Volatility Modeling Archives - Page 4 of 4 - QUANTITATIVE RESEARCH AND TRADING

September 13, 2018

Enhancing Mutual Fund Returns With Market Timing

Summary

In this article, I will apply market timing techniques to several popular mutual funds.

The market timing approach produces annual rates of return that are 3% to 7% higher, with lower risk, than an equivalent buy and hold mutual fund investment.

Investors could in some cases have earned more than double the return achieved by holding a mutual fund investment over a 10-year period.

Hedging strategies that use market timing signals are able to sidestep market corrections, volatile conditions and the ensuing equity drawdowns.

Hedged portfolios typically employ around 12% less capital than the equivalent buy and hold strategy.

Background to the Market Timing Approach

In an earlier article, I discussed how to use marketing timing techniques to hedge an equity portfolio correlated to the broad market. I showed how, by using signals produced by a trading system modeled on the CBOE VIX index, we can smooth out volatility in an equity portfolio consisting of holdings in the SPDR S&P 500 ETF (NYSEARCA:SPY). An investor will typically reduce their equity holdings by a modest amount, say 20%, or step out of the market altogether during periods when the VIX index is forecast to rise, returning to the market when the VIX is likely to fall. An investment strategy based on this approach would have avoided most of the 2000-03 correction, as well as much of the market turmoil of 2008-09.

A more levered version of the hedging strategy, which I termed the MT aggressive portfolio, uses the VIX index signals to go to cash during high volatility periods, and then double the original equity portfolio holdings (using standard Reg-T leverage) during benign market conditions, as signaled by the model. The MT aggressive approach would have yielded net returns almost three times greater than that of a buy and hold portfolio in the SPY ETF, over the period from 1999-2014. Even though this version of the strategy makes use of leverage, the average holding in the portfolio would have been slightly lower than in the buy and hold portfolio because, in a majority of days, the strategy would have been 100% in cash. The result is illustrated in the chart in Fig. 1, which is reproduced below.

Fig. 1: Value of $1,000 – Long-Only Vs. MT Aggressive Portfolio

Source: Yahoo Finance.

Note that this approach does not entail shorting any stock. And for investors who prefer to buy and hold, I would make the point that the MT aggressive approach would have enabled you to buy almost three times as much stock in dollar terms by mid-2014 than would be the case if you had simply owned the SPY portfolio over the entire period.

Market Timing and Mutual Funds

With that background, we turn our attention to how we can use market timing techniques to improve returns from equity mutual funds. The funds selected for analysis are the Vanguard 500 Index Admiral (MUTF:VFIAX), Fidelity Spartan 500 Index Advtg (MUTF:FUSVX) and BlackRock S&P 500 Stock K (MUTF:WFSPX). This group of popular mutual funds is a representative sample of available funds that offer broad equity market exposure, with a high degree of correlation to the S&P 500 index. In what follows, we will focus attention on the MT aggressive approach, although other more conservative hedging strategies are equally valid.

We consider performance over the 10-year period from 2005, as at least one of the funds opened late in 2004. In each case, the MT aggressive portfolio is created by exiting the current mutual fund position and going 100% to cash, whenever the VIX model issues a buy signal in the VIX index. Conversely, we double our original mutual fund investment when the model issues a sell signal in the VIX index. In calculating returns, we make an allowance for trading costs of $3 cents per share for all transactions.

Returns for each of the mutual funds, as well as for the SPY ETF and the corresponding MT aggressive hedge strategies, are illustrated in the charts in Fig. 2. The broad pattern is similar in each case – we see significant outperformance of the MT aggressive portfolios relative to their ETF or mutual fund benchmarks. Furthermore, in most cases the hedge strategy tends to exhibit lower volatility, with less prolonged drawdowns during critical periods such as 2000/03 and 2008/09.

Fig. 2 – Value of $1,000: Mutual Fund Vs. MT Aggressive Portfolio January 2005 – June 2014

Source: Yahoo Finance.

Looking at the performance numbers in more detail, we can see from the tables shown in Fig. 3 that the MT aggressive strategies outperformed their mutual fund buy and hold benchmarks by a substantial margin. In the case of VFIAX and WFSPX, the hedge strategies produce a total net return more than double that of the corresponding mutual fund. With one exception, FUSVX, annual volatility of the MT aggressive portfolio was similar to, or lower than, that of the corresponding mutual fund, confirming our reading of the charts in Fig. 2. As a consequence, the MT aggressive strategies have higher Sharpe Ratios than any of the mutual funds. The improvement in risk adjusted returns is significant – more than double in the case of two of the funds, and about 40% higher in the case of the third.

Finally, we note that the MT aggressive strategies have an average holding that is around 12% lower than the equivalent long-only fund. That’s because of the periods in which investment proceeds are held in cash.

Fig. 3: Mutual Fund and MT Aggressive Portfolio Performance January 2005 – June 2014

Source: Yahoo Finance.

Conclusion

The aim of market timing is to smooth out the returns by hedging, and preferably avoiding altogether periods of market turmoil. In other words, the objective is to achieve the same, or better, rates of return, with lower volatility and drawdowns. We have demonstrated that this can be done, not only when the underlying investment is in an ETF such as SPY, but also where we hold an investment in one of several popular equity mutual funds. Over a 10-year period the hedge strategies produced consistently higher returns, with lower volatility and drawdown, while putting less capital at risk than their counterpart buy and hold mutual fund investments.

September 12, 2018

How to Bulletproof Your Portfolio

Summary

How to stay in the market and navigate the rocky terrain ahead, without risking hard won gains.

A hedging program to get you out of trouble at the right time and step back in when skies are clear.

