In an earlier post, I have been quickly exploring adjoint differentiation in the context of analytical Black-Scholes. Today, I tried to mix it in a simple Black-Scholes Monte-Carlo as described in L. Capriotti paper, and measured the performance to compute delta compared to a numerical single sided finite difference delta.

I was a bit surprised that even on a single underlying, without any real optimization, adjoint delta was faster by a factor of nearly 40%. I suspect this is mostly due to exp evaluations being /2.

On a basket of 4 assets, the adjoint method was 3.25x faster.

It's quick to have such results on basic payoffs: it took me a few hours and worked on the first run, even though my Monte-Carlo is slightly different from the Capriotti paper. It is much more challenging to have it working across a wide variety of payoffs, and to automatize some of it.

## Tuesday, February 25, 2014

### Adjoint Delta for Monte-Carlo

In an earlier post, I have been quickly exploring adjoint differentiation in the context of analytical Black-Scholes. Today, I tried to mix it in a simple Black-Scholes Monte-Carlo as described in L. Capriotti paper, and measured the performance to compute delta compared to a numerical single sided finite difference delta.

I was a bit surprised that even on a single underlying, without any real optimization, adjoint delta was faster by a factor of nearly 40%. I suspect this is mostly due to exp evaluations being /2.

On a basket of 4 assets, the adjoint method was 3.25x faster.

It's quick to have such results on basic payoffs: it took me a few hours and worked on the first run, even though my Monte-Carlo is slightly different from the Capriotti paper. It is much more challenging to have it working across a wide variety of payoffs, and to automatize some of it.

I was a bit surprised that even on a single underlying, without any real optimization, adjoint delta was faster by a factor of nearly 40%. I suspect this is mostly due to exp evaluations being /2.

On a basket of 4 assets, the adjoint method was 3.25x faster.

It's quick to have such results on basic payoffs: it took me a few hours and worked on the first run, even though my Monte-Carlo is slightly different from the Capriotti paper. It is much more challenging to have it working across a wide variety of payoffs, and to automatize some of it.

## Thursday, February 20, 2014

### SVI on top of SABR

Several papers show that the limit for large strikes of Heston is SVI.

Interestingly, I stumbled onto a surface where the Hagan SABR fit was perfect as well as the SVI fit, while the Heston fit was not.

Originally, I knew that, on this data, the SVI fit was perfect. Until today, I just never tried to fit a lognormal SABR on the same data. I did a small test with random values of the SABR parameters alpha, rho, nu, and found out that in deed, the SVI fit is always perfect on SABR.

Is this just because the Taylor expansion of SVI will match the Taylor expansion of SABR up to the fourth order? It seems that the wings are also not too far off in general so there could be more to it.

Gatheral actually devised an SVI formula that uses SABR like variables in a talk here.

Of course, the reverse is not true. SVI has more parameters and provides in general a better fit than SABR.

Interestingly, I stumbled onto a surface where the Hagan SABR fit was perfect as well as the SVI fit, while the Heston fit was not.

Originally, I knew that, on this data, the SVI fit was perfect. Until today, I just never tried to fit a lognormal SABR on the same data. I did a small test with random values of the SABR parameters alpha, rho, nu, and found out that in deed, the SVI fit is always perfect on SABR.

Is this just because the Taylor expansion of SVI will match the Taylor expansion of SABR up to the fourth order? It seems that the wings are also not too far off in general so there could be more to it.

Gatheral actually devised an SVI formula that uses SABR like variables in a talk here.

Of course, the reverse is not true. SVI has more parameters and provides in general a better fit than SABR.

### SVI on top of SABR

Several papers show that the limit for large strikes of Heston is SVI.

Interestingly, I stumbled onto a surface where the Hagan SABR fit was perfect as well as the SVI fit, while the Heston fit was not.

Originally, I knew that, on this data, the SVI fit was perfect. Until today, I just never tried to fit a lognormal SABR on the same data. I did a small test with random values of the SABR parameters alpha, rho, nu, and found out that in deed, the SVI fit is always perfect on SABR.

