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Write $T=t_0$ and $B_t=e^{-\rho (T-t)}$ for the risk-free asset paying one at $T$. Under the physical measure, $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, with $\sigma >0$, while $d{B}_t=\rho B_{t}dt$. A self-financing portfolio with predictable stock and bond holdings $g_t,h_t$ has value $V_t=g_t{S}_t+h_t{B}_t$ and gainsRebalancing therefore adds no outside money and withdraws none. This gains equation, rather than merely the value decomposition, defines self-financing.

Let $p(x,t)=xg(x,t)+B_{t}h(x,t)$. The Itô formula givesThe prescribed portfolio gains instead have diffusion coefficient $\sigma S_{t}g$ and drift $\mu S_{t}g+\rho B_{t}h$. Equality of diffusion coefficients requires $p_x=g$, equivalentlyUnder this condition, $p_{xx}=g_x$. Equality of drifts is then $p_t+\frac{1}{2}{\sigma }^2{x}^2{g}_x=\rho B_{t}h$. Since $p_t=x{g}_t+B_t{h}_t+\rho B_{t}h$, this becomesThese conditions are also sufficient by the same Itô formula. Necessity initially holds along the stock process. Its positive transition density on $(0,\infty )$ for every positive time, together with continuity of the coefficient functions, extends the identities to all $x>0$ and then to the time endpoints by continuity.

For a self-financing portfolio, substitute $g=p_x$ and $B_{t}h=p-x{p}_x$ into the drift identity to obtain the Black-Scholes equationConversely, a smooth solution of this equation gives a self-financing portfolio by choosing the delta hedgeIts value is $p$, and both coefficient identities follow directly.

The printed deduction needs this holdings qualification: the Black-Scholes value equation needs delta-compatible holdings. A value function alone cannot make an arbitrary prescribed decomposition self-financing. As a counterexample, take $p=0$, $g=1$, $h=-x/B_t$. The value function solves the Black-Scholes equation, but this zero-value portfolio has gains $d{S}_t-\rho S_{t}dt$, which contain the nonzero term $\sigma S_{t}d{W}_t$. Thus for the originally specified $g,h$, the exact equivalence is the pricing equation together with $g=p_x$, not the pricing equation alone.

For a claim paying the holder at rate $k(S_t)$, let $p$ denote its remaining, ex-dividend value. A replicating portfolio pays out $k(S_t)dt$, so its net wealth equation is $dp=gdS+hdB-k(S_t)dt$. The delta hedge is still $g=p_x$, but the Black-Scholes equation becomesIf there is a terminal payoff $f$, set $p(x,T)=f(x)$; with no terminal payment set it to zero. The corresponding risk-neutral valuation isEach intervening payment is discounted from its own payment time.

Let $T=t_0$, $\Delta =T-t$ and assume the payoff has sufficient integrability for risk-neutral valuation. Under the risk-neutral measure $Q$, the Black-Scholes model has $d{S}_u=\rho S_{u}du+\sigma S_{u}d{W}_u^Q$, so conditionally on $S_t=x$,Absence of arbitrage gives the time-$t$ priceFor a regular payoff this is the solution of the Black-Scholes equation with terminal value $f$, and the delta hedge $g=V_x$ replicates it. Suitable limits or conditional-expectation representations extend the price formula beyond smooth terminal payoffs.

For the logarithmic stock payoff, put $Y=(\rho -{\sigma }^2/2)\Delta +\sigma \sqrt{\Delta }Z$. Its Gaussian moment-generating function is $\mathbb{E}{e}^Y=e^{\rho \Delta }$, while differentiating that moment gives $\mathbb{E}[Y{e}^Y]=(\rho +{\sigma }^2/2)\Delta e^{\rho \Delta }$. ThereforeThe option delta is $\log x+1+(\rho +{\sigma }^2/2)\Delta$, and the second stock derivative is $1/x$. The Black-Scholes equation can also be checked directly from $V_t=-x(\rho +{\sigma }^2/2)$.