Even a modest ability to time the market can produce enormous dividends over the long haul.

Investors can benefit by using quantitative market timing techniques to strategically adjust their market exposure.

Market timing can be a useful tool to avoid major corrections, increasing investment returns, while reducing volatility and drawdowns.

The Role of Market Timing

Investors have enjoyed record returns since the market lows in March 2009, but sentiment is growing that we may be in the final stages of this extended bull run. The road ahead could be considerably rockier. How do you stay the course, without risking all those hard won gains?

The smart move might be to take some money off the table at this point. But there could be adverse tax effects from cashing out and, besides, you can’t afford to sit on the sidelines and miss another 3,000 points on the Dow. Hedging tools like index options, or inverse volatility plays such as the VelocityShares Daily Inverse VIX Short-Term ETN (NASDAQ:XIV), are too expensive. What you need is a hedging program that will get you out of trouble at the right time – and step back in when the skies are clear. We’re talking about a concept known as market timing.

Market timing is the ability to switch between risky investments such as stocks and less-risky investments like bonds by anticipating the overall trend in the market. It’s extremely difficult to do. But as Nobel prize-winning economist Robert C. Merton pointed out in the 1980s, even a modest ability to time the market can produce enormous dividends over the long haul. This is where quantitative techniques can help – regardless of the nature of your underlying investment strategy.

Let’s assume that your investment portfolio is correlated with a broad US equity index – we’ll use the SPDR S&P 500 Trust ETF (NYSEARCA:SPY) as a proxy, for illustrative purposes. While the market has more than doubled over the last 15 years, this represents a modest average annual return of only 7.21%, accompanied by high levels of volatility of 20.48% annually, not to mention sizeable drawdowns in 2000 and 2008/09.

Fig. 1 SPY – Value of $1,000 Jan 1999 – Jul 2014

Source: Yahoo! Finance, 2014

The aim of market timing is to smooth out the returns by hedging, and preferably avoiding altogether, periods of market turmoil. In other words, the aim is to achieve the same, or better, rates of return, with lower volatility and drawdown.

Market Timing with the VIX Index

The mechanism we are going to use for timing our investment is the CBOE VIX index, a measure of anticipated market volatility in the S&P 500 index. It is well known that the VIX and S&P 500 indices are negatively correlated – when one rises, the other tends to fall. By acting ahead of rising levels of the VIX index, we might avoid difficult market conditions when market volatility is high and returns are likely to be low. Our aim would be to reduce market exposure during such periods and increase exposure when the VIX is in decline.

Forecasting the VIX index is a complex topic in its own right. The approach I am going to take here is simpler: instead of developing a forecasting model, I am going to use an algorithm to “trade” the VIX index. When the trading model “buys” the VIX index, we will assume it is anticipating increased market volatility and lighten our exposure accordingly. When the model “sells” the VIX, we will increase market exposure.

Don’t be misled by the apparent simplicity of this approach: a trading algorithm is often much more complex in its structure than even a very sophisticated forecasting model. For example, it can incorporate many different kinds of non-linear behavior and dynamically adjust its investment horizon. The results from such a trading algorithm, produced by our quantitative modeling system, are set out in the figure below.

Fig. 2a -VIX Trading Algorithm – Equity Curve

Source: TradeStation Technologies Inc.

Fig. 2b -VIX Trading Algorithm – Performance Analysis

Source: TradeStation Technologies Inc.

Not only is the strategy very profitable, it has several desirable features, including a high percentage of winning trades. If this were an actual trading system, we might want to trade it in production. But, of course, it is only a theoretical model – the VIX index itself is not tradable – and, besides, the intention here is not to trade the algorithm, but to use it for market timing purposes.

Our approach is straightforward: when the algorithm generates a “buy” signal in the VIX, we will reduce our exposure to the market. When the system initiates a “sell”, we will increase our market exposure. Trades generated by the VIX algorithm are held for around five days on average, so we can anticipate rebalancing our portfolio approximately weekly. In what follows, we will assume that we adjust our position by trading the SPY ETF at the closing price the day following a signal from the VIX model. We will apply trading commissions of $1c per share and a further $1c per share in slippage.

Hedging Strategies

Let’s begin our evaluation by looking at the outcome if we adjust the SPY holding in our market portfolio by 20% whenever the VIX model generates a signal. When the model buys the VIX, we will reduce our original SPY holding by 20%, and when it sells the VIX, we will increase our SPY holding by 20%, using the original holding in the long only portfolio as a baseline. We refer to this in the chart below as the MT 20% hedge portfolio.

Fig. 3 Value of $1000 – Long only vs MT 20% hedge portfolio

Source: Yahoo! Finance, 2014

The hedge portfolio dominates the long only portfolio over the entire period from 1999, producing a total net return of 156% compared to 112% for the SPY ETF. Not only is the rate of return higher, at 10.00% vs. 7.21% annually, volatility in investment returns is also significantly reduced (17.15% vs 20.48%). Although it, too, suffers substantial drawdowns in 2000 and 2008/09, the effects on the hedge portfolio are less severe. It appears that our market timing approach adds value.

The selection of 20% as a hedge ratio is somewhat arbitrary – an argument can be made for smaller, or larger, hedge adjustments. Let’s consider a different scenario, one in which we exit our long-only position entirely, whenever the VIX algorithm issues a buy order. We will re-buy our entire original SPY holding whenever the model issues a sell order in the VIX. We refer to this strategy variant as the MT cash out portfolio. Let’s look at how the results compare.

