# Fake or JPEG? Revealing Common Biases in Generated Image Detection Datasets

Patrick Grommelt Fraunhofer ITWM, Competence Center High Performance Computing
Kaiserslautern, Germany  Institute of Machine Learning and Analysis (IMLA),
Offenburg University, Germany  Equal contribution Louis Weiss Fraunhofer ITWM,
Competence Center High Performance Computing Kaiserslautern, Germany
Institute of Machine Learning and Analysis (IMLA), Offenburg University,
Germany  Equal contribution Franz-Josef Pfreundt Fraunhofer ITWM, Competence
Center High Performance Computing Kaiserslautern, Germany  Janis Keuper
Institute of Machine Learning and Analysis (IMLA), Offenburg University,
Germany  University of Mannheim, Germany

###### Abstract

The widespread adoption of generative image models has highlighted the urgent
need to detect artificial content, which is a crucial step in combating
widespread manipulation and misinformation. Consequently, numerous detectors
and associated datasets have emerged. However, many of these datasets
inadvertently introduce undesirable biases, thereby impacting the
effectiveness and evaluation of detectors.

In this paper, we emphasize that many datasets for AI-generated image
detection contain biases related to JPEG compression and image size. Using the
GenImage dataset, we demonstrate that detectors indeed learn from these
undesired factors. Furthermore, we show that removing the named biases
substantially increases robustness to JPEG compression and significantly
alters the cross-generator performance of evaluated detectors. Specifically,
it leads to more than 11 percentage points increase in cross-generator
performance for ResNet50 and Swin-T detectors on the GenImage dataset,
achieving state-of-the-art results.

We provide the dataset and source codes of this paper on the anonymous
website:
https://www.unbiased-genimage.org

_Keywords_ Generation Detection $\cdot$ Diffusion Model $\cdot$ Bias $\cdot$
Dataset

##  1 Introduction

In recent years, generative models have improved significantly in creating
photorealistic images, making it hard for humans to distinguish between
natural and generated images [1]. Even though these advancements are a
significant achievement in many computer vision applications, it does not take
much to realize how the misuse of these models can threaten public safety. To
tackle this issue, extensive research has been conducted on the development of
robust detection mechanisms and techniques for identifying manipulated or
synthetic content that is not actively fingerprinted by the generator side.

Generative Adversarial Networks (GANs) [2] have marked a major breakthrough in
generating realistic synthetic images. Nevertheless, spectral analysis reveals
that images produced by GANs inherently contain distinctive artifacts, making
them recognizable. Zhang et al. [3] observed that GAN-generated images exhibit
periodic, grid-like patterns in their frequency spectrum, a clear deviation
from natural image spectra. Furthermore, Durall et al. [4] pointed out that
GANs fail to replicate the spectral distribution of training data, probably
due to the inherent transposed convolution operations of these models. This
discrepancy between natural and generated images has enabled the research
community to develop highly effective, generator-agnostic detection tools,
achieving near-perfect accuracy in identifying synthetic images.

However, the landscape of synthetic content generation has evolved rapidly.
One significant advancement came with the introduction of Denoising Diffusion
Probabilistic Models (DDPM) [5], which represented a paradigm shift in
generative modelling. DDPMs employ a fundamentally different approach compared
to GANs, utilizing a diffusion process to generate images, which results in a
smoother and more realistic appearance. These generative models have also
shown better capabilities in approximating the frequency spectra of natural
images [6, 7, 8], which caused a lot of detectors developed for the
identification of GAN generated images to perform poorly on these images [6].
To the best of our knowledge, the problem of finding generator-agnostic and
robust detection methods for DDPM-generated images has not been solved yet.
Even though it seems trivial to correctly classify images which are generated
by the same model the detector has seen in training, it does not transfer well
to other generative models [6, 7, 9, 10].

To effectively assess and compare the generalization capabilities of
detectors, it is imperative to establish a consensus within the community
regarding the benchmark to be employed. Ideally, such benchmarks should
closely mimic real-world scenarios by being large-scale and encompassing a
diverse array of classes and images produced by a range of distinct
generators. To address this need, GenImage [9] introduced an extensive dataset
that includes all natural images from the ImageNet1k dataset [11], as well as
approximately an equal number of generated images, originating from various
generators with differing architectures. Nevertheless, the evaluation of a
detector based on the raw GenImage Benchmark is not reliable yet since, as we
will show, JPEG compression and image size biases are used by detectors during
the training.

This paper’s primary goal is to raise awareness of JPEG compression and image
size biases present in most datasets for generated image detection and
emphasize the need for heightened scrutiny to ensure that detectors do not
inadvertently learn from undesirable variables. Current evaluation benchmarks
are somewhat limited in their interpretability, because they do not ensure
whether the detection is really based on generation-specific artifacts and
thus applicable for real world usage. Consequently, it becomes challenging to
assess the effectiveness of various approaches and determine which ideas
warrant further research. We firmly believe that identifying and mitigating
biases in datasets for generated image detection is crucial, as it lays the
foundation for establishing a robust and transparent research environment for
generated image detection.

In summary, our main contributions are as follows:

• We demonstrate, using the GenImage dataset as an example, that many datasets
for generated image detection contain JPEG and image size biases, which are
subsequently used by the detectors during inference.

• We show that removing these biases significantly enhances cross-generator
performance, achieving state-of-the-art results on GenImage and increasing the
average accuracy by more than 11 percentage points for ResNet50 and Swin-T
detectors. Additionally, detectors become more robust against distortions due
to now learning the actual task of detecting generation specific artifacts.

##  2 Related Works

###  2.1 Detecting Artifically Generated Images

The problem of detecting AI-generated images is traditionally approached as a
binary classification task [4, 3, 12, 13, 14]. However, for practical use in
real-world scenarios, detector models not only need to perform well on the
training and validation datasets but also exhibit two critical
characteristics. Firstly, they must generalize effectively to generative
models that were not part of their training data, as the specific generator is
typically unknown. This concept is referred to as cross-generator performance.
Secondly, these models need to be robust against various transformations such
as resizing, compression, or noise, as attackers may attempt to manipulate
content to deceive the detector and social media platforms often apply these
transformations by default.

#### Detecting GAN Generated Images

Research has indicated that GAN-generated images contain distinctive patterns
in the frequency spectrum of their Discrete Cosine Transformation [3]. These
patterns arise due to the presence of upsampling layers [4], making GAN-
generated content detectable. Wang et al. [12] have demonstrated that even a
basic CNN classifier, ResNet50 [15], can successfully identify images
generated by specific GANs. They have also shown that augmentations like
random JPEG compression or Gaussian Noise significantly enhance cross-
generator performance. To encourage the CNN classifier to focus on the
frequency information, which is a generator-agnostic characteristic,
Gragnaniello et al. [16] suggest removing the initial downsampling layers of
ResNet50. Cozzolino et al. [17] have combined these concepts but opted for
cropping instead of resizing during training. This approach prevents the loss
of frequency information and enhances robustness against resizing
augmentations during inference. These detectors are often referred to as
universal GAN detectors due to their impressive ability to perform well on
unseen GANs and their robustness against various transformations, making them
suitable for real-world scenarios.

#### Detecting Diffusion Model Generated Images

Unfortunately, several existing studies [6, 7, 9] have indicated that
universal GAN detectors do not effectively classify images generated by
Diffusion Models (DM). While the frequency spectrum of DM-generated images
still exhibits differences compared to natural images, these artifacts differ
from those present in GAN-generated images. Furthermore, these artifacts tend
to be more specific to the particular Diffusion Model generator, making the
achievement of good cross-generator performance a considerably more
challenging task. It is also worth noting that Diffusion Models are capable of
replicating transformation artifacts found in the images they have been
trained on with great accuracy [7]. For instance, if the DM generator has been
exposed to JPEG-compressed images during training, the generated images will
contain artifacts that are similar to JPEG-artifacts. Research also indicates
that transformation artifacts in frequency space can closely resemble those
generated by Diffusion Models. This complicates the task of ensuring
robustness to transformations and as our research will demonstrate, it
underscores the importance of carefully selecting the training dataset for the
detection model.

Nevertheless, it still appears relatively easy to detect images generated by a
specific generator, even with small detector models and training datasets [9,
18]. Prior work has shown that Diffusion Models, for example, exhibit
systematic errors in projective geometry [14]. However, to the best of our
knowledge, no detection model has achieved good cross-generator performance
and robustness in realistic application scenarios.Promising methods have
emerged that achieve outstanding results on small datasets, but they have yet
to prove their mettle on larger and more diverse benchmarks like the GenImage
dataset [9]. DIRE [13], for instance, measures the disparity between an image
and its reconstructed version of a Diffusion Model to detect images from both
Diffusion Models and GANs. They report nearly perfect results in cross-
generator performance and robustness on the DIRE dataset, but performance
drastically declines when evaluated on GenImage [10]. GenDet [10]
significantly improves results by approaching the task as an outlier detection
problem instead of binary classification. Effectively, they train a
Teacher/Student-network and, to the best of our knowledge, present state-of-
the-art results on GenImage.

###  2.2 GenImage Dataset

The GenImage dataset [9], stands out as one of the biggest and most diverse
datasets for generated image detection. Its primary goal is to establish a
unified benchmark within the research community, facilitating the evaluation
of detection methods on a standardized dataset. Built upon the ImageNet
dataset [11], GenImage incorporates the original ImageNet images as the
natural-image class. For the AI-generated class, GenImage includes images
generated by eight generative models, comprising seven Diffusion Models and
one GAN. These models are specifically Midjourney (MJ) [19], Stable Diffusion
V1.5 (SD1.5) [20], Stable Diffusion V1.4 (SD1.4) [20], Wukong [21], VQDM [22],
ADM [23], GLIDE [24], and BigGAN [25]. They are either Text-to-Image (MJ, SD,
Wukong, VQDM, GLIDE), utilizing ImageNet-classes as text prompts, or class-
conditional (ADM, BigGAN), to guarantee a consistent content distribution
between natural and AI-generated images. The dataset is organized into eight
distinct subsets, one for every generative model. Each subset contains
training and validation data, with a nearly equal number of natural images
from ImageNet and generated images from the respective generative model. To
assess the performance of a detection method, the standard approach involves
training on one GenImage subset and evaluating on the others. This method
enables the measurement of a detection method’s cross-generator performance,
providing insights into its ability to generalize across different generative
models. The evaluation process is quantified by a cross-generator matrix. Fig.
1 illustrates the results presented in the original paper using a basic
ResNet50 classifier, with accuracy (in %) serving as the key metric.

##  3 Common Biases in Datasets for AI-Generated Image Detection

When examining various datasets used for AI-generated image detection, it is
evident that a significant disparity exists in terms of compression
techniques. As demonstrated in Table 1, a common practice involves storing all
artificially AI-generated images within the dataset as PNG files, which
involves lossless compression, while natural images are stored in JPEG format,
involving lossy compression and introducing noticeable artifacts. This prompts
us to investigate whether detectors trained on such datasets rely on detecting
JPEG compression artifacts for classification.

Table 1: Overview over common AI-Generated Image Detection Datasets and their image compression and size properties. Dataset | Natural Images | Synthetic Images
---|---|---
|  Compression |  Size |  Source |  Compression |  Size |  Source
GenImage [9] |  JPEG |  diverse |  ImageNet |  PNG |  128x128, 256x256, 512x512, 1024x1024 |  LAION, ImageNet, unknown
Wang et al. [12] |  mostly JPEG (saved as PNG) |  diverse (resized to 256x256) |  LSUN, ImageNet, CelebA, COCO |  PNG |  256x256 |  trained on same data as natural images
DIRE [13] |  JPEG |  diverse |  LSUN, CelebA-HQ, ImageNet |  PNG |  256x256, 512x512, 1024x1024 |  LSUN, ImageNet, unknown
Epstein et al. [26] |  JPEG |  diverse |  LAION |  PNG |  256x256, 512x512, 1024x1024 |  LSUN, ImageNet, LAION, unknown
Ricker et al. [6] |  JPEG |  256x256 (resized to) |  LSUN-Bedroom |  PNG |  256x256 |  LSUN-Bedroom
Ojha et al. [8] |  JPEG |  diverse |  ImageNet, LAION |  PNG |  256x256 |  ImageNet, unknown

Furthermore, considering that the generators employed for most of AI-generated
image detection datasets generate images of a fixed size, as opposed to the
diverse size distribution found in natural images, we investigated whether
this disparity in size distribution could potentially cause detectors to
distinguish between natural and generated images based on their dimensions.
The reason why this is problematic is twofold: firstly, it can lead to an
overestimation of the performance in detecting generation-specific artifacts,
since evaluation data often contains the same biases from which the detector
learned during training. Secondly, detectors having such a reliance on
undesirable variables do not generalize well to datasets where biases related
to these variables are absent, which is the case in real world scenarios. This
lack of robustness hinders the adaptability of detectors to changes in these
variables.

(a) GenImage Results

(b) Reproduced Results

Figure 1: Reproduced Results. Cross-generator performance of a ResNet50
classifier from the GenImage paper (left), and our reproduced results (right).
The matrix shows the accuracy (in %) of a model trained on a GenImage training
subset (row), when evaluated on a GenImage validation subset (column).

We conducted experiments using the GenImage dataset to assess whether
detectors, trained on datasets containing such biases, inadvertently acquire
information from these undesirable variables. For these experiments, we
trained ResNet50 detectors since this is the baseline methodology in the
GenImage paper. We used the provided code from GenImage and successfully
replicated the reported results, as depicted in Fig. 1. The results show near
perfect accuracy when the detectors are tested on the subset containing
generated images from the same generative model they were trained on, but poor
accuracy on others. By sorting the columns and rows according to the output
size of the respective generative models, as we have done, it becomes apparent
that better generalization tends to occur near the diagonal. This observation
raises the question of whether similarity in generator size may enhance cross-
generator performance, a topic we discuss in section 3.2 and section 4.2.

###  3.1 JPEG Compression Bias

As mentioned, the natural images used by GenImage are sourced from the
ImageNet dataset. Appendix A illustrates the distribution of JPEG quality
factors employed in ImageNet on a logarithmic scale, with the majority of
images compressed using a quality factor of 96 [27]. On the other hand,
generated images from GenImage are uncompressed. This clear disparity of
compression between natural and generated images is common in many datasets,
as shown by Table 1.

To investigate whether detectors trained on datasets containing such
compression disparities partially function as JPEG detectors, we conducted two
experiments on the GenImage dataset. Initially, we examined whether
compressing the dataset’s generated images influences their classification as
natural. To do so, we used the ResNet50 detector trained on raw GenImage
dataset (cf. Fig. 1) and evaluated the cross-generator performance of these
detectors on test data that was progressively compressed with lower quality
factors.

Figure 2: Cross-generator performance of detectors trained on raw GenImage for
different compression quality factors, given in accuracy (in %).

Fig. 2 illustrates a strong decline in accuracy with increased compression,
even for high quality factors like 95. Analyzing the confusion-matrix of the
results revealed that the precision in detecting AI-generated images
consistently remained close to one, whereas the recall dropped significantly
(cf. App. B). This suggests that compressing a generated image considerably
increases the likelihood of the model classifying it as natural.

However, this experiment alone does not confirm that the detector learned to
identify JPEG compression, as compression might destroy generation-specific
artifacts. To address this possibility, we used uncompressed natural PNG
images from the FFHQ dataset [28] to observe how compression impacts their
classification. Utilizing the same detectors, we evaluated their performance
on the FFHQ images as they were JPEG-compressed with different quality
factors.

Table 2 shows the results for the detector trained on the Midjourney subset of
GenImage. We observe that the detectors’ ability to accurately classify a
natural image improves as the level of compression increases. Specifically,
the detector’s accuracy improves from 80.4% for uncompressed PNG images up to
100% for natural images compressed with a low-quality factor of 60. Even a
small compression with qualtity factor 95 leads to a much better accuracy of
94.8%. This experience underscores the direct influence of compression towards
the classification as natural, as compression, in this scenario, can´t result
in the potential destruction of generation-specific artifacts.

###  3.2 Size Distribution Bias

The generated images in the GenImage dataset originate from eight different
generators, producing images of four different sizes: 1024×1024 (MJ), 512×512
(SD4, SD5, Wukong), 256×256 (GLIDE, ADM, VQDM), and 128×128 (BigGAN). In
contrast, the natural images sourced from ImageNet contain images of various
sizes, as shown in Appendix A.

Table 2: Accuracy (in %) of the ResNet50 detector trained on the Midjourney subset on 1024×1024 natural FFHQ images when progressively increasing compression. Compression |  PNG |  JPEG95 |  JPEG90 |  JPEG80 |  JPEG70 |  JPEG60
---|---|---|---|---|---|---
Accuracy |  80.45 |  94.84 |  98.93 |  99.95 |  99.99 |  100.0

We anticipated that this fundamental difference in size distribution could be
exploited by some detectors to discern whether an image is generated or not.
For instance, consider a ResNet50 detector which initially resizes all the
input data to 224×224 pixels, as the ResNet50 detector evaluated in GenImage,
and is trained with generated images from the GLIDE generator. Given that most
natural images from the GenImage dataset have dimensions around 450×450
pixels, they would typically undergo more resizing than the 256×256 images
from GLIDE. Consequently, the detector could potentially extract information
about the nature of an image by detecting the strength of resizing artifacts.
Furthermore, for non-square images, resizing results in significant
information loss along one axis, leading to frequency artifacts. However, this
problem extends beyond detection methods that rely solely on resizing as a
preprocessing step. Detectors employing cropping to achieve a uniform input
size are still subject to bias in terms of object size. Discrete-Cosine-
Transormation based detectors will contain similar biases as well, as larger
images will have more data points (pixels), which affects the frequency
spectrum towards having higher frequencies.

To investigate whether detectors trained on the GenImage dataset indeed
acquire information about image size, we conducted an experiment to evaluate
how well a detector performs on natural images of various sizes. A decrease in
performance for natural images that closely match the dimensions of the
generated images the detector was trained on might suggest that the detector
is, to some extent, a size detector. Fig. 3 displays the accuracy of ResNet50
detectors on natural images across different size intervals. For clarity, we
show the performance of detectors trained on one subset for each generator-
size available in the GenImage dataset. Each diagram illustrates the
performance of a detector trained on a specific GenImage subset. The analysis
included all natural ImageNet images not seen by the detector during training,
effectively using the training as well as validation data from other GenImage
subsets. Appendix A illustrates the number of ImageNet images in each
interval. It is important to highlight that some intervals contain very few
images, which is why the single cells are not as informative as the global
trend. Note that we used the models trained in section 4.1, since otherwise
the classification of natural images is highly affected by JPEG-artifacts and
thus the prediction is near perfect for all intervals.

Figure 3: Accuracy (in %) of detectors trained on different GenImage subsets
when evaluated on natural images of varying sizes. Black crosses indicate
intervals without any data. The red cross marks the size interval of generated
images corresponding to this subset.

Our findings indicate a correlation between decreased detector performance and
natural images sized similarly to those of the generated images the detector
was trained on. These results demonstrate that detectors indeed perform better
on natural images that have significantly different sizes compared to the size
of the generated images used during detector training. For instance, the
BigGAN detector accurately classifies most natural images, except from very
small images with one side between 100-150, as BigGAN images are of size
128×128. ADM images are of size 256×256, which is why the effect is also
visible for slightly bigger images. Conversely, the Midjourney detector shows
reduced performance on larger natural images, given that Midjourney images are
much bigger than most natural images in ImageNet. For Stable Diffusion, no
significant trend emerges, as the generated images are of similar size as most
of the natural images. This detector has to learn more from other
discriminative patterns in the interval of the generated images, which does
not mean that the the detector did not learn from the bias to classify images
outside of the interval.

##  4 Removing The Biases

We have emphasized that disparities in size distribution and compression
between natural and generated data can cause detectors to learn from
differences that may not be pertinent in real-world contexts. Consequently, we
retrained the same detectors using a constrained dataset and reevaluated them
to investigate whether the removal of compression and size biases would affect
the final evaluation of detectors. Our findings reveal significantly divergent
results and notably enhanced cross-generator performance and robustness.

###  4.1 JPEG Constraint

Figure 4: Cross-generator performance given in accuracy (in %) of detectors
trained only with images compressed with JPEG quality factor of 96 from
GenImage for different Compression Rates.

To ensure that the detector does not learn to differentiate between natural
and generated data based on JPEG compression artifacts, we constrained the
quality factor of all images used in training. Specifically, we constructed a
training set by exclusively selecting natural images compressed with a quality
factor of 96. We then selected an equivalent number of generated images and
compressed them using the same quality factor. Subsequently, we trained a
ResNet50 detector using this constrained dataset and reevaluated it’s cross-
generator and robustness performance. This approach is expected to make the
detector’s classification less sensitive to JPEG compression than before.

Table 3: Comparison of robustness to compression between training on raw GenImage and only JPEG96 GenImage: Accuracy in % averaged cross the generator matrix. | Training dataset |
---|---|---
Compression |  Classic GenImage |  JPEG96 (ours) |  Difference
JPEG95 |  53.91 |  67.17 |  +13.26
JPEG80 |  50.62 |  59.37 |  +8.75
JPEG60 |  50.58 |  55.07 |  +4.49

Fig. 4 illustrates the cross-generator performance of these detectors when the
test data is compressed with increasing quality factors. Unlike the training
data, the test data is not subject to any constraints. It is important to note
that for JPEG96111We mean with JPEGX a JPEG compression with a quality factor
of X., compression was applied solely to the generated images, as the majority
of natural images is already compressed to JPEG96 or a lower quality.
Consequently, for compression rates other than JPEG96, natural images undergo
a second round of compression.

Comparing the results shown in Fig. 4 with those in Fig. 2, which presents the
same evaluation for detectors trained on the raw GenImage dataset, we observe
a strong enhancement in robustness against JPEG compression. Specifically, as
illustrated in Table 3, we observe an overall improvement of 13.26 accuracy
points to JPEG95 compression, 8.75 for JPEG80, and 4.49 for JPEG60. For
example, the BigGAN detectors performance remains nearly intact up to JPEG60,
whereas previously, the detector incorrectly classified all JPEG95 images as
natural.

These results prove our assumption that biases in compression lead to the
detectors learning wrong causalities and not being robust to changes in
compression. Previous research [12] has already indicated that introducing
random JPEG augmentation into the training set enhances detector robustness
and generalization to other generators. We firmly believe that this
improvement does not solely stem from increasing the training set’s variance
but also partially from reducing the bias arising from differences in JPEG
compression during training. Nevertheless, random-augmentation does not
completely remove the bias, since it merely shifts the distributions of
quality factors to have a bigger overlap instead of equalizing them. Moreover,
when using augmentation, already compressed images are compressed a second
time, while uncompressed images are only compressed once. This introduces a
bias, as JPEG compression is not an idempotent transformation.

Surprisingly, mitigating the JPEG bias marginally improves the generalization
across different generators. This contradicts our initial expectation that the
bias, being present in all GenImage subsets, should facilitate
transferability. To explain this we propose two hypothesis: Firstly, a
training dataset with reduced bias compels the detectors to focus on learning
from generation-specific artifacts. These artifacts are not only more
resilient to compression but also to other transformations like resizing,
thereby enhancing generalization to generators of various sizes. Secondly,
research indicates that Diffusion Models, when trained on JPEG-compressed
images, produce artifacts that mimic those of compression artifacts [7]. This
suggests that biased detectors do not actually distinguish between compressed
and uncompressed images, but rather between genuine compression artifacts and
their approximations. This distinction may generalize less effectively to
other generative models.

###  4.2 Size Constraint

To minimize the detector’s reliance on variations in size, we aimed to control
this variable by selecting natural images within a specific size range for
training, such that the natural images in training are of similar size as
those of the corresponding generator. As this selection drastically reduces
the number of training samples, we only executed the experiment on subsets
with generated images of dimension 512×512, because most natural GenImage
images have both height and width within the range [450, 550]. In order to
obtain sufficient training data, we also utilized the natural training data
from all GenImage training subsets in this interval. Furthermore we still only
used JPEG96 images to mitigate the compression bias. We then sampled the same
number of generated images for each 512×512 generator. To avoid disparities in
content distribution between natural and generated images, we ensured an equal
number of natural and generated images per ImageNet class. This data selection
in total reduced the number of training samples from approximately 320,000 to
75,000. For the evaluation, we utilized the unconstrained validation sets, to
maintain consistency with other experiments. To guarantee that all images
images contain the same amount of resize artifacts during training, we center-
cropped images to 450 - the lower bound of the size interval - before resizing
to 224 the final inputsize of the ResNet50 detectors. During inference, for
images of any size, we first resize the input to 512 to allow cropping to 450
and then resizing.

(a) Accuracy (in %) for size-constrained training on full validation datