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If holding one unit of a commodity costs a constant cash amount $\delta dt$ for storage, its gain is $dS-\delta dt$. A delta hedge long a derivative and short $V_S$ commodity units has riskless drift $V_t+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}+\delta V_S$ and value $V-S{V}_S$. Equating the drift to the bank return gives the displayed Black-Scholes equation. This is an additive cash carry term, unlike a proportional dividend or storage yield.

The corresponding pricing diffusion is $d\hat{S}=(r\hat{S}+\delta )dt+\sigma \hat{S}dW$. A fixed-strike forward has value $S+\delta (1-e^{-r\tau })/r-K{e}^{-r\tau }$, and put-call parity gives that same value for a matching call minus put. Replication buys the commodity and a bank reserve equal to the present value of its future storage payments. For $r=0$, the reserve is $\delta \tau$.

The holder chooses at a specified time between a European call option and a European put option with the same later maturity and strike. Put-call parity makes its value equal to a call of the original maturity plus a put of the choice-time maturity with appropriately discounted strike.

At its choice time this claim permits purchase of either of two later-expiring European options for an exercise cost $K$, or abandonment. Its payoff is $\max (p-K,c-K,0)$. For common strike $K_3$ and remaining time $\tau$, put-call parity places the branch crossing at $D=K_3{e}^{-r\tau }$. In the Black-Scholes model, both branch values there equal $D[2\Phi (\sigma \sqrt{\tau }/2)-1]$. An exercise cost above this minimum creates a central interval in which neither choice is worth exercising.

For a non-dividend-paying stock and constant interest rate $r$, discounted call prices satisfy $C_T=\frac{1}{2}{K}^2\sigma (T,K{)}^2{C}_{KK}-rK{C}_K$. Strike differentiation recovers the discounted terminal density as $C_{KK}$; differentiating the discounted payoff identity supplies the maturity derivative. Put-call parity implies the same equation for put prices.

A European put option gives its holder the right, without an obligation, to sell one unit of an underlying financial asset at the strike price $K$ only on the expiry date $T$. For one stock share, its terminal financial payoff is the positive partFor strike price $K=100$, terminal stock prices $80$, $100$, and $120$ give financial payoffs $20$, $0$, and $0$, respectively. At $S_T=80$, selling for $100$ rather than the market price $80$ supplies the difference $20$; at $S_T=120$, the holder can let the right expire. Equality at $S_T=100$ gives zero net exercise value. The European call option example plots both financial payoffs on the same axes.

Combining a European put option with the underlying stock share produces the terminal amount $S_T+(K-S_T{)}^{+}=\max (S_T,K)$. Its floor at the strike price is why a European put option can describe protection against a fall in terminal stock value. The protection has an initial cost and lasts to the specified expiry; the terminal floor alone says nothing about the financial return. Put-call parity replicates this same amount with a European call option and a zero-coupon bond.

Using the put-call parity example's price $P_0=10+100e^{-0.05}-100\approx 5.12294245$, the cost financed for one year at interest rate $r=0.05$ grows to $P_0{e}^{0.05}=100-90e^{0.05}\approx 5.38560133$. The terminal net amount is therefore approximately $14.61439867$ when $S_T=80$, and $-5.38560133$ when $S_T=100$ or $120$. These computations use the unrounded price and distinguish the financial payoff from the financed gain. Unlike an American put option, a European put option supplies no right to exercise before $T$.

put-call parity and nonnegativity of the European put option giveFor $r\ge 0$ this is at least $S-K$, and the call value is also nonnegative, so $C^E$ dominates the immediate call exercise contingent claim payoff. Its discounted price is a martingale. At any admissible exercise time, optional stopping theorem therefore bounds the expected discounted call option by the European call option price. Exercising at maturity attains that price. Hence American and European calls coincide for this non-dividend model with nonnegative rates.

For the strict put comparison assume $r>0$, $\sigma >0$, $S>0$ and $t<T$. Choose $u$ strictly between $t$ and $T$. A European put option at $u$ has value at most $K{e}^{-r(T-u)}$. On the eventimmediate exercise gives $K-S_u>K{e}^{-r(T-u)}\ge P^E(u,S_u)$. This event has positive probability because the lognormal stock density is positive on $(0,\infty )$. Exercise there at time $u$, and otherwise hold to maturity. This strategy strictly improves the European put option expected value, provingunder these positive-rate, pre-maturity assumptions.

The source's strict statement needs those qualifications. At maturity the prices agree, and at $r=0$ the put option is a convex function of the martingale stock and therefore is a submartingale, so waiting to maturity is optimal and the American and European puts coincide. Negative rates also invalidate the nonnegative-rate call argument. These are mathematical assumptions of the comparison, rather than changes to the printed question.