Fig. 4 Value of $1,000 – Long only vs MT cash out portfolio

Source: Yahoo! Finance, 2014

The MT cash out portfolio appears to do everything we hoped for, avoiding the downturn of 2000 almost entirely and the worst of the market turmoil in 2008/09. Total net return over the period rises to 165%, with higher average annual returns of 10.62%. Annual volatility of 9.95% is less than half that of the long only portfolio.

Finally, let’s consider a more extreme approach, which I have termed the “MT aggressive portfolio”. Here, whenever the VIX model issues a buy order we sell our entire SPY holding, as with the MT cash out strategy. Now, however, whenever the model issues a sell order on the VIX, we invest heavily in the market, buying double our original holding in SPY (i.e. we are using standard, reg-T leverage of 2:1, available to most investors). In fact, our average holding over the period turns out to be slightly lower than for the original long only portfolio because we are 100% in cash for slightly more than half the time. But the outcome represents a substantial improvement.

Fig. 5 Value of $1,000 – Long only vs. MT aggressive portfolio

Source: Yahoo! Finance, 2014

Total net returns for the MT aggressive portfolio at 330% are about three times that of the original long only portfolio. Annual volatility at 14.90% is greater than for the MT cash out portfolio due to the use of leverage. But this is still significantly lower than the 20.48% annual volatility of the long only portfolio, while the annual rate of return of 21.16% is the highest of the group, by far. And here, too, the hedge strategy succeeds in protecting our investment portfolio from the worst of the effects of downturns in 2000 and 2008.

Conclusion

Whatever the basis for their underlying investment strategy, investors can benefit by using quantitative market timing techniques to strategically adjust their market exposure. Market timing can be a useful tool to avoid major downturns, increasing investment returns while reducing volatility. This could be especially relevant in the weeks and months ahead, as we may be facing a period of greater uncertainty and, potentially at least, the risk of a significant market correction.

Disclosure: The author has no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article.

September 8, 2018

A Scalping Strategy in E-Mini Futures

This is a follow up post to my post on the Mathematics of Scalping. To illustrate the scalping methodology, I coded up a simple strategy based on the techniques described in the post.

The strategy trades a single @ES contract on 1-minute bars. The attached ELD file contains the Easylanguage code for ES scalping strategy, which can be run in Tradestation or Multicharts.

This strategy makes no attempt to forecast market direction and doesn’t consider market trends at all. It simply looks at the current levels of volatility and takes a long volatility position or a short volatility position depending on whether volatility is above or below some threshold parameters.

By long volatility I mean a position where we buy or sell the market and set a loose Profit Target and a tight Stop Loss. By short volatility I mean a position where we buy or sell the market and set a tight Profit Target and loose Stop Loss. This is exactly the methodology I described earlier in the post. The parameters I ended up using are as follows:

Long Volatility: Profit Target = 8 ticks, Stop Loss = 2 ticks
Short Volatility: Profit Target = 2 ticks, Stop Loss = 30 ticks

I have made no attempt to optimize these parameters settings, which can easily be done in Tradestation or Multicharts.

What do we mean by volatility being above our threshold level? I use a very simple metric: I take the TrueRange for the current bar and add 50% of the increase or decrease in TrueRange over the last two bars. That’s my crude volatility “forecast”.

The final point to explain is this: let’s suppose our volatility forecast is above our threshold level, so we know we want to be long volatility. Ok, but do we buy or sell the ES? One approach ia to try to gauge the direction of the market by estimating the trend. Not a bad idea, by any means, although I have argued that volatility drowns out any trend signal at short time frames (like 1 minute, for example). So I prefer an approach that makes no assumptions about market direction.

In this approach what we do is divide volatility into upsideVolatility and downsideVolatility. upsideVolatility uses the TrueRange for bars where Close > Close[1]. downsideVolatility is calculated only for bars where Close < Close[1]. This kind of methodology, where you calculate volatility based on the sign of the returns, is well known and is used in performance measures like the Sortino ratio. This is like the Sharpe ratio, except that you calculate the standard deviation of returns using only days in which the market was down. When it’s calculated this way, standard deviation is known as the (square root of the) semi-variance.

Anyway, back to our strategy. So we calculate the upside and downside volatilities and test them against our upper and lower volatility thresholds.

The decision tree looks like this:

LONG VOLATILITY
If upsideVolatilityForecast > upperVolThrehold, buy at the market with wide PT and tight ST (long market, long volatility)
If downsideVolatilityForecast > upperVolThrehold, sell at the market with wide PT and tight ST (short market, long volatility)

SHORT VOLATILITY
If upsideVolatilityForecast < lowerVolThrehold, sell at the Ask on a limit with tight PT and wide ST (short market, short volatility)
If downsideVolatilityForecast < lowerVolThrehold, buy at the Bid on a limit with tight PT and wide ST (long market, short volatility)

NOTE THE FOLLOWING CAVEATS. DO NOT TRY TO TRADE THIS STRATEGY LIVE (but use it as a basis for a tradable strategy)

1. The strategy makes the usual TS assumption about fill rates, which is unrealistic, especially at short intervals like 1-minute.
2. The strategy allows fees and commissions of $3 per contract, or $6 per round turn. Your trading costs may be higher than this.
3. Tradestation is unable to perform analysis at the tick level for a period as long at the one used here (2000 to 2014). A tick by tick analysis would likely show very different results (better or worse).
4. The strategy is extremely lop-sided: the great majority of the profits are made on the long side and the Win Rates and Profit Factors are very different for long trades vs short trades. I suspect this would change with a tick by tick analysis. But it also may be necessary to add more parameters so that long trades are treated differently from short trades.
5. No attempt has been made to optimize the parameters.
6 This is a daytading strategy that will exit the market on close.

So with all that said here are the results.

