Quantitative Finance Collector

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.
  1. Recent developments of option pricing models
    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.

    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.


    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.


    The fine structure of equity-index option dynamics
    We analyze the high-frequency dynamics of S&P 500 equity-index option prices by constructing an assortment of implied volatility measures. This allows us to infer the underlying fine structure behind the innovations in the latent state variables driving the evolution of the volatility surface. In particular, we focus attention on implied volatilities covering a wide range of moneyness (strike/underlying stock price), which load differentially on the different latent state variables. We conduct a similar analysis for high-frequency observations on the VIX volatility index as well as on futures written on it. We find that the innovations over small time scales in the risk-neutral intensity of the negative jumps in the S&P 500 index, which is the dominant component of the short-maturity out-of-the-money put implied volatility dynamics, are best described via non-Gaussian shocks, i.e., jumps. On the other hand, the innovations over small time scales of the diffusive volatility, which is the dominant component in the short-maturity at-the-money option implied volatility dynamics, are best modeled as Gaussian with occasional jumps.


    Leverage and feedback effects on multifactor Wishart stochastic volatility for option pricing
    The paper proposes a general asymmetric multifactor Wishart stochastic volatility (AMWSV) diffusion process which accommodates leverage, feedback effects and multifactor for the covariance process. The paper gives the closed-form solution for the conditional and unconditional Laplace transform of the AMWSV models. The paper also suggests estimating the AMWSV model by the generalized method of moments using information not only of stock prices but also of realized volatilities and co-volatilities. The empirical results for the bivariate data of the NASDAQ 100 and S&P 500 indices show that the general AMWSV model is preferred among several nested models.


    What’s beneath the surface? Option pricing with multifrequency latent states
    We introduce a tractable class of multi-factor price processes with regime-switching stochastic volatility and jumps, which flexibly adapt to changing market conditions and permit fast option pricing. A small set of structural parameters, whose dimension is invariant to the number of factors, fully specifies the joint dynamics of the underlying asset and options implied volatility surface. We develop a novel particle filter for efficiently extracting the latent state from joint S&P 500 returns and options data. The model outperforms standard benchmarks in- and out-of-sample, and remains robust even in the wake of seemingly large discontinuities such as the recent financial crisis.


    Model-based pricing for financial derivatives
    Assume that St is a stock price process and Bt is a bond price process with a constant continuously compounded risk-free interest rate, where both are defined on an appropriate probability space P. Let yt=log(St/St−1). yt can be generally decomposed into a conditional mean plus a noise with volatility components, but the discounted St is not a martingale under P. Under a general framework, we obtain a risk-neutralized measure Q under which the discounted St is a martingale in this paper. Using this measure, we show how to derive the risk neutralized price for the derivatives. Special examples, such as NGARCH, EGARCH and GJR pricing models, are given. Simulation study reveals that these pricing models can capture the “volatility skew” of implied volatilities in the European option. A small application highlights the importance of our model-based pricing procedure.

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  2. Sell in May and Go Away: Evidence from China
    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".

    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.

    “Sell in May and Go Away” puzzle means that stocks have higher returns in the November-April period than the May-October period, in this paper we first run a dummy regression that assign dummy=0 when the date t is in the May-October period, and dummy=1 when otherwise. We find the dummy variable is highly significant, not driven by a specific industry, and cannot be explained by well-known January or February effect, nor by time-varying risk, nevertheless, time-varying risk aversion approximated by the SAD (seasonal affective disorder) effect by Kamstra, et al. (2003) subsumes part of the Sell in May effect.

    Then we test whether such a phenomenon could generate any economic benefit, We construct a trading strategy that buys the Chinese stock market at the beginning of November and sells it at the end of April of the next year. We save the capital in a bank earning a risk-free floating deposit rate from the beginning of May to the end of October . Our benchmark is a buy-and-hold strategy. This simple trading strategy is shown to outperform the buy-and-hold strategy and can protect investors from dramatic losses during large recession, as shown in belowing Table and Figure.

      Sell in May strategy  Buy-and-hold strategy
    Return  13.03%  7.50%
    Sharpe ratio  0.6002  0.2199
    Maximum drawdown  27.00%  69.30%
    Downside deviation  2.98%  5.34%
    Historical VaR (95%)  6.86%  11.20%
    Leland’s alpha  8.69%  

    The short paper is at http://www.sciencedirect.com/science/article/pii/S1544612314000579
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  3. Performance of Trend Factor in Chinese market
    Han, Y.F., and Zhou, G.F. have an interesting working paper on the performance of a trend factor they proposed:
    In this paper, we propose a trend factor to capture cross-section stock price trends. In contrast to the popular momentum factor constructed by sorting stocks based on a single criterion of past year performance, we form our trend factor with a cross-section regression approach that makes use of multiple trend indicators containing daily, weekly, monthly and yearly information. We find that the average return on the trend factor is 1.61% per month, more than twice of the momentum factor. The Sharpe ratio is more than twice too. Moreover, during the recent financial crisis, the trend factor earns 1.65% per month while the momentum factor loses 1.33% per month. The trend factor return is robust to a variety of control variables including size, prior month return, book-to-market, idiosyncratic volatility, liquidity, etc., and is greater under greater information uncertainty. In addition, the trend factor explains well the cross-section decile portfolio returns sorted by short-term reversal, momentum, and long-term reversal as well as various price ratios (e.g. E/P), and performs much better than the momentum factor.

