Put option

A put option gives its holder the right to sell an underlying asset at a specified strike price. A European put option may be exercised only at maturity, whereas an American put option permits earlier exercise.

A power put has financial payoff $(K-S{)}^n$ below the strike price $K$ and zero at or above it, where $n$ is a positive integer. For $n=1$ this is the ordinary put option. For every $n\ge 1$ the payoff is decreasing and is a convex function of the stock price. A European option permits exercise only at expiry; a perpetual power put option permits exercise at any finite time without a fixed expiry.

For a dividend-free Black-Scholes model with $r>0$, $\sigma >0$, $K>0$ and positive integer $n$, the optimal stopping value of the power put option isIn the continuation region the stationary Black-Scholes equation has powers $S$ and $S^{-\gamma }$. Boundedness at infinity removes $S$. Value matching gives the coefficient $(K-b{)}^n{b}^{\gamma }$, maximized at the displayed $b$; equivalently this is the smooth fit boundary.

For a verification beyond constant-trigger policies, let $g(S)=(K-S{)}_{+}^n$. The maximum property of $b^{\gamma }(K-b{)}^n$ gives $V\ge g$. For the infinitesimal generator $\mathcal{L}=rS{\partial }_S+{\sigma }^2{S}^2{\partial }_{SS}/2$, $(\mathcal{L}-r)V=0$ above $b$. Below $b$, set $y=S/(K-S)\le \gamma /n$; thenThe candidate is continuously differentiable across $b$, so the generalized Itô formula has no derivative-jump local time of a semimartingale term. The discounted candidate is a nonnegative supermartingale, bounding every stopping reward by optional stopping. First hitting of $b$ attains the bound; the remainder at infinite time vanishes since $V\le K^n$ and $r>0$. For $n=1$ this recovers the perpetual put option. At $r=0$ the value is $K^n$ for $S>0$, approached but not attained at a finite exercise time. At $r<0$ the perpetual value is infinite, because exercise at deterministic times tending to infinity has unbounded discounted expected value.

In the dividend-free Black-Scholes model, let $\tau =T-t>0$, $K>0$, $S>0$, $\sigma >0$ andThen the displayed finite sum prices the European option with power put option payoff. Here $\Phi$ is the standard normal distribution function. Indeed, $\log S_T$ conditionally has a normal distribution with mean $\log S+(r-{\sigma }^2/2)\tau$ and variance ${\sigma }^2\tau$. Expand $(K-S_T{)}^n$ by the binomial theorem on $S_T<K$ and apply the lower truncated lognormal moment to each term. Equivalently,This expectation representation proves monotonicity and convexity in $S$. It also gives $P_n(0,t)=K^n{e}^{-r\tau }$ and $P_n(S,t)\to 0$ as $S\to \infty$ by the dominated convergence theorem. For positive $S$ the lognormal distribution smoothing makes the price smooth and strictly positive.

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 $T$.

An American put has exercise payoff $(K-S_n{)}^{+}$. In a finite binomial market, backward option pricing compares that payoff with the discounted risk-neutral continuation value at every node.

A put with a prescribed exercise rule: exercise at the first touch of $B<K$, or at expiry if no touch occurs. Its value is bounded above by the American put option, which optimizes over exercise times. With zero interest rate, its value equals the down-and-out put value plus $(K-B)$ times a cash-at-expiry one-touch option value. At positive interest rate the rebate is paid at touch, so replacing it by an expiry payment would generally be incorrect.

For an alive barrier-triggered put, the difference from the corresponding European put option is the displayed discounted conditional expectation, evaluated under the risk-neutral measure. To prove it, condition the European payoff on the first hitting time or expiry and use the Strong Markov property. At a hit, continuation of the European claim is worth $P(B,{\tau }_B;K)$, whereas the fixed exercise rule pays $K-B$ immediately. On survival to expiry the two payoffs coincide. The relevant stopped quantities are bounded, so the conditional-expectation argument is justified.

For risk-neutral stock dynamics $dS=(r-\delta )Sdt+\sigma SdW$, with $r>0$ and $\sigma >0$, let ${\beta }_{-}<0$ solve ${\sigma }^2\beta (\beta -1)/2+(r-\delta )\beta -r=0$. A fixed downward trigger $L\in (0,X)$ gives continuation value $(X-L)(S/L{)}^{{\beta }_{-}}$ for $S>L$. Optimizing over the trigger, or imposing smooth fit, gives the displayed boundary. The optimal value equals $X-S$ below $S^{\ast }$ and the continuation expression above. A verification argument uses its payoff majorization and nonpositive discounted generator. At $r=0$ and $\delta \ge 0$, the perpetual value is $X$ for positive spot, approached by triggers tending to zero rather than attained at a positive finite trigger.

Here $\Delta =\sqrt{(r-\delta -{\sigma }^2/2{)}^2+2r{\sigma }^2}$. Implicit differentiation of the negative-root equation gives $d{\beta }_{-}/d\delta =-{\beta }_{-}/\Delta >0$. Differentiating $S^{\ast }=X{\beta }_{-}/({\beta }_{-}-1)$ gives the displayed negative derivative. A higher proportional dividend yield lowers risk-neutral stock growth and makes waiting for lower stock prices more attractive to the put holder.