Market Implied Volatility Formula

Implied volatility is a critical metric used in financial markets to gauge the market's expectations of future volatility. It is often derived from the price of financial options and reflects the market's forecast of the likely movement of an asset's price. This article delves into the formula for calculating market implied volatility, its significance, and practical applications.

Implied volatility (IV) is not directly observable but is inferred from the prices of options using models like the Black-Scholes model. The Black-Scholes formula provides a theoretical price for options based on various factors, including the asset's current price, strike price, time to expiration, risk-free rate, and volatility. By rearranging the formula to solve for volatility, we derive the implied volatility.

The Black-Scholes Formula

The Black-Scholes formula for a call option is expressed as:

$C=S_0\cdot N\left(d_1\right)-X\cdot e^{-r\cdot T}\cdot N\left(d_2\right)$C=S0​⋅N(d1​)−X⋅e−r⋅T⋅N(d2​)

where:

• $C$C = Call option price
• $S_0$S0​ = Current stock price
• $X$X = Strike price of the option
• $r$r = Risk-free interest rate
• $T$T = Time to expiration in years
• $N\left(d\right)$N(d) = Cumulative distribution function of the standard normal distribution
• $d_1$d1​ = $\frac{\ln \left(S_0/X\right)+(r+{\sigma }^2/2)\cdot T}{\sigma \cdot \sqrt{T}}$σ⋅T​ln(S0​/X)+(r+σ2/2)⋅T​
• $d_2$d2​ = $d_1-\sigma \cdot \sqrt{T}$d1​−σ⋅T​

For a put option, the formula is:

$P=X\cdot e^{-r\cdot T}\cdot N(-d_2)-S_0\cdot N(-d_1)$P=X⋅e−r⋅T⋅N(−d2​)−S0​⋅N(−d1​)

where:

• $P$P = Put option price

Implied Volatility Calculation

To find implied volatility, we use the market price of the option and input it into the Black-Scholes formula. The implied volatility is the value of $\sigma$σ that equates the theoretical price to the actual market price of the option.

Steps to Calculate Implied Volatility:

1. Input Known Values: Insert the known values into the Black-Scholes formula—stock price, strike price, risk-free rate, and time to expiration.

2. Set up the Equation: Use the Black-Scholes formula to set up an equation where the option price from the formula equals the actual market price.

3. Solve for Volatility: Use numerical methods, such as the Newton-Raphson method or other iterative techniques, to solve for the implied volatility.

Since implied volatility does not have a closed-form solution, it is typically calculated using computational tools and software.

Example Calculation

Let’s assume an option has the following characteristics:

• Current stock price ($S_0$S0​) = $100
• Strike price ($X$X) = $100
• Time to expiration ($T$T) = 0.5 years
• Risk-free rate ($r$r) = 5% per annum
• Market price of the call option ($C$C) = $10

To find implied volatility ($\sigma$σ):

1. Substitute Known Values into the Black-Scholes formula.

2. Iterate to Solve for $\sigma$σ: Use a financial calculator or software to find the value of $\sigma$σ that satisfies the equation $C=10$C=10.

Practical Application and Importance

Implied volatility is crucial for several reasons:

• Pricing and Trading: Traders use implied volatility to gauge market sentiment and to price options more accurately. Higher implied volatility typically leads to higher option premiums.

• Risk Management: Investors use implied volatility to assess the risk associated with an asset and to adjust their portfolios accordingly.

• Market Forecasting: Implied volatility can be a leading indicator of market trends and potential price movements.

Summary

Implied volatility is a key concept in financial markets that reflects market expectations of future volatility based on option prices. The Black-Scholes formula, though originally developed for pricing options, can be adapted to calculate implied volatility by solving for the volatility that equates the theoretical option price with the observed market price. Understanding and calculating implied volatility can provide valuable insights into market conditions and help in making informed trading and investment decisions.