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We are a boutique financial service firm specializing in quantitative analysis, derivatives valuation and risk management.

We are a boutique financial service firm specializing in quantitative analysis, derivatives valuation and risk management.
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We are a boutique financial service firm specializing in quantitative analysis, derivatives valuation and risk management.




Mean Reverting and Trending Properties of SPX and VIX

In the previous post, we looked at some statistical properties of the empirical distributions of spot SPX and VIX. In this post, we are going to investigate the mean reverting and trending properties of these indices. To do so, we are going to calculate their Hurst exponents. http://blog.harbourfronts.com/2017/12/29/mean-reverting-trending-properties-spx-vix/


Statistical Distributions of the Volatility Index

VIX related products (exchange traded notes, futures and options) are becoming popular financial instruments for both hedging and speculation. The volatility index VIX was developed in the early 90’s. In its early days, it led the derivative markets. Today the dynamics has changed. Now there is strong evidence that the VIX futures market leads the cash index. In this installment we are going to look at some statistical properties of the spot VIX index. We used data from January 1990 to May...


Are Short Out-of-the-Money Put Options Risky? Part 2: Dynamic Case

This installment is the continuation of the previous one on the riskiness of out-of-the-money vs. at-the-money short put options and the effect of leverage on the risk measures. Here we’re going to perform similar studies with the only exception that from inception until maturity the short options are dynamically hedged. The simulation methodology and parameters are the same as in the previous study. http://blog.harbourfronts.com/2017/09/28/short-money-put-options-risky-part-2-dynamic-case/


Are Short Out-of-the-Money Put Options Risky?

We quantify and compare the risks of short out-of-the-money and at-the-money put options. We do so by performing Monte Carlo simulations and calculating the Value at Risk (VaR at 95% confidence interval) and variance of the return distribution. http://blog.harbourfronts.com/2017/08/23/are-short-out-of-the-money-put-options-risky/


Using a Market Timing Rule to Size an Option Position, A Static Case

In the previous installment, we discussed the use of a popular market timing rule to size a short option position. The strategy did not work well as it was the case in traditional asset allocation. As a follow up, we will apply the 10M SMA rule to a static, unhedged position. http://blog.harbourfronts.com/2017/06/30/using-market-timing-rule-size-option-position-static-case/


Volatility Trading Strategies, a Comparison of Volatility Risk Premium and Roll Yield Strategies

We previously presented 2 volatility trading strategies: one strategy is based on the volatility risk premium and the other on the volatility term structure, or roll yield. Here we present a detailed comparison of these 2 strategies and analyze their performance. http://blog.harbourfronts.com/2016/12/20/volatility-trading-strategies-a-comparison-of-volatility-risk-premium-and-roll-yield-strategies/


Using a Market Timing Rule to Size an Option Position

Position sizing and portfolio allocation have not received much attention in the options trading community. Here, we are going to apply a simple position sizing rule and see how it performs within the context of volatility trading. http://blog.harbourfronts.com/2017/04/30/use-market-timing-rule-size-option-position/


Stationarity and Autocorrelation Functions of VXX-Time Series Analysis in Python

In the previous post, we presented a system for trading VXX, a volatility Exchange Traded Note. The trading system was built based on simple moving averages. In this post, we are going to examine the time series properties of VXX in more details. http://tech.harbourfronts.com/trading/stationarity-autocorrelation-functions-vxx-time-series-analysis-python/


Differences Between the VIX Index and At-the-Money Implied Volatility

When trading options, we often use the volatility index, VIX, as a measure of volatility in order to enter and manage positions. This works most of the time. However, there exist some differences between the VIX index and at-the-money implied volatility. In this post, we are going to show such a difference through an example. Specifically, we study the relationship between the implied volatility and forward realized volatility of the SP500...


Is Asset Dynamics Priced In Correctly by Black-Scholes-Merton Model?

A lot of research has been devoted to answering the question: do options price in the volatility risks correctly? The most noteworthy phenomenon (or bias) is called the volatility risk premium, i.e. options implied volatilities tend to overestimate future realized volatilities. Much less attention is paid, however, to the underlying asset dynamics, i.e. to answering the question: do options price in the asset dynamics correctly?...


