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If $V$ satisfies the pricing equation for a local volatility model, hold $h_t^S=V_S(t,S_t)$ stocks and $h_t^B=[V(t,S_t)-S_t{V}_S(t,S_t)]/B_t$ units of the bank account. Their value is $V(t,S_t)$. The pricing equation for a local volatility model and Itô formula giveThus this is a self-financing strategy, with terminal value $g(S_T)$. It is a delta hedge, and nonnegative $V$ makes its wealth an admissible trading strategy. The stock and bank wealth amounts are $S_t{V}_S$ and $V-S_t{V}_S$, rather than the asset-unit holdings.

A model specifying traded financial assets, their price and payment processes, available information, and allowed admissible trading strategies. A self-financing portfolio generates wealth using gains on these assets. Absence of arbitrage and market completeness respectively concern whether trading can create a riskless profit from zero capital and whether every specified contingent claim can be replicated.

The law of one price requires traded assets or portfolios with the same terminal payoff to have the same earlier price. In a model where every discounted traded price is a true martingale under a common equivalent measure, it follows by taking the conditional expectation of that terminal payoff. With only local martingale prices and a lower-bound restriction on admissible trading strategies, it can fail: the seemingly profitable short position may have unbounded possible losses before maturity.

Use a self-financing portfolio with ${\Delta }_t$ units of the traded asset and the remainder in the continuous-time bank account. If its value equals $V(S_t,t)$, its self-financing condition isMeanwhile Itô's formula givesMatching the Brownian motion coefficients gives the delta hedge ${\Delta }_t=V_S$. Matching drifts then cancels the physical drift $\alpha$, leaving the pricing equation for a linear asset with additive noise:The domain is the full real line: additive noise can take the asset through zero. The diffusion coefficient is $\sigma$, so the second-derivative term has no factor $S^2$. As usual, the replication argument assumes suitable differentiability and an admissible trading strategy.

Assume a frictionless market with no taxes or transaction costs, unrestricted short selling and borrowing/lending at the same constant rate $r$, continuous trading, and absence of arbitrage. The non-dividend-paying stock has geometric Brownian motion dynamics $dS=\mu Sdt+\sigma SdW$ with constant $\sigma >0$, and the bank account satisfies $dB=rBdt$. A contingent claim with no interim cash flows has sufficiently regular value $f(t,S)$, meaning $f\in C^{1,2}$ before maturity; terminal financial payoff kinks can be handled by the smooth value for earlier times. Trading strategies are adapted, self-financing portfolios, and admissible trading strategies so that doubling strategies are excluded.

The Itô formula givesChoose $\Delta =f_S$ shares and put the remaining value $f-S{f}_S$ in the bank account. Its self-financing gain isMatching the contingent claim's gain to this replicating gain cancels the random increment and equates the drifts:This is the Black-Scholes equation. The argument uses the self-financing gain $\Delta dS$, not a product differential that incorrectly treats a changing $\Delta$ as free cash. Rebalancing purchases are financed from the bank account. Conversely, a smooth solution with suitable growth and boundary conditions defines a delta hedge whose value replicates the contingent claim; law of one price identifies this value with the contingent claim price. Here $S$ is the current stock price, $t$ is time, $\sigma$ is proportional volatility, $r$ is the risk-free rate, and $f$ is the contingent claim price. The physical expected return $\mu$ cancels because exposure to price risk has been hedged.

Let $I=(U^{\prime }{)}^{-1}$ be the inverse marginal utility. The derivative $U^{\prime }$ is continuous and strictly decreasing: concavity makes it nonincreasing, and equality at two distinct points would make $U$ affine between them, contradicting strict concavity. The Inada conditions therefore make $U^{\prime }$ a bijection from $(0,\infty )$ onto $(0,\infty )$. For each $y>0$, the derivative of $U(x)-xy$ is positive below $I(y)$ and negative above it. Consequently its unique global maximum is finite andThere is no need to differentiate $I$ to find the first derivative of this utility conjugate. Comparing the maxima at $y$ and $y+h$ gives, for $h>0$,Continuity of $I$, together with the corresponding inequalities for $h<0$, givesThus $V$ is strictly decreasing and strictly convex, since its derivative is strictly increasing. This proves all the asserted properties except the second differentiability, which does not follow from the printed hypotheses.

