The Heston Recipe: A Picture, from Fabrice Rouah

This is an update to my first article about the Heston Recipe on the site.

The Heston model is a mathematical formula for the call price. Similar to the Black-Scholes formula, it requires a set of observable inputs such as the spot price and strike price, the risk free rate, and the time to expiry. In place of the single parameter 'sigma' in  Black-Scholes, however, the Heston model requires a number of parameters that must be estimated from market data. These parameters drive the shape of the implied volatility surface extracted from call prices generated with the model. This surface is simply a three-dimensional plot of  implied volatility as it varies according to strike and to maturity.

In FX markets the implied volatility is usually symmetric in the strike direction and the pattern is often referred to as a "smile". In equities markets, however, the implied volatility is usually asymmetric and is referred to as a "smirk" or "sneer." A number of explanations have been proposed for the asymmetrical smirk, but one of the most plausible is "crash-o-phobia." According to this explanation, following the October 1987 crash, investors in equity options have been willing to pay relatively more for downside protection than for upside speculation. Consequently, out-of-the-money (OTM) puts are relatively more expensive than OTM calls. Since there is a one-to-one, monotonic, relationship between the option price and its implied volatility, the smirk is simply a reflection of this price mismatching. The Heston model is a convenient way to capture the smirk and other features of implied volatility, along a continuum of strikes and maturities.

The most common way to estimate the parameters of the model is to construct a function that calculates the distance between implied volatilities observed in the market, and implied volatilities generated by the model and matched by strike and maturity. The distance is often chosen as the squared difference between the model and market implied volatilities, and the function is the sum of all the squared differences. The parameter set is chosen as that which leads to the smallest sum, and consequently, which provides the closest fit of market implied volatilities to their model counterparts. The figure illustrates the implied volatility surface for a subset of SPY options on April 13, 2012. The market implied volatilities are represented by black dots, and the implied volatility surface generated by the Heston model, by the colored mesh. The Heston model is able to fit the smirk, and to account for the mitigation of the smirk at long maturities.

The Heston Recipe, from Fabrice Rouah

June 13, 2013 | 1 Comment

Most traders are intimately familiar the implied volatilities of equity options. These implied volatilities are often smoothed to avoid the temporary spikes in the strike/maturity surface that can lead to butterfly and calendar arbitrage. Many trading desks and market makers use the Heston stochastic volatility model for smoothing.

To understand the genesis behind Heston model, and why it is so important, we must revisit an event that shook financial markets around the world: the stock market crash of October 1987. The consequence on the options market was an exacerbation of smiles and skews in the implied volatility surface which has persisted to this day. This brought into question the restrictive assumptions behind the Black-Scholes option pricing model, the most tenuous of which is that continuously compounded stock returns be normally distributed with constant volatility. A number of researchers since then have sought to eliminate the constant volatility assumption in their models, by allowing volatility to be time-varying.

One popular time-varying approach is to allow volatility to be stochastic. The Heston model, developed in 1993, was not the first stochastic volatility model for pricing equity options, but for mathematical and practical reasons it is by far the most popular and the most successful. It is used throughout the world by option trading desks and market makers, banks, hedge funds, and academics. It forms a crucial part of the options curriculum of financial engineering programs offered by universities across the world. The model has been refined and extended in many ways, to overcome some of the shortcomings of its original formulation. The top option valuation software companies, such as Numerix, SuperDerivatives, and Fincad, all incorporate the model into their pricing routines. My forthcoming book is devoted entirely to the model.

In short, the Heston model is one of the great success stories of mathematical finance, yet most financial professionals have never heard of it. The next time someone mentions the Heston model, you won't be wondering whether it refers to Charlton Heston, the late American actor, or Heston Blumenthal, the quirky British chef, but to Steve Heston, one of the most influential financial engineers of the modern era.

Fabrice Rouah is the author of the forthcoming book "The Heston Model in Matlab
and C#" from John Wiley & Sons.and a consultant on option pricing models.

Paper for Non-Statisticians, from Victor Niederhoffer

There's a rather direct and rather readily understandable paper that's the standard for non-statisticians trying to asses the difference between such things as test-retest correlations, difference between means of binary splits, and agreement of two successive measurement of individuals. Completely related and very similar to the binary split cart essay of Doc and me. But no consideration of sampling without replacement, small number of observations requiring a t distribution, simulation of actual distributions of differences, or adjustments for highest of n differences. Bland statistical methods for assessing agreement. Lancet 1986.

Fabrice Rouah writes:

Another great resource in this area is the textbook by Joseph Fleiss, Statistical Methods for Rates and Proportions, now in its 3rd edition.

Thoughts on Lance Armstrong, from Fabrice Rouah

February 8, 2013 | 4 Comments

I recently saw this article: "Armstrong Sued Over Prize Money" . Sure to be the first of many lawsuits against him.

It's worth remembering that doping constitutes only a tiny part of an athlete's activities. Most of their time is spent training, in the gym, exercising, dieting, etc. You cannot transform an out-of-shape and overweight cycling wannabe (me!) overnight into a Tour winner simply by giving him dope.

We all know Armstrong's story but it's worth repeating. Testicular cancer that spread to his brain and to 12 golf ball-sized tumors in his lungs; given less than 50% chance of survival; refused a cancer treatment that would damage his lungs; insisted to his doctors that he would survive return to competitive cycling when they were telling him that he would be confined to a wheelchair.

His remarkable recovery wasn't due to doping, but to his determination and will.