For general convex $f$, let $Y_{+}=\exp ((\rho -{\sigma }^2/2)\Delta +\sigma \sqrt{\Delta }Z)>0$. For $0\le \lambda \le 1$, convexity givesTaking the positive discounted expectation proves that $V(\cdot ,t)$ is convex. Its derivative, where defined, is therefore nondecreasing, proving that the stock holding in the delta hedge rises weakly with the stock price. For twice differentiable $f$, differentiation gives $V_{xx}=e^{-\rho \Delta }\mathbb{E}[Y_{+}^2{f}^{\prime \prime }(x{Y}_{+})]\ge 0$. For a merely convex payoff, Gaussian smoothing at $t<T$ and the usual growth assumptions give the same conclusion; one-sided derivatives are monotone even without smoothing. Strict increase is not implied by convexity alone: an affine payoff has constant option delta.

For the random constant Gaussian interest-rate mixture, represent the rate as $\rho ={\rho }_0+\tau Y$, with $Y,Z$ independent standard normal variables. This independence simply represents averaging the constant-rate price function over its Gaussian parameter. Completing the square in $Y$ yields the identityApply it inside the time-zero price formula:The combined normal term has variance ${\tau }^2{T}^2+{\sigma }^{2}T$. Define ${\rho }_{\ast }={\rho }_0-{\tau }^{2}T/2$ and ${\sigma }_{\ast }^2={\sigma }^2+{\tau }^{2}T$. Then the discount factor is $e^{-{\rho }_{\ast }T}$, and the log-price mean is $({\rho }_{\ast }-{\sigma }_{\ast }^2/2)T$. Thus the expectation is exactly the constant-parameter Black-Scholes model price with those parameters:The argument applies to nonnegative payoffs by Tonelli theorem, and to signed payoffs whenever absolute integrability justifies the expectations. It averages constant-rate prices; it does not assume a stochastic short-rate process.

Use the natural filtration ${\mathcal{F}}_t=\sigma (Z_1,\dots ,Z_t)$ and the usual integrable definition of the Snell envelope. Since the factors are positive, its terminal value is $U_T=Y_T$. If $U_{t+1}=Y_{t+1}{c}_{t+1}$, independence of random variables givesThe backward recursion therefore becomesThis proves the assertion by backward induction. All constants are finite and at least one. More explicitly,where the empty product is one. This is the Snell envelope of a multiplicative process. The standard Snell envelope hypothesis includes integrable rewards; independence of random variables and positivity then give finite first moments of the factors. If that hypothesis is dropped, an infinite-mean factor gives an infinite continuation value, so finite constants need not exist.

For the discrete-time binomial market, write $p=(1+r/\epsilon )/2$. The assumptions give $0<p<1$, positive bank account and stock prices, andThusThe finite discrete-time fundamental theorem of asset pricing states that a market with a strictly positive numéraire has no arbitrage if there is an equivalent probability measure under which all numéraire-denominated asset prices are martingales. The original probability measure is already such an equivalent martingale measure, proving no arbitrage.

The complete-market American-option pricing theorem identifies the discounted replication or superhedging value with the Snell envelope of the discounted exercise rewards. HereThe independent factors have common meanThe Snell envelope recursion consequently gives $c_t={\lambda }^{T-t}$. At every time before $T$, the continuation value strictly exceeds immediate exercise, so optimal exercise is at maturity. The undiscounted value process is $V_t=S_t^2{\lambda }^{T-t}$, and the time-zero replication cost isIn fact $V_t/B_t$ is a martingale, not just a supermartingale, because the Snell envelope has no exercise compensator before maturity.

To hedge the optimally exercised payoff, subtract the two next-state portfolio equations. The first-period delta hedge isThe corresponding bank account holding, since $B_0=1$, is $V_0-{\Delta }_0{S}_0=S_0^2{\lambda }^{T-1}(\lambda -2)$. Substitution into either next-state equation verifies both terminal values at time one. Repeating this two-state calculation at every node gives a self-financing strategy replicating the maturity payoff. It also covers earlier exercise: $V_t\ge S_t^2$ at all dates, so the seller can meet any earlier payment with the available wealth.

