In a previous post, we wrote about Employee Stock Options, a form of financial compensation that a company uses to reward its employees. In this post, we are going to discuss another form of compensation, Performance Share Units.

*Performance share units (PSUs) are hypothetical share units that are granted to you based mainly on corporate and/or individual performance. Structurally, they are very similar to restricted stock units except these are more focused on your performance. These notional units fluctuate in value based on the underlying company stock but do not represent actual share ownership until you convert them to shares. They are designed to mirror share ownership and you will generally be granted additional units having the same value as dividends being paid on the regular shares.*

*Companies typically use PSUs as a form of mid-term compensation as the units usually vest after three years. It converts an amount that would normally be paid as a bonus or other cash remuneration to share ownership. It is meant to encourage employees to meet certain performance targets and maximize share value over the medium-term. If the performance target is not met, the shares the employees could have received are forfeited to the company [1].*

The valuation of PSUs is based on the same principles similar to the valuation of stock options. However, more often than not, the payoff of a PSU is more complex and is usually tied to a relative performance measure. Therefore, Monte Carlo simulation is a preferred choice for pricing PSUs.

To price a PSU, we first simulate the price paths using the following Stochastic Differential Equation:

where

*S*is the stock price at time_{t}*t,*- σ denotes the stock volatility,
- µ is the drift which equals the risk-free rate, and
*dW*represents the standard normal random variable._{t}

The simulation is carried out until the PSU’s maturity. We then implement the payoff function and calculate the mean value of the payoff. Finally, we discount the mean value to the present and obtain the PSU value.

The Monte Carlo program used for pricing stock options can be modified without difficulty to implement the particular payoff of a PSU. If a relative performance with respect to an index or other stocks is required, then we need to perform Monte Carlo simulations involving multiple assets [2].

Follow the link below to download the Python program for performing Monte Carlo simulations.

**References**

[1] *The Navigator*, RBC Wealth Management, June 2018

[2] Glasserman, Paul; *Monte Carlo Methods in Financial Engineering,* Springer; 2003

Article Source Here: Performance Share Units-Derivative Valuation in Python

Employee Stock Option (ESO) is a form of compensation that a company uses to reward, motivate, and retain its employees.

*An employee stock option (ESO) is a label that refers to compensation contracts between an employer and an employee that carries some characteristics of financial options.*

*Employee stock options are commonly viewed as a complex call option on the common stock of a company, granted by the company to an employee as part of the employee's remuneration package. Regulators and economists have since specified that ESOs are compensation contracts.*

*These nonstandard contracts exist between employee and employer, whereby the employer has the liability of delivering a certain number of shares of the employer stock, when and if the employee stock options are exercised by the employee. The contract length varies, and often carries terms that may change depending on the employer and the current employment status of the employee.* *Read more*

An ESO is a financial option, but it differs from a regular stock option in the following,

- There is usually a vesting period during which the option cannot be exercised
- When the employees leave their jobs (voluntary or involuntary) during the vesting period they forfeit the unvested options.
- When employees leave (voluntarily or involuntarily) after the vesting period they forfeit options that are out of the money and they have to exercise vested options that are in the money immediately.
- Employees are not permitted to sell their employee stock options. They must exercise the options and sell the underlying shares in order to realize a cash benefit or diversify their portfolios. This tends to lead to employee stock options being exercised earlier than similar regular options.
- There is some dilution arising from the issue of employee stock options because if they are exercised, then new common shares are issued.

Because of these characteristics, the valuation of ESOs is different from regular stock options. In this post, we are going to implement the approach proposed by Hull and White [1]. Specifically, we are going to implement the vesting and forfeiture rate features. Other features can also be implemented without difficulty.

The input parameters are as follows,

*Stock price: 50*

*Strike: 50*

*Maturity: 5 years*

*Risk-free rate: 2%*

*Volatility: 40%*

*Vesting period: 2 years*

*Forfeiture rate: 2%*

We implemented the Hull and White approach in Python, and we obtained a price of 17.9

Follow the link below to download the Python program.

**References**

[1] J. Hull and A. White, *How to Value Employee Stock Options*, Financial Analysts Journal, Vol. 60, No. 1 (Jan. - Feb., 2004), pp. 114-119

Post Source Here: Employee Stock Options-Derivative Pricing in Python

In a previous post, we presented the binomial tree method for pricing American options. Recall that an American option is an option that can be exercised any time before maturity.

