According to the Palgrave Dictionary of Economics “Option pricing theory is the most successful theory not only in finance but also in economics”

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

An explicit inversion formula of Black-Scholes formula for implied volatility have found in

A semi-explicit equation for implied volatility has been developed by using a predefined function for option price, in

Some of the literature aimed at developing an asymptotic expansion of the implied volatility, either by using some extreme parameter values (ex; small maturity or large strike) through perturbations of the associated partial differential equations or estimates of semi groups. In certain conditions, asymptotic methods may not converge well outside those extreme parameter limits. Some recent literature

A remarkable breakthrough has been done in

So, the above theoretical and practical requirements have encouraged to implement the closed-form formula for the implied volatility in terms of current observed option prices.

The contribution of this research can be listed as follows,

This research intends to introduce detailed implementation of a closed-form formula for the market to calculate implied volatility by using market observed call option prices.

This implied volatility formula is well defined and it has been derived in terms of known functions and constants.

The traditional method of calculating the implied volatility involves an iterative approach which associates with high computational cost. Hence, the constructed equation calculates implied volatility without any time- consuming iterations.

Practically, thousands of options are converted to implied volatility in the real-time market circumstances. Thus, the method which comprises a high computational cost is not appropriate for the practical use of calculating implied volatility.

This study is mainly based on

In 1970 Fischer Black, Myron Scholes, and Robert Merton Established the Black-Scholes-Merton theory of pricing European stock options and later it was developed as a formula for option pricing. There are two versions of Black- Scholes formula, one for the call options and rest for the put options

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

Operator calculus or Operational Analysis is a technique of solving differential equations by transforming them into an algebraic problem. According to

Let

Then this method seems to be fine since this satisfies the following rules of calculus.

D(cf)(t) = cDf(t) where c is an arbitrary constant

D(f+g)(t) = Df(t) + Dg(t)

D^{k}(D^{l}f)(t) = D^{k+l}f(t) where k,

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.

According to the study ^{th} order derivative of the composition

where,

Here, the summation is taken over all combinations of j_{1}, j_{2}, . . . , j_{n−k}_{+1} of non-negative integers such that,

Let f be an analytic function at a point

where,

Here, the Bell polynomials can be calculated by using

Suppose that function

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

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

Here,

According to

The component of the partial derivative operator corresponding to

Transformation of

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

Moreover,

The derivation of the

According to

For each

where, spatial derivatives are given by the

Although, the major issue in this equation is if the term

Here, the above result was obtained by a recent study

The Taylor series of the European call option price with respect to σ around a positive level

Here,

Moreover, the radius of convergence of this series is

The Black-Scholes formula can be considered as a function of

Let V (σ) be the Black-Schole function. Then,

Since

where,

Here,

The radius of convergence of the series (17) is not given by the Lagrange inversion theorem and according to

Let

such that,

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,

the

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

According to the previous chapter the convergence of the constructed implied volatility formula depends on the pre-determined positive value (

The computational cost could be reduced using this new (

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

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.

Option name |
Strike ($) |
Maturity |
Current price ($) |
Market price ($) |
Risk free rate |
σ |
C. I. V |
Error |

CGC |
20 |
35/252 |
19.90 |
1.73 |
0.017880 |
0.6509 |
0.5960 |
0.0024 |

CGC |
20 |
260/252 |
19.90 |
4.06 |
0.017880 |
0.5444 |
0.4975 |
0.0024 |

IBM |
135 |
35/252 |
134.34 |
3.69 |
0.017880 |
0.2141 |
0.1936 |
0.0010 |

IBM |
135 |
260/252 |
134.34 |
9.35 |
0.017880 |
0.1810 |
0.1574 |
0.0016 |

FB |
210 |
35/252 |
208.67 |
8.50 |
0.017880 |
0.3237 |
0.2884 |
0.0020 |

FB |
210 |
260/252 |
208.67 |
25.50 |
0.017880 |
0.3159 |
0.2903 |
0.0011 |

TSLA |
460 |
34/252 |
458.09 |
33.05 |
0.018130 |
0.5213 |
0.4989 |
4.9339e-04 |

TSLA |
460 |
259/252 |
462.36 |
85.40 |
0.01807 |
0.4550 |
0.4361 |
4.3345e-04 |

GOOG |
1395 |
34/252 |
1395.11 |
46.90 |
0.018270 |
0.2316 |
0.2212 |
2.3611e-04 |

Option name Strike ($ ) Maturity Current price ($) Market price ($) Risk free rate σ C. I. V Error CGC 17.50 35/252 19.90 3.08 0.017880 0.6787 0.5715 0.0036 CGC 17.50 260/252 19.90 4.85 0.017880 0.5151 0.4499 0.0041 IBM 125 35/252 134.34 11.16 0.017880 0.2598 0.2544 1.6660e-05 IBM 125 260/252 134.34 15.59 0.017880 0.1821 0.1651 4.0532e-04 FB 190 35/252 208.67 22.25 0.017880 0.3725 0.3346 1.2505e-04 FB 190 260/252 208.67 36.70 0.017880 0.3455 0.3039 0.0021 TSLA 445 34/252 458.09 40.45 0.018130 0.5251 0.4988 6.4019e-04 TSLA 420 259/252 104.60 85.40 0.01811 0.4655 0.4317 0.0012 GOOG 1370 33/252 1395.11 60.00 0.018270 0.2410 0.2203 8.6243e-04

_{0}

Option name Strike ($) Maturity Current price ($) Market price ($) Risk free rate σ C. I. V Error CGC 22.50 35/252 19.90 0.95 0.017880 0.6851 0.6285 0.0013 CGC 22.50 260/252 19.90 3.34 0.017880 0.5498 0.5183 9.8554e-04 IBM 145 35/252 134.34 0.65 0.017880 0.1946 0.1807 7.0141e-04 IBM 145 260/252 134.34 5.45 0.017880 0.1776 0.1596 7.4456e-04 FB 220 35/252 208.67 4.40 0.017880 0.3106 0.2768 0.0011 FB 230 260/252 208.67 16.75 0.017880 0.3018 0.2787 8.1694e-04 TSLA 475 34/252 458.09 27.23 0.018130 0.5190 0.5067 1.4001e-04 TSLA 500 259/252 463.67 68.87 0.01809 0.4471 0.4272 4.7512e-04 GOOG 1415 34/252 1395.11 37.12 0.018270 0.2263 0.2184 1.4192e-04

_{0}

Here, the dividend rate was considered as zero.

^{−}^{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

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.

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”).