    The basic idea is to first calculate the month-end price moving average time series of different lags, then regress cross-sectionally monthly returns at date t on all moving average series at date t-1, finally predict monthly returns at date t+1 using the regression estimates and the moving average series at date t. This procedure guarantees we forecast stock returns at t+1 with information set only up to t. We then rank all stocks based on the forecasts into five quintiles, long the quintile with highest forecast returns and short the quintile with lowest, and rebalance once per month. This strategy generates, on average, 1.61% monthly return and 0.29 sharpe ratio using all US stocks, performs especially good during recession, and outperforms several existing factors. Moreover, the good performance of this strategy cannot be explained by firm fundamentals.

    I implement this strategy with Chinese stock data, adjust the rebalance frequency to weekly for convenience, and trade in extreme by always long the one stock with the highest forecast return, no short is allowed, stop loss is set at 5%. The result is amazing, it yields an annualized return at 97.15% from March, 2013 to Feb, 2014, with maximum drawdown at 30.01%. The fund curve is as follows (note: I didn't use all Chinese stocks but only 840 stocks in my stock pool with good liquidity, so there is selection bias and please accept the result cautiously...)

    Nice shot. It seems to be better than the simple strategy between A-shares and H-shares.
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  4. A simple strategy between A-shares and H-shares
    A similar article was posted at the sub-personal blog before and I paste it here in case someone is interested.

    At the moment there are 84 firms listed at both A (Shanghai and Shenzhen) and H (Hongkong) stock markets, according to the law of one price, the stock prices of these firms should be at similar level. However, there are huge differences, without considering exchange rate (1 RMB = 1.28 HK$), the ratio of the price in A market to the price in H market for a same firm is as low as 52.72% and as high as 617.59% as of 02/03/2014. Is the difference mean reverting? If yes, we would expect the stock traded cheaper in A market to go up, and vice versa. So can we make profit by long the stocks with large differences?

    Rigorous statistical method should be undertaken to examine whether the ratio is indeed mean reverting. For simplicity, I construct a trading strategy that each week, I go long at the opening price the stock in A market that has the smallest price ratio  of previous week, hold it one week and sell it at the weekly closing price. Short trading is not allowed for individual investor in A market. Stop loss is set arbitrarily at 5%. Transaction cost is 0.18% per trading.

    The results for this simple strategy from 02.2013 to 01.2014 are:
    Annualized Return         0.2070
    Annualized Std Dev        0.2545
    Annualized Sharpe          0.8133
    Maximum Drawdown
            From     Trough         To   Depth Length To Trough Recovery
    1 2013-09-13 2013-12-13           -0.1275     19        12       NA
    2 2013-08-16 2013-08-23 2013-09-06 -0.0566      4         2        2
    3 2013-03-22 2013-04-19 2013-05-03 -0.0488      5         3        2
    4 2013-07-12 2013-07-12 2013-07-19 -0.0374      2         1        1
    5 2013-05-31 2013-05-31 2013-07-05 -0.0229      6         1        5
    The fund curve
    Open in new window
    Lower line is the return for a buy-and-hold strategy of all 84 firms.

    Considering the fact that 2013 is a gloomy year for A market and this strategy is long only, the performance is not bad at all. Comments are welcomed
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  5. Thinknum Platform
    Justin, the founder of Thinknum, contacted me a few days ago about his site, I am very glad to share on this blog since it looks interesting and close related to the blog content.  

    Thinknum is a web platform that enables investors to collaborate on financial analysis, it aggregates the abundance of financial data and insights on the web and presents it to our users in an intuitive format, indexing the world’s financial information in the process.

    A few samples of what you can do on Thinknum:
    Thinknum’s Cashflow Model allows users to value companies based on fundamentals just like Wall Street research analysts do.  All the assumptions that go into the valuation models are visible and editable.  The data for the models is also updated automatically when companies publish their quarterly filings.

    The Plotter allows users to track financial data, analyze trends, and perform expressions such as regressions and correlations without having to write code.  Thinknum currently provides data from over 2,000 sources.

    A few experts have written about Thinknum:
    •  Jason Voss of the CFA Institute published a comprehensive overview of Thinknum’s mission.
    •  Francis Smart discussed Thinknum’s integration with R on R-Bloggers.  

    Thinknum was founded in 2013 by Gregory Ugwi and Justin Zhen, two friends who met at Princeton University in 2006.  After graduation, Gregory went to work for Goldman Sachs and Justin worked at a hedge fund, where they both discovered the major flaws with existing financial data analysis tools.  That’s when they decided to create a superior platform for all types of investors.