On a fixed finite horizon, a self-financing strategy consists of predictable stock holdings $h^i$, integrable against the price processes, and a bank account holding $h^0$. Its wealth and gains satisfyAn admissible trading strategy has discounted wealth $X_t/B_t$ bounded below by a deterministic constant throughout the horizon. With the bounded short rate in this part, an undiscounted deterministic lower bound is equivalent after adjusting the constant. An arbitrage is an admissible zero-initial-cost strategy with $X_T\ge 0$ almost surely and $\mathbb{P}(X_T>0)>0$.

Put $1=(1,\dots ,1{)}^{\top }$ and define the market price of riskThe bounded coefficients and bounded inverse make $\lambda$ bounded. Define the stochastic exponentialThe Novikov condition states that $\mathbb{E}\exp (\frac{1}{2}{\int }_0^T\Vert {\lambda }_s{\Vert }^{2}ds)<\infty$ makes this exponential a true martingale with expectation one. Boundedness verifies the condition. Therefore $dQ=L_{T}dP$ defines an equivalent probability measure. The Girsanov theorem states thatis a $d$-dimensional Brownian motion under $Q$. Substituting into the asset equations givesThus $Q$ is an equivalent local martingale measure. The sufficient direction of the fundamental theorem of asset pricing says that such a measure excludes arbitrage among admissible self-financing strategies. Here the reason can be seen directly: $X/B$ is a local martingale under $Q$, and its lower bound makes it a supermartingale. Consequently $X_0=0$ implies ${\mathbb{E}}_Q[X_T/B_T]\le 0$, incompatible with the nonnegative terminal payoff of an arbitrage. Therefore the bounded-inverse market has no arbitrage.

For the requested law of one price counterexample, take a broader continuous-time local-martingale model on $[0,T]$. Let $X$ be a Brownian motion, let $\tau =\inf \{u\ge 0:X_u=-1\}$, and set $q(t)=t/(T-t)$ for $t<T$. DefineThe given hitting-time fact gives $\tau <\infty$ almost surely. The corresponding physical hitting time is $T\tau /(1+\tau )<T$, so $S$ reaches one and then stays one. In particular it is continuous at maturity and $S_t\ge 1$.

For a rigorous closed-horizon localization, stop additionally when $S$ hits $n+2$, or at $T$ if it never does. The stopped process is Brownian motion stopped on first exiting $(-1,n)$ and subjected to the deterministic clock; it is a bounded martingale, including at $T$. The stopping times increase to $T$, since every Brownian path on the finite random interval $[0,\tau ]$ has a finite maximum. Thus $S$ is a positive local martingale on the whole horizon. With the time-changed Brownian filtration, the original probability measure is an equivalent local martingale measure. The admissible-wealth supermartingale argument above proves no admissible arbitrage.

Nevertheless $S_T=B_T=1$ while $S_0=2\ne B_0=1$, so the law of one price fails. Shorting one share and buying two units of cash has zero initial cost and terminal gain one, but its wealth is $2-S_t$. It has no deterministic lower bound: for every $n$, the probability of Brownian motion hitting $n$ before $-1$ is $1/(n+1)>0$, obtained by optional stopping of the bounded stopped Brownian motion. That apparently profitable strategy is therefore inadmissible. The counterexample deliberately uses a strict local martingale; the bounded-coefficient class in the first part has true discounted asset martingales and does satisfy the law of one price.

The boundedness of $a^{\prime }$ makes $a$ globally Lipschitz, so the stock stochastic differential equation has a nonexplosive strong solution. Since $a$ is bounded, $S_t-S_0={\int }_0^{t}a(S_u)d{W}_u$ is a square-integrable martingale on every finite horizon. Apply Itô formula to the option price:Thus both the stock and the option are local martingales under the original probability measure, while cash is constant. The relevant no-arbitrage theorem is the sufficient direction of the fundamental theorem of asset pricing: a common equivalent measure under which all numéraire-denominated traded prices are local martingales excludes arbitrage by admissible trading strategies. In this case the original measure itself is an equivalent local martingale measure. Hence the augmented market has no admissible arbitrage. The nonnegative option-value assumption also makes the replicating wealth below admissible.

