Boundary for European Put Option

Question

As an entry level financial engineer, I'm learning about call-put parity, which helps us to get the boundary for call option: $S-Ke^{-rT}\leq c\leq S$, what about put option? Should its upper bound be $Ke^{-rT}$ or $K$?

Answer 1

Let's carefully distinguish which exercise type we consider.

• European-style call option $$ \max\{S_0-Ke^{-rT},0\}\leq C_E \leq S_0.$$

• European-style put option $$\max\{Ke^{-rT}-S_0,0\}\leq P_E\leq Ke^{-rT}.$$

• American-style call option $$\max\{S_0-K,C_E\}\leq C_A\leq S_0.$$

• American-style put option $$\max\{K-S_0,P_E\}\leq P_A\leq K.$$

Because American-style options offer the early-exercise feature, they are more valuable and have sharper inequalities. In particular, American-style options are dearer than their European-style counterparts and worth at least as much as their intrinsic value (immediate payoff).

If you have a European-style put option, an upper bound is $Ke^{-rT}$ simply by no arbitrage: the highest possible payoff occurs if $S_T=0$ in which case a put pays $K$. Thus, a put option can never cost more than $Ke^{-rT}$ as you need to discount the payoff. The lower bound of zero results from options being rights only -- you're never obliged to a negative payoff.

You can also use the put-call parity. We have for European-style options, $$C_E=P_E+S_0-Ke^{-rT}.$$ This implies the two inequalities for $C_E$. There is no put-call parity for American-style options, just an inequality, $$S_0-K\leq C_A-P_A\leq S_0-Ke^{-rT}.$$

The formulae above can be adjusted for dividend payments by replacing $S_0$ with $S_0e^{-qT}$ (continuous dividend rate) or $S_0-D$ (discrete payments). If there are no dividend payments, an American-style call option equals a European-style call option.

All the above relationships are derived from no-arbitrage arguments and are independent of any probabilistic model for the stock price. There exist further no-arbitrage properties of options (monotone increasing in time-to-maturity or convexity in strike price).