Implementation and validation of a closed form formula for implied volatility

Introduction: Implied volatility is one of the most commonly used estimator to predict future stock market behaviours. Accordingly, millions of option prices are used to compute the implied volatility in the stock market frequently. Traditionally, this is calculated by inverting Black-Scholes option prices with iterative numerical methods but this associates high computational cost. Objectives: This research was mainly focused on implementing and validating an explicit closed-form formula for the implied volatility by using market observed call option prices. Methods: In order to obtain the explicit formula, the Taylor series of the Black-Schole call option price with respect to the volatility around a pre-determined initial value was obtained using the operator calculus and the Faa’ di Bruno’s formula. Taylor series of the implied volatility was acquired using the Lagrange inversion theorem. Here, all the coeﬃcients were explicitly determined using known functions and constants. Findings: The developed equation was tested using real time market call option prices with the corresponding market listed implied volatilities for options were used as the initial values. Numerical examples illustrate a signiﬁcant accuracy of the formula. Novelty : It is a closed form formula where the coeﬃcients are explicitly determined and free of numerical iterations making it suitable for industrial implementations and adaptation.


Introduction
According to the Palgrave Dictionary of Economics "Option pricing theory is the most successful theory not only in finance but also in economics" (1) . In 1970 Fischer Black, Myron Scholes, and Robert Merton invented the theory of the pricing of European stock options. Later this was developed as the Black-Scholes-Merton model for option pricing. According to the Black-Scholes-Merton formula, European stock option price depends on strike price, current stock price, time to maturity risk-free rate, dividend rate and the stock price volatility (2) . Black-Scholes formula is rarely used in the original direction because of the observed market volatility fluctuations, but it is frequently used in the opposite direction to quote implied volatility through option prices (3) .
As stated, implied volatility can be calculated by inverting Black-Scholes formula, but it's challenging and time consuming. Therefore, it is long believed that there is no exact closed-form formula for the implied volatility. Hence in practice, the implied volatility is typically determined by iterative numerical root finding algorithms such as Newton Raphson method or bisection method (3) . According to (3) , numerical root finding methods suffer from divergence issues, slow speed of convergence, and biases which may lead to incorrect interpretations. Therefore, from a practical viewpoint, the calculation of implied volatility is an important part of any financial tool-box and is central to pricing, risk management and model calibration involving market option prices (3) . Practically, millions of real-time option prices are converted to implied volatility at any given time, hence there is a requirement for a quick method to compute implied volatility. Therefore, there is a rising interest on developing productive non-iterative methods to estimate implied volatility to speed up the computational process. So, here are some previous literature that are related to this scenario.
for the call options and rest for the put options (2) . where, The variables can be defined as follows.
• S -stock price • K -strike price • T -maturity time • r -risk-free rate • q -dividend rate • σ -volatility • N(x) -cumulative probability distribution function of Standard Normal distribution.
Here, the time to maturity and the strike price can be defined as similar to the section 1 and the risk-free interest rate is the rate of 10-year bonds in the market. The Equation (1) represents the call option price and the Equation (2) stands for the put option. Here, the call option version of the Black-Scholes formula has been used to derive the closed-form formula of the implied volatility in this research.

Operator Calculus
Operator calculus or Operational Analysis is a technique of solving differential equations by transforming them into an algebraic problem. According to (10) , Oliver Heaviside (1850-1925) was a pioneer in promoting operational calculus methods through his research papers. He applied operator calculus in finding the solutions of differential equations, especially in the theory of electricity.
Let D = d dt be the operator of differentiation. According to Heaviside, D can be defined in an algebraic manner such that, D 0 = I,D k = d k dt k where k ∈ Z + and I is the identity operator. Then this method seems to be fine since this satisfies the following rules of calculus.
1. D(cf)(t) = cDf(t) where c is an arbitrary constant 2. D(f+g)(t) = Df(t) + Dg(t) 3. D k (D l f)(t) = D k+l f(t) where k, l ∈ Z + According to this definition, derivatives in differential equations can be replaced by operator D and then the relevant differential equation becomes a function of D. Moreover, it can be considered as a polynomial of D. Then one can use algebraic properties to solve differential equation by using above transformation and this is the simple idea behind operator calculus.

