Introduction & Definition
Implied Volatility (IV) serves as a crucial indicator in the realm of options trading, encapsulating the market’s perception of potential future fluctuations in an asset’s price. This metric isn’t directly observable in the market but is inferred from an option’s price, reflecting market sentiments about the asset’s future volatility. It differs from historical volatility (realized volatility), which represents past price movements.
When employing the Black-Scholes-Merton (BSM) model, options traders estimate the theoretical price of an option using several key parameters: the stock price, strike price, time until expiration, risk-free interest rate, and the stock’s volatility.
By rearranging the BSM model to solve for volatility instead of the option price, we unearth what’s known as implied volatility. This implied metric showcases the level of volatility expected by the market, enabling traders to gain insights into future price swings and the dynamics of supply and demand for options contracts. It’s a vital component for evaluating and pricing options, aiding traders in making informed decisions based on market expectations.
Let’s understand Mathematically!
The rationale behind option pricing models, such as the Black-Scholes model, lies in their utilization of various parameters to compute a theoretical value for an option. The specific inputs required by these models differ based on the type of option and the chosen pricing model. However, fundamentally, an option’s value hinges significantly on the anticipated future realized price volatility, denoted as σ, of the underlying asset.
Mathematically, the theoretical value (C) of an option is determined by a function f, reliant on σ and other pertinent inputs. This function f exhibits a monotonically increasing behavior concerning σ. In simpler terms, higher estimated volatility leads to a greater theoretical value for the option.
Conversely, according to the inverse function theorem, there exists at most one value for σ that, when inputted into the function f, yields a specific value for C. In essence, let’s assume the existence of an inverse function g = f−1
In this context, the market price (C_market) for an option corresponds to a specific value C, thereby resulting in the implied volatility σ_market. This implied volatility represents the value derived from the market price, signifying the anticipated future volatility embedded within the option’s pricing.
Implications
Here’s the implication:
- If the implied volatility is high, it suggests that the market expects a greater potential for larger price swings in the underlying stock before the option’s expiration.
- Conversely, if the implied volatility is low, it suggests that the market anticipates relatively smaller price movements in the underlying stock.
Traders and investors use implied volatility for various purposes:
- Assessing potential price movement: Higher implied volatility might suggest higher risk but also higher potential returns.
- Pricing other options: Implied volatility serves as a crucial input for pricing other options and creating trading strategies.
- Evaluating market sentiment: Changes in implied volatility can indicate shifts in market sentiment or expectations regarding the stock’s future movements.
Implied volatility is an essential metric in options trading because it provides insights into market expectations about future price fluctuations, helping traders make informed decisions about their options positions and strategies.
Factors Affecting Implied Volatility
Implied volatility, much like the broader market, is susceptible to unforeseeable fluctuations. It is significantly influenced by the dynamics of supply and demand within the options market. When an asset experiences heightened demand, its price escalates, consequently driving up implied volatility. This surge in implied volatility elevates the option premium, reflecting the increased risk associated with the option.
Conversely, an abundance of supply coupled with insufficient market demand leads to a decline in implied volatility. Consequently, the option price diminishes as a result of decreased risk perception.
Another pivotal factor influencing the option premium is the concept of time value, representing the duration until the option reaches its expiration. Short-term options typically exhibit lower implied volatility, while long-term options tend to manifest higher implied volatility. This disparity stems from the time available before the contract expiration. With an extended duration, there exists a greater opportunity for the asset price to move favorably concerning the strike price, thereby impacting the option’s price dynamics.
Benefits and Limitations
Benefits
- Market Expectations: IV reflects the market’s anticipation of future price movements. It provides insights into the expected level of volatility, aiding traders in assessing potential risks and returns.
- Options Pricing: IV is a vital component in options pricing models like Black-Scholes. It assists traders in determining fair prices for options, considering future market volatility.
- Trading Strategies: It enables the creation of diverse trading strategies. Traders use IV to gauge option mispricing, identify potential opportunities, and construct strategies like straddles or strangles.
