# Black scholes implied volatility matlab torrent

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It also makes sense of the finding of Andersen and Bollerslev that the daily range has approximately the same informational content as sampling intra-daily returns every four hours. Except for the model of Chou , GARCH-type volatility models rely on squared or absolute returns which have the same information content to capture variation in the conditional volatility h t. Since the range is a more informative volatility proxy, it makes sense to consider range-based GARCH models, in which the range is used in place of squared or absolute returns to capture variation in the conditional volatility.

This is particularly true for the EGARCH framework of Nelson , which describes the dynamics of log volatility of which the log range is a linear proxy. However, for stock indices the in-sample evidence reported by Hentschel and the forecasting performance presented by Pagan and Schwert show a slight superiority of the EGARCH specification. The parameters q, k q , f q and d q determine the long-run mean, sensitivity of the long run mean to lagged absolute returns, and the asymmetry of absolute return sensitivity respectively.

The intuition is that when the lagged absolute return is large small relative to the lagged level of volatility, volatility is likely to have experienced a positive negative innovation. Unfortunately, as we explained above, the absolute return is a rather noisy proxy of volatility, suggesting that a substantial part of the volatility variation in GARCH-type models is driven by proxy noise as opposed to true information about volatility.

In other words, the noise in the volatility proxy introduces noise in the implied volatility process. In a volatility forecasting context, this noise in the implied volatility process deteriorates the quality of the forecasts through less precise parameter estimates and, more importantly, through less precise estimates of the current level of volatility to which the forecasts are anchored.

These arise from remarks by one commentator as follows:. This, of course, is used to describe to the higher implied vols seen in the Black-Scholes prices of OTM options. But the forecasting tests referenced in the paper are tests of the ability of the model to predict the direction of volatility, i. Thus we are looking at, not a LogNormal distribution, but the difference in two LogNormal distributions with equal mean — and this, of course, has an expectation of zero.

In other words, the expected level of volatility for the next period is the same as the current period and the expected change in the level of volatility is zero. You can test this very easily for yourself by generating a large number of observations from a LogNormal process, taking the difference and counting the number of positive and negative changes in the level of volatility from one period to the next.

You will find, on average, half the time the change of direction is positive and half the time it is negative. For instance, the following chart shows the distribution of the number of positive changes in the level of a LogNormally distributed random variable with mean and standard deviation of 0. The sample mean 5, So, a naive predictor will forecast volatility to remain unchanged for the next period and by random chance approximately half the time volatility will turn out to be higher and half the time it will turn out to be lower than in the current period.

This, then, is one of the key benchmarks you use to assess the ability of the model to predict market direction. The other benchmark we use is the change of direction predicted by the implied volatility of ATM options. On its face, it is because of this exceptional direction prediction accuracy that a simple strategy is able to generate what appear to be abnormal returns using the change of direction forecasts generated by the model, as described in the paper.

In fact, the situation is more complicated than that, once you introduce the concept of a market price of volatility risk. There has been a good deal of interest in the market timing ideas discussed in my earlier blog post Using Volatility to Predict Market Direction , which discusses the research of Diebold and Christoffersen into the sign predictability induced by volatility dynamics.

The ideas are thoroughly explored in a QuantNotes article from , which you can download here. There is a follow-up article from in which Christoffersen, Diebold, Mariano and Tay develop the ideas further to consider the impact of higher moments of the asset return distribution on sign predictability and the potential for market timing in international markets download here. We assume that the position is held for 30 days and rebalanced at the end of each period. In this test we make no allowance for market impact, or transaction costs.

The average gain is 7. The compound annual return is The under-performance of the strategy in is explained by the fact that direction-of-change probabilities were rising from a very low base in Q4 and do not reach trigger levels until the end of the year. Further tests are required to determine whether the failure of the strategy to produce an exceptional performance on par with was the result of normal statistical variation or due to changes in the underlying structure of the process requiring model recalibration.

