No-Arbitrage Bounds for European and American Put Options
Summary
The document explains theoretical price bounds for European and American options, focusing on why a European put’s upper bound is the present value of its strike while an American put’s upper bound is the strike itself. It lists lower and upper bounds for calls and puts of both exercise styles. For a European put, the maximum terminal payoff is the strike if the underlying falls to zero, so discounting that payoff gives the upper bound; the option’s limited liability gives a zero floor, strengthened by intrinsic value when applicable.
The response also uses European put-call parity to connect call and put prices and gives an inequality relating American call and put prices. It notes adjustments for continuous or discrete dividends and that an American call on a non-dividend-paying asset has the same value as its European counterpart. These are no-arbitrage results, not predictions from a price model. The discussion is introductory and assumes standard pricing conditions; it does not cover transaction costs or other market frictions.
Key ideas
- A European put is bounded below by its intrinsic value and above by the discounted strike.
- An American put can be worth up to the undiscounted strike because early exercise is available.
- European put-call parity implies bounds on European call and put prices.
- Dividend payments modify the standard option bounds and parity relationships.
- The stated bounds follow from no-arbitrage reasoning rather than a probabilistic price model.
Tags
Full text
# Boundary for European Put Option
# 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 $K$?
## Answer by Kevin (score 8)
https://quant.stackexchange.com/a/50846
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.