As you can see, the strategy produces a smooth, upward sloping equity curve, the slope of which increases markedly during the period of high market volatility in 2008.
Net profits after commissions for a single ES contract amount to $243,000 ($3.42 per contract) with a win rate of 76% and Profit Factor of 1.24.

This basic implementation would obviously require improvement in several areas, not least of which would be to address the imbalance in strategy profitability on the short vs long side, where most of the profits are generated.

September 6, 2018

Quantitative Analysis of Fat Tails – JonathanKinlay.com

In this quantitative analysis I explore how, starting from the assumption of a stable, Gaussian distribution in a returns process, we evolve to a system that displays all the characteristics of empirical market data, notably time-dependent moments, high levels of kurtosis and fat tails. As it turns out, the only additional assumption one needs to make is that the market is periodically disturbed by the random arrival of news.

NOTE: if you are unable to see the Mathematica models below, you can download the free Wolfram CDF player and you may also need this plug-in.

You can also download the complete Mathematica CDF file here.

Stationarity

A stationary process is one that evolves over time, but whose probability distribution does not vary with time. As the word implies, such a process is stable. More formally, the moments of the distribution are independent of time.

Let’s assume we are dealing with such a process that have constant mean μ and constant volatility (standard deviation) σ.

Φ=NormalDistribution[μ,σ]

Here are some examples of Normal probability distributions, with constant mean μ = 0 and standard deviation σ ranging from 0.75 to 2

Plot[Evaluate@Table[PDF[Φ,x],{σ,{.75,1,2}}]/.μ→0,{x,-6,6},Filling→Axis]

The moments of Φ are given by:

Through[{Mean, StandardDeviation, Skewness, Kurtosis}[Φ]]

{μ, σ, 0, 3}

They, too, are time – independent.

We can simulate some observations from such a process, with, say, mean μ = 0 and standard deviation σ = 1:

ListPlot[sampleData=RandomVariate[Φ /.{μ→0, σ→1},10^4]]

Histogram[sampleData]

If we assume for the moment that such a process is an adequate description of an asset returns process, we can simulate the evolution of a price process as follows :

ListPlot[prices=Accumulate[sampleData]]

An Empirical Distribution

Lets take a look at a real price series, comprising 1 – minute bar data in the June ‘ 14 E – Mini futures contract.

As with our simulated price process, it is clear that the real price process for Emini futures is also non – stationary.

What about the returns process?

ListPlot[returnsES]

Notice the banding effect in returns, which results from having a fixed, minimum price move of $12 .50, rather than a continuous scale.

Histogram[returnsES]

Through[{Min,Max,Mean,Median,StandardDeviation,Skewness,Kurtosis}[returnsES]]

{-0.00867214, 0.0112353, 2.75501×10^{-6, 0., 0.000780895, 0.35467, 26.2376}}

The empirical returns distribution doesn’ t appear to be Gaussian – the distribution is much more peaked than a standard Normal distribution with the same mean and standard deviation. And the higher moments don’t fit the Normal model either – the empirical distribution has positive skew and a kurtosis that is almost 9x greater than a Gaussian distribution. The latter signifies what is often referred to as “fat tails”: the distribution has much greater weight in the tails than a standard Normal distribution, indicating a much greater likelihood of an extreme value than a Normal distribution would predict.

A Quantitative Analysis of Non-Stationarity: Two States

Non – stationarity arises when one or more of the moments of a distribution vary over time. Let’s take a look at how that can arise, and its effects.Suppose we have a Gaussian returns process for which the mean, or drift, or trend, fluctuates over time.

Let’s consider a simple example where the process drift is μ1 and volatility σ1 for most of the time and then for some proportion of time k, we get addition drift μ2 and volatility σ2. In other words we have:

Φ1=NormalDistribution[μ1,σ1]

Through[{Mean,StandardDeviation,Skewness,Kurtosis}[Φ1]]

{μ1, σ1, 0, 3}

Φ2=NormalDistribution[μ2,σ2]

Through[{Mean,StandardDeviation,Skewness,Kurtosis}[Φ2]]

{μ2, σ2, 0, 3}

This simple model fits a scenario in which we suppose that the returns process spends most of its time in State 1, in which is Normally distributed with drift is μ1 and volatility σ1, and suffers from the occasional “shock” which propels the systems into a second State 2, in which its distribution is a combination of its original distribution and a new Gaussian distribution with different mean and volatility.

Let’ s suppose that we sample the combined process y = Φ1 + k Φ2. What distribution would it have? We can represent this is follows :

y=TransformedDistribution[(x1+k x2),{x1~Φ1,x2~Φ2}]

Through[{Mean,StandardDeviation,Skewness,Kurtosis}[y]]

Plot[PDF[y,x]/.{μ1→0,μ2→0,σ1 →1,σ2 →2, k→0.5},{x,-6,6},Filling→Axis]

The result is just another Normal distribution. Depending on the incidence k, y will follow a Gaussian distribution whose mean and variance depend on the mean and variance of the two Normal distributions being mixed. The resulting distribution in State 2 may have higher or lower drift and volatility, but it is still Gaussian, with constant kurtosis of 3.

In other words, the system y will be non-stationary, because the first and second moments change over time, depending on what state it is in. But the form of the distribution is unchanged – it is still Gaussian. There are no fat-tails.

Non – Stationarity : Random States

In the above example the system moved between states in a known, predictable way. The “shocks” to the system were not really shocks, but transitions. But that’s not how financial markets behave: markets move from one state to another in an unpredictable way, with the arrival of news.

We can simulate this situation as follows. Using the former model as a starting point, lets now relax the assumption that the incidence of the second state, k, is a constant. Instead, let’ s assume that k is itself a random variable. In other words we are going to now assume that our system changes state in a random way. How does this alter the distribution?