A drawback of the binomial tree method is that the implementation of a more complex option payoff is difficult, especially when the payoff is path-dependent. For example, for an American double-average option with periodic sampling time points, the strike price is not known at the start of the option. It can only be determined in the future and is therefore path-dependent. Another example is an American forward start option. These options cannot be valued using the binomial tree approach.

In this post, we are going to present a method for valuing American options using Monte Carlo simulation. This method will allow us to implement more complex option payoffs with greater flexibility, even if the payoffs are path-dependent. Specifically, we use the Least-Squares Method of Longstaff and Schwartz [1] in order to take into account the early exercise feature. The stock price is assumed to follow the Geometrical Brownian Motion and the dividend is simulated continuously.

Using this approach, it would be optimal to exercise the option if the immediate payment is larger than the expected future cash flows, otherwise it should be kept. Specifically, for each generated path, we regress the future payoffs on the basis functions of *S* and *S ^{2}*[2]. The regression equation provides us with estimation for the expected value of future payoffs as a function of

We implemented the Least-Squares Method of Longstaff and Schwartz in Python and priced the option presented in the previous post. The main input parameters are as follows,

The picture below shows the results obtained by using the Python program.

Follow the link below to download the Python program.

**References**

[1] F. Longstaff and E. Schwartz, *Valuing American options by simulation: A simple least-squares approach*, Review of Financial Studies, Spring 2001, pp. 113–147.

[2] *S* denotes the stock price. Other basis functions can also be used.

Article Source Here: Valuing American Options Using Monte Carlo Simulation –Derivative Pricing in Python

In the previous post, we introduced the Garman-Klass volatility estimator that takes into account the high, low, open, and closing prices of a stock. In this installment, we present an extension of the Garman-Klass volatility estimator that also takes into consideration overnight jumps.

Garman-Klass-Yang-Zhang (GKYZ) volatility estimator consists of using the returns of open, high, low, and closing prices in its calculation. It also uses the previous day's closing price. It is calculated as follows,

where *h _{i}* denotes the daily high price,

We implemented the above equation in Python. We downloaded SPY data from Yahoo finance and calculated the GKYZ historical volatility using the Python program. The picture below shows the GKYZ historical volatility of SPY from March 2015 to March 2020.

We note that the GKYZ volatility estimator takes into account overnight jumps but not the trend, i.e. it assumes that the underlying asset follows a GBM process with zero drift. Therefore the GKYZ volatility estimator tends to overestimate the volatility when the drift is different from zero. However, for a GBM process, this estimator is eight times more efficient than the close-to-close volatility estimator.

Follow the link below to download the Python program.

Post Source Here: Garman-Klass-Yang-Zhang Historical Volatility Calculation – Volatility Analysis in Python

In the previous post, we introduced the Parkinson volatility estimator that takes into account the high and low prices of a stock. In this follow-up post, we present the Garman-Klass volatility estimator that uses not only the high and low but also the opening and closing prices.

Garman-Klass (GK) volatility estimator consists of using the returns of the open, high, low, and closing prices in its calculation. It is calculated as follow,

where *h _{i}* denotes the daily high price,

We implemented the above equation in Python. We downloaded SPY data from Yahoo finance and calculated GK historical volatility using the Python program. The picture below shows the GK historical volatility of SPY from March 2015 to March 2020.

The GK volatility estimator has the following characteristics [1]

__Advantages__

- It is up to eight times more efficient than the close-to-close estimator
- It makes the best use of the commonly available price information

__Disadvantages__

- It is even more biased than the Parkinson estimator

Follow the link below to download the Python program.

**References**

[1] E. Sinclair, *Volatility Trading,* John Wiley & Sons, 2008

Article Source Here: Garman-Klass Volatility Calculation – Volatility Analysis in Python

In the previous post, we discussed the close-to-close historical volatility. Recall that the close-to-close historical volatility (CCHV) is calculated as follows,

where *x _{i}* are the logarithmic returns calculated based on closing prices, and

A disadvantage of using the CCHV is that it does not take into account the information about intraday prices. The Parkinson volatility extends the CCHV by incorporating the stock’s daily high and low prices. It is calculated as follow,

where *h _{i}* denotes the daily high price, and

We implemented the above equation in Python. We downloaded SPY data from Yahoo finance and calculated the Parkinson volatility using the Python program. The picture below shows the Parkinson historical volatility of SPY from March 2015 to March 2020.

