# Quantitative Finance Collector is a blog on Quantitative finance analysis, financial engineering methods in mathematical finance focusing on derivative pricing, quantitative trading and quantitative risk management. Random thoughts on financial markets and personal staff are posted at the sub personal blog.

Dec
8

Choosing an appropriate performance measure is important for fund investors, nevertheless, many researchers find empirically that the choice of measures does not matter because those measures generate identical rank ordering, even though the distribution of fund returns is non-normal. In this paper we certify their findings by proving the monotonicity of several widely used performance measures when the distribution is a location-scale family. The mutual fund monthly return data from 1997 to 2015, together with simulation results, collaborate with our proof.

An adequate risk-adjusted return performance measure to select investment funds is crucial for financial analysts and investors. Sharpe ratio has become a standard measure by adjusting the return of a fund by its standard deviation (Sharpe, 1966), nevertheless, practitioners often question this measure mainly for its invalidity if the distribution of fund returns is beyond normal (Kao, 2002; Amin and Kat, 2003; Gregoriou and Gueyie, 2003, Cavenaile, et al, 2011, Di Cesare, et al, 2014). Several new measures have been proposed and investigated to overcome this limitation of the Sharpe ratio, however, Eling (2008)

finds choosing a performance measure is not critical to mutual fund evaluation, Eling and Schuhmacher (2007) compare the Sharpe ratio with 12 other measures for hedge funds and conclude that the Sharpe ratio and other measures generate virtually identical rank ordering, despite the significant deviations from normal distribution. Similar evaluation includes Eling and Faust (2010) on funds in emerging markets, Auer and Schuhmacher (2013) on hedge funds, and Auer (2015) on commodity investments.

This paper proves that several widely used performance measures are monotonic if the distribution of asset returns is a LS family, a family of univariate probability distributions parametrized by a location and a non-negative scale parameters that is commonly applied in finance (Levy and Duchin, 2004). Our proof certifies the empirical findings in other studies on the indifference of choosing a performance measure when valuing a fund. We show that those measures generate virtually the same rank ordering using monthly mutual fund return data from 1997 to 2005 and Monte-Carlo simulations. Therefore this paper contributes to both the academia and industry by clarifying the phenomenon.

For example, the below figure plots the correlation and confidence intervals based on 2000 simulations for each sample size. For simplicity, we show the results for the Sharpe (ρ1), the Sharpe-Omega (ρ2) and the Sortino ratio (ρ3) only. Consistent with the previous finding, the rank correlation among these performance measures is roughly equal, and is approaching one with the increase of sample size.

An adequate risk-adjusted return performance measure to select investment funds is crucial for financial analysts and investors. Sharpe ratio has become a standard measure by adjusting the return of a fund by its standard deviation (Sharpe, 1966), nevertheless, practitioners often question this measure mainly for its invalidity if the distribution of fund returns is beyond normal (Kao, 2002; Amin and Kat, 2003; Gregoriou and Gueyie, 2003, Cavenaile, et al, 2011, Di Cesare, et al, 2014). Several new measures have been proposed and investigated to overcome this limitation of the Sharpe ratio, however, Eling (2008)

finds choosing a performance measure is not critical to mutual fund evaluation, Eling and Schuhmacher (2007) compare the Sharpe ratio with 12 other measures for hedge funds and conclude that the Sharpe ratio and other measures generate virtually identical rank ordering, despite the significant deviations from normal distribution. Similar evaluation includes Eling and Faust (2010) on funds in emerging markets, Auer and Schuhmacher (2013) on hedge funds, and Auer (2015) on commodity investments.

This paper proves that several widely used performance measures are monotonic if the distribution of asset returns is a LS family, a family of univariate probability distributions parametrized by a location and a non-negative scale parameters that is commonly applied in finance (Levy and Duchin, 2004). Our proof certifies the empirical findings in other studies on the indifference of choosing a performance measure when valuing a fund. We show that those measures generate virtually the same rank ordering using monthly mutual fund return data from 1997 to 2005 and Monte-Carlo simulations. Therefore this paper contributes to both the academia and industry by clarifying the phenomenon.

For example, the below figure plots the correlation and confidence intervals based on 2000 simulations for each sample size. For simplicity, we show the results for the Sharpe (ρ1), the Sharpe-Omega (ρ2) and the Sortino ratio (ρ3) only. Consistent with the previous finding, the rank correlation among these performance measures is roughly equal, and is approaching one with the increase of sample size.

Dec
7

Pawel wrote a great article on predicting heavy and extreme losses in real-time for portfolio holders, the goal is to calculate the probability of a very rare event (e.g. a heavy and/or extreme loss) in the trading market (e.g. of a stock plummeting 5% or much more) in a specified time-horizon (e.g. on the next day, in one week, in one month, etc.). The probability. Not the certainty of that event.