An appropriate model for λ might be a Poisson process, which is often used as a model for unpredictable, discrete events, ranging from bus arrivals to earthquakes. PDFs of Poisson distributions with means λ=5, 10 and 20 are shown in the chart below. These represent probability distributions for processes that have mean arrivals of 5, 10 or 20 events.

DiscretePlot[Evaluate@Table[PDF[PoissonDistribution[λ],k],{λ,{5,10,20}}],{k,0,30},PlotRange→All,PlotMarkers→Automatic]

Our new model now looks like this :

y=TransformedDistribution[{x1+k*x2},{x1⎡Φ1,x2⎡Φ2,k⎡PoissonDistribution[λ]}]

The first two moments of the distribution are as follows :

Through[{Mean,StandardDeviation}[y]]

As before, the mean and standard deviation of the distribution are going to vary, depending on the state of the system, and the mean arrival rate of shocks, . But what about kurtosis? Is it still constant?

Kurtosis[y]

Emphatically not! The fourth moment of the distribution is now dependent on the drift in the second state, the volatilities of both states and the mean arrival rate of shocks, λ.

Let’ s look at a specific example. Assume that in State 1 the process has volatility of 7.5 %, with zero drift, and that the shock distribution also has zero drift with volatility of 65 %. If the mean incidence rate of shocks λ = 10 %, the distribution kurtosis is close to that seen in the empirical distribution for the E-Mini.

Kurtosis[y] /.{σ1→0.075,μ2→0,σ2→0.65,λ→0.1}

{35.3551}

More generally :

ListLinePlot[Flatten[Kurtosis[y]/.Table[{σ1→0.075,μ2→0,σ2→0.65,λ→i/20},{i,1,20}]],PlotLabel→Style[“Kurtosis vs Mean Shock Arrival Rate”, FontSize→18],AxesLabel->{“Incidence Rate (%)”, “Kurtosis”},Filling→Axis, ImageSize→Large]

Thus we can see how, even if the underlying returns distribution is Gaussian in form, the random arrival of news “shocks” to the system can induce non – stationarity in overall drift and volatility. It can also result in fat tails. More specifically, if the arrival of news is stochastic in nature, rather than deterministic, the process may exhibit far higher levels of kurtosis than in its original Gaussian state, in which the fourth moment was a constant level of 3.

Quantitative Analysis of a Jump Diffusion Process

Nobel – prize winning economist Robert Merton extended this basic concept to the realm of stochastic calculus.

In Merton’s jump diffusion model, the stock price follows the random process

∂S_t / S_t =μdt + σdW_t+(J-1)dN_t

The first two terms are familiar from the Black–Scholes model : drift rate μ, volatility σ, and random walk W_t (Wiener process).The last term represents the jumps :J is the jump size as a multiple of stock price, while N_t is the number of jump events that have occurred up to time t.is assumed to follow the Poisson process.

PDF[PoissonDistribution[λt]]

where λ is the average frequency with which jumps occur.

The jump size J follows a log – normal distribution

PDF[LogNormalDistribution[m, ν], s]

where m is the average jump size and v is the volatility of the jump size.

In the jump diffusion model, the stock price S_t follows the random process dS_t/S_t=μ dt+σ dW_t+(J-1) dN(t), which comprises, in order, drift, diffusive, and jump components. The jumps occur according to a Poisson distribution and their size follows a log-normal distribution. The model is characterized by the diffusive volatility σ, the average jump size J (expressed as a fraction of S_t), the frequency of jumps λ, and the volatility of jump size ν.

The Volatility Smile

The “implied volatility” corresponding to an option price is the value of the volatility parameter for which the Black-Scholes model gives the same price. A well-known phenomenon in market option prices is the “volatility smile”, in which the implied volatility increases for strike values away from the spot price. The jump diffusion model is a generalization of Black–Scholes in which the stock price has randomly occurring jumps in addition to the random walk behavior. One of the interesting properties of this model is that it displays the volatility smile effect. In this Demonstration, we explore the Black–Scholes implied volatility of option prices (equal for both put and call options) in the jump diffusion model. The implied volatility is modeled as a function of the ratio of option strike price to spot price.

September 5, 2018

The Mathematics of Scalping

NOTE: if you are unable to see the Mathematica models below, you can download the free Wolfram CDF player and you may also need this plug-in.

You can also download the complete Mathematica CDF file here.

In this post I want to explore aspects of scalping, a type of strategy widely utilized by high frequency trading firms.

I will define a scalping strategy as one in which we seek to take small profits by posting limit orders on alternate side of the book. Scalping, as I define it, is a strategy rather like market making, except that we “lean” on one side of the book. So, at any given time, we may have a long bias and so look to enter with a limit buy order. If this is filled, we will then look to exit with a subsequent limit sell order, taking a profit of a few ticks. Conversely, we may enter with a limit sell order and look to exit with a limit buy order.
The strategy relies on two critical factors:

(i) the alpha signal which tells us from moment to moment whether we should prefer to be long or short
(ii) the execution strategy, or “trade expression”

In this article I want to focus on the latter, making the assumption that we have some kind of alpha generation model already in place (more about this in later posts).

There are several means that a trader can use to enter a position. The simplest approach, the one we will be considering here, is simply to place a single limit order at or just outside the inside bid/ask prices – so in other words we will be looking to buy on the bid and sell on the ask (and hoping to earn the bid-ask spread, at least).