The approach also has potential for asset allocation, portfolio theory and risk management applications. Magicicada is the genus of the year and year periodical cicadas of eastern North America. Magicicada species spend most of their and year lives underground feeding on xylem fluids from the roots of deciduous forest trees in the eastern United States.

After 13 or 17 years, mature cicada nymphs emerge in the springtime at any given locality, synchronously and in tremendous numbers. Within two months of the original emergence, the lifecycle is complete, the eggs have been laid, and the adult cicadas are gone for another 13 or 17 years.

The emergence period of large prime numbers 13 and 17 years has been hypothesized to be a predator avoidance strategy adopted to eliminate the possibility of potential predators receiving periodic population boosts by synchronizing their own generations to divisors of the cicada emergence period. If, for example, the cycle length was, say, 12 years, then the species would be exposed to predators regenerating over cycles of 2, 3, 4, or 6 years. Limiting their cycle to a large prime number reduces the variety of predators the species is likely to face.

What has any of this to do with trading? When building a strategy in a particular market we might start by creating a model that works reasonably well on, say, 5-minute bars. Then, in order to improve the risk-adjusted returns we might try create a second sub-strategy on a different frequency. This will hopefully result in a new series of signals, an increase in the number of trades, and corresponding improvement in the risk-adjusted returns of the overall strategy.

This phenomenon is referred to as temporal diversification. What time frequency should we select for our second sub-strategy? There are many factors to consider, of course, but one of them is that we would like to see as few duplicate signals between the two sub-strategies. Otherwise we will simply be replicating trades, rather than reducing the overall level of strategy risk through temporal diversification.

The best way to minimize the overlap in signals generated by multiple sub-strategies is to use prime number bar frequencies 5 minute, 7 minute, 11 minute, etc. This strategy is actually a combination of several different sub-strategies that operate on 5-minute, minute, minute and minute bars. The resulting increase in trade frequency and temporal diversification produces very attractive risk-adjusted performance: after an exceptional year in which saw a Investors can auto-trade the E-Mini Swing Trading strategy and many other strategies in their own account — see the Leaderboard for more details.

Recently I have been discussing possible areas of collaboration with an RIA contact on LinkedIn, who also happens to be very familiar with the hedge fund world. He outlined the case of a high net worth investor in equities long only , who wanted to remain invested, but was becoming increasingly concerned about the prospects for a significant market downturn, or even a market crash, similar to those of or I am guessing he is not alone: hardly a day goes by without the publication of yet another article sounding a warning about stretched equity valuations and the dangerously elevated level of the market.

Typically, conservative investors would have simply moved more of their investment portfolio into fixed income securities, but with yields at such low levels this is hardly an attractive option today. Besides, many see the bond market as representing an even more extreme bubble than equities currently.

The problem with traditional hedging mechanisms such as put options, for example, is that they are relatively expensive and can easily reduce annual returns from the overall portfolio by several hundred basis points. Even at current low level of volatility the performance drag is noticeable, since the potential upside in the equity portfolio is also lower than it has been for some time. A further consideration is that many investors are not mandated — or are simply reluctant — to move beyond traditional equity investing into complex ETF products or derivatives.

And while a short hedge may provide some downside protection it is unlikely to fully safeguard the investor in a crash scenario. Furthermore, the cost of a hedge fund investment is typically greater than for a long-only product, entailing the payment of a performance fee in addition to management fees that are often higher than for standard investment products. But no buy-and-hold strategy could ever be expected to prosper during times of severe market stress.

A more sophisticated approach is required. The idea, simply, is to increase or reduce risk exposure according to the prospects for the overall market. For a very long time the concept has been dismissed as impossible, by definition, given that markets are mostly efficient.

But analysts have persisted in the attempt to develop market timing techniques, motivated by the enormous benefits that a viable market timing strategy would bring. And gradually, over time, evidence has accumulated that the market can be timed successfully and profitably. The rate of progress has accelerated in the last decade by the considerable advances in computing power and the development of machine learning algorithms and application of artificial intelligence to investment finance.

