# Boundary for European Put Option

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 $$K$$?

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).

• This answer is mostly not correct. Among other things, european-style put options are bound below by max(0, Ke-rT-S0) and american ones by max(0, K-S0). Please correct your answer as this result comes up high in google. Jun 7 '20 at 8:40
• @qbodart Thanks for spotting the typo. :) It's corrected! Jun 7 '20 at 9:21