At maturity debt receives the smaller of face value and firm assets:Thus it is equivalent to a zero-coupon payment $F$ minus a European put option on the firm value. Its time-$t$ price isEquity receives the residual $(V_T-F{)}_{+}$, hence is a European call option:put-call parity verifies $E_t+D_t=V_t$, consistent with the stated balance-sheet identity. As with the replication arguments elsewhere, the physical drift $\mu$ does not enter these option prices.

Write $p(s)=p(T,s;K_3,T_3)$ and $c(s)=c(T,s;K_3,T_3)$. Because the right can be left unexercised, the chooser with positive exercise cost has payoffLet $\tau =T_3-T$ and $D=K_3{e}^{-r\tau }$. Put-call parity gives $c(s)-p(s)=s-D$. Thus the put is selected for $s<D$ and the call for $s>D$; the two are equal at $s=D$. Before truncation at zero, the payoff is decreasing on the left and increasing on the right.

At the meeting point, the Black-Scholes formula givesIf $K<M$, the payoff has a positive minimum $M-K$ at $D$. If $K=M$, it touches zero there. If $M<K<D$, each side has a distinct exercise threshold and the payoff is zero between them. If $K\ge D$, even the left-end put value cannot cover the exercise cost, and only the right call branch can be positive. The illustration shows the middle case, including the competing underlying option values.

At expiry, the difference of a European call and put with matching strike is $(S_T-K{)}^{+}-(K-S_T{)}^{+}=S_T-K$. Linearity of pricing therefore makes their difference equal the forward-contract value from part (c):A direct put-call parity replication gives the same result. Buy one unit of the commodity and deposit $\delta {\int }_t^T{e}^{-r(u-t)}du$ in the bank. Withdraw the fixed cash storage payments from this reserve; the reserve is exhausted at expiry, leaving one commodity unit. Its initial cost is $S+\delta (1-e^{-r\tau })/r$. Borrow $K{e}^{-r\tau }$ as well, leaving terminal value $S_T-K$. The law of one price proves the parity identity. If $r=0$, replace the storage reserve by $\delta \tau$.

For matching European strike and expiry with no dividends, put-call parity gives $C_0-P_0=S_0-K{e}^{-2r}$. Using the rate and put price from part (c),As a check, only the $uu$ call payoff is positive, at $27$. Its risk-neutral expectation is $27(2/3{)}^2=12$, and discounting by $1.1^2$ gives the same value.

For strike $K>0$ and maturity $T$, a European call option and a European put option have boundary conditionsThe Black-Scholes equation is solved backward from these conditions. At zero stock price, the zero state is absorbing in geometric Brownian motion, soAt the other boundary, with $\tau =T-t>0$,These conditions and a suitable growth bound select the financial solution and provide boundary data for numerical truncations. In the constant-coefficient model, under the risk-neutral measure,For $d_1=[\log (S/K)+(r+{\sigma }^2/2)\tau ]/(\sigma \sqrt{\tau })$ and $d_2=d_1-\sigma \sqrt{\tau }$, the exercise event is $Z>-d_2$. Completing the square in $e^{\sigma \sqrt{\tau }Z}$ times the normal density gives the truncated first moment and hencewhere $\Phi$ is the standard normal cumulative distribution function. These are the Black-Scholes formula values and satisfy the specified terminal and boundary conditions.

Put-call parity is the relationA call minus a put has terminal financial payoff $S_T-K$. A portfolio of one share minus a zero-coupon bond paying $K$ has the same financial payoff, so absence of arbitrage and law of one price equate their current values. The formula here assumes the same strike and maturity and no dividends; known dividends modify the stock-side value.

An American option allows exercise at a stopping time $\tau \in [t,T]$. For its exercise financial payoff $H$, the pricing problem isThe discounted value is the Snell envelope of discounted exercise payoffs. Define the pricing operatorIn the continuation region the Black-Scholes equation holds, $\mathcal{L}V=0$. In the exercise region $V=H$, while the ability to wait and the supermartingale property give $\mathcal{L}V\le 0$. Together these conditions are the obstacle problemThus an unknown exercise boundary must be found alongside the value. At a regular boundary in the nondegenerate Itô diffusion, value matching is $V=H$, and smooth pasting is $V_S=H^{\prime }$ where the financial payoff is differentiable. A finite difference method or a binomial options pricing model steps backward taking the larger of continuation and immediate exercise values at each node.

For a non-dividend-paying American call option with $r\ge 0$, no early exercise of a call without dividends follows directly from the European lower boundAt every candidate exercise time, keeping the European call has value at least the immediate exercise financial payoff, so early exercise cannot improve the value; the American and European prices coincide. The first inequality follows from Jensen inequality or put-call parity and nonnegative put value. With $r>0$ an American put option can benefit from early receipt of its strike at sufficiently low stock prices, so its exercise boundary is generally nontrivial and its value is at least the European put value. Dividends can make call exercise worthwhile, and negative interest invalidates the stated no-early-exercise call argument.