One of the problems with this approach is that it is highly latency sensitive. Limit orders join the limit order book at the back of the queue and slowly works their way towards the front, as earlier orders get filled. Buy the time the market gets around to your limit buy order, there may be no more sellers at that price. In that case the market trades away, a higher bid comes in and supersedes your order, and you don’t get filled. Conversely, yours may be one of the last orders to get filled, after which the market trades down to a lower bid and your position is immediately under water.

This simplistic model explains why latency is such a concern – you want to get as near to the front of the queue as you can, as quickly as possible. You do this by minimizing the time it takes to issue and order and get it into the limit order book. That entails both hardware (co-located servers, fiber-optic connections) and software optimization and typically also involves the use of Immediate or Cancel (IOC) orders. The use of IOC orders by HFT firms to gain order priority is highly controversial and is seen as gaming the system by traditional investors, who may end up paying higher prices as a result.

Another approach is to layer limit orders at price points up and down the order book, establishing priority long before the market trades there. Order layering is a highly complex execution strategy that brings addition complications.

Let’s confine ourselves to considering the single limit order, the type of order available to any trader using a standard retail platform.

As I have explained, we are assuming here that, at any point in time, you know whether you prefer to be long or short, and therefore whether you want to place a bid or an offer. The issue is, at what price do you place your order, and what do you do about limiting your risk? In other words, we are discussing profit targets and stop losses, which, of course, are all about risk and return.

Risk and Return in Scalping

Lets start by considering risk. The biggest risk to a scalper is that, once filled, the market goes against his position until he is obliged to trigger his stop loss. If he sets his stop loss too tight, he may be forced to exit positions that are initially unprofitable, but which would have recovered and shown a profit if he had not exited. Conversely, if he sets the stop loss too loose, the risk reward ratio is very low – a single loss-making trade could eradicate the profit from a large number of smaller, profitable trades.

Now lets think about reward. If the trader is too ambitious in setting his profit target he may never get to realize the gains his position is showing – the market could reverse, leaving him with a loss on a position that was, initially, profitable. Conversely, if he sets the target too tight, the trader may give up too much potential in a winning trade to overcome the effects of the occasional, large loss.

It’s clear that these are critical concerns for a scalper: indeed the trade exit rules are just as important, or even more important, than the entry rules. So how should he proceed?

Theoretical Framework for Scalping

Let’s make the rather heroic assumption that market returns are Normally distributed (in fact, we know from empirical research that they are not – but this is a starting point, at least). And let’s assume for the moment that our trader has been filled on a limit buy order and is looking to decide where to place his profit target and stop loss limit orders. Given a current price of the underlying security of X, the scalper is seeking to determine the profit target of p ticks and the stop loss level of q ticks that will determine the prices at which he should post his limit orders to exit the trade. We can translate these into returns, as follows:

to the upside: Ru = Ln[X+p] – Ln[X]

and to the downside: Rd = Ln[X-q] – Ln[X]

This situation is illustrated in the chart below.

The profitable area is the shaded region on the RHS of the distribution. If the market trades at this price or higher, we will make money: p ticks, less trading fees and commissions, to be precise. Conversely we lose q ticks (plus commissions) if the market trades in the region shaded on the LHS of the distribution.

Under our assumptions, the probability of ending up in the RHS shaded region is:

probWin = 1 – NormalCDF(Ru, mu, sigma),

where mu and sigma are the mean and standard deviation of the distribution.

The probability of losing money, i.e. the shaded area in the LHS of the distribution, is given by:

probLoss = NormalCDF(Rd, mu, sigma),

where NormalCDF is the cumulative distribution function of the Gaussian distribution.

The expected profit from the trade is therefore:

Expected profit = p * probWin – q * probLoss

And the expected win rate, the proportion of profitable trades, is given by:

WinRate = probWin / (probWin + probLoss)

If we set a stretch profit target, then p will be large, and probWin, the shaded region on the RHS of the distribution, will be small, so our winRate will be low. Under this scenario we would have a low probability of a large gain. Conversely, if we set p to, say, 1 tick, and our stop loss q to, say, 20 ticks, the shaded region on the RHS will represent close to half of the probability density, while the shaded LHS will encompass only around 5%. Our win rate in that case would be of the order of 91%:

WinRate = 50% / (50% + 5%) = 91%

Under this scenario, we make frequent, small profits and suffer the occasional large loss.

So the critical question is: how do we pick p and q, our profit target and stop loss? Does it matter? What should the decision depend on?

Modeling Scalping Strategies

We can begin to address these questions by noticing, as we have already seen, that there is a trade-off between the size of profit we are hoping to make, and the size of loss we are willing to tolerate, and the probability of that gain or loss arising. Those probabilities in turn depend on the underlying probability distribution, assumed here to be Gaussian.

Now, the Normal or Gaussian distribution which determines the probabilities of winning or losing at different price levels has two parameters – the mean, mu, or drift of the returns process and sigma, its volatility.

Over short time intervals the effect of volatility outweigh any impact from drift by orders of magnitude. The reason for this is simple: volatility scales with the square root of time, while the drift scales linearly. Over small time intervals, the drift becomes un-noticeably small, compared to the process volatility. Hence we can assume that mu, the process mean is zero, without concern, and focus exclusively on sigma, the volatility.

What other factors do we need to consider? Well there is a minimum price move, which might be 1 tick, and the dollar value of that tick, from which we can derive our upside and downside returns, Ru and Rd. And, finally, we need to factor in commissions and exchange fees into our net trade P&L.

Here’s a simple formulation of the model, in which I am using the E-mini futures contract as an exemplar.