SetDifferentiating the pricing equation in the stock-price variable givesThe limiting terminal derivative is $1_{\{S>K\}}$ away from the strike. Its value at the strike may be assigned as the stated $1_{\{S\ge K\}}$: the smooth positive diffusion coefficient gives a continuous terminal distribution, so that point has probability zero. This is the call delta equation for a driftless local volatility diffusion.

Hold ${\pi }_t$ shares and ${\eta }_t=V(t,S_t)-{\pi }_t{S}_t$ units of cash. The portfolio value is $V(t,S_t)$, and the preceding Itô calculation gives $dV={\pi }_{t}d{S}_t+{\eta }_{t}d{B}_t$, with $d{B}_t=0$. It is therefore a self-financing portfolio, with initial wealth $V(0,S_0)$ and terminal wealth $(S_T-K{)}^{+}$. This proves replication by the stated delta hedge.

A further application of Itô formula, now to $U$, yieldsThe density process is the strictly positive stochastic exponentialIts integrand is bounded, so the Novikov condition proves that it is a true martingale. The product formula for $M=Z\pi$ givesThe drift in $Zd\pi$ cancels this quadratic covariation term, leavingThus $M$ is a local martingale; the expression also identifies its stochastic integral integrand explicitly.

Under the stated assumption that $M$ is a true martingale on $[0,T]$, its terminal value givesSince $0\le Z_T{1}_{\{S_T\ge K\}}\le Z_T$ and $\mathbb{E}[Z_T\mid {\mathcal{F}}_t]=Z_t>0$, division givesThis is the derivative-weighted call delta martingale argument. It uses true martingality through maturity, not just local martingality before the terminal date.

For $v>0$ and $m>0$, let $\Phi$ and $\varphi$ be the standard normal distribution function and density, and putThe exercise event is $Z>(\log m+v/2)/\sqrt{v}=-d_2$. Complete the square in the first truncated expectation:The probability of exercise is $\Phi (d_2)$. Hence the normalized Black-Scholes call function isAt $v=0$ it is $(1-m{)}^{+}$. If $m\le 0$, the positive underlying always exceeds $m$, so $F(v,m)=1-m$; the reciprocal identity below is understood on its natural domain $m>0$.

For the reciprocal strike, $d_1(v,1/m)=-d_2(v,m)$ and $d_2(v,1/m)=-d_1(v,m)$. ThusThereforeThe $v=0$ case follows either by continuity or by checking the two positive parts directly.

For the stock model, the original probability measure is already risk-neutral: $e^{-rt}{S}_t=S_0{e}^{\sigma W_t-{\sigma }^{2}t/2}$ is a true martingale. Write $\tau =T-t$. Conditional on ${\mathcal{F}}_t$, the stock at maturity iswith $Z$ independent of current information and standard normal distribution. ThereforeEquivalently,which is a true martingale, since the payoff is integrable. The sufficient direction of the fundamental theorem of asset pricing says that an equivalent measure making all discounted traded prices local martingales excludes arbitrage by admissible trading strategies. The original measure works simultaneously for the bank account, stock and call. Thus the call-augmented market has no arbitrage.

For the put, hold one call, short one share, and hold $K{e}^{-rT}$ units of the bank account. This self-financing portfolio has current value $C_t-S_t+K{e}^{-r\tau }$ and terminal payoffIts discounted value is also the conditional expectation of that payoff. Thus the put-call parity priceadds another discounted martingale to the same market, proving no arbitrage with both call and put included.

