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Write $h=T-t$. Conditional on ${\mathcal{F}}_t^S$, independent increments of the Brownian motion givewhere $Y$ has the standard normal distribution and is independent of ${\mathcal{F}}_t^S$. Factoring $S_t$ out of the positive part identifies the normalized Black-Scholes call function:The European call option payoff is integrable, because $0\le (S_T-K{)}^{+}\le S_T$ and $\mathbb{E}{S}_T=S_0$. The tower property of conditional expectation therefore proves that $C$ is a true martingale. At $t=T$, the boundary value $F(0,m)=(1-m{)}^{+}$ gives $C_T=(S_T-K{)}^{+}$. The answer is the conditional payoff martingale

A martingale with the specified terminal value must equal its conditional expectation. If ${\hat{S}}_t=0$, default has already occurred and the terminal European call option payoff is zero. If ${\hat{S}}_t>0$, put $h=T-t$. Conditional on survival to $T$,where $Y$ has the standard normal distribution; the conditional probability of this survival is $e^{-\lambda h}$. The conditional expectation of the payoff in the enlarged filtration used in part (c) is consequentlyThis expression is already measurable with respect to the natural filtration of $\hat{S}$, so the tower property of conditional expectation gives the same value there. The survival factor cancels the compensating growth factor:The separate zero branch avoids division by zero. At $t=T$ it gives the required payoff, using $F(0,m)=(1-m{)}^{+}$; integrability follows from $({\hat{S}}_T-K{)}^{+}\le {\hat{S}}_T$.

Let $h$ be the stock holding and let $c$ be the initial cost, so the cash holding is $b=c-10h$. Terminal portfolio wealth at stock price $s$ is $c+h(s-10)$. Dominating the European call option payoff at the three possible prices givesAdding the two endpoint inequalities yields $2c\ge 1$. Equality is attained by $h=1/2$ and $c=1/2$, which imply $b=-9/2$. The corresponding terminal wealth is $0,1/2,1$ at $s=9,10,11$, respectively, compared with the required payoff $0,0,1$. The cheapest super-replication strategy isThe middle-state excess shows why this is superhedging rather than exact claim replication. The endpoint bound proves global minimality, without relying on the physical state probabilities.

Under the usual no-arbitrage interpretation from part (a), let $Q^N$ be an equivalent martingale measure for the strictly positive numéraire $N$. Then $M_t=S_t/N_t$ is a martingale. Define the discounted European call option payoff $Y_t=(S_t-K{)}^{+}/N_t=(M_t-K/N_t{)}^{+}$. Because $N_{t+1}\ge N_t$ and $K>0$, we have $K/N_{t+1}\le K/N_t$. The conditional Jensen inequality therefore givesFor the middle step, the strike $K/N_t$ is fixed conditionally on ${\mathcal{F}}_t$; alternatively use $\mathbb{E}[X^{+}\mid {\mathcal{F}}_t]\ge \max \{\mathbb{E}[X\mid {\mathcal{F}}_t],0\}$. Thus $Y$ is a submartingale. Risk-neutral valuation in units of $N$ gives $C(T,K)=N_0{\mathbb{E}}^{Q^N}{Y}_T$, with the usual fixed initial prices. ConsequentlyThe source's word “increasing” means nondecreasing: a European call option can have zero payoff at successive maturities, so strict increase is not guaranteed. Integrability is understood in the claim class for which the stated replication prices exist.

Let $b$ and $h$ be the numbers of units of the numéraire and the stock in a replicating strategy. The two terminal payoffs of the European call option are $2$ and $0$, soSolving these simultaneous equations gives $b=-6/35$ and $h=8/35$. The initial replication cost isAs an independent pricing check, let $q$ be the risk-neutral probability of the state $(15,20)$ under the numéraire measure. The discounted stock must have initial value one and terminal mean one:Risk-neutral valuation then gives $10(3/7)(2/15)=4/7$. The physical probabilities $1/2,1/2$ are not the numéraire-measure probabilities.

The positive-part function is convex, so pathwise Jensen inequality givesTo compare the Asian option price with European call option prices at different dates, carry each earlier payoff forward using the numéraire. Purchase $1/T$ of each replicating strategy for maturity $t$, and, when its payoff is received, reinvest it in $N$ until $T$. This is a self-financing portfolio, with terminal wealthThe first inequality uses $N_T\ge N_t$ and nonnegative payoffs. Since the Asian option is replicable in the complete market, absence of arbitrage makes its replication cost no larger than this superhedging cost. ThereforeEquivalently, divide the first pathwise bound by $N_T$, use $(S_t-K{)}^{+}/N_T\le (S_t-K{)}^{+}/N_t$, and take expectations under the numéraire equivalent martingale measure. Merely averaging earlier payoffs without reinvestment would miss the difference in payment dates.

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 $t_0=T$.