Read this excellent post and accompanying Pathon codes at http://www.quantatrisk.com/2015/06/14/predicting-heavy-extreme-losses-portfolio-1/

In this Part 1, first, we look at the tail of an asset return distribution and compress our knowledge on Value-at-Risk (VaR) to extract the essence required to understand why VaR-stuff is not the best card in our deck. Next, we move to a classical Bayes’ theorem which helps us to derive a conditional probability of a rare event given… yep, another event that (hypothetically) will take place. Eventually, in Part 2, we will hit the bull between its eyes with an advanced concept taken from the Bayesian approach to statistics and map, in real-time, for any return-series its loss probabilities. Again, the probabilities, not certainties.

Read this excellent post and accompanying Pathon codes at http://www.quantatrisk.com/2015/06/14/predicting-heavy-extreme-losses-portfolio-1/

Nov
13

I have written a working paper on CDS (credit default swap) implied stock volatility and found some interesting results. Post it here just in case someone is interested.

Both CDS and out-of-money put option can protect investors against downside risk, so they are related while not being mutually replaceable. This study provides a straightforward linkage between corporate CDS and equity option by inferring stock volatility from CDS spread and, thus, enables a direct analogy with the implied volatility from option price. I find CDS inferred volatility (CIV) and option implied volatility (OIV) are complementary, both containing some information that is not captured by the other. CIV dominates OIV in forecasting stock future realized volatility. Moreover, a trading strategy based on the CIV-OIV mean reverting spreads generates significant risk-adjusted return. These findings complement existing empirical evidence on cross-market analysis.

Mar
12

Journal of Econometrics accepts several papers on option pricing, some are quite interesting and represent the recent developments of this field. I list them here just in case you are also interested.

http://www.sciencedirect.com/science/article/pii/S0304407615000615

http://www.sciencedirect.com/science/article/pii/S0304407615000585

**Smile from the Past: A general option pricing framework with multiple volatility and leverage components**In the current literature, the analytical tractability of discrete time option pricing models is guaranteed only for rather specific types of models and pricing kernels. We propose a very general and fully analytical option pricing framework, encompassing a wide class of discrete time models featuring multiple-component structure in both volatility and leverage, and a flexible pricing kernel with multiple risk premia. Although the proposed framework is general enough to include either GARCH-type volatility, Realized Volatility or a combination of the two, in this paper we focus on realized volatility option pricing models by extending the Heterogeneous Autoregressive Gamma (HARG) model of Corsi et al. (2012) to incorporate heterogeneous leverage structures with multiple components, while preserving closed-form solutions for option prices. Applying our analytically tractable asymmetric HARG model to a large sample of S&P 500 index options, we demonstrate its superior ability to price out-of-the-money options compared to existing benchmarks.

http://www.sciencedirect.com/science/article/pii/S0304407615000615

**Option pricing with non-Gaussian scaling and infinite-state switching volatility**Volatility clustering, long-range dependence, and non-Gaussian scaling are stylized facts of financial assets dynamics. They are ignored in the Black & Scholes framework, but have a relevant impact on the pricing of options written on financial assets. Using a recent model for market dynamics which adequately captures the above stylized facts, we derive closed form equations for option pricing, obtaining the Black & Scholes as a special case. By applying our pricing equations to a major equity index option dataset, we show that inclusion of stylized features in financial modeling moves derivative prices about 30% closer to the market values without the need of calibrating models parameters on available derivative prices.

http://www.sciencedirect.com/science/article/pii/S0304407615000585

Oct
27

I have co-authored a short paper with a friend in Zhejiang University, forthcoming in the Finance Research Letters, titled "Sell in May and Go Away: Evidence from China".

As the abstract suggests, basically we aim to examine whether the sell-in-may phenomenon existed in developed country also happens in China, and if Yes, if there is any special reason to explain it, which has implications for those international investors as MSCI plans to add Chinese A shares to its emerging index from May 2015, and as the recent China's stock market plan that permits Hong Kong investors to trade designated stocks in Shanghai Exchange market directly. People would expect investing in China provides a diversified strategy.

Using the Chinese stock market data from 1997 to 2013, this paper examines the “Sell in May and Go Away” puzzle first identified by Bouman and Jacobsen (2002). We find strong existence of the Sell in May effect, robust to different regression assumptions, industries, and after controlling for the January or February effect. However, part of the puzzle is subsumed by the seasonal affective disorder effect. We then construct a trading strategy based on this puzzle, and find that it outperforms the buy-and-hold strategy and could resist the market downside risk during large recession periods.

As the abstract suggests, basically we aim to examine whether the sell-in-may phenomenon existed in developed country also happens in China, and if Yes, if there is any special reason to explain it, which has implications for those international investors as MSCI plans to add Chinese A shares to its emerging index from May 2015, and as the recent China's stock market plan that permits Hong Kong investors to trade designated stocks in Shanghai Exchange market directly. People would expect investing in China provides a diversified strategy.