What is Implied Volatility?
Implied volatility (IV) represents the market’s expectations of the future volatility of the underlying asset over the life of an option. It is derived from the market price of the option, rather than historical prices, and is an essential component of option pricing models. Unlike historical volatility, which measures past price movements, implied volatility reflects investors’ forward-looking sentiment.
IV is a key output of the Black-Scholes option pricing model, which calculates the theoretical price of an option based on several factors, including:
•Current price of the underlying asset
•Strike price of the option
•Time to expiration
•Risk-free interest rate
•Volatility of the underlying asset
While the model uses volatility as an input, implied volatility is found by reversing the process: we observe the market price of the option and solve for the volatility that would produce that price in the Black-Scholes model.
The Black-Scholes Formula
One of my favourite subjects at UCL was applied mathematics. During this course, we extensively covered the subject of Partial Differential Equations. The Black-Scholes is a partial differential equation governing the price evolution of an European call or put option formula for the price of a European call or put option is:
C = S N(d_1) – K e^{-rT} N(d_2)
Where:
•C : Price of the call option (observed in the market)
•S : Current price of the underlying asset
•K : Strike price of the option
•T : Time to maturity (in years)
•r : Risk-free interest rate
•\sigma : Volatility of the underlying asset (to be solved for)
•N : Cumulative distribution function of the standard normal distribution
•d_1 = \frac{\ln(S/K) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}
•d_2 = d_1 – \sigma \sqrt{T}
Calculating Implied Volatility
To calculate implied volatility, we must solve for \sigma (volatility) in the Black-Scholes equation given the market price of the option ( C ):
1.Input Market Data:
•Market price of the option ( C )
•Current price of the underlying asset ( S )
•Strike price ( K )
•Time to expiration ( T )
•Risk-free interest rate ( r )
2.Initial Guess:
•Start with an initial guess for \sigma , often based on historical volatility.
3.Iterative Solving (Newton-Raphson):
•Use numerical methods like the Newton-Raphson algorithm to adjust \sigma iteratively until the theoretical price matches the market price.
Newton-Raphson Method for Solving IV
The Newton-Raphson method refines \sigma as follows:
\sigma_{new} = \sigma_{old} – \frac{f(\sigma_{old})}{f{\prime}(\sigma_{old})}
Where:
•f(\sigma) = C_{theoretical}(\sigma) – C_{market}
•f{\prime}(\sigma) = \text{Vega} , the sensitivity of the option price to changes in volatility.
\text{Vega} = S \sqrt{T} N{\prime}(d_1)
Here, N{\prime}(d_1) is the probability density function of the standard normal distribution.
Steps:
1.Compute f(\sigma) (difference between theoretical and market prices).
2.Calculate f{\prime}(\sigma) (Vega).
3.Update \sigma using the Newton-Raphson formula.
4.Repeat until convergence (i.e., when |f(\sigma)| < \epsilon , where \epsilon is a small threshold).
Practical Applications of Implied Volatility
1.Option Pricing:
•IV is essential for accurately pricing options and assessing their value.
2.Market Sentiment Analysis:
•High IV indicates increased uncertainty or expected volatility.
•Low IV suggests stability or reduced price movement expectations.
3.Volatility Trading Strategies:
•Traders can profit from changes in IV through strategies like straddles or strangles.
4.Risk Management:
•IV helps in estimating the risk and potential price range of the underlying asset.
Tools for Calculating IV
•Programming Libraries:
•Python: Use libraries like SciPy (for optimization) and NumPy (for numerical calculations).
•R: Libraries such as fOptions and nloptr.
•Financial Software:
•Platforms like Bloomberg Terminal or financial calculators with built-in IV solvers.
Conclusion
Implied volatility is a cornerstone of modern options trading and risk management. Its calculation using the Black-Scholes model and numerical methods like Newton-Raphson enables traders and analysts to derive insights into market sentiment and prepare strategies based on expected price movements.
In my next blog, I will cover how to translate implied volatility into your trading strategy by extracting it from a Vix option.
Caio Marchesani