Implied Volatility of Options-Volatility Analysis in Python

Volatility measures market expectations regarding how the price of an underlying asset is expected to move in the future. There are two types of volatility: historical volatility and implied volatility. In a series of previous posts, we presented methods and provided Python programs for calculating historical volatilities. In this post, we are going to discuss implied volatility and provide a concrete example of implied volatility calculation in Python.

In financial mathematics, the implied volatility (IV) of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model (such as Black–Scholes), will return a theoretical value equal to the current market price of said option. A non-option financial instrument that has embedded optionality, such as an interest rate cap, can also have an implied volatility. Implied volatility, a forward-looking and subjective measure, differs from historical volatility because the latter is calculated from known past returns of a security. To understand where implied volatility stands in terms of the underlying, implied volatility rank is used to understand its implied volatility from a one-year high and low IV. Read more

Since there is no analytical formula for calculating the implied volatility of an option, we must use numerical root-finding techniques. While there are many techniques for finding roots, two of the most commonly used are Newton's method and Brent's method.  More details about these methods can be found in Reference [1].

In this post, we utilize a Python program to calculate the implied volatility of a European call option. The parameters of the option are as follows.

Valuation date: August 31, 2020

Expiry: August 31, 2021

Option type: Call

Exercise type: European

Option price: 10

Strike: 100

Underlying asset price: 100

Risk-free rate: 2%

Dividend yield: 1%

The picture below shows the implied volatility calculated by the Python program

We note that the implied volatility calculated by the Python program agrees well with the result obtained by using a third-party program as shown below.

Click on the link below to download the Python program.

References

[1] William H. Press, Saul A. Teukolsky, William T. Vetterling , Brian P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, 3rd Edition, 2007.

Post Source Here: Implied Volatility of Options-Volatility Analysis in Python

Are AI-Powered Hedge Funds Outperforming?

Last month, institutionalinvestor.com reported that AI-powered hedge funds outperformed their peers,

Hedge funds with artificial intelligence capabilities showed a huge competitive edge over investors that didn’t use AI, new research indicates.

AI-led hedge funds produced cumulative returns of 34 percent in the three years through May, a report Tuesday from consulting and research firm Cerulli shows. That compares with a 12 percent gain for the global hedge fund industry over the same period. Read more

However, things are not so bright under the surface. It was recently reported that one of the largest AI hedge funds is underperforming,

Voleon Group, one of the world’s biggest and best-performing artificial intelligence hedge funds, has suffered large losses after being hit by choppy markets, making it one of a number of computer-driven fund managers to struggle this year

...like many computer-driven funds, Voleon was caught out by this year’s market turmoil. That had left the Investors fund down by double digits earlier this year, and the Institutional Strategies fund down single digits, say people familiar with its returns.

Voleon is unusual in that, while many quantitative fund firms use a small element of machine learning in their investment process, very few focus solely on this approach. Machine learning involves letting algorithms learn and extrapolate rules from the data, rather than a human setting the rules an algorithm should follow. The firm trades a strategy known as statistical arbitrage, which involves betting that short-term discrepancies in prices will revert to a mean. Read more

Article Source Here: Are AI-Powered Hedge Funds Outperforming?

How to Calculate Stock Beta in Excel-Replicating Yahoo Stock Beta

In a previous post, we presented a method for calculating a stock beta and implemented it in Python. In this follow-up post, we are going to implement the calculation in Excel. We continue to use Facebook as an example. Recall that,

In finance, the beta (market beta or beta coefficient) is a measure of how an individual asset moves (on average) when the overall stock market increases or decreases. As such, beta is a useful measure of the contribution of an individual asset to the risk of the market portfolio when it is added in small quantity. Thus, beta is referred to as an asset's non-diversifiable risk, its systematic risk, market risk, or hedge ratio. Beta is not a measure of idiosyncratic risk. Read more

We utilize the following equation to calculate the stock beta,

where

• r_S is the stock return,
• r_M is the market return,
• Cov denotes the return covariance and,
• Var denotes the return variance.

We downloaded 5 years of monthly Facebook data from Yahoo Finance. We implemented the above equation in Excel and obtained a beta of 1.21

The picture below shows Facebook beta calculated by Yahoo Finance. It is 1.26. Our result agrees well with Yahoo’s result, although there is a small difference.

Click on the link below to download the Excel workbook.

Also, let us know what you think where the difference comes from.

Article Source Here: How to Calculate Stock Beta in Excel-Replicating Yahoo Stock Beta

Valuation of Executive Stock Options Using a Closed-Form Formula

Executive stock options usually have complex payoffs. To price them, we often use the binomial tree method or Monte Carlo simulations.

... a lattice model can be designed to accommodate dynamic assumptions of expected volatility and dividends over the option's contractual term, and estimates of expected option exercise patterns during the option's contractual term, including the effect of blackout periods. Therefore, the design of a lattice model more fully reflects the substantive characteristics of a particular employee share option or similar instrument. Nevertheless, both a lattice model and the Black–Scholes–Merton formula, as well as other valuation techniques that meet the requirements ... can provide a fair value estimate that is consistent with the measurement objective and fair-value-based method. Read more

Recently, Hsiao et al. [1] derived a close-form solution to a particular class of executive stock options.

This study derives a closed-form approximate solution for Bermudan reset executive stock options with cash dividends. The complicated payoff characteristic of this exotic option is caused by its particular stylistic features, including the reset mechanism, cash dividends, early exercises, and increasing volatilities. Using the integral representation method, a closed-form approximate solution was successfully derived. Moreover, this study discusses how specific critical factors affect executive stock option values and arrives at some interesting results by way of a numerical calibration.

 

References

[1]  Y.L. Hsiao, L.L. Chen, C.J. Ting,  The Valuation of Executive Stock Option Using the Integral Representation Method, Journal of Mathematical Finance, 2020, 10, 431-447

Article Source Here: Valuation of Executive Stock Options Using a Closed-Form Formula