Merton Model for Credit Risk Management and a Case Study

Robert Merton published a seminal paper that laid the foundation for the development of structural credit risk models. We’re going to provide an example of how it can be used for managing credit risks. We are going to present a case study based on the Merton credit risk model. http://tech.harbourfronts.com/risk-management/merton-credit-risk-model-case-study/


Valuation of European and American Options in Python

We have provided examples of pricing European and American options in Excel. For pricing the European option, we utilized the Black-Scholes formula, and for pricing the American option we utilized the binomial approach. Here, we are going to implement these methods in Python. http://tech.harbourfronts.com/derivatives/valuation-european-american-options-derivative-pricing-python/


Interest Rate Swap-Derivative Pricing in Python

We are going to provide an example of interest rate swap pricing in Python. We are going to use the USD Libor swap curve as at December 31 2018. Note that we utilize the deposit and swap rates only and ignore the futures prices in the bootstrapping process. The values of the fixed, floating legs and the interest rate swap are calculated using a Python program. http://tech.harbourfronts.com/derivatives/interest-rate-swap-derivative-pricing-python/


Weighted Average Cost of Capital (WACC)-Business Valuation Calculator in Excel

The weighted average cost of capital (WACC) is the rate that a company is expected to pay on average to all its security holders to finance its assets. The WACC is commonly referred to as the firm's cost of capital. For illustration purposes, we are going to calculate the WACC of Barrick Gold, a major Canadian mining company. http://tech.harbourfronts.com/derivatives/weighted-average-cost-of-capital-wacc-business-valuation-calculator-in-excel/


Derivative Valuation-How to Price a Convertible Bond

A convertible bond (or preferred share) is a hybrid security, part debt and part equity. Its valuation is derived from both the level of interest rates and the price of the underlying equity. Several modeling approaches are available to value these complex hybrid securities such as Binomial Tree, Partial Differential Equation and Monte Carlo simulation. One of the earliest approaches was the Binomial Tree model originally developed by Goldman Sachs and this model allows for an efficient...


Valuing a Fixed Rate Bond-Derivative Pricing in Python

Debt instruments are an important part of the capital market. In this post, we are going to provide an example of pricing a fixed-rate bond. We are going to price a hypothetical bond as at October 31, 2018. We first build a zero coupon curve, then use it to price a hypothetical fixed rate bond. http://tech.harbourfronts.com/derivatives/valuing-fixed-rate-bond-derivative-pricing-python/


Interest Rate Swap-Derivative Pricing in Excel

An interest rate swap is a financial derivative instrument that involves an exchange of a fixed interest rate for a floating interest rate. Interest rate swaps are often used to hedge the fluctuation in the interest rate. To value an interest rate swap, fixed and floating legs are priced separately using the discounted cash flow approach. http://tech.harbourfronts.com/derivatives/interest-rate-swap-derivative-pricing-excel/


Valuing an American Option Using Binomial Tree-Derivative Pricing in Excel

We’re going to use the binomial pricing model to value an American equity option. Essentially, the model uses a “discrete-time” model of the varying price over time of the underlying financial instrument. Valuation is performed iteratively, starting at each of the final nodes, and then working backwards through the tree towards the first node. http://tech.harbourfronts.com/derivatives/valuing-american-option-using-binomial-tree-derivative-pricing-excel/


Valuing an American Option Using Barone-Andesi-Whaley Approximation

We are going to provide an example of valuing American options. We’re going to use the Barone-Andesi-Whaley approximation. The Barone-Adesi and Whaley Model has the advantages of being fast, accurate and inexpensive to use. It is most accurate for options that will expire in less than one year. http://tech.harbourfronts.com/derivatives/valuing-american-option-derivative-pricing-excel/


Valuing a European Option-Derivative Pricing in Excel

We’re going to price a put option on Barrick Gold, a Canadian mining company publicly traded on the Toronto Stock Exchange under the symbol ABX.TO. For this exercise, we assume that the option is of European style with a strike price of $13. (American style option will be dealt with in the next installment). The option expires in 3 years, and the valuation date is August 22, 2018. http://tech.harbourfronts.com/derivatives/valuing-european-option/