Volatility and the Value of American Options

Summary

The document discusses the familiar claim that higher volatility raises the value of standard American calls and puts, and considers why a formal reference may be hard to find. The response suggests reasoning from the relationship between American and European option values and the known volatility sensitivity of European options. It also notes that American options allow exercise decisions over time, making their valuation a dynamic optimization problem.

The answer describes American option valuation as generally requiring numerical methods, such as finite differences or Monte Carlo, rather than a simple closed-form solution. It offers a comparison argument as intuition, not a cited proof or a formal derivation. The discussion provides no model assumptions, theorem, or numerical evidence, so the monotonicity claim should be treated as a result to verify under the chosen contract and market assumptions rather than as a fully established result here.

Key ideas

• The discussion concerns how volatility affects the value of standard American calls and puts.
• American options permit exercise decisions over time, making valuation a dynamic optimization problem.
• The response proposes comparison with European options as intuition for the volatility effect.
• It gives no formal reference, proof, or detailed assumptions for the claimed monotonicity.

Tags

# Reference for "an increase in volatility increases value/price of american options"


# Reference for "an increase in volatility increases value/price of american options"












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.

## Answer by siou0107 (score 0, accepted)

https://quant.stackexchange.com/a/53321

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.