For European options with the same strike and maturity, the terminal payoff identity isThus long the call and short the put has the same terminal payoff as one stock less $X$ units of the maturity-$T$ zero-coupon bond. With no dividends, the stock strategy has no intervening payments. The law of one price gives put-call parity:Equivalently the two nonnegative terminal-payoff portfolios, call plus strike bond and put plus stock, agree in every state. The parity proof does not require lognormal stock prices or positive interest.

For American options the exercise times can differ, so equality of their maturity payoffs is not an identity between their full rights. Under the nonnegative-interest, non-dividend assumptions of part (a), $C_A=C_E$ while $P_A=P_E+{\Pi }_P$ with early-exercise premium ${\Pi }_P\ge 0$. Hencewhich generally differs from European parity. For a concrete limiting state, if the stock is zero and remains zero and interest is positive, then $C_A=C_E=0$, $P_A=X$ and $P_E=XB(t,T)<X$. European parity is not a general American-option equality. It holds in a particular case only when the relevant early-exercise premiums cancel, or both vanish.

Substitute the nonzero product $V=f(S)g(t)$ into the Black-Scholes equation with continuous stock dividends and divide by $fg$ on intervals where it is nonzero. The result isThe left side depends only on time and the right side only on spot, so both equal a constant $\lambda$. ThusThese are the separated solutions of the Black-Scholes equation. The zero solution is included separately, so division loses no relevant case.

For the unheaded parity request, the terminal difference of the power-call and power-put payoffs is $S_T^{\beta }-K^{\beta }$. Under the risk-neutral measure, $S_T=S\exp [(r-q-{\sigma }^2/2)\tau +\sigma (W_T-W_t)]$, where $\tau =T-t$. The Gaussian exponential moment gives ${\mathbb{E}}_Q[S_T^{\beta }\mid S_t=S]=S^{\beta }{e}^{{\lambda }_{\beta }\tau }$ with ${\lambda }_{\beta }=\beta (r-q)+{\sigma }^2\beta (\beta -1)/2$. Discounting the difference proves power put-call parity:For $\beta =1$ this is the familiar dividend-adjusted put-call parity, $S{e}^{-q\tau }-K{e}^{-r\tau }$.

At $B=K$, homogeneity givesThe price is therefore $C(S_0,K)-P(S_0,K)$. Zero-rate put-call parity yields

European put-call parity givesFor the American call, there are no dividends and the riskless rate is positive, so early exercise sacrifices the benefit of deferring the strike. Directly, at stock 30 its continuation value is $(4/5)[(1/6)30+(5/6)21]=18$, exceeding its exercise value 15; at stock 12 it is $(4/5)(5/6)=2/3$, exceeding exercise value zero. At time zero continuation is $86/15>0$. Thus $C_A(0)=C_E(0)=86/15$, with no optimal early exercise. European parity must not be applied using the American put price $3/2$.

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.

The Brownian martingale representation theorem argument in part (a) also replicates the bounded put payoff. The discounted stock is a true martingale, so the elementary terminal payoff identity yields put-call paritySet $H(T,K)=-S_0+K{e}^{-rT}$. Then $H_T=-rK{e}^{-rT}$, $H_K=e^{-rT}$, and $H_{KK}=0$. ThusLinearity of the Dupire equation and $P=C+H$ giveThe initial payoff is $P(0,K)=(K-S_0{)}^{+}$, and $P(T,0)=0$. Therefore calls and puts obey the same maturity-strike differential equation, with their respective initial and boundary data.

The difference of the terminal European call option and European put option payoffs is $S_T^1-K$. Replicating that difference by one share and a borrowing position gives conditional put-call parityThus the call is more valuable exactly when the right side is positive. With a put selected at equality, the chooser option payoff isAt equality both choices have the same price, even though their eventual payoffs can differ. Put-call parity assumes no dividends or other intermediate stock cash flows, as in the stated asset-price model.

Put $K^{\prime }=K(1+r{)}^{t_0-T}$. At the choice time,Buy a European call option of strike $K$ and maturity $T$, together with a European put option of strike $K^{\prime }$ and maturity $t_0$. If the call is selected, the short-maturity put expires worthless. If the put is selected, sell the call and combine its value with the short-maturity put's cash payment; put-call parity makes this exactly enough to buy the maturity-$T$ put. This self-financing replication provesIt requires tradability of the priced options at the choice time, but not completeness of every other market claim. The identity also covers $t_0=0$ and $t_0=T$.