A Tau Measure of Serial Dependendence, from Victor Niederhoffer

It is interesting and useful to measure the tendency to continuation or reversal in a series. It's particularly useful for markets because many traders like to go with or against in a period. And some measure of whether this works or not, and how it's changing provides a rudder.

The usual methods of measuring it rely on the serial correlation coefficient, but this tends to be disrupted by extreme or missing observations and doesn't have stable properties for many non-normal distributions. Non-parametric measures that rely on ranks, runs, or moves above and below the median, or curve fitting for consecutive observations have often been used. Cowles started the whole subject for stock prices by looking at sequences and reversals in consecutive prices.

A measure that I have been working with that is relatively new and has many advantages is to consider the concordances and discordances in a series. This method is based on work done by Kendall in rank correlations with his statistic, Kendall's Tau. A key article in this area that provides an excellent foundation is Ferguson, Genet and Hallin, "Kendall's tau for serial dependence" and "Bandt Ordinal Time series analysis".

The method of concordances and discordances starts with looking at 3 consecutive observations in a series. Let's call them p1, p2, and p3. If p2 > 1 and P3 > p2, that's a positive concordance. If p2< 1 and P3 < p2, that's a negative concordance. All the other rises followed by declines, or declines followed by rises are discordances. (Note that there are 6 permutations of the 3 numbers and only 2 yield concordances.)

To make it more tangible consider the levels in stocks from Friday 1/4/2013 to  Friday 1/11/2013

day        Date                  Level       change    rank of change

```Fri           1/4/2013         1458
```
```Mon          1/7/2013       1456       -2          2
```
```Tues         1/8/2013        1452       -4         1
```
`Wed          1/9/2013       1456        4         4                                      `
`Thurs        1/10/2013      1467        11        5                                     `
`Fri            1/11/2013      1467      0         3`

To measure the momentum in the series of changes, one must compute all the consecutive one day discordances, + the number of consecutive 2 day discordances + the number of 3 day discordances. It is best to focus on the ranks. If the consecutive pairs of ranks reverse there is a discordance. If they are in the same direction, there is a concordance.

Comparing Mon to Tue and Tue to Wed, one notes a discordance.

Comparing Tues to Wed, and Wed to Thur, there is a concordance.

Comparing Wed to Thurs, and Thurs to Friday, there is a discordance.

Mon and Wed and Tue and Thur are in concord.

Tues and Thur and Wed and Friday are in discord as Thur rank is higher than Tues and Friday's rank is lower than Wed.

There is one 3 day comparison. Mon and Thur, and Tuesday to Friday are in concordance. Thus, there were 3 discordances and 3 concordances. It turns out that the expected number of discordances for a time series is ( n-2) ( 3n-1 ) / 12. since n is 5 , the expected number of discordances is 3.5. An exact calculation is possible and shows that 3 or less discordances has a prob of 20%.

How can this measure be used? First, it provides a nice estimate of the degree of correlation between the consecutive values of a series. The question then arises, how can one predict subsequent momentum based on past momentum. It turns out that that there is a tendency in the series that we have looked at , for periods with high concordances to be followed by periods with high discordances, i.e. momentum changes from period to period. This would have to be quantified with the period one is interested in, a week, a month etc.

I will report further work on this in future. I would like to thank Doc Castaldo and Mike Chuprin for their kind assistance on this project.

Fabrice Rouah writes:

Very good point. Non-parametric methods are definitely preferable for financial time series that rarely meet the normality or linearity assumptions required of many parametric methods. Another example of parametric methods are t-tests and ANOVA. To compare returns between different groups one is better off using their non-parametric counterparts, namely the tests of Wilcoxon, Mann-Whitney, Kruskal-Wallis and many others.

Jan

31

Hi everyone,

I've created www.Volopta.com, which contains free derivatives pricing code in a variety of languages, such as Matlab, C++, etc.

Feel free to distribute the link to anyone you think might be interested.

Thanks!

Fabrice

What Vol for an American Option? from Fabrice Rouah

January 25, 2013 | 1 Comment

The next time your dealer quotes the implied volatility of an American option, be sure to ask "is that trinomial, BAW, or something else?" Implied volatilities are obtained by matching the market price of an option to its corresponding model price, and finding the value of volatility that equates both. Implied volatilities are therefore dependent on the model used. For European options the industry-standard implied volatility model is the Black-Scholes model. Implied volatilities from European options should be denoted "Black-Scholes implied volatilities" but since there is no ambiguity about the model used to extract them they are simply known as "implied volatilities." Consequently, given the correct inputs such as the strike price and underlying spot price an operator should, in principle, be able to replicate the same implied volatility as that which has been quoted, up to rounding-off error.

For American options (example) the situation is different as there is no industry-standard model for pricing American options, even under Black-Scholes assumptions. The available models include binomial trees and trinomial trees of various sorts, the Barone-Adesi and Whaley approximation (BAW) and its variants, the Longstaff and Schwartz algorithm, and many others. Hence, without the choice of model being used to extract an implied volatility, a trader cannot obtain the same quoted implied volatility. Even given the choice of model, approximation error (such as the number of time steps used in the tree, or simulation noise, for example) would preclude an exact replication of the quoted implied volatility.

One very promising method to valuing American puts has been proposed by Alexey Medvedev and Olivier Scaillet in their 2010 paper, published in the Journal of Financial Economics (link). Their method takes the form of an infinite expansion with analytic terms, truncated in practice to usually five terms or less. Being of closed form, prices are produced very fast. An implied volatility extracted with this model could be replicated, provided the number of expansion terms is specified.

Fabrice Rouah is the author of the book "The Heston Model in Matlab and C#" to be published in early 2013 by John Wiley & Sons.