Is this just because the Taylor expansion of SVI will match the Taylor expansion of SABR up to the fourth order? It seems that the wings are also not too far off in general so there could be more to it.

Gatheral actually devised an SVI formula that uses SABR like variables in a talk here.

Of course, the reverse is not true. SVI has more parameters and provides in general a better fit than SABR.

Interestingly, I stumbled onto a surface where the Hagan SABR fit was perfect as well as the SVI fit, while the Heston fit was not.

Originally, I knew that, on this data, the SVI fit was perfect. Until today, I just never tried to fit a lognormal SABR on the same data. I did a small test with random values of the SABR parameters alpha, rho, nu, and found out that in deed, the SVI fit is always perfect on SABR.

Is this just because the Taylor expansion of SVI will match the Taylor expansion of SABR up to the fourth order? It seems that the wings are also not too far off in general so there could be more to it.

Gatheral actually devised an SVI formula that uses SABR like variables in a talk here.

Of course, the reverse is not true. SVI has more parameters and provides in general a better fit than SABR.

## Wednesday, February 19, 2014

### Smart Initial Guess for Schobel-Zhu

With a small time expansion, it is easy to derive a reasonable initial guess, without resorting to some global minimizer.

Like Forde & al did for Heston, one can find the 5 Schobel-Zhu parameters through 5 points at coordinates (0,0), (x0,t1), (-x0,t1), (x0,t2), (-x0,t2), where x0 is a chosen the log-moneyness, for example, 0.1 and t1, t2 relatively short expiries (for example, 0.1, 0.25).

We can truncate the small time expansion so that the polynomial in (x,t) is fully captured by those 5 points. In practice, I have noticed that using a more refined expansion with more terms resulted not only in more complex formulas to lookup the original stochastic volatility parameters, but also in an increased error, because of the redundancy of parameters in the polynomial expansion. My previous Schobel-Zhu expansion becomes just:

In practice, I have found that the procedure works rather well.

On some more extreme surfaces, where theta=0, the error in kappa and theta is higher. Interestingly, I received a few real world surfaces like this, where theta=0, which I found a bit puzzling. I wondered if it was because those surfaces were preprocessed with SABR, that has no mean reversion, but I could not fit those exactly with SABR.

Like Forde & al did for Heston, one can find the 5 Schobel-Zhu parameters through 5 points at coordinates (0,0), (x0,t1), (-x0,t1), (x0,t2), (-x0,t2), where x0 is a chosen the log-moneyness, for example, 0.1 and t1, t2 relatively short expiries (for example, 0.1, 0.25).

We can truncate the small time expansion so that the polynomial in (x,t) is fully captured by those 5 points. In practice, I have noticed that using a more refined expansion with more terms resulted not only in more complex formulas to lookup the original stochastic volatility parameters, but also in an increased error, because of the redundancy of parameters in the polynomial expansion. My previous Schobel-Zhu expansion becomes just:

In practice, I have found that the procedure works rather well.

On some more extreme surfaces, where theta=0, the error in kappa and theta is higher. Interestingly, I received a few real world surfaces like this, where theta=0, which I found a bit puzzling. I wondered if it was because those surfaces were preprocessed with SABR, that has no mean reversion, but I could not fit those exactly with SABR.

**Update March 2014**- this is now described in my paper Fourier Integration and Stochastic Volatility Calibration.### Smart Initial Guess for Schobel-Zhu

With a small time expansion, it is easy to derive a reasonable initial guess, without resorting to some global minimizer.

Like Forde & al did for Heston, one can find the 5 Schobel-Zhu parameters through 5 points at coordinates (0,0), (x0,t1), (-x0,t1), (x0,t2), (-x0,t2), where x0 is a chosen the log-moneyness, for example, 0.1 and t1, t2 relatively short expiries (for example, 0.1, 0.25).