Let $p_K=C(T,K+1)-2C(T,K)+C(T,K-1)>0$ and $D_K=C(T+1,K)-C(T,K)$. Given $S_T=K$, the next value is one of $K-1,K,K+1$. The martingale property makes its upward and downward conditional probabilities equal; their sum is the conditional variance of the increment, ${\sigma }^2(T,K)=2D_K/p_K$. ThusThe discrete Dupire equation therefore recovers all positive-probability one-step conditional transitions from the European call option price surface. No Markov property is required: these are conditional probabilities given the current value, not necessarily given the whole past.

Recovery is impossible at a zero-probability state. For example, $S_t\equiv S_0$ satisfies every stated assumption, but at $T\ge 1$ a different integer $K$ can satisfy $|K-S_0|\le T$ while $p_K=0$. Its conditional transitions can be specified arbitrarily without altering $C$. Thus the literal all-states claim needs the positive-probability qualification.

By the bound in part (a), Fubini's theorem applies. Conditional on $S$,where the Characteristic function of the Cauchy distribution was used. Since $\mathbb{E}S=1$,The formula expresses a European call option value through complex moments of $S$. In an affine stochastic-volatility model such as the Heston model, those moments are available from an explicit transform, so call prices reduce to a one-dimensional Fourier expectation or integral.

For a European call option, the pointwise inequality $(S_T-K{)}^{+}\ge S_T-K$ and risk-neutral pricing give

At time $t$, immediate exercise is worth $(S_t-K{)}^{+}$, while waiting is worth the European call option price with remaining maturity $\tau =T-t$. The conditional version of part (iii) givesUnder the standard Black-Scholes assumption $r\ge 0$, this is at least $S_t-K$, while a call value is nonnegative. Hence $C(t,S_t)\ge (S_t-K{)}^{+}$, and the investor should wait until $T$; with $\sigma >0$ and $\tau >0$, waiting has strictly positive time value.

If negative interest rates are allowed, the unconditional instruction “wait” need not be correct. The exact decision is to exercise whenand otherwise to continue.

For the European call option, risk-neutral valuation givesDefine the stock-numeraire measure in a binomial market byIt is a probability measure because the risk-neutral martingale property gives ${\mathbb{E}}_Q{S}_N=S_0(1+r{)}^N$. Consequently

Tilting one up move by its stock factor changes its probability toSimilarly $1-\hat{q}=(1-q)(1+a)/(1+r)$. Thus under $\hat{Q}$ the number of up moves is binomial with parameter $\hat{q}$, and

The European call option with strike $5$ pays $2,0,0$ at terminal stock prices $7,5,3$. Its lower-node value is zero, while its upper-node value isA second step of backward option pricing givesBy the fundamental theorem of asset pricing, this discounted expectation under the unique risk-neutral probabilities is the unique no-arbitrage price.

A European call option with maturity $N$ and strike $K$ paysat time $N$. Let $q$ be the risk-neutral conditional probability of the factor $1+b$. The discounted stock must be a martingale, soand thereforeBoth probabilities are strictly positive. At every node these are the unique probabilities satisfying the martingale condition, so the fundamental theorem of asset pricing gives a unique no-arbitrage price.

If exactly $n$ of the $N$ moves use the factor $1+a$, thenand there are $\left(\genfrac{}{}{0ex}{}{N}{n}\right)$ such paths. Discounted risk-neutral expectation now giveswhere

Put-call parity fixes the price difference between a European call option and a European put option on the same underlying stock, with the same strike price $K$ and expiry $T$. Assume no dividends or other payments from the stock before or at expiry, a known constant continuously compounded interest rate $r$, and a frictionless market. Borrowing and lending are available at that same interest rate; short selling of the stock, European call option, European put option, and zero-coupon bond is permitted. Settlement is certain, and the buy-and-hold portfolios below may be held until $T$ without interim funding restrictions or forced liquidation. These assumptions exclude transaction costs, taxes, borrowing fees, and execution or settlement risks from the arbitrage argument.

At expiry, the two positive parts cancel their corners:Thus a European call option and a short European put option together have the same terminal financial payoff as one stock share with a debt of $K$. A particularly readable claim replication uses two portfolios with nonnegative terminal amounts: portfolio A holds one European call option and a risk-free zero-coupon bond paying $K$ at $T$; portfolio B holds one European put option and one stock share. The zero-coupon bond costs $K{e}^{-rT}$ today, since the discount factor is $e^{-rT}$. There are no intermediate cash flows. For every terminal stock price,For the strike price $K=100$, even the following very different outcomes leave the two portfolios equal:

The law of one price forces $A_0=B_0$ in this setting: otherwise buy the cheaper portfolio and sell the more expensive portfolio using short selling. Their terminal obligations cancel in every scenario, while the initial price difference can grow in the bank account. Hence absence of arbitrage givesWith a constant per-period interest rate $i$ over $N$ discrete periods, the same reasoning gives $C_0-P_0=S_0-K(1+i{)}^{-N}$. This is a different compounding convention; $i$ is not the continuously compounded interest rate $r$.

