pricing.

The Black-Scholes parameter sensitivities follow directly from the same expectation. Differentiation in the interest rate gives the option rhoFor spot volatility, differentiation first givesThe Gaussian integration by parts identity $\mathbb{E}[N\psi (N)]=\mathbb{E}[{\psi }^{\prime }(N)]$, applied to $\psi (N)=f^{\prime }(xM)xM$, cancels the term containing $f^{\prime }$ and gives the option vegaConsequently increasing spot volatility increases a convex payoff's value and decreases a concave payoff's value, with weak inequalities in affine or degenerate cases. A short bond position gives positive option rho, while a long bond position gives negative option rho. The physical stock drift $\mu$ does not enter these prices: the risk-neutral measure replaces it by $\rho$.

The option theta is the calendar-time derivative at fixed current stock price. The Black-Scholes equation givesFor $\rho \ge 0$, a convex payoff with $\beta <0$ therefore has $V_t\le 0$, while a concave payoff with $\beta >0$ has $V_t\ge 0$. The signs are strict when $\rho >0$. At $\rho =0$, an affine payoff can give equality. Remaining-maturity sensitivity is $V_T=-V_t$ when the payoff is independent of $T$. All these time comparisons hold at fixed current stock price.

The stated time-monotonicity conclusion needs the nonnegative-rate condition. For example, the convex affine payoff $f(x)=x-K$ hasThus with negative interest rates it provides an explicit counterexample even though the replica is short in bonds.

Subtract the boundary slope from the Brownian motion: $X_t=B_t-bt$. Then $T_{a,b}$ is the first-passage time of $X$ to the fixed level $a$. The zero-drift Brownian reflection principle givesHere $\Phi$ is the standard normal distribution function. To change the drift to $-b$, use the Girsanov theorem on each finite horizon, with density $L_t=\exp (-b{B}_t-b^{2}t/2)$. On $\{T_a\le t\}$, conditional expectation at the stopped time gives $\mathbb{E}[L_t\mid {\mathcal{F}}_{T_a}]=L_{T_a}$, because the density is a true martingale. Since $B_{T_a}=a$, the drifted Brownian first-passage density is consequentlyThere may be additional mass at infinite hitting time; the displayed density only describes finite passage. With $e^{-\theta \infty }=0$, apply the given zero-drift Brownian first-passage Laplace transform:The zero-discount limit is $1$ for $b\le 0$ and $e^{-2ab}$ for $b>0$, so the hitting probability is less than one precisely when the line moves away with positive slope.

For a direct verification of the linear-boundary Brownian first-passage distribution, defineBoth terms tend to zero as $t\downarrow 0$. Let $\varphi$ be the standard normal density. The identityfollows by expanding the two squares. Differentiating the two normal probabilities then givesSince this is the previously derived hitting density and both distribution functions start at zero,This proof applies to every real $b$, including the defective distribution when $b>0$.

For the barrier digital put, use the dividend-free Black-Scholes model with constant interest rate $\rho$ and spot volatility $\sigma >0$. Under the risk-neutral measure,Set $T=t_0$ andReaching the barrier $c$ is equivalent to $W_t^Q$ reaching $a+bt$. The contingent claim pays one only on the survival event $\{T_{a,b}>T\}$. Risk-neutral pricing therefore givesThis is a path-survival price, not the price of a terminal digital put option. Continuity of the stock paths and the absence of an atom in the finite-horizon maximum imply that alternative touching conventions change no price. As $c\to \infty$, the value tends to $e^{-\rho T}$; as $c\downarrow S_0$, it tends to zero. Both checks agree with the survival interpretation.

Let $T=t_0$, $\tau =T-t$, $x=S_t>0$, and $\sigma >0$. Use a dividend-free Black-Scholes model with constant interest rate $\rho$. The terminal payoff is assumed independent of the parameters being varied. In addition to twice differentiability, impose enough integrability to differentiate the pricing formula; for example, $f,f^{\prime },f^{\prime \prime }$ of polynomial growth suffice. Twice differentiability alone is not enough: the smooth payoff $f(x)=e^{x^2}$ has infinite expectation under every nondegenerate lognormal distribution.

