According to a recent article in Risk magazine Korea's Financial Services Commission is planning to introduce volatility futures on VKOSPI index sometime before the end of the year. It has been for quite sometime in the planning stage ( I first wrote about it 2.5 years ago ) but the product may finally come to the market. Given lack of liquidity in other Asian volatility futures I am pessimistic about the product attracting volume.

# Volatility Futures & Options

## Jul 29, 2014

## Jul 28, 2014

### Reliability of the Maximum Drawdown

I recently came across an old article titled Reliability of the Maximum Drawdown, and suggest that trades familiarize themselves with the ideas mentioned there. I would like to suggest an idea that may be a way forward.

The mathematics of maximum drawdown (expected maximum drawdown, and its distribution) are far from trivial, and many other related measures (for example expected length of drawdown) afaik have not been seriously studied at all.

The problem with max-measures, like mentioned in the article above, is that they are extreme measures, and thus are at the very corner of distribution charts. If you have a strategy that suffered a maximum drawdown of x% you know that such drawdown is possible, but if you observe another strategy with smaller drawdown, you can hardly be sure that it will not suffer from a greater drawdown in the future.

One possible way forward is to use "average measures" - average drawdown and average drawdown length instead of their max counterparts. Intuition suggests (and my extensive monte carlo confirms) that these measures have smaller variability, smaller skewness, and more predictive (as measured by linear and nonlinear correlations) from one period to the next. These measures seem to scale linearly with time, with the scale coeffcient depending on kurtosis. I don't have the maths to take this much further on my own, but if you have some ideas please leave a comment or send me an email.

The mathematics of maximum drawdown (expected maximum drawdown, and its distribution) are far from trivial, and many other related measures (for example expected length of drawdown) afaik have not been seriously studied at all.

The problem with max-measures, like mentioned in the article above, is that they are extreme measures, and thus are at the very corner of distribution charts. If you have a strategy that suffered a maximum drawdown of x% you know that such drawdown is possible, but if you observe another strategy with smaller drawdown, you can hardly be sure that it will not suffer from a greater drawdown in the future.

One possible way forward is to use "average measures" - average drawdown and average drawdown length instead of their max counterparts. Intuition suggests (and my extensive monte carlo confirms) that these measures have smaller variability, smaller skewness, and more predictive (as measured by linear and nonlinear correlations) from one period to the next. These measures seem to scale linearly with time, with the scale coeffcient depending on kurtosis. I don't have the maths to take this much further on my own, but if you have some ideas please leave a comment or send me an email.

## Jun 27, 2014

### Weekend Reading

Risk: Asia-specific Vix indexes fail to ignite market interest.

The title pretty much says it all, that outside US and VSTOXX on Eurex volatility futures have not taken off. Will this change now since CBOE introduced (almost) 24-hour trading in VIX futures? Will arbitrageurs add liquidity to Asian products, or will Asian hedgers flock to VIX to manage risk? Only time will tell.

The title pretty much says it all, that outside US and VSTOXX on Eurex volatility futures have not taken off. Will this change now since CBOE introduced (almost) 24-hour trading in VIX futures? Will arbitrageurs add liquidity to Asian products, or will Asian hedgers flock to VIX to manage risk? Only time will tell.

## Jun 19, 2014

### More Intuition On Volatility And Square Root Of Time

Vance at Six Figure Investing recently wrote an excellent post titled Volatility and the Square Root of Time. I would like to try to add a little more intuition about this particular theoretical property of gaussian random walk process, or Wiener process.

If we gloss over technical details Wiener process can be described as a sum of random draws from normal distribution - that is starting at 0 we keep adding a new random number at each time step. However we are going simplify this even further, and will use even more basic process - starting at 0, at each time step we flip a coin, and with heads we add 1, with tails we subtract one. And although this may sound like a grade-school version of Wiener process this is actually a valid "substitute" for our purposes.

Here I plotted some sample paths. And yes, this binomial walk looks quite like gaussian random walk.

Now we will look at the process in greater detail, and specifically we will look at the volatility of the process at each time step.

So, we take the first step, and calculate expected variance as sum of squared values multiplied by probabilities. In this case we get (1)

After 2 steps we have 3 values, and 4 distinct paths: +1 + 1 = 2, +1 -1 = 0, -1 + 1 = 0, and -1 -1 = -2. We use the same method to calculate variance: (2)

After 3 steps we have 4 values, and 8 distinct paths which I am not going to enumerate; count for yourself. We use the same method to calculate variance: (3)

As you can see variance scales with time, while volatility scales with its square root. I should point out that volatility is not the only quantity to scale with the √t, the property also holds true for expected high, expected low, expected range, expected drawdown and drawup - they all scale the same way, as I wrote in another post.

Finally, Vance writes that the relationship approximately holds for options (assuming no rates and no dividends). I would clarify: it holds approximately assuming small rates, small dividends, and small volatility. Also it holds much better for ATM options, but not so much when you move away from the money.

If we gloss over technical details Wiener process can be described as a sum of random draws from normal distribution - that is starting at 0 we keep adding a new random number at each time step. However we are going simplify this even further, and will use even more basic process - starting at 0, at each time step we flip a coin, and with heads we add 1, with tails we subtract one. And although this may sound like a grade-school version of Wiener process this is actually a valid "substitute" for our purposes.

Here I plotted some sample paths. And yes, this binomial walk looks quite like gaussian random walk.

Now we will look at the process in greater detail, and specifically we will look at the volatility of the process at each time step.

t=0 | t=1 | probability |
---|---|---|

+1 | ½ | |

0 | ||

-1 | ½ |

^{2}* ½ + (-1)^{2}* ½ = 1. So, t=1, variance=1, volatility=√1=1 . Ok, onet=0 | t=1 | t=2 | probability |
---|---|---|---|

+2 | ¼ | ||

+1 | |||

0 | 0 | ½ | |

-1 | |||

-2 | ¼ |

After 2 steps we have 3 values, and 4 distinct paths: +1 + 1 = 2, +1 -1 = 0, -1 + 1 = 0, and -1 -1 = -2. We use the same method to calculate variance: (2)

^{2}* ¼ + (0)^{2}* ½ + (-2)^{2}* ¼ = 1 + 1 = 2. So, t=2, variance=2, volatility=√2 . Ok, let's do one more step:t=0 | t=1 | t=2 | t=3 | probability |
---|---|---|---|---|

+3 | ⅛ | |||

+2 | ||||

+1 | +1 | ⅜ | ||

0 | 0 | |||

-1 | -1 | ⅜ | ||

-2 | ||||

-3 | ⅛ |

After 3 steps we have 4 values, and 8 distinct paths which I am not going to enumerate; count for yourself. We use the same method to calculate variance: (3)

^{2}* ⅛ + (1)^{2}* ⅜ + (-1)^{2}* ⅜ + (-3)^{2}* ⅛ = 9/8 + 3/8 + 3/8 + 9/8 = 24/8 = 3. So, t=3, variance=3, volatility=√3As you can see variance scales with time, while volatility scales with its square root. I should point out that volatility is not the only quantity to scale with the √t, the property also holds true for expected high, expected low, expected range, expected drawdown and drawup - they all scale the same way, as I wrote in another post.

Finally, Vance writes that the relationship approximately holds for options (assuming no rates and no dividends). I would clarify: it holds approximately assuming small rates, small dividends, and small volatility. Also it holds much better for ATM options, but not so much when you move away from the money.

Subscribe to:
Posts (Atom)