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.

http://tech.harbourfronts.com/trading/garman-klass-volatility-calculation-volatility-analysis-python/

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.

http://tech.harbourfronts.com/trading/parkinson-historical-volatility-calculation-volatility-analysis-python/

In this post, we are going to discuss historical volatilities of a stock in more details. There are various types of historical volatilities such as close to close, Parkinson, Garman-KIass, Yang-Zhang, etc. Here we will discuss the close-to-close historical volatility. It’s observed that the volatility is a mean-reverting process.

http://tech.harbourfronts.com/trading/close-close-historical-volatility-calculation-volatility-analysis-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. 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.

http://tech.harbourfronts.com/trading/stock-beta-calculate-stock-beta-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.

http://blog.harbourfronts.com/2020/03/25/correlation-vvix-vix-indices/

Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. The method presented in this post is suitable for calculating the VaR in a normal market condition. More advanced approaches such as Expected Tail Loss have been developed that can better take into account the tail risk.

http://tech.harbourfronts.com/risk-management/value-risk-financial-risk-management-python/

To price the options, we first simulate the price paths using the following Stochastic Differential Equation. The simulation is carried out until the options’ maturity. We then apply the terminal payoff functions and calculate the mean values of all the payoffs. Finally, we discount the mean values to the present and thus obtain the option values.

http://tech.harbourfronts.com/derivatives/valuing-european-options-using-monte-carlo-simulation-derivative-pricing-python/

In this post, we focus on the implementation of the Black-Scholes-Merton option pricing model in Python. Closed-form formula for European call and put are implemented in a Python code. We compare the results to the ones obtained by using third-party software and notice that they are in good agreement.

http://tech.harbourfronts.com/derivatives/black-scholes-merton-option-pricing-model-derivative-pricing-python/

In a previous post, we presented a theoretical framework for pricing convertible bonds and preferred shares. We also provided an example of pricing a convertible bond in Excel. In this installment, we present an example of pricing a convertible bond in Python.

http://tech.harbourfronts.com/derivatives/valuing-convertible-bond-derivative-pricing-python/

As negative interest rates started popping up around the world, quantitative analysts and traders have been asking a mundane but fundamental question: How to price trillions of dollars of financial instruments when their complex pricing models don’t work with negative numbers?

http://tech.harbourfronts.com/derivatives/how-negative-interest-rates-affect-derivative-pricing-models/