- Risk Assessment: IV helps in risk assessment by indicating the level of uncertainty or risk priced into options. High IV implies higher expected price fluctuations and, therefore, higher risk.
Limitations
- Derived solely from market prices, not fundamentals: IV is exclusively determined from option prices and doesn’t encompass fundamental factors like company performance, economic indicators, or underlying asset fundamentals. Consequently, IV might not align with the intrinsic value or fundamental analysis of the underlying asset.
- Highly sensitive to unexpected factors, and news events: Implied volatility is extremely responsive to unforeseen events, news releases, or sudden market shocks. Unanticipated developments such as geopolitical events, economic data releases, or corporate announcements can significantly impact IV, causing abrupt fluctuations or spikes.
- Predicts movement, but not direction: IV indicates the expected extent of future price movements but doesn’t specify their direction. It portrays the potential range of price fluctuations without forecasting whether the price will ascend or descend. Hence, traders need supplementary analysis to ascertain the probable direction of price movements.
- Potential for Overestimation or Underestimation: IV might occasionally overstate or underestimate actual future volatility due to market inefficiencies, transient distortions, or imperfect pricing models. This can lead to misinterpretations and erroneous trading decisions.
Methods to calculate Implied Volatility
Several methods exist for calculating Implied Volatility (IV), including the Newton-Raphson Method, Bisection Method, Secant Method, Least Squares Monte Carlo (LSMC), and Implied Volatility Surface Interpolation, among others. Let’s delve into the mathematical principles behind two prominent IV calculation techniques:
- Newton-Raphson Method.
- Bisection Method.
Newton -Raphson method
The Newton-Raphson method is an iterative technique used to find the root of a function by utilizing its derivative. It relies on the derivative concept, representing the function’s slope at a given point, which forms the tangent line to the function curve. The method estimates the subsequent value of ‘x’ (denoted as ‘x_1’) by moving along this tangent line toward the point where the function intersects the x-axis (where ‘y = 0’).
The formula to update the estimate ‘x_1’ from the current value ‘x’ using the Newton-Raphson method is:
x_1 = x – f(x)/f'(x)
Here, ‘f(x)’ represents the function, and ‘f'(x) is its derivative.
The process continues iteratively until a stopping condition is met, such as reaching a maximum number of iterations or when the difference between successive approximations (x_1 and x) becomes sufficiently small (within an acceptable tolerance).
Advantages of the Newton-Raphson method include its rapid convergence, particularly for well-behaved functions, owing to its quadratic convergence rate. However, this method necessitates an initial guess for ‘x’, and the function must be differentiable within the given interval. Additionally, the Newton-Raphson method might fail to converge when handling multiple roots or encountering local extrema, potentially leading to divergence or inaccurate results in such cases.
For example:
S = 100 # Underlying price
K = 100 # Strike price
T = 1 # Time to expiration
r = 0.02 # Risk-free rate
Option price = 10 # Market price of the option
After four iterations, the implied volatility (IV) approaches approximately 22.78%.
The following graph is the result of finding volatility from the method:
Bisection Method
The bisection method stands out as a straightforward yet robust algorithm for finding roots within a given interval. Imagine we have two distinct points, 𝑎 and 𝑏, on a continuous function, satisfying 𝑎<𝑏 and 𝑓(𝑎)<0 and 𝑓(𝑏)>0. These points form an interval. Calculating the midpoint, 𝑐 =(𝑎+𝑏)/2, allows us to evaluate 𝑓(𝑐) within this interval.
Through an iterative process, we interchange 𝑐 with either 𝑎 or 𝑏, effectively narrowing down the interval to pinpoint the root. If 𝑓(𝑐) equals zero or reaches an acceptable value, we’ve discovered a root. Bisection methods boast stability and assure convergence. Despite not relying on derivative information, they consume more computational time compared to other methods.
After doing twenty-six iterations the IV came close to 22.77%
The following graph is the result of finding volatility from the method:
“The Python code for the above examples is accessible in the Python code category.“
