# Properties of the American price function in the Heston-type models

## TL;DR

This paper investigates properties of American option prices within the Heston stochastic volatility model, highlighting how convex payoffs influence the value function and extending classical results from Black-Scholes to this more complex setting.

## Contribution

It establishes the monotonicity of the American option value with respect to volatility and extends key classical results to the Heston model using probabilistic methods.

## Key findings

- Value function increases with volatility for convex payoffs
- Exercise boundary properties are characterized in the Heston model
- Early exercise premium formula is extended to the Heston setting

## Abstract

We study some properties of the American option price in the stochastic volatility Heston model. We first prove that, if the payoff function is convex and satisfies some regularity assumptions, then the option value function is increasing with respect to the volatility variable. Then, we focus on the standard put option and we extend to the Heston model some well known results in the Black and Scholes world, most by using probabilistic techniques. In particular, we study the exercise boundary, we prove the strict convexity of the value function in the continuation region, we extend to this model the early exercise premium formula and we prove a weak form of the smooth fit property.