WinRate[currentPrice_,annualVolatility_,BarSizeMins_, nTicksPT_, nTicksSL_,minMove_, tickValue_, costContract_]:=Module[{ nMinsPerDay, periodVolatility, tgtReturn, slReturn,tgtDollar, slDollar, probWin, probLoss, winRate, expWinDollar, expLossDollar, expProfit},
nMinsPerDay = 250*6.5*60;
periodVolatility = annualVolatility / Sqrt[nMinsPerDay/BarSizeMins];
tgtReturn=nTicksPT*minMove/currentPrice;tgtDollar = nTicksPT * tickValue;
slReturn = nTicksSL*minMove/currentPrice;
slDollar=nTicksSL*tickValue;
probWin=1-CDF[NormalDistribution[0, periodVolatility],tgtReturn];
probLoss=CDF[NormalDistribution[0, periodVolatility],slReturn];
winRate=probWin/(probWin+probLoss);
expWinDollar=tgtDollar*probWin;
expLossDollar=slDollar*probLoss;
expProfit=expWinDollar+expLossDollar-costContract;
{expProfit, winRate}]

For the ES contract we have a min price move of 0.25 and the tick value is $12.50. Notice that we scale annual volatility to the size of the period we are trading (15 minute bars, in the following example).

Scenario Analysis

Let’s take a look at how the expected profit and win rate vary with the profit target and stop loss limits we set. In the following interactive graphics, we can assess the impact of different levels of volatility on the outcome.

Expected Profit by Bar Size and Volatility

Expected Win Rate by Volatility

Notice to begin with that the win rate (and expected profit) are very far from being Normally distributed – not least because they change radically with volatility, which is itself time-varying.

For very low levels of volatility, around 5%, we appear to do best in terms of maximizing our expected P&L by setting a tight profit target of a couple of ticks, and a stop loss of around 10 ticks. Our win rate is very high at these levels – around 90% or more. In other words, at low levels of volatility, our aim should be to try to make a large number of small gains.

But as volatility increases to around 15%, it becomes evident that we need to increase our profit target, to around 10 or 11 ticks. The distribution of the expected P&L suggests we have a couple of different strategy options: either we can set a larger stop loss, of around 30 ticks, or we can head in the other direction, and set a very low stop loss of perhaps just 1-2 ticks. This later strategy is, in fact, the mirror image of our low-volatility strategy: at higher levels of volatility, we are aiming to make occasional, large gains and we are willing to pay the price of sustaining repeated small stop-losses. Our win rate, although still well above 50%, naturally declines.

As volatility rises still further, to 20% or 30%, or more, it becomes apparent that we really have no alternative but to aim for occasional large gains, by increasing our profit target and tightening stop loss limits. Our win rate under this strategy scenario will be much lower – around 30% or less.

Non – Gaussian Model

Now let’s address the concern that asset returns are not typically distributed Normally. In particular, the empirical distribution of returns tends to have “fat tails”, i.e. the probability of an extreme event is much higher than in an equivalent Normal distribution.

A widely used model for fat-tailed distributions in the Extreme Value Distribution. This has pdf:

PDF[ExtremeValueDistribution[,],x]

Plot[Evaluate@Table[PDF[ExtremeValueDistribution[,2],x],{,{-3,0,4}}],{x,-8,12},FillingAxis]

Mean[ExtremeValueDistribution[,]]

+EulerGamma 

Variance[ExtremeValueDistribution[,]]

In order to set the parameters of the EVD, we need to arrange them so that the mean and variance match those of the equivalent Gaussian distribution with mean  = 0 and standard deviation . hence:

The code for a version of the model using the GED is given as follows

WinRateExtreme[currentPrice_,annualVolatility_,BarSizeMins_, nTicksPT_, nTicksSL_,minMove_, tickValue_, costContract_]:=Module[{ nMinsPerDay, periodVolatility, alpha, beta,tgtReturn, slReturn,tgtDollar, slDollar, probWin, probLoss, winRate, expWinDollar, expLossDollar, expProfit},
nMinsPerDay = 250*6.5*60;
periodVolatility = annualVolatility / Sqrt[nMinsPerDay/BarSizeMins];
beta = Sqrt[6]*periodVolatility / Pi;
alpha=-EulerGamma*beta;
tgtReturn=nTicksPT*minMove/currentPrice;tgtDollar = nTicksPT * tickValue;
slReturn = nTicksSL*minMove/currentPrice;
slDollar=nTicksSL*tickValue;
probWin=1-CDF[ExtremeValueDistribution[alpha, beta],tgtReturn];
probLoss=CDF[ExtremeValueDistribution[alpha, beta],slReturn];
winRate=probWin/(probWin+probLoss);
expWinDollar=tgtDollar*probWin;
expLossDollar=slDollar*probLoss;
expProfit=expWinDollar+expLossDollar-costContract;
{expProfit, winRate}]

WinRateExtreme[1900,0.05,15,2,30,0.25,12.50,3][[2]]

0.21759

We can now produce the same plots for the EVD version of the model that we plotted for the Gaussian versions :

Expected Profit by Bar Size and Volatility – Extreme Value Distribution

Expected Win Rate by Volatility – Extreme Value Distribution

Next we compare the Gaussian and EVD versions of the model, to gain an understanding of how the differing assumptions impact the expected Win Rate.

Expected Win Rate by Stop Loss and Profit Target

As you can see, for moderate levels of volatility, up to around 18 % annually, the expected Win Rate is actually higher if we assume an Extreme Value distribution of returns, rather than a Normal distribution.If we use a Normal distribution we will actually underestimate the Win Rate, if the actual return distribution is closer to Extreme Value.In other words, the assumption of a Gaussian distribution for returns is actually conservative.