We can truncate the small time expansion so that the polynomial in (x,t) is fully captured by those 5 points. In practice, I have noticed that using a more refined expansion with more terms resulted not only in more complex formulas to lookup the original stochastic volatility parameters, but also in an increased error, because of the redundancy of parameters in the polynomial expansion. My previous Schobel-Zhu expansion becomes just:

In practice, I have found that the procedure works rather well.

On some more extreme surfaces, where theta=0, the error in kappa and theta is higher. Interestingly, I received a few real world surfaces like this, where theta=0, which I found a bit puzzling. I wondered if it was because those surfaces were preprocessed with SABR, that has no mean reversion, but I could not fit those exactly with SABR.

Like Forde & al did for Heston, one can find the 5 Schobel-Zhu parameters through 5 points at coordinates (0,0), (x0,t1), (-x0,t1), (x0,t2), (-x0,t2), where x0 is a chosen the log-moneyness, for example, 0.1 and t1, t2 relatively short expiries (for example, 0.1, 0.25).

We can truncate the small time expansion so that the polynomial in (x,t) is fully captured by those 5 points. In practice, I have noticed that using a more refined expansion with more terms resulted not only in more complex formulas to lookup the original stochastic volatility parameters, but also in an increased error, because of the redundancy of parameters in the polynomial expansion. My previous Schobel-Zhu expansion becomes just:

In practice, I have found that the procedure works rather well.

On some more extreme surfaces, where theta=0, the error in kappa and theta is higher. Interestingly, I received a few real world surfaces like this, where theta=0, which I found a bit puzzling. I wondered if it was because those surfaces were preprocessed with SABR, that has no mean reversion, but I could not fit those exactly with SABR.

**Update March 2014**- this is now described in my paper Fourier Integration and Stochastic Volatility Calibration.## Wednesday, February 12, 2014

### A Look at Small Time Expansions for Heston

Small time expansions for Heston can be useful during the calibration of the implied volatility surface, in order to find an initial guess for a local minimizer (for example, Levenberg-Marquardt). Even if they are not so accurate, they capture the dynamic of the model parameters, and that is often enough.

In 2011, Forde et al. proposed a second order small time expansion around the money, which I found to work well for calibration. More recently, Lorig et al. proposed different expansions up to order-3 around the money. I already looked at the later in my previous post, applying the idea to Schobel-Zhu.

I noticed, however, that on some surfaces, the Lorig expansion was quickly very inaccurate (LPP1 for order-1, LPP2 for order-2, LPP3 for order-3). Those surfaces seem to be the ones were the Feller condition is largely violated. In practice, in my set of volatility surfaces for 10 different equities/indices, the best fit is always produced by Heston parameters where the Feller condition is violated.

Out of curiosity, I calibrated my surfaces feeding the order-1 approximation to the differential evolution, in order to find my initial guess, and it worked for all surfaces.

The order-3 formula, even though it is more precise at the money, was actually more problematic for calibration: it failed to find a good enough initial guess in some cases, maybe because the reference data should be truncated, to possibly keep the few shortest expiries, and close to ATM strikes.

In 2011, Forde et al. proposed a second order small time expansion around the money, which I found to work well for calibration. More recently, Lorig et al. proposed different expansions up to order-3 around the money. I already looked at the later in my previous post, applying the idea to Schobel-Zhu.

I noticed, however, that on some surfaces, the Lorig expansion was quickly very inaccurate (LPP1 for order-1, LPP2 for order-2, LPP3 for order-3). Those surfaces seem to be the ones were the Feller condition is largely violated. In practice, in my set of volatility surfaces for 10 different equities/indices, the best fit is always produced by Heston parameters where the Feller condition is violated.

T=0.5, Feller condition largely violated |

T=0.5, Feller condition slightly violated |

Out of curiosity, I calibrated my surfaces feeding the order-1 approximation to the differential evolution, in order to find my initial guess, and it worked for all surfaces.