Here is an explicit Inada utility with vanishing curvature. PutThe positive integral defining $p$ makes $U$ strictly increasing. Also $p^{\prime }=-w$ is negative except at one point, so $p$ is strictly decreasing and $U$ is strictly concave. Near zero $w(s)$ is asymptotic to $s^{-2}$, and at infinity it is asymptotic to $s^{-2}$; thus $p(0+)=\infty$ and $p(\infty )=0$. This smooth utility function satisfies every printed positive-domain assumption, but $U^{\prime \prime }(1)=0$. At $y_0=p(1)=3-4\log 2$, differentiability of $I$ would imply, by differentiating $p(I(y))=y$,which is impossible. Hence $V^{\prime }=-I$ has no finite derivative there. More explicitly, $p(x)-p(1)=-(x-1{)}^3/12+o((x-1{)}^3)$, so its inverse function has a cube-root singularity. The twice-differentiable conclusion is false as printed. With the additional assumption $U^{\prime \prime }(x)<0$ everywhere, the inverse function theorem proves the intended dual differentiability with nonvanishing utility curvature:

For the financial assertions, use the positive-wealth admissible trading strategy class appropriate to the positive-domain utility function, and expectations for which the expressions are defined. A normalized terminal state-price density satisfies $Z_N>0$, $\mathbb{E}{Z}_N=1$, and defines an equivalent martingale measure $Q$ by $dQ=Z_{N}dP$. A self-financing portfolio obeyswhere the holdings are measurable before the next price increment. Under $Q$ the gains are a martingale when integrable; for the usual lower-bounded admissible class, localization gives a supermartingale. In particular the state-price budget constraint is $\mathbb{E}[Z_N{X}_N]\le x$. The positive-domain qualification matters: assumptions only on $U$ at positive arguments do not control its values at nonpositive wealth.

The definition of the utility conjugate gives the pointwise inequalityTaking expectations, using the state-price budget constraint, and then taking the supremum over strategies gives $u(x)\le \mathbb{E}V(y{Z}_N)+yx$ for every $y>0$ and every state-price density. Taking the two infima proves the utility duality with martingale deflators upper bound:If an expectation is infinite, use the same inequality in the extended-real sense whenever the negative parts make it well defined; no subtraction of two infinities is needed.

A complete market is one in which every terminal contingent claim in the specified claim class is attainable by a self-financing portfolio. Bounded terminal claims suffice for the uniqueness argument. In the standard finite-horizon discrete-time market, absence of arbitrage gives existence of an equivalent martingale measure by the fundamental theorem of asset pricing. To prove uniqueness, let $Z_N$ and ${\tilde{Z}}_N$ be two normalized state-price densities. Every bounded claim $H$ has a replicating price $p(H)$, and its replication has the same price under either equivalent martingale measure:Take $H=1_A$ for every terminal event $A$. The integrable difference $Z_N-{\tilde{Z}}_N$ has zero integral over every event, hence vanishes almost surely: its positive and negative sets would otherwise give a nonzero integral. The normalized terminal state-price density is unique up to null sets. Its density process is then uniquely $Z_n=\mathbb{E}[Z_N\mid {\mathcal{F}}_n]$. This uses the usual pricing replication class, where bounded claims have bounded replicating value processes; arbitrary doubling strategies are not an admissible definition of completeness.

For logarithmic utility, direct maximization gives $V(y)=-\log y-1$. In the complete market, write $Z=Z_N$. The unique-density utility conjugate bound is minimized at $y=1/x$, givingThe candidate terminal portfolio wealth is $H^{\ast }=x/Z$, whose cost is exactly $\mathbb{E}[Z{H}^{\ast }]=x$. If completeness includes nonnegative $Q$-integrable claims, it is replicable and attains the bound. There is also an approximation proof using only bounded-claim completeness. DefineThen $0<b_n\le 1$, $b_n\uparrow 1$, $H_n$ is positive and bounded, and $\mathbb{E}[Z{H}_n]=x$. Replicating $H_n$ therefore gives an admissible terminal claim of initial cost $x$. Since $\mathbb{E}(\log Z{)}^{+}\le \mathbb{E}Z=1$, the negative parts of $-\log (Z\vee n^{-1})$ have an integrable common bound. Monotone convergence after adding this lower bound givesConsequently the logarithmic terminal wealth in a complete market isThe value may be $+\infty$ if $\mathbb{E}(\log Z_N{)}^{-}=\infty$; the displayed formula remains meaningful, while attainment by $H^{\ast }$ depends on the precise admissible claim class.

Write $A=\sigma {\sigma }^T$ and let $V(x,t)$ denote the optimal remaining expected reward from positive portfolio wealth $x$ at time $t$. The Hamilton-Jacobi-Bellman equation and terminal condition areThe controls are progressively measurable, the investment integrals exist, consumption is nonnegative and locally integrable, and the resulting portfolio wealth stays in the utility domain. At zero consumption use the right limit of the utility function if needed. This is the usual admissible trading strategy formulation; without restrictions preserving the domain the reward need not be defined.