Now, on the other hand, it is also the case that at higher levels of volatility the assumption of Normality will tend to over – estimate the expected Win Rate, if returns actually follow an extreme value distribution. But, as indicated before, for high levels of volatility we need to consider amending the scalping strategy very substantially. Either we need to reverse it, setting larger Profit Targets and tighter Stops, or we need to stop trading altogether, until volatility declines to normal levels.Many scalpers would prefer the second option, as the first alternative doesn’t strike them as being close enough to scalping to justify the name.If you take that approach, i.e.stop trying to scalp in periods when volatility is elevated, then the differences in estimated Win Rate resulting from alternative assumptions of return distribution are irrelevant.

If you only try to scalp when volatility is under, say, 20 % and you use a Gaussian distribution in your scalping model, you will only ever typically under – estimate your actual expected Win Rate.In other words, the assumption of Normality helps, not hurts, your strategy, by being conservative in its estimate of the expected Win Rate.

If, in the alternative, you want to trade the strategy regardless of the level of volatility, then by all means use something like an Extreme Value distribution in your model, as I have done here.That changes the estimates of expected Win Rate that the model produces, but it in no way changes the structure of the model, or invalidates it.It’ s just a different, arguably more realistic set of assumptions pertaining to situations of elevated volatility.

Monte-Carlo Simulation Analysis

Let’ s move on to do some simulation analysis so we can get an understanding of the distribution of the expected Win Rate and Avg Trade PL for our two alternative models. We begin by coding a generator that produces a sample of 1,000 trades and calculates the Avg Trade PL and Win Rate.

Gaussian Model

GenWinRate[currentPrice_,annualVolatility_,BarSizeMins_, nTicksPT_, nTicksSL_,minMove_, tickValue_, costContract_]:=Module[{ nMinsPerDay, periodVolatility, randObs, tgtReturn, slReturn,tgtDollar, slDollar, nWins,nLosses, perTradePL, probWin, probLoss, winRate, expWinDollar, expLossDollar, expProfit},
nMinsPerDay = 250*6.5*60;
periodVolatility = annualVolatility / Sqrt[nMinsPerDay/BarSizeMins];
tgtReturn=nTicksPT*minMove/currentPrice;tgtDollar = nTicksPT * tickValue;
slReturn = nTicksSL*minMove/currentPrice;
slDollar=nTicksSL*tickValue;
randObs=RandomVariate[NormalDistribution[0,periodVolatility],10^3];
nWins=Count[randObs,x_/;x>=tgtReturn];
nLosses=Count[randObs,x_/;xslReturn];
winRate=nWins/(nWins+nLosses)//N;
perTradePL=(nWins*tgtDollar+nLosses*slDollar)/(nWins+nLosses);{perTradePL,winRate}]

GenWinRate[1900,0.1,15,1,-24,0.25,12.50,3]

{7.69231,0.984615}

Now we can generate a random sample of 10, 000 simulation runs and plot a histogram of the Win Rates, using, for example, ES on 5-min bars, with a PT of 2 ticks and SL of – 20 ticks, assuming annual volatility of 15 %.

Histogram[Table[GenWinRate[1900,0.15,5,2,-20,0.25,12.50,3][[2]],{i,10000}],10,AxesLabel{“Exp. Win Rate (%)”}]

Histogram[Table[GenWinRate[1900,0.15,5,2,-20,0.25,12.50,3][[1]],{i,10000}],10,AxesLabel{“Exp. PL/Trade ($)”}]

Extreme Value Distribution Model

Next we can do the same for the Extreme Value Distribution version of the model.

GenWinRateExtreme[currentPrice_,annualVolatility_,BarSizeMins_, nTicksPT_, nTicksSL_,minMove_, tickValue_, costContract_]:=Module[{ nMinsPerDay, periodVolatility, randObs, tgtReturn, slReturn,tgtDollar, slDollar, alpha, beta,nWins,nLosses, perTradePL, probWin, probLoss, winRate, expWinDollar, expLossDollar, expProfit},
nMinsPerDay = 250*6.5*60;
periodVolatility = annualVolatility / Sqrt[nMinsPerDay/BarSizeMins];
beta = Sqrt[6]*periodVolatility / Pi;
alpha=-EulerGamma*beta;
tgtReturn=nTicksPT*minMove/currentPrice;tgtDollar = nTicksPT * tickValue;
slReturn = nTicksSL*minMove/currentPrice;
slDollar=nTicksSL*tickValue;
randObs=RandomVariate[ExtremeValueDistribution[alpha, beta],10^3];
nWins=Count[randObs,x_/;x>=tgtReturn];
nLosses=Count[randObs,x_/;xslReturn];
winRate=nWins/(nWins+nLosses)//N;
perTradePL=(nWins*tgtDollar+nLosses*slDollar)/(nWins+nLosses);{perTradePL,winRate}]

Histogram[Table[GenWinRateExtreme[1900,0.15,5,2,-10,0.25,12.50,3][[2]],{i,10000}],10,AxesLabel{“Exp. Win Rate (%)”}]

Histogram[Table[GenWinRateExtreme[1900,0.15,5,2,-10,0.25,12.50,3][[1]],{i,10000}],10,AxesLabel{“Exp. PL/Trade ($)”}]

Conclusions

The key conclusions from this analysis are:

Scalping is essentially a volatility trade
The setting of optimal profit targets are stop loss limits depend critically on the volatility of the underlying, and needs to be handled dynamically, depending on current levels of market volatility
At low levels of volatility we should set tight profit targets and wide stop loss limits, looking to make a high percentage of small gains, of perhaps 2-3 ticks.
As volatility rises, we need to reverse that position, setting more ambitious profit targets and tight stops, aiming for the occasional big win.