# Greeks and Implied Volatility

## Key Concepts

### Option Pricing

As derivatives, Options are subject to a zero-sum game dynamic, necessitating counterparties to take opposing positions. Consequently, it is essential to accurately determine the fair value of an Option to avoid potential disadvantages resulting from counterparties. It's possible to calculate the theoretical value of an Option using only its own terms and the underlying price series expectations.

$$P = F(S_t, K, \sigma, T, r, q)$$

where $P$ is the Option price, $S_t$ is the current underlying price, $K$ is the strike price, $\sigma$ is the volatility, $T$ is the expiry, $r$ is the continuous interest rate, and $q$ is the continuous dividend yield.

With a correct Option pricing model, arbitrage traders capitalize on market inefficiencies, speculation traders can trade on the underlying price expectation or volatility, and risk/portfolio managers can hedge their positions to perfectly reduce risk exposure. The following sections describe popular Option pricing models.

#### Black-Scholes-Merton (BSM) Framework

This framework operates on the assumption of a log-normal stochastic process governing underlying price dynamics. Its appeal stems from its continuous modeling approach and provision of closed-form analytical solutions, rendering it particularly amenable to real-world scenarios. Nevertheless, it has faced scrutiny for its reliance on assumptions of constant volatility and interest rates. It's primarily employed in the valuation of European Options, so utility wanes in situations where early exercise risk is significant.

#### Lattice Models

These encompass prevalent models such as the Binomial Tree and Trinomial Tree. Unlike the BSM model, lattice models adopt a discrete process framework lacking an analytical solution, necessitating numerical methods and substantial computational resources for the solution. However, they offer the advantage of facilitating the calculation of early exercise payoffs at each time step, making it a suitable tool on pricing American Options.

#### Monte Carlo Simulation

Particularly relevant for the pricing of exotic Options devoid of explicit analytical frameworks, Monte Carlo simulations serve as indispensable tools for estimating the expected fair Option prices. Despite their computational complexity and time-intensive nature, these simulations, including methods like Markov-Chain Monte Carlo, play a pivotal role in pricing certain Options.

### 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 an Iron Condor.

### Greeks

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:

GreekDescription
DeltaSensitivity of the option price by $1 change of the underlying security. GammaChange in Delta's value by$1 change of the underlying security.
VegaSensitivity of the option price by 1% change of the implied volatility.
ThetaSensitivity of the option price by daily time decay.
RhoSensitivity of the option price by 1% change of the interest rate.

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.

### Volatility Smile

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

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.

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