Suppose, if the price of a European option (say a put) can be shown to be monotone in volatility (say for any maturity), does it follow that American options has to be monotone in volatility?
CLARIFICATION: Monotonicity in volatility means, assuming all other paramters is fixed, the option price is increasing or decreasing in volatility level (at time 0)
^ Presumably, for Black-Scholes model, we can explicitly demonstrate this. (Though I have not tried myself)
what I am intersted in is, does anyone know any counter examples for this?
I work on optimal stopping problems. I am currently working on Barndoff-Nielson Shephard model and I am trying to prove the American put (or a more general pay off) under a pricing measure is monotone in volatility. While I think it should not be too difficult to show monotonicity in volatility if the option is European, it is a lot harder to do it for an American option. I am just trying to get some intuition for this.