I'm looking for a textbook/journal article reference for the well-known result that an increase in volatility increases the value/price of a standard American (call and put) option. In the case of continuous time with geometric brownian motion dynamics, I know how to prove the result directly from standard comparative-statics on the value function; however, I'm looking for a formal reference as every time I read something related on an article it is taken as given with no formal reference to the fact or only numerical examples are provided.
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I don’t think that you will ever find a “demonstration” of it, for two reasons. First, as you mentioned, it can be obviously deduced from comparison arguments: American option price is always no less than European option price, which monotonically increases with volatility. Second, because there are no analytic framework for American option. Their valuation is a dynamic programming problem that pretty always needs to be solved numerically (either by finite differences methods or by a smart Monte Carlo). If you want to convince yourself of that result, just take the “comparison argument with European options”, should be enough.
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$\begingroup$ Thanks, that is in fact enlightening. I was curious because I see it everywhere as a hard-known fact but I've never seen an explicit argument about it, and I was wondering if someone established it as a general fact. As I did it was by computing the value function for the infinite-time horizon in explicit form from the dynamic programming problem and just taking a partial derivative with respect to $\sigma$, so I guess it is as good as it gets. $\endgroup$ Apr 15, 2020 at 20:46
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$\begingroup$ It seems that in the context of the Black-Scholes model, "Hong-Yi Chen, Cheng-Few Lee and Weikang Shih (2010). Derivations and Applications of Greek Letters: Review and Integration, Handbook of Quantitative Finance and Risk Management, III:491–503." has a chapter on the greeks, where it is established that Vega is positive. $\endgroup$ Apr 15, 2020 at 20:52