Now put $m=K{e}^{-r\tau }/S_t$. The reciprocal identity givesThe resulting Put-call symmetry in the Black-Scholes model isThe final formula printed in the PDF has two errors: its discount-rate symbol is undefined, and the reciprocal argument has the wrong sign in the interest-rate exponential. Both are resolved by the displayed calculation. Even replacing the undefined symbol by $r$ leaves an incorrect expression with $e^{-r\tau }$ inside $F$. For $\tau >0$, $r>0$ and $\sigma >0$, $F(v,m)$ is strictly decreasing in $m$, because ${\partial }_{m}F=-\mathbb{P}(e^{-v/2+\sqrt{v}Z}>m)<0$. The printed argument is smaller than the reciprocal argument and hence gives a strictly larger put price. Thus the literal printed symmetry cannot be established; the boxed expression is the corrected identity.

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.

Interpret the exponential distribution change time as an exogenous time independent of the stock's Brownian motion, or, equivalently for this calculation, as having constant conditional intensity $\lambda$ while the rate remains high. Its marginal distribution alone would not imply that conditional intensity if the change time carried information about stock returns. Write $r_1=r+\epsilon$, $p=1-R$, $B_1=B(r_1)$, and let $V_1$ be the prechange value function.

Over a short interval $dt$, a rate change occurs with probability $\lambda dt+o(dt)$ and changes continuation value from $V_1(w)$ to $V_0(w)$. The dynamic programming equation is consequentlyThis jump term distinguishes the exponential interest-rate switch from a permanent higher bank rate.

By the homogeneity of CRRA utility, put $V_1(w)=K_1{w}^p/p$, with $K_1>0$. Differentiating the strictly concave expressions in $c$ and $\pi$ givesThe optimized consumption terms contribute $R{K}_1^{1-1/R}{w}^p/p$, and the optimized investment terms contribute $K_1{B}_1{w}^p$. With $K_0={\gamma }_0^{-R}$, the Hamilton-Jacobi-Bellman equation reduces toEquivalently,The left side is continuous and strictly increasing for ${\gamma }_1>0$, with limits zero and infinity. Thus $D_1>0$ gives one and only one positive root. This scalar equation is an explicit characterization for arbitrary real $R$; elementary closed forms are not generally available. For example, when $R=2$,Use $({\pi }_1^{\ast },{\gamma }_1)$ until the rate changes, and $({\pi }_0^{\ast },{\gamma }_0)$ afterwards. If the two rates coincide, the root is ${\gamma }_1={\gamma }_0$; as $\lambda$ tends to infinity, it also tends to ${\gamma }_0$.

For a direct attainability check, use the proposed prechange constant fractions. The geometric Brownian motion moment is $E[w_t^p]=w^p\exp [p(B_1-{\gamma }_1)t]$ on the prechange evolution. Conditioning on the independent exponential distribution time therefore givesThe denominator is positive because the scalar equation gives $D_1+p{\gamma }_1={\gamma }_1+\lambda K_0/K_1>0$. Hence the candidate policy actually achieves the stated value.

One can also verify the upper bound for both signs of $p$ without relying on a sign-sensitive terminal-value argument. Let $j_t$ be the current regime, $w^{\ast }$ the candidate portfolio wealth, andIf $N_t=1_{\{\tau \le t\}}$ and ${\tilde{N}}_t=N_t-\lambda (t\wedge \tau )$, the Itô formula and the scalar equation givewhere the jump term stops after the switch and continuous coefficients are understood predictably. Thus $M$ is a positive martingale deflator: for any admissible portfolio wealth $w$, the process $M_t{w}_t+{\int }_0^t{M}_s{c}_{s}ds$ is a nonnegative local martingale. It follows thatFor the candidate policy, $E[M_t{w}_t^{\ast }]\to 0$. Indeed its surviving prechange term decays at rate $D_1+p{\gamma }_1>0$, and its postchange terms decay at rate ${\gamma }_0>0$; integrating over the switch time preserves convergence to zero. The finite-time moment computations also make its budget equality exact. Consequently $E{\int }_0^{\infty }{M}_t{c}_t^{\ast }dt=M_0{w}_0$. Apply the concave supporting-tangent inequalityand integrate in time and expectation. The budget inequality proves that no admissible policy exceeds the candidate. This verifies the value function, including when $R>1$.