Under the risk-neutral measure $Q$, future stock values conditional on $S_t=x$ have the formClaim replication, or equivalently the martingale property of discounted value in the complete market, givesThe replicating strategy holds the option delta $p_x$ shares and bank-account value $\beta =p-x{p}_x$. No physical drift parameter appears: the Girsanov theorem replaces the physical drift by $\rho$ under the pricing measure, and the replication argument cancels it directly. The risk-neutral pricing operator is linear in the payoff and preserves pointwise payoff inequalities.

Here is a verification of the Black-Scholes equation using the integral itself. Differentiation in spot givesDifferentiating with respect to $\tau$ givesFor $G(Z)=S_T{f}^{\prime }(S_T)$, the Gaussian integration by parts identity $\mathbb{E}[ZG]=\mathbb{E}[G^{\prime }]$ yieldsSubstitution cancels the $-{\sigma }^2/2$ term and provesSince $p_t=-p_{\tau }$, this is precisely the Black-Scholes equation. As $\tau \downarrow 0$, $M\to 1$ and growth control permits passage to the limit, giving $p(x,T)=f(x)$. The delta hedge then supplies the admissible replicating gains under the same regularity assumptions.

The Black-Scholes parameter sensitivities describe the dependence on the model parameters without assuming a particular payoff. If $f$ is increasing, then $p_x\ge 0$. If $f$ is convex, then $p_{xx}\ge 0$, so its price is convex in spot. The option gamma $p_{xx}$ measures the response of the option delta to spot changes.

For the interest rate, the distribution of the future stock also changes with $\rho$, so looking only at discounting gives the wrong general answer. Differentiation gives the option rhoThus a short bank-account position gives positive rate sensitivity, while a positive bank-account position gives negative rate sensitivity. For a cash payoff the result is negative; for the payoff $f(x)=x$ it is zero.

For spot volatility, differentiate the lognormal multiplier and then use Gaussian integration by parts:Consequently a convex payoff has nonnegative option vega. This monotonicity can also be understood by a mean-preserving increase of dispersion in the discounted stock. Strict positivity requires nonzero curvature seen with positive probability; an affine payoff has zero option vega.

Time enters through the remaining maturity $\tau$. Holding spot fixed, the Black-Scholes equation givesUnder the conventional assumption $\rho \ge 0$, a convex payoff and a short bond position $\beta <0$ imply $p_t\le 0$. The inequality is strict if $\rho >0$, or if $p_{xx}>0$ with positive volatility. Thus the price at fixed stock value is nonincreasing in calendar time, and nondecreasing in maturity, under these hypotheses. It is not a claim that the realized stochastic price path decreases: that path still has Brownian exposure $\sigma S_t{p}_{x}d{W}_t^Q$.

The printed conclusion needs the nonnegative-rate qualification; it is false for arbitrary negative rates. For $K>0$, the twice-differentiable convex payoff $f(x)=x-K$ hasThis is an exact counterexample, not a failure of the pricing formula. Also, with $\rho =0$ and an affine payoff, “decreasing” can only mean nonincreasing. Without payoff monotonicity or the specified bond-position sign, neither the spot nor rate nor time sensitivity has a universal sign. Any additional payoff parameter, such as a strike, is differentiated through $f$; pointwise payoff ordering, rather than a generic parameter rule, determines its price effect.

Put $p=1-R$, $b=\mu -r1$, $A=\sigma {\sigma }^T$ and $q=R+\epsilon$. First assume the covariance matrix $A$ is invertible and initial portfolio wealth is positive. Introduce the accumulated fee processThe Itô formula and the self-financing portfolio equation giveThis converts the path-dependent payment into a terminal-utility problem with state $y$. The volatility-penalized terminal fee changes its drift as well as its spot volatility.

Let $J(t,y)$ be the manager's value function. Its Hamilton-Jacobi-Bellman equation, with terminal condition $J(T,y)=y^p/p$, isThe homogeneity of CRRA utility suggests $J(t,y)=f(t)y^p/p$. Since $y{J}_y=f(t)y^p$ and $y^2{J}_{yy}=-Rf(t)y^p$, the portfolio-dependent expression isIts coefficient is positive even when $p<0$. Completing the square givesThus the stock fractions are constant. Substitution into the Hamilton-Jacobi-Bellman equation yieldsIn particular, multiplying the fee by $a$ does not affect the optimal investment portfolio.