The Parkinson volatility has the following characteristics [1]

__Advantages__

- Using daily ranges seems sensible and provides completely separate information from using time-based sampling such as closing prices

__Disadvantages__

- It is really only appropriate for measuring the volatility of a GBM process. It cannot handle trends and jumps
- It systematically underestimates volatility.

Follow the link below to download the Python program.

**References**

[1] E. Sinclair, *Volatility Trading,* John Wiley & Sons, 2008

Post Source Here: Parkinson Historical Volatility Calculation – Volatility Analysis in Python

International Financial Reporting Standard -2 deals with the recognition, measurement, and disclosure of Employee Stock Options. In this article, we will offer examples of accounting for Employee Stock Options. At the end of this article, we will present methods for valuing Employee Stock Options.

**What is an Employee Stock Option?**

A company often has the policy to make its employees the shareholders; therefore they offer a certain number of shares to eligible employees as an incentive.

To retain and motivate the workforce and sometimes to comply with the regulatory requirement, the company’s management can opt to issue share options to its employees. The Employee Stock Option plan is not meant to apply to all employees, rather to those who meet certain prescribed criteria.

Employees are normally required to meet the performance as well as service criteria to be eligible for the Employee Stock Option plan. Suppose that the management imposes a service condition of five years and an employee, Mr. A, opted for this option, then after five years of service, he would become eligible to exercise his options. The company often fixes a strike price for the option holders to exercise their rights.

**Eligibility of employees**

It’s up to the company’s management to decide what criteria they should use when issuing Employee Stock Options, but typically it involves the fulfillment of performance obligation and service period.

**Example**

Company A has offered 500 share options to each of 5 managers, subject to achievement of their sales targets and continuous services of 5 years with the company. Until these conditions are not fulfilled, the company cannot book expenses for those shares in its accounting books. When the options expire, the employees can choose to settle the transaction either in equity or in cash.

**Equity-settled options**

In this case, the employees reserve the right to convert their share options into equity by paying only the option exercise price. The employees then become the shareholders of the company.

**Cash settled options**

In this case, the company offers the employees the option of selling the shares or to get cash equivalent to the market value of those shares.

From the accounting perspective, the company has to make accounting adjustments for both the equity-settled and cash-settled transactions.

**How to account for the Employee Stock Options in the financial statement**

At the time of offering share options, the company would need to determine the fair value of options or intrinsic value of those options. Then every year after, an expense for compensation shall be debited to employee compensation account and credit entry shall be made in outstanding balance for the compensation plan. The same pattern of entries shall be repeated until the completion of the vesting period. The vesting period is the period after which the employee would be eligible to exercise his/her right to purchase the common stock of the company.

At the end of the vesting period, the employee is offered two options: either to buy common stocks by exercising his/her options, or to get cash equivalent to the number of shares. If an employee chooses to buy shares then the balance in the outstanding account shall be debited and credit shall be made in the capital account as common stocks. If he opts for the cash option, then the outstanding account shall be debited and cash account shall be credited.

**How to value Employee Stock Options**

Monte Carlo and the Binomial Tree methods are the most common approaches used to price the Employee Stock Options.

Post Source Here: Accounting for Employee Stock Options, Examples and Valuation Methods

In a previous post, we touched upon a stock’s volatility through its beta. In this post, we are going to discuss historical volatilities of a stock in more details.

*Also referred to as statistical volatility, historical volatility gauges the fluctuations of underlying securities by measuring price changes over predetermined periods of time. It is the less prevalent metric compared to implied volatility because it isn't forward-looking.*

*When there is a rise in historical volatility, a security's price will also move more than normal. At this time, there is an expectation that something will or has changed. If the historical volatility is dropping, on the other hand, it means any uncertainty has been eliminated, so things return to the way they were.* *Read more*

There are various types of historical volatilities such as close to close, Parkinson, Garman-KIass, Yang-Zhang, etc. In this post, we will discuss the close-to-close historical volatility.

