Greeks and Implied Volatility

Failure of Historical Volatility

It is worth noting that all these pricing models have a volatility input, as to account for the future price expectation. However, the conventional approach of using historical volatility to predict future price uncertainty has faced criticism due to its assumption of time-invariant volatility, which fails to capture the dynamic nature of stock price distributions. This assumption overlooks the observed fat-tailed and evolving characteristics of price distributions over time. Consequently, employing such assumptions in Option pricing models derived from the underlying stock may lead to pricing inaccuracies. For example, examining the Brownian Motion stochastic differential formula within the Black-Scholes-Merton (BSM) framework: $$ dS_t = \mu S_t dt + \sigma S_t dW_t^S $$ In this equation, the parameter $\sigma$ denotes volatility, assumed to be constant. However, this assumption is unrealistic, as any movement in the underlying asset's price naturally induces fluctuations.

Local and Stochastic Volatility

One effective approach involves employing an alternative measure of volatility. This strategy is intuitive, as deviations of a stock's price from its current expected value typically correspond to increased variance in the series of price changes. This alternative measure can be broken down into stochastic and local volatility components.

Mathematically, stochastic volatility can be represented by the following stochastic differential equation: $$ d\sigma_t=\alpha(t,\sigma_t)dt + \beta(t,\sigma_t)dW_t^\sigma $$ On the other hand, local volatility, which is deterministic, serves as a parameterization of the volatility term within the original price Brownian Motion. Notable examples of local volatility models include the Dupire local volatility model (details). It is worth noting that many models utilize past volatility as a predictive input for future volatility. This approach is justified by the observation of volatility clustering, where volatility tends to persist at similar levels over short time frames.

Implied Volatility and Its Importance

If we view volatility modeling as a prospective approach, implied volatility (IV) can be understood as a retrospective, backward-looking measure derived from a chosen option pricing model. Following the no-arbitrage assumption, if we consider the market option price as the fair, theoretical price, we can use the current option price, along with other known variables, to determine the "fair" volatility value implied by the specific pricing model.

Implied volatility serves not only as the output of the pricing formula's inverse but also as an input for option Greeks, which we will discuss subsequently, as it represents a "correct" and risk-adjusted measure of volatility. Accurate implied volatility is crucial for calculating the correct Greeks for both trading and hedging purposes. Additionally, implied volatility offers insights into the likelihood of a stock moving toward a specific price level, thereby serving as valuable data points for regression analysis in the modeling process or to assess the accuracy of the model.

While modeling implied volatility can enhance pricing accuracy, some traders leverage implied volatility for volatility trading, as it tends to be more predictable due to volatility clustering. This concept can be implemented through trading composite option strategies such as Straddles and Iron Condor.

Option Greeks represent the changes of an option's price concerning changes in specific parameters. They serve as indicators of the sensitivity of an option's value to variations in these parameters, effectively quantifying the impact of each component on the final option price. The table below outlines some of the most commonly used Option Greeks:

Option Greeks are the partial derivatives of the option price. Therefore, their values are also a function of the IV. This stresses the importance of an accurate IV model to obtain accurate Greeks, hence accurate hedge size calculation for option hedging methods like Delta hedging.

The "volatility smile" is a term used in finance to describe a particular shape that can be observed when plotting IV against strike prices for options that share the same expiration date. Typically, when you plot IVs against strike prices, the curve tends to be concave, resembling a smile. This means that options with strike prices significantly above or below the current market price of the underlying asset tend to have higher IV than options with strike prices closer to the current market price.

Several theories attempt to explain the existence of the volatility smile. One explanation is that it arises from adjustments to the classic Black-Scholes-Merton (BSM) model, which assumes that asset prices follow a log-normal distribution. The volatility smile suggests that the market anticipates a higher probability of extreme price movements (fat tails) than the BSM model predicts. Another explanation is related to market dynamics and the demand for hedging. During periods of heightened market volatility, the volatility smile tends to become steeper, indicating a greater disparity in implied volatilities across different strike prices. This means that options, particularly those with out-of-the-money strike prices, become more expensive as investors seek protection against potential market downturns or increased price fluctuations. As a result, the volatility smile can serve as a valuable tool for investors and traders to assess market sentiment and gauge the level of risk in their portfolios.