To prove global optimality rather than merely solve a formal Hamilton-Jacobi-Bellman equation, write $v_t={\sigma }^T{\pi }_t$, $v^{\ast }={\sigma }^T{\pi }^{\ast }$ and let $y^{\ast }$ be the fee under the proposed constant policy. Then $b^T\pi =q{v}^{\ast }\cdot v$. Define a probability $Q$ byThis density is a true martingale because $v^{\ast }$ is constant. By the Girsanov theorem, $W_t^Q=W_t-p{v}^{\ast }t$ is a Brownian motion under $Q$. The geometric Brownian motion formula for the fee ratio givesFor admissible, locally square-integrable exposures with finite accumulated quadratic variation, setThis stochastic exponential is a positive local martingale, so $E_Q{Z}_T\le 1$. Moreover,For $p>0$, the exponent $p/(1+\epsilon )$ lies between zero and one. The Jensen inequality for this concave function gives $E_Q[Z_T^{p/(1+\epsilon )}]\le 1$. For $p<0$, the power is convex and decreasing, so the Jensen inequality gives $E_Q[Z_T^{p/(1+\epsilon )}]\ge (E_Q{Z}_T{)}^{p/(1+\epsilon )}\ge 1$; an infinite moment only makes this inequality stronger. Dividing by $p$, with its corresponding sign, proves $E[U(y_T)]\le E[U(y_T^{\ast })]$ in both cases. Equality is achieved by the constant policy.

For an ordinary terminal-wealth agent with relative risk aversion $q$, the same Hamilton-Jacobi-Bellman equation calculation, with no fee penalty, maximizes $b^T\pi -q{\pi }^{T}A\pi /2$. It therefore gives exactly the same optimal investment portfolio. For $q\ne 1$, the preceding stochastic exponential and Jensen inequality verification applies with penalty zero and exponent $1-q$, proving global optimality for this agent too. The spot volatility penalty adds $\epsilon$ to effective relative risk aversion. If $q=1$, the printed expression for $U_0$ has a zero denominator; its appropriate limit, after subtracting an irrelevant constant, is $U_0(x)=\log x$, which still gives ${\pi }^{\ast }=A^{-1}b$. In this logarithmic case, the terminal log-wealth difference from the candidate equals $\log Z_T$ for a positive stochastic exponential $Z$, so $E[\log Z_T]\le \log E[Z_T]\le 0$ proves its optimality.

If $A$ is singular, the same conclusion holds in the absence of arbitrage, provided $b$ lies in the image of a linear map $x\mapsto Ax$: replace $A^{-1}$ by the Moore-Penrose inverse, and add any vector in the kernel of $A$ to ${\pi }^{\ast }$. These additions change neither returns nor Brownian portfolio exposures. If $b$ has a nonzero component in that kernel, a zero-volatility position has a nonzero excess return. Scaling its profitable sign gives arbitrage, so a finite attained optimum as claimed requires this degeneracy to be excluded.

Use the primitive $U(c)=c^{1-R}/(1-R)$, as in Question 3, and the usual positive spot volatility convention $\sigma >0$. An additive utility function constant shifts both agents' value functions equally and does not affect their policies or the comparison. SetThe Itô formula applied to the stock price and the binary Brownian drift filter givesLet ${\theta }_t$ denote the dollar stock holding. The separated state equations areBoth states use the same observed Brownian motion. In particular, their quadratic covariation is $d[w,X{]}_t=\sigma {\theta }_{t}dt$.

Let $J(w,x)$ be the Bayesian value function. Its Hamilton-Jacobi-Bellman equation isThe cross derivative is necessary: learning and portfolio wealth are correlated through the same observation shock. At a smooth, finite, strictly concave value, the optimal consumption and stock holding areThe number of shares is ${\theta }^{\ast }/S$, and the wealth fraction is ${\theta }^{\ast }/w$.