Faa' di Bruno's formula, Bell polynomial version (Riordan's formula)
According to the study (11) , if f and g are two functions then the n th order derivative of the composition f (g (t)) can be defined as, where, B n,k (x 1 , x 2 , . . . , x n−k+1 ) are the Bell polynomials defined by, Here, the summation is taken over all combinations of j 1 , j 2 , . . . , j n−k+1 of non-negative integers such that,

Lagrange Inversion Theorem
Let f be an analytic function at a point x 0 and d f (x 0 ) dx ̸ = 0 Then we can express the function f in its Taylor series. Moreover, the inverse function g = f −1 can also be performed as a formal power series which has a non-zero radius of convergence (3) . Moreover, if we are given a power series of the form y = ∑ ∞ n=0 a n x n then the inverse series has the form of x = ∑ ∞ n=0 A n y n where, Here, the Bell polynomials can be calculated by using Equation (5).

Theorem of radius of convergence
Suppose that function f is analytic at z 0 ∈ C with power series expansion f (z) = ∑ ∞ n=0 a n (z − z 0 ) n centered at z 0 . Then the radius of convergence of the power series is given by (12) ,

Spatial Derivatives
The higher-order derivatives of the European call option price with respect to the volatility are required to obtain a Taylor series of Black-Scholes call option price with respect to the volatility. Therefore, an explicit equation is needed to calculate these partial derivatives and pursuant to (3) the direct calculation of these derivatives is profound in computation and hardly leads to an explicit formula. Therefore, all these partial derivatives of the call option price with respect to the volatility were linked to spatial derivatives by using operator calculus. The derivatives of the option price with respect to the log stock price are called the spatial derivatives. So, the following variable transformation has been done in the Black-Scholes call option formula in order to obtain the spatial derivatives.
Denote log stock price as x = ln(s) then the Black-Scholes call option formula can be re expressed as: Here, V (S, T ) stands for the non-transforming Black-Scholes formula and U (x, T ) denotes the variable transformed version of Black-Scholes formula. According to (3) , there are two desirable properties associated with using the log stock price as the variable: 1. The component of the partial derivative operator corresponding to x commute to each other. 2. Transformation of S to x, the Black-Scholes formula bears no singularity on x.
So, by using above transformation, the following equation was obtained to calculate the spatial derivatives. For n=1,2,..., the n th order derivative of the European call option price with respect to x is given by, Moreover, H n (.) are called Hermit polynomials given by, The derivation of the Equation (10) is given in (3) , appendix A.1.

2.2.2
Higher order partial derivatives of the option price with respect to the volatility According to (3) , using the operator calculus (see 2.1.2), Faa' di Bruno's formula (see 2.1.3) and spatial derivatives, the following equation has been acquired to calculate the partial derivatives of the option price with respect to σ . For each n = 1, 2, 3, . . . , where, spatial derivatives are given by the Equation (10) . Although, the major issue in this equation is if the term 2k −n becomes negative then the factorial of negative integers cannot be defined in the usual way by using Gamma function, and that was not mentioned in (3) . Hence, a reasonable way of defining the factorial of a negative integer was needed to guarantee the validity of this equation and finally it has been defined as: Here, the above result was obtained by a recent study (13) . The derivation of the Equation (14) is given in (3) , appendix A.2.

Taylor series of the European call option price
The Taylor series of the European call option price with respect to σ around a positive level σ 0 can be expressed as: Here, V (S, T, σ ) denotes the European call option price and ∂ n V (S,T,σ o ) ∂ σ n is given by Equation (14) Moreover, the radius of convergence of this series is σ o and it can be obtained using the theorem in the section (2.1.5).

