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In financial mathematics, the implied volatility of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model (such as Black–Scholes) will return a theoretical value equal to the current market price of the option. A non-option financial instrument that has embedded optionality, such as an interest rate cap, can also have an implied volatility. Implied volatility, a forward-looking and subjective measure, differs from historical volatility because the latter is calculated from known past returns of a security. ==Motivation== An option pricing model, such as Black–Scholes, uses a variety of inputs to derive a theoretical value for an option. Inputs to pricing models vary depending on the type of option being priced and the pricing model used. However, in general, the value of an option depends on an estimate of the future realized price volatility, σ, of the underlying. Or, mathematically: : where ''C'' is the theoretical value of an option, and ''f'' is a pricing model that depends on σ, along with other inputs. The function ''f'' is monotonically increasing in σ, meaning that a higher value for volatility results in a higher theoretical value of the option. Conversely, by the inverse function theorem, there can be at most one value for σ that, when applied as an input to , will result in a particular value for ''C''. Put in other terms, assume that there is some inverse function ''g'' = ''f''−1, such that : where is the market price for an option. The value is the volatility implied by the market price , or the implied volatility. In general, it is not possible to give a closed form formula for implied volatility in terms of call price. However, in some cases (large strike, low strike, short expiry, large expiry) it is possible to give an asymptotic expansion of implied volatility in terms of call price.〔(Asymptotic Expansions of the Lognormal Implied Volatility ), Grunspan, C. (2011)〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「implied volatility」の詳細全文を読む スポンサード リンク
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