More explicitly, homogeneity gives $J(w,x)=K(x)w^p/p$, where $p=1-R$. ThenThe additional term is the learning hedge in a binary-drift investment model, a form of intertemporal hedging demand. Merely putting the Bayesian posterior mean drift into a constant-coefficient investment formula would omit it. Eliminating the controls gives the coefficient equationThe economic solution is selected by admissibility and the infinite-horizon value condition, rather than by an arbitrary solution of this ordinary differential equation.

For the comparison, now suppose $0<R<1$ and $\mu >r$. Write $m=\mu -r>0$, $b_{+}=m+\sigma a$, andWhen ${\gamma }_{+}>0$, the informed agent's Merton consumption-investment problem has valueThese are the policies available to an agent who knows the true drift is $+a$.

For any admissible Bayesian policy, not only its optimum, defineThe integration variable is $s$ throughout the accumulated utility function. Applying the Itô formula in the Bayesian observation filtration giveswhereThe informed-agent Hamilton-Jacobi-Bellman equation says ${\sup }_{c,\theta }{H}_{b_{+}}=0$. Maximizing the stock term at any excess drift $b$ givesSince $V_{+}^{\prime \prime }<0$, this maximum increases with $b^2$. The binary Brownian drift filter satisfies $|h(x)|\le a$, and thereforeIt follows that $H_{b(x)}(w,c,\theta )\le {\sup }_{c,\theta }{H}_{b(x)}\le {\sup }_{c,\theta }{H}_{b_{+}}=0$. Thus $Y$ is a local supermartingale. Because $0<R<1$, both accumulated utility function and $V_{+}$ are nonnegative. A nonnegative local supermartingale is a genuine supermartingale, by localization and the conditional Fatou lemma. This proves the requested assertion under the Bayesian probability law, without conditioning that law on the favorable drift.

ConsequentlyThe monotone convergence theorem lets $t$ tend to infinity. Taking the supremum over Bayesian policies proves the maximal squared Sharpe ratio value boundIn particular this holds at the equal-prior state $x=0$. No vanishing-terminal-value assumption is needed for this upper bound, because the terminal term is nonnegative and can simply be dropped.

The finite-value assumption ${\gamma }_{+}>0$ is missing from the printed comparison. Without it, $V_{+}=+\infty$ and the value inequality is trivial, but $Y$ is not a finite supermartingale. For a concrete allowed example, take $r=0$, $\mu =1$, $\sigma =a=1$, $R=1/2$ and $\rho =1$. Then $b_{+}=2$ and ${\gamma }_{+}=-2$. For the informed agent, the constant stock fraction $\pi =4$ and consumption $c=w/2$ giveThe time integral diverges, so his value is infinite. This is why the nontrivial finite-supermartingale proof above needs the stated qualification.

Assume the usual positive initial stock price and strike, so the logarithm is defined, and interpret $U$ as a classical solution of the displayed partial differential equation. For a fixed complex $z$, put $F_t(z)=S_t^{z}U(t,{\sigma }_t,z)$, defining $S_t^z=\exp (z\log S_t)$. The Itô formula for the two Brownian motions with correlation coefficient $\rho$ givesThe partial differential equation cancels the entire drift. By the permission to treat the resulting local martingales as true martingales, and the terminal condition, $F_t(z)=\mathbb{E}[S_T^z\mid {\mathcal{F}}_t]$.

The bounded spot volatility ensures $\mathbb{E}{S}_T^{x_0}<\infty$: the Itô formula for a real power and localization bound its moment by $S_0^{x_0}\exp (x_0(x_0-1)\Vert \sigma {\Vert }_{\infty }^{2}T/2)$. Rewrite the proposed integrand asNow $|F_t(x_0+iy)|\le \mathbb{E}[S_T^{x_0}\mid {\mathcal{F}}_t]$, an integrable bound independent of $y$. Conditional Fubini's theorem and part (b) therefore implyThis route justifies the contour exchange without assuming bounds on $U$ uniform in all complex $z$. Cash is constant, $S$ is a true martingale under the stated allowance, and the displayed $C$ is a true martingale. The original measure is thus an equivalent martingale measure for all three assets. The fundamental theorem of asset pricing gives the market has no arbitrage under the usual admissible trading convention.