For the verification bound, assume $V$ is a nonnegative classical solution, continuous through the terminal time, with one time derivative and two wealth derivatives in the interior. Fix any admissible controls and putThe Itô formula givesThe drift is nonpositive because the actual controls cannot exceed the supremum in the Hamilton-Jacobi-Bellman equation. Localize by stopping times on which the stochastic integral is a true martingale, the derivatives and wealth stay in their interior domain, and the accumulated reward is finite. For each such stop ${\tau }_n$,Both utilities and $V$ are nonnegative. Thus the Fatou lemma, continuity, and the terminal condition giveFor admissible paths strictly positive on the closed horizon, the domain-localizing stops eventually reach $T$. If zero wealth is allowed, the same proof uses the continuous boundary extension of $V$ and the utilities there. Nonnegativity is what permits removing the stops without assuming the uncontrolled stochastic term already has zero expectation. This is verification by a nonnegative control supermartingale.

The square-root case needs an additional no-arbitrage qualification because the printed volatility matrix need not be invertible. If there is $v\in \ker {\sigma }^T$ with $v^T\mu >0$, investment $\theta =Kv$ has zero volatility and arbitrarily large positive drift. With any fixed positive consumption, sufficiently large $K$ preserves positive wealth and makes terminal square-root utility tend to infinity. For example, in one dimension $\sigma =0$, $\mu =1$, $\theta =K$, $C=1$ gives $X_s=x+(K-1)(s-t)$, so the value is infinite when $T>t$. A finite separated solution does not exist for all matrices allowed by the printed hypotheses.

The needed condition is $\mu \in \mathrm{Range}A$, equivalent to $\mu$ being orthogonal to $\ker {\sigma }^T$, because $A$ is a symmetric matrix and $\ker A=\ker {\sigma }^T$. Assume this condition and let $B=A^{+}$ be its Moore-Penrose inverse; if $A$ is invertible, simply use $B=A^{-1}$. Put $q={\mu }^{T}B\mu \ge 0$. Seek the square-root investment-consumption value in the formThen $V_x=g/\sqrt{x}$ and $V_{xx}=-g/(2x^{3/2})$. Completing the quadratic investment expression, using $AB\mu =\mu$, shows that its maximum is attained at ${\theta }^{\ast }=2xB\mu$ and has value $qg\sqrt{x}$:Any investment component in $\ker A$ has no effect. For consumption, set $z=\sqrt{c}$. Its contribution is the concave quadratic $2z-(g/\sqrt{x})z^2$, whose maximum is $\sqrt{x}/g$ at $c^{\ast }=x/g^2$. Substituting both maxima into the Hamilton-Jacobi-Bellman equation givesWith $h=g^2$, this becomes $h^{\prime }+qh+1=0$, $h(T)=1$. Solving this linear ordinary differential equation givesIn particular $h\ge 1$, so $g=\sqrt{h}$ stays positive, and the two controls areThese controls are admissible and actually attain the value. Indeed their portfolio wealth satisfiesSince $\Vert 2{\mu }^{T}B\sigma {\Vert }^2=4q$, the explicit positive solution starting at $(x,t)$ isIts finite-horizon moments of every positive order are finite. More directly,Differentiating $g(s)\mathbb{E}\sqrt{X_s}$ and using the equation for $g$ gives $\frac{d}{ds}(g(s)\mathbb{E}\sqrt{X_s})=-\mathbb{E}\sqrt{X_s}/g(s)$. Integrating from $t$ to $T$ therefore yieldsThis proves attainment without relying on an unjustified expectation of a local martingale. When $\mu$ is outside $\mathrm{Range}A$, the singular-volatility drift arbitrage counterexample applies instead.

Use the usual stochastic-integrability assumptions implicit in the price stochastic differential equation: the coefficients have progressively measurable versions, the volatility is locally square integrable, and initial prices are finite deterministic positive numbers. A no-consumption admissible trading strategy consists of predictable risky share holdings $\pi$ for which the price stochastic integral exists and the portfolio wealth of the self-financing portfolio isThe remainder $X_t-{\pi }_t^T{S}_t$ is held in the unit cash asset. Admissibility requires $X_t\ge -a$ throughout the horizon for some deterministic finite $a$; the more restrictive nonnegative-wealth convention also works. A finite-horizon arbitrage is such a strategy with $x=0$, $X_T\ge 0$ almost surely and $P(X_T>0)>0$. The lower bound is essential to exclude doubling strategies.