I have written several articles on the subject of market timing that the reader might be interested to review see below. In this article, however, I want to focus firstly on the work on another investment strategist, Blair Hull. The goal to achieve long-term growth from investments in the U. How well has the Hull Tactical strategy performed? How does the Hull Tactical team achieve these results?

A couple of years ago I and my colleagues carried out an investigation of long-only equity strategies as part of a research project. Our primary focus was on index replication, but in the course of our research we came up with a methodology for developing long-only strategies that are highly crash-resistant.

These forecasts are derived from machine learning algorithms that are specifically tuned to minimize the downside risk in the investment portfolio. This not only makes strategy returns less volatile, but also ensures that the strategy is very robust to market downturns. In fact, even better than that, not only does the LOMT strategy tend to avoid large losses during periods of market stress, it is capable of capitalizing on the opportunities that more volatile market conditions offer.

The reason is clear from the charts: during the periods and again in , when the market crashed and returns in the SPY ETF were substantially negative, the LOMT strategy managed to produce positive returns. He explained that his analysis had shown that volatility was often underpriced due to an under-estimation of tail risk, which the fund would seek to exploit by purchasing cheap out-of-the-money options.

My response was that this struck me a great idea for an insurance product, but not a hedge fund — his investors, I explained, were going to hate seeing month after month of negative returns and would flee the fund. And so it proved. What investors have been seeking is a strategy that can yield positive returns during normal market conditions while at the same time offering protection against the kind of market gyrations that typically decimate several years of returns from investment portfolios, such as we saw after the market crashes in and With the new breed of long-only strategies now being developed using machine learning algorithms, it appears that investors finally have an opportunity to get what they always wanted, at a reasonable price.

One of the most widely used risk measures is the Value-at-Risk, defined as the expected loss on a portfolio at a specified confidence level. In other words, VaR is a percentile of a loss distribution. But despite its popularity VaR suffers from well-known limitations: its tendency to underestimate the risk in the left tail of the loss distribution and its failure to capture the dynamics of correlation between portfolio components or nonlinearities in the risk characteristics of the underlying assets.

One method of seeking to address these shortcomings is discussed in a previous post Copulas in Risk Management. Another approach known as Conditional Value at Risk CVaR , which seeks to focus on tail risk, is the subject of this post. We look at how to estimate Conditional Value at Risk in both Gaussian and non-Gaussian frameworks, incorporating loss distributions with heavy tails and show how to apply the concept in the context of nonlinear time series models such as GARCH.

A perennial favorite with investors, presumably because they are easy to understand and implement, are trades based on a regularly occurring pattern, preferably one that is seasonal in nature. A well-known example is the Christmas effect, wherein equities generally make their highest risk-adjusted returns during the month of December and equity indices make the greater proportion of their annual gains in the period from November to January.

As we approach the Easter holiday I thought I might join in the fun with a trade of my own. If so, I apologize in advance if this is duplicative. The first question is whether there are significant differences economic and statistical in index returns in the weeks before and after Easter, compared to a regular week.

It is perhaps not immediately apparent from the smooth histogram plot above, but a whisker plot gives a clearer indication of the disparity in the distributions of returns in the post-Easter week vs. It is evident that chief distinction is not in the means of the distributions, but in their variances.

On its face, it is because of this exceptional direction prediction accuracy that a simple strategy is able to generate what appear to be abnormal returns using the change of direction forecasts generated by the model, as described in the paper. In fact, the situation is more complicated than that, once you introduce the concept of a market price of volatility risk.

I am planning a series of posts on the subject of asset volatility and option pricing and thought I would begin with a survey of some of the central ideas. The attached presentation on Modeling Asset Volatility sets out the foundation for a number of key concepts and the basis for the research to follow.

Perhaps the most important feature of volatility is that it is stochastic rather than constant, as envisioned in the Black Scholes framework. The presentation addresses this issue by identifying some of the chief stylized facts about volatility processes and how they can be modelled. However, there are many other typical features that are less often rehearsed and these too are examined in the presentation.