    Thinknum is constantly adding new features, so join their community and sign up for free today at thinknum.com.
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  6. Bloomberg Businessweek 15% Off Coupon

    Bloomberg Businessweek
    Bloomberg Businessweek, commonly and formerly known as BusinessWeek, is a weekly business magazine published by Bloomberg L.P. Founded in 1929, the magazine was created to provide information and interpretation about what was happening in the business world. BusinessWeek was first published in September 1929, only weeks before the stock market crash of 1929. The magazine provided information and opinions on what was happening in the business world at the time. Early sections of the magazine included marketing, labor, finance, management and Washington Outlook, which made BusinessWeek one of the first publications to cover national political issues that directly impacted the business world.

    I am offered a 15% off coupon for Bloomberg Businessweek, so should you are interested, you can order 16 Issues of Bloomberg Businessweek for $15! (That's an 81% Savings)!.

    Bloomberg Businessweek

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  7. Time-Varying Fund Manager Skill
    Another interesting paper forthcoming in Journal of Finance investigates the stock picking and market timing abilities of mutual fund managers.

    We propose a new definition of skill as a general cognitive ability to either pick stocks or time the market at different times. We find evidence for stock picking in booms and for market timing in recessions. Moreover, the same fund managers that pick stocks well in expansions also time the market well in recessions. These fund managers significantly outperform other funds and passive benchmarks. Our results suggest a new measure of managerial ability that gives more weight to a fund’s market timing in recessions and to a fund’s stock picking in booms. The measure displays far more persistence than either market timing or stock picking alone and can predict fund performance.

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  8. European Option Price with Excess Skewness and Kurtosis
    Stock returns however exhibit nonormal skewness and kurtosis as pointed out by Hull (1993) and Nattenburg (1994). Moreover, the volatility skews are a consequence of the empirical normality assumption violation. For this reason, Corrado and Su (1996) extend the Black-Scholes formula to account for nonnormal skewness and kurtosis in stock returns.

    This package calculates the European put and call option prices using the Corrado and Su (1996) model. This method explicitly allows for excess skewness and kurtosis in an expanded Black-Scholes option pricing formula. The approach adapts a Gram-Charlier series expansions of the standard normal density function to yield an option price formula that is the sum of a Black–Scholes option price plus adjustment terms for nonnormal skewness and kurtosis (Corrado and Su, 1997).
    For skewness = 0 and kurtosis = 3, the Corrado-Su option prices are equal to the prices obtained using the Black and Scholes (1973) model.

    You can download the Matlab code at Corrado and Su (1996) European Option Prices.

    Corrado, C.J., and Su T. Skewness and kurtosis in S&P 500 Index returns implied by option prices. Financial Research 19:175–92, 1996.

    Corrado, C.J., and Su T. Implied volatility skews and stock return skewness and kurtosis implied by stock option prices. European Journal of Finance 3:73–85, 1997.

    Hull, J.C., "Options, Futures, and Other Derivatives", Prentice Hall, 5th edition, 2003.

    Luenberger, D.G., "Investment Science", Oxford Press, 1998.
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  9. Liquidity-Driven Dynamic Asset Allocation
    A paper published in The Journal of Portfolio Management, 2013, 39 (3), pp 102-111, by James X. Xiong, Rodney N. Sullivan, and Peng Wang.

    We propose a model of portfolio selection that adjusts an investors’ portfolio allocation in accordance with changing market liquidity environments and market conditions. We found that market liquidity provides a useful “leading indicator” in dynamic asset allocation. Specifically, market liquidity risk premium cycles anticipate economic and market cycles. Investors can therefore act to avoid markets with low liquidity premiums, waiting to extract liquidity risk premiums when the likelihood of extracting a liquidity premium improves. The result, meaningfully enhanced portfolio performance through economic and market cycles, and is robust to transactions costs and alternate specifications.

    Basically this article examines a portfolio strategy that buys stocks and sells bonds when the market is less liquid, thus enjoying a higher liquidity premium, this strategy outperforms a benchmark with equal weights on stocks and bonds by generating a higher sharpe ratio and positive alpha.

    Journal paper Working paper
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  10. Mutual Funds R2 as Predictor of Performance
    Improving the accuracy of mutual funds' performance prediction is an interesting and endless topic. A paper published in Review of Financial Studies by Amihud and Goyenko (2013) No. 26 (3) investigates this issue at a new angle: Lower R2 indicates greater selectivity, and it significantly predicts better performance. Nice.

    We propose that fund performance can be predicted by its R2, obtained from a regression of its returns on a multifactor benchmark model. Lower R2 indicates greater selectivity, and it significantly predicts better performance. Stock funds sorted into lowest-quintile lagged R2 and highest-quintile lagged alpha produce significant annual alpha of 3.8%. Across funds, R2 is positively associated with fund size and negatively associated with its expenses and manager's tenure.

    Journal paper, Working paper.
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