Interpret the stated uniform ellipticity as ${\sigma }_t{\sigma }_t^T\ge \epsilon I$ for a deterministic $\epsilon >0$. If $|{\mu }_t|\le K$, the market price of risk is bounded:The authoritative PDF puts both integrals inside the exponent. With that interpretation,On every finite horizon $T$, boundedness of $\lambda$ gives the Novikov condition:Hence the positive stochastic exponential $Z$ is a true martingale with $\mathbb{E}{Z}_T=1$. Define the equivalent probability measure $Q_T$ by $d{Q}_T=Z_{T}dP$. The Girsanov theorem says that $W_t^Q=W_t+{\int }_0^t{\lambda }_{s}ds$ is a Brownian motion under $Q_T$, up to time $T$. Consequentlyso the prices are local martingales under $Q_T$.

Every admissible zero-initial-capital self-financing portfolio has portfolio wealth which is therefore a $Q_T$-local martingale bounded below by $-a$. The shifted process $X+a$ is a nonnegative local martingale, hence a supermartingale by localization and the Conditional Fatou lemma. In particular ${\mathbb{E}}_{Q_T}{X}_T\le 0$. If the strategy were an arbitrage, nonnegativity of $X_T$ would force $X_T=0$ $Q_T$-almost surely. Equivalence of $P$ and $Q_T$ would then force $P(X_T>0)=0$, a contradiction. The market has no admissible arbitrage on any finite horizon. This proves the exclusion argument itself, rather than just asserting the fundamental theorem of asset pricing.

For the deflated price calculation, let ${\sigma }_{t,i}$ denote the column vector whose transpose is the $i$th row of ${\sigma }_t$. The Itô product rule includes the quadratic covariation termSince ${\sigma }_{t,i}^T{\lambda }_t={\mu }_t^i$, the full product equation isLocalization makes the last stochastic integral a true martingale on each stopped interval, proving each component of the deflated price vector is a local martingale under the original measure. This is the bounded-coefficient asset deflator identity; bounded volatility has not yet been used.

If $\sigma$ is bounded, then ${\sigma }_{t,i}-{\lambda }_t$ is bounded as well. Solving the last linear stochastic differential equation givesThe Novikov condition now holds for this stochastic exponential on every finite horizon, because its bracket is bounded by a deterministic constant times the horizon. ThereforeThe entire deflated price vector is a true martingale when volatility is bounded. This finite-horizon argument does not assert uniform integrability as time tends to infinity, which is not required. Integrable random initial prices can also be handled by the conditional martingale argument; without any integrability assumption on initial prices, a true-martingale assertion would not be meaningful.

The delta replication from a local-volatility pricing equation uses the asset-unit holdingsTheir wealth is exactly $V(t,S_t)$. Moreover,where the final equality is the calculation in part (i). Thus the holdings are a self-financing strategy with terminal wealth $g(S_T)$, proving replication. Since $V\ge 0$, this replicating strategy has nonnegative wealth and is an admissible trading strategy. The equality of trading gains is unchanged by replacing the drift with its physical-measure value: the stock drift in Itô formula and in the stock holding changes by the same amount. If ${\pi }_t$ denotes wealth fractions, then, wherever $V>0$, its stock fraction is $S_t{V}_S/V$ and its bank fraction is $1-S_t{V}_S/V$.

Let $Q$ be the given risk-neutral measure, and write $B_t=B_0{e}^{rt}$. Take $V$ to be classical on $t<T$ and continuous at its prescribed terminal value. By Itô formula and the pricing equation for a local volatility model,soThe discounted added asset is a local martingale, as is $S_t/B_t$, while $B_t/B_t=1$. Consequently $Q$ is an equivalent local martingale measure for all three assets.

Here is the arbitrage argument, including the usual admissible trading strategy condition. If $X$ is the discounted wealth of a self-financing strategy, it is a stochastic integral against the discounted asset prices, plus its initial value, and hence a local martingale. Assume $X_t\ge -c$ for a fixed $c$, as required for an admissible trading strategy. For a localizing sequence ${\tau }_n$, the stopped process $X^{{\tau }_n}$ is a martingale. Apply the conditional Fatou lemma to $X_{t\wedge {\tau }_n}+c\ge 0$: for $s\le t$, continuity and the stopped martingale identity give ${\mathbb{E}}_Q[X_t+c\mid {\mathcal{F}}_s]\le X_s+c$. Thus $X$ is a supermartingale. An arbitrage would have $X_0=0$, $X_T\ge 0$ and positive probability of $X_T>0$. Equivalence of $P,Q$ makes that probability positive under $Q$, contradicting ${\mathbb{E}}_Q[X_T]\le 0$. Therefore the augmented market has no admissible arbitrage.

Nonnegativity of $V$ gives the needed price process, but does not by itself make its discounted local martingale a true martingale. If $V$ is bounded, the latter follows by bounded localization, and its price is also $B_t{\mathbb{E}}_Q[g(S_T)/B_T\mid {\mathcal{F}}_t]$. The arbitrage proof above applies to the stated nonnegative classical solution without assuming that additional bound.

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.