Long Memory For example, while it is true that GARCH models do a fine job of modeling the clustering effect they typically fail to capture one of the most important features of volatility processes — long term serial autocorrelation. In the typical GARCH model autocorrelations die away approximately exponentially, and historical events are seen to have little influence on the behaviour of the process very far into the future.

In volatility processes that is typically not the case, however: autocorrelations die away very slowly and historical events may continue to affect the process many weeks, months or even years ahead. There are two immediate and very important consequences of this feature. The first is that volatility processes will tend to trend over long periods — a characteristic of Black Noise or Fractionally Integrated processes, compared to the White Noise behavior that typically characterizes asset return processes.

Secondly, and again in contrast with asset return processes, volatility processes are inherently predictable, being conditioned to a significant degree on past behavior. The presentation considers the fractional integration frameworks as a basis for modeling and forecasting volatility. Mean Reversion vs.

Momentum A puzzling feature of much of the literature on volatility is that it tends to stress the mean-reverting behavior of volatility processes. This appears to contradict the finding that volatility behaves as a reinforcing process, whose long-term serial autocorrelations create a tendency to trend. This leads to one of the most important findings about asset processes in general, and volatility process in particular: i. One way to understand this is to think of volatility, not as a single process, but as the superposition of two processes: a long term process in the mean, which tends to reinforce and trend, around which there operates a second, transient process that has a tendency to produce short term spikes in volatility that decay very quickly.

In other words, a transient, mean reverting processes inter-linked with a momentum process in the mean. The presentation discusses two-factor modeling concepts along these lines, and about which I will have more to say later. Cointegration One of the most striking developments in econometrics over the last thirty years, cointegration is now a principal weapon of choice routinely used by quantitative analysts to address research issues ranging from statistical arbitrage to portfolio construction and asset allocation.

In fact, this modeling technique provided the basis for the Caissa Capital volatility fund, which I founded in Dispersion Dynamics Finally, one topic that is not considered in the presentation, but on which I have spent much research effort in recent years, is the behavior of cross-sectional volatility processes, which I like to term dispersion.

It turns out that, like its univariate cousin, dispersion displays certain characteristics that in principle make it highly forecastable. Given an appropriate model of dispersion dynamics, the question then becomes how to monetize efficiently the insight that such a model offers. Again, I will have much more to say on this subject, in future.

I am in the process of updating the research, but in the meantime a copy of the original paper is available here. The ARFIMA-GARCH model, which uses high frequency data comprising 5-minute returns, makes volatility the subject process of interest, to which innovations are introduced via a volatility-of-volatility kurtosis process. Despite performing robustly in- and out-of-sample, an encompassing regression indicates that the model is unable to add to the information already contained in market forecasts.

However, unlike model forecasts, implied volatility forecasts show evidence of a consistent and substantial bias. This suggests that either option markets may be inefficient, or that the option pricing model is mis-specified. To examine this hypothesis, an empirical test is carried out in which at-the-money straddles are bought or sold and delta-hedged depending on whether the model forecasts exceed or fall below implied volatility forecasts. This simple strategy generates an annual compound return of Our findings suggest that, over the period of analysis, investors required an additional risk premium of 88 basis points of incremental return for each unit of volatility risk.

We can decompose the returns process R t as follows:. While the left hand side of the equation is essentially unforecastable, both of the right-hand-side components of returns display persistent dynamics and hence are forecastable. Both the signs of returns and magni tude of returns are conditional mean dependent and hence forecastable, but their product is conditional mean independent and hence unforecastable. Although asset returns are essentially unforecastable, the same is not true for asset return signs i.

As long as expected returns are nonzero, one should expect sign dependence, given the overwhelming evidence of volatility dependence. Even in assets where expected returns are zero, sign dependence may be induced by skewness in the asset returns process. Hence market timing ability is a very real possibility, depending on the relationship between the mean of the asset returns process and its higher moments.

The highly nonlinear nature of the relationship means that conditional sign dependence is not likely to be found by traditional measures such as signs autocorrelations, runs tests or traditional market timing tests. Sign dependence is likely to be strongest at intermediate horizons of months, and unlikely to be important at very low or high frequencies.

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