The order-3 formula, even though it is more precise at the money, was actually more problematic for calibration: it failed to find a good enough initial guess in some cases, maybe because the reference data should be truncated, to possibly keep the few shortest expiries, and close to ATM strikes.

### A Look at Small Time Expansions for Heston

Small time expansions for Heston can be useful during the calibration of the implied volatility surface, in order to find an initial guess for a local minimizer (for example, Levenberg-Marquardt). Even if they are not so accurate, they capture the dynamic of the model parameters, and that is often enough.

In 2011, Forde et al. proposed a second order small time expansion around the money, which I found to work well for calibration. More recently, Lorig et al. proposed different expansions up to order-3 around the money. I already looked at the later in my previous post, applying the idea to Schobel-Zhu.

I noticed, however, that on some surfaces, the Lorig expansion was quickly very inaccurate (LPP1 for order-1, LPP2 for order-2, LPP3 for order-3). Those surfaces seem to be the ones were the Feller condition is largely violated. In practice, in my set of volatility surfaces for 10 different equities/indices, the best fit is always produced by Heston parameters where the Feller condition is violated.

Out of curiosity, I calibrated my surfaces feeding the order-1 approximation to the differential evolution, in order to find my initial guess, and it worked for all surfaces.

The order-3 formula, even though it is more precise at the money, was actually more problematic for calibration: it failed to find a good enough initial guess in some cases, maybe because the reference data should be truncated, to possibly keep the few shortest expiries, and close to ATM strikes.

In 2011, Forde et al. proposed a second order small time expansion around the money, which I found to work well for calibration. More recently, Lorig et al. proposed different expansions up to order-3 around the money. I already looked at the later in my previous post, applying the idea to Schobel-Zhu.

I noticed, however, that on some surfaces, the Lorig expansion was quickly very inaccurate (LPP1 for order-1, LPP2 for order-2, LPP3 for order-3). Those surfaces seem to be the ones were the Feller condition is largely violated. In practice, in my set of volatility surfaces for 10 different equities/indices, the best fit is always produced by Heston parameters where the Feller condition is violated.

T=0.5, Feller condition largely violated |

T=0.5, Feller condition slightly violated |

Out of curiosity, I calibrated my surfaces feeding the order-1 approximation to the differential evolution, in order to find my initial guess, and it worked for all surfaces.

The order-3 formula, even though it is more precise at the money, was actually more problematic for calibration: it failed to find a good enough initial guess in some cases, maybe because the reference data should be truncated, to possibly keep the few shortest expiries, and close to ATM strikes.

## Monday, February 10, 2014

### A Small-Time Schobel-Zhu Expansion

The paper "implied vol for any local stochastic vol model" from Lorig et al. presents a very generic and simple formula to compute implied volatility expansions up to order-2 (there is actually an order-3 formula available in their Mathematica CDF file).

I tried it on the Schobel-Zhu stochastic volatility model. This model is an interesting alternative to Heston. I found that, in practice, the implied volatility surface fit was as good, while the simulation under the QE scheme is quite faster (and simpler) than Heston. Here is the result of applying their technique on Schobel-Zhu:

And this is how it behaves on some realistic input:

In practice, while not extremely good, it seems to be enough for Calibration to find an initial guess via differential evolution.

I tried it on the Schobel-Zhu stochastic volatility model. This model is an interesting alternative to Heston. I found that, in practice, the implied volatility surface fit was as good, while the simulation under the QE scheme is quite faster (and simpler) than Heston. Here is the result of applying their technique on Schobel-Zhu:

And this is how it behaves on some realistic input:

T=0.1 |

T=0.5 |

**Update March 2014**- this is now described in my paper Fourier Integration and Stochastic Volatility Calibration.### A Small-Time Schobel-Zhu Expansion

The paper "implied vol for any local stochastic vol model" from Lorig et al. presents a very generic and simple formula to compute implied volatility expansions up to order-2 (there is actually an order-3 formula available in their Mathematica CDF file).