According to some volatility models, such as the Heston model, the IV smile/surface should exhibit a smooth pattern. Any deviations from this smoothness, such as non-smooth spikes or irregularities, may signal potential arbitrage opportunities for traders.

Volatility Smirk

However, in some cases, rather than a traditional U-shaped curve, traders may observe what's known as a "smirk." This smirk occurs when the implied volatilities for higher-strike options are lower than those for ATM and lower-strike options. This deviation from the typical volatility smile pattern indicates a market expectation of a significant downward movement in the underlying asset's price. Traders interpret this as a signal that investors are allocating funds towards options with higher probabilities of such downward movements, resulting in increased demand for lower-strike options.

Volatility smoothing refers to the creation of a smooth volatility smile/surface. Derivative like options are favorable for maket-making and arbitration trading, due to the existence of fair price modeling. In the contemporary domain of high-frequency trading, the efficiency of calculating IV and Greeks assumes paramount significance. Thus, the quest for swift and refined IV surface computation becomes an inevitable quest for traders. Rather than relying solely on conventional methods involving slow-paced sampling and continuous parameter recalibration, the adoption of approximation techniques and stationary state analyses gains traction, especially in the context of IV estimation.

Accuracy Consideration

The Law of One Price stipulates the existence of a singular fair price for any given underlying asset. Consequently, according to the principles of put-call parity, each strike price for European options at expiration should correspond to a unique IV value. Within the realm of the Black-Scholes-Merton (BSM) framework, for instance, the concept of put-call parity yields insights into the equivalence of IVs for call and put options at identical strike prices and expirations. This equivalence serves as the foundation for formulating objective function: $$ C(t,K) - P(t,K)=e^{(q-r)t} (S_T - K) \\\Rightarrow e^{(q-r)t} [S_T (\Phi(d_{+}^{call})+\Phi(-d_{+}^{put})) - K (\Phi(d_{-}^{call})+\Phi(-d_{-}^{put}))] = e^{(q-r)t} (S_T - K) \\\Rightarrow \Phi(d_{+}^{call})+\Phi(-d_{+}^{put}) = 1\ \text{and}\ \Phi(d_{-}^{call})+\Phi(-d_{-}^{put}) = 1 \\\Rightarrow d_{\pm}^{call}=d_{\pm}^{put} $$ Thus, we can formulate an objective function of $C_{market} + P_{market} = C_{BSM} + P_{BSM}$.

Alternatively, certain schools of thought contend that the bulk of historical volatility information is encapsulated within at-the-money (ATM) options, with any residual volatility attributable to the extrinsic value of the option. Consequently, a strategy focused on out-of-the-money (OTM) options for IV calculations gains traction.

Speed Consideration

In the realm of high-frequency trading, where split-second decisions are the norm, the traditional process of sampling, evaluating, recalibrating, and deriving implied volatility expectations is simply unfeasible. In response, traders and analysts increasingly turn to polynomial regression techniques as a pragmatic solution for IV modeling.

For example, $$ IV_t = \beta_0 + \beta_1 M + \beta_2 M^2 + \beta_3 (T-t) + \beta_4 (T-t)^2 + \beta_5 M(T-t) \\\text{where} M = ln(S_t/K) $$

Additionally, the utilization of advanced mathematical algorithms, such as the Fast Fourier Transform (FFT), enables the rapid computation of the entire Greek surface, enhancing speed and efficiency. While these approaches may entail minor trade-offs in terms of accuracy, the overarching objective remains the attainment of a seamlessly smooth implied volatility surface that fosters effective arbitrage strategies.