Taylor series of the implied volatility
The Black-Scholes formula can be considered as a function of σ thus, it satisfies the following properties.
Since V (σ ) satisfies all the conditions in the Lagrange inversion theorem (see the section 2.1.4) the Taylor series of the implied volatility was obtained using the series (16) and the Lagrange inversion theorem as: https://www.indjst.org/ where, T, σ o ) , V market is the market observed European call option price and σ implied is the implied volatility. Additionally, these V k s were acquired using the Equation (14) and these A n s were expressed according to the section (2.1.4).
The radius of convergence of the series (17) is not given by the Lagrange inversion theorem and according to (3) , it cannot be found explicitly. Therefore, a numerical method was determined using Cauchy Hadamard formula to calculate this radius of convergence and it is expressed as follows.
Let {R k } be a sequence of real numbers defined as: such that, A k s are the coefficients of the series (18). Then by calculating R k up to an appropriate number of terms, the radius of convergence R of the series (18) can be obtained approximately, and furthermore, the convergence interval of the series (18) can be expressed as

Validation using error calculation
The Matlab version 2017 was used to conduct all the numerical calculations. The real time trading option in the market was extracted randomly by using "Yahoo Finance". The absolute value of the approximated error between the calculated implied volatility and the true implied volatility can be calculated as follows: Here, σ true denotes the true implied volatility. Moreover, this equation can be simply derived using the Equations (16) and (17).

Results and Discussion
Implied Volatility is a widely used estimator to forecast the future stock volatility. As per the real circumstance in the stock market, millions of option prices are converted to the implied volatility in every moment. Traditionally, the implied volatility is calculated using options with iterative numerical methods involving high computational cost. Hence, this study was mainly focused on implementing and validating an accurate explicit closed form formula for the implied volatility by using market observed call option prices. In order to achieve the 2 nd objective different methodologies were examined thus, the Taylor series method which was introduced in (3) was found to be the most accurate and the apprehensible method for the general public. Hence the same methodology of (3) , was followed while adjusting the formulas appropriately.

Identifying the σ 0 value
According to the previous chapter the convergence of the constructed implied volatility formula depends on the pre-determined positive value (σ 0 ) which was used for the Taylor series expansions, so when calculating the implied volatility using the developed model, the most important step can be identified as determining this (σ 0 ) value. The value of the market listed implied volatility for a given option was used as (σ 0 ) in this study. After studying on Ask and Bid prices of an option it can be assumed that initial implied volatility is calculated using the gap between Ask, Bid prices. Moreover, one can identify this market listed implied volatility value in "Yahoo Finance" according to the following Figure 1.
The computational cost could be reduced using this new (σ 0 ) value instead of the upper bound value that has been used in (3) for (σ 0 ). https://www.indjst.org/

Computational Results
Under this section, the developed explicit formula for the implied volatility has been implemented in order to investigate the convergence and the accuracy. According to the Equation (17) , the implied volatility is given as an infinite sum. In order to convert this equation as a closed form formula, the summation should be taken up to a pre-determined truncation order, hence, n=10 was used as the truncation order. The analysis was mainly divided into 3 categories under option sensitivity, "At the Money", "In the Money" and the "Out of the Money".
Randomization was done to represent the strike in the range of 0-1500. The implied volatility for the chosen options were observed by Matlab calculations. Furthermore, respective errors were calculated between the observed values and the true implied volatility values.    Figure 2 displays the fluctuation between calculated log error and the strike price. It can be noticed that the minimum calculated error for the implied volatility was found from the "Out of the money" options which was approximately 10 −4 . The calculated error for the implied volatility for the majority of the cases in all 3 types of options were resulted as approximately between 10 −3 to 10 −4 . Most importantly the same accuracy of the model in (3) was obtained using a non-calculated σ 0 value hence, it reduces the computational cost.
Also, it can be introduced some suggestions for future research regarding this study. The development model can be extended for put options to calculate implied volatility within a wider range. Furthermore, a different programming platform can be used for numerical calculations in order to reduce the execution time. Moreover, the error can be reduced increasing the truncation order.

Conclusion
The explicit closed form formula was implemented for the implied volatility using Taylor series expansion. The formula contains known functions, constants and, the coefficients of the model were determined accurately and explicitly. The market listed implied volatility value for a given option was utilized as the initial expansion point of the formula. Moreover, it was found that the formula performs exceptionally well for "out of the money" options. Although, overall accuracy produced by the formula is significantly high for all 3 types of options ("At the money", "Out of the money" and "In the money").