I tried it on the Schobel-Zhu stochastic volatility model. This model is an interesting alternative to Heston. I found that, in practice, the implied volatility surface fit was as good, while the simulation under the QE scheme is quite faster (and simpler) than Heston. Here is the result of applying their technique on Schobel-Zhu:

And this is how it behaves on some realistic input:

In practice, while not extremely good, it seems to be enough for Calibration to find an initial guess via differential evolution.

I tried it on the Schobel-Zhu stochastic volatility model. This model is an interesting alternative to Heston. I found that, in practice, the implied volatility surface fit was as good, while the simulation under the QE scheme is quite faster (and simpler) than Heston. Here is the result of applying their technique on Schobel-Zhu:

And this is how it behaves on some realistic input:

T=0.1 |

T=0.5 |

**Update March 2014**- this is now described in my paper Fourier Integration and Stochastic Volatility Calibration.## Tuesday, February 04, 2014

### An SSD instead of a laptop

Last week, my work laptop died after spilling out some water on its keyboard inadvertently. Fortunately, its SSD was intact.

As the laptop SSD connector (SATA) and power follow the desktop computers standards, and as I use Linux, I just plugged the SSD to my home desktop and booted off the SSD. I had to change slightly the X configuration but otherwise everything worked. Linux is great for that.

The same way, I just plugged the SSD to the desktop at work and it worked. Instead of carrying a laptop, I carry now my small & ultra-light SSD.

The experience is so good, that it seems to me it should be an option to everybody. Before, hard drives were very sensitive to travel, but now with SSDs, I don't really see why we should carry a screen and a keyboard unless we use them (for example in a travel). Also I find that I actually notice the speed difference between my desktops and former laptop, which I never really paid too much attention to before.

As the laptop SSD connector (SATA) and power follow the desktop computers standards, and as I use Linux, I just plugged the SSD to my home desktop and booted off the SSD. I had to change slightly the X configuration but otherwise everything worked. Linux is great for that.

The same way, I just plugged the SSD to the desktop at work and it worked. Instead of carrying a laptop, I carry now my small & ultra-light SSD.

The experience is so good, that it seems to me it should be an option to everybody. Before, hard drives were very sensitive to travel, but now with SSDs, I don't really see why we should carry a screen and a keyboard unless we use them (for example in a travel). Also I find that I actually notice the speed difference between my desktops and former laptop, which I never really paid too much attention to before.

### An SSD instead of a laptop

Last week, my work laptop died after spilling out some water on its keyboard inadvertently. Fortunately, its SSD was intact.

As the laptop SSD connector (SATA) and power follow the desktop computers standards, and as I use Linux, I just plugged the SSD to my home desktop and booted off the SSD. I had to change slightly the X configuration but otherwise everything worked. Linux is great for that.

The same way, I just plugged the SSD to the desktop at work and it worked. Instead of carrying a laptop, I carry now my small & ultra-light SSD.

The experience is so good, that it seems to me it should be an option to everybody. Before, hard drives were very sensitive to travel, but now with SSDs, I don't really see why we should carry a screen and a keyboard unless we use them (for example in a travel). Also I find that I actually notice the speed difference between my desktops and former laptop, which I never really paid too much attention to before.

As the laptop SSD connector (SATA) and power follow the desktop computers standards, and as I use Linux, I just plugged the SSD to my home desktop and booted off the SSD. I had to change slightly the X configuration but otherwise everything worked. Linux is great for that.

The same way, I just plugged the SSD to the desktop at work and it worked. Instead of carrying a laptop, I carry now my small & ultra-light SSD.

The experience is so good, that it seems to me it should be an option to everybody. Before, hard drives were very sensitive to travel, but now with SSDs, I don't really see why we should carry a screen and a keyboard unless we use them (for example in a travel). Also I find that I actually notice the speed difference between my desktops and former laptop, which I never really paid too much attention to before.

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