The close-to-close historical volatility (CCHV) is calculated as follows,

where *x _{i}* are the logarithmic returns calculated based on the stock's closing prices, and

We implemented the above equation in Python. We downloaded SPY data from Yahoo finance and calculated CCHV using the Python program. The picture below shows the close-to-close historical volatility of SPY from March 2015 to March 2020.

It’s observed that the volatility is a mean-reverting process. The CCHV has the following characteristics [1]

__Advantages__

- It has well-understood sampling properties
- It is easy to correct bias
- It is easy to convert to a form involving typical daily moves

__Disadvantages__

- It is a very inefficient use of data and converges very slowly

Follow the link below to download the Python program.

**References**

[1] E. Sinclair, *Volatility Trading,* John Wiley & Sons, 2008

Post Source Here: Close-to-Close Historical Volatility Calculation – Volatility Analysis in Python

In finance, beta measures a stock's volatility with respect to the overall market. It is used in many areas of financial analysis and investment, for example in the calculation of the Weighted Average Cost of Capital, in the Capital Asset Pricing Model and market-neutral trading.

*Beta of an investment is a measure of the risk arising from exposure to general market movements as opposed to idiosyncratic factors.*

*The market portfolio of all investable assets has a beta of exactly 1. A beta below 1 can indicate either an investment with lower volatility than the market, or a volatile investment whose price movements are not highly correlated with the market. An example of the first is a treasury bill: the price does not fluctuate significantly, so it has a low beta. An example of the second is gold. The price of gold fluctuates significantly, but not in the same direction or at the same time as the market.*

*A beta greater than 1 generally means that the asset both is volatile and tends to move up and down with the market. An example is a stock in a big technology company. Negative betas are possible for investments that tend to go down when the market goes up, and vice versa. There are few fundamental investments with consistent and significant negative betas, but some derivatives like put options can have large negative betas.* *Read more*

In this post, we present a concrete example of calculating the beta of Facebook, a technology stock. As for the market benchmark, we utilize SPY.

The beta of a financial instrument is calculated as follows,

where

*r*is the stock return,_{S}*r*is the market return,_{M}- Cov denotes the return covariance and,
- Var denotes the return variance.

We downloaded 5 years of data from Yahoo Finance and implemented equation (1) in Python. The picture below shows the result returned by the Python program

It’s observed that the beta of Facebook is 1.19, which means that Facebook is more volatile than the market.

The next picture shows the stock returns regressed against market returns. Note that the slope of the linear regression line equals the beta of the stock.

Follow the link below to download the Python program.

Article Source Here: What is Stock Beta and How to Calculate Stock Beta in Python

The VIX index is an important market indicator that everyone is watching. VVIX, on the other hand, receives less attention. In this post, we are going to take a look at the relationship between the VIX and VVIX indices.

While the VIX index measures the volatility risks, VVIX measures the volatility-of-volatility risks. Its calculation methodology is similar to the VIX’s except that instead of using SPX options it uses VIX options.

To study the relationship between these 2 indices, we first calculated the rolling 20-days correlation of the VIX and VVIX returns from January 2007 to March 2020. The median value of correlation is 0.807 and 25% quantile is 0.66

The figure below presents the rolling 20-days VIX/VVIX correlation for the last 2 years. We also superimposed SPY on the chart. We observe that the correlation is usually high but there are periods where it decreases significantly. The current period is one of those.

[caption id="attachment_393" align="alignnone" width="628"] Correlation between the VVIX and VIX indices[/caption]

The next figure shows the scatter plot of VVIX returns vs. VIX returns. It’s observed that there is a significant population where VIX and VVIX returns are of opposite signs. We subsequently calculated the number of instances where VIX and VVIX move in the opposite direction. This indeed happens 22% of the time.

[caption id="attachment_394" align="aligncenter" width="563"] VVIX returns vs. VIX returns[/caption]

Some implications of this study are:

- Although the correlation between VIX and VVIX appears to be high, there is a significant number of instances where VIX and VVIX move in the opposite direction. So it’s fair to say that VVIX follows separate price dynamics which is different from the VIX. In other words, VVIX prices in different risks.
- Long VIX options or SPX back spreads are not always a good hedge for an equity portfolio. The hedge can break down.
- At times it’s cheaper to hedge a long equity portfolio using SPX options; at times it’s cheaper using VIX options.

See More Here: Correlation Between the VVIX and VIX indices