For an illustrative stock price $S_0=100$, strike price $K=100$, expiry one year away, interest rate $r=0.05$ per year, and given European call option price $C_0=10$, put-call parity requiresBoth portfolios therefore cost approximately $105.12294245$. The inputs specify no stock price dynamics: $C_0=10$ is supplied for illustration, not calculated from the Black-Scholes model. Put-call parity determines the European put option price conditional on that European call option price; it does not determine both prices by itself. At zero interest rate, the same $S_0=K=100$ and $C_0=10$ give the exact result $P_0=10$.

To compute both directions of arbitrage, keep all the preceding inputs except the quoted European put option price. If it is $6$, portfolio B costs $106$ while portfolio A costs $105.12294245$. Buy A, sell B, and place the initial surplus $6-(10+100e^{-0.05}-100)\approx 0.87705755$ in the bank account. At expiry, A and the obligations from selling B cancel exactly, leaving $96e^{0.05}-100\approx 0.92202525$ in every scenario. If the European put option instead costs $4$, buy B for $104$ and sell A for $105.12294245$; the initial surplus is approximately $1.12294245$, growing to $100-94e^{0.05}\approx 1.18051694$. Both constructions start with zero external capital. They are model calculations under the stated frictionless market assumptions.

The beauty of put-call parity is that uncertain individual financial payoffs produce an exact relation between today's prices. No forecast of the terminal stock price and no probability distribution are needed. It provides a consistency check and lets claim replication exchange one contract for a combination of others. Dividends introduce extra cash flows, and American put options can be exercised early, so the displayed European equality cannot simply be transferred to those contracts. K. Foster's ^[ref] explain the matching portfolios; Jun Pan's ^[ref] discuss put-call parity and its dividend adjustment.

An original modern research source is Hans R. Stoll's ^[ref], The Journal of Finance 24(5), 801-824 (December 1969). Stoll studied the relation between contract prices and tested it against market data; this was a relation between prices rather than a formula fixing each price individually. Robert C. Merton's ^[ref], 28(1), 183-184, highlighted the importance of excluding premature exercise for an exact arbitrage equality. This distinction explains why the exercise assumption belongs in the statement of put-call parity.

Let $c_0$ be the initial monetary balance in the bank account, negative for borrowing. Matching successor financial payoffs gives the linear systemSubtracting determines the stock holding $\Delta$ in the header; substitution determines $c_0$. The initial cost is $V_0=\Delta S_0+c_0$. For the European call option with $S_u=120$, $S_d=80$, strike price 100 and payoffs 20 and zero,The debt becomes 40 at maturity: half a share minus 40 yields $60-40=20$ in the up state and $40-40=0$ in the down state. Holding these positions until maturity is self-financing. Repeating this two-equation construction at earlier nodes gives backward option pricing and the unique price in the complete market.

The discrete replication method was developed in Cox, Ross and Rubinstein's 1979 paper, whose Sections 2–3 explain borrowing, statewise matching and why physical probability drops out of relative option valuation. Their symbols for physical and pricing probabilities differ from the $p,q$ convention here. ^[ref]

In a finite-state discrete-time financial market, a replicable terminal financial payoff $X$ has the displayed value under any equivalent martingale measure $Q$ for the bank account numéraire. Here ${\mathcal{F}}_t$ represents current information and conditional expectation averages over future states. The self-financing portfolio's discounted portfolio wealth is a martingale, so its current value equals the discounted expectation of its terminal value. In an incomplete market, nonreplicable claims need not have a unique price; continuous-time extensions require further conditions on admissible trading strategies and integrable discounted wealth.

For the European call option in the example,Discounting the physical expected value instead gives $200/21\approx 9.52381$ when $p=1/2$, or $320/21\approx 15.23810$ when $p=4/5$. Those amounts describe discounted physical expectations, not the replicating portfolio's cost. Holding $S_0,u,d,R$ fixed is crucial: changing beliefs can influence traded prices in a richer financial market model.

Differentiating the displayed transformation shows that it preserves the Black-Scholes equation. The first derivative coefficient fixes the exponent $1-2r/{\sigma }^2$, which also makes the zero-order coefficient correct. Subtracting this reflected image of a European call option from the call itself enforces a zero boundary at $S=B$. For strike price $K\ge B$, the reflected maturity contingent claim payoff vanishes on $S>B$, giving the no-rebate down-and-out option formula. Prices using this reflection are conditional on the barrier not having been hit already.

The fixed price parameter at which an option permits an underlying trade. A European call option pays $(S_T-K{)}^{+}$ at maturity; a European put option pays $(K-S_T{)}^{+}$. Specifying $K$ therefore determines how terminal prices translate into the financial payoff.