Implied volatility

In financial mathematics, the implied volatility (IV) 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. To understand where Implied Volatility stands in terms of the underlying, implied volatility rank is used to understand its implied volatility from a one year high and low IV. In financial mathematics, the implied volatility (IV) 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. To understand where Implied Volatility stands in terms of the underlying, implied volatility rank is used to understand its implied volatility from a one year high and low IV. 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 f ( σ , ⋅ ) {displaystyle f(sigma ,cdot ),} , 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 C ¯ {displaystyle scriptstyle {ar {C}},} is the market price for an option. The value σ C ¯ {displaystyle sigma _{ar {C}},} is the volatility implied by the market price C ¯ {displaystyle scriptstyle {ar {C}},} , 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. A European call option, C X Y Z {displaystyle C_{XYZ}} , on one share of non-dividend-paying XYZ Corp. The option is struck at $50 and expires in 32 days. The risk-free interest rate is 5%. XYZ stock is currently trading at $51.25 and the current market price of C X Y Z {displaystyle C_{XYZ}} is $2.00. Using a standard Black–Scholes pricing model, the volatility implied by the market price C X Y Z {displaystyle C_{XYZ}} is 18.7%, or: To verify, we apply the implied volatility back into the pricing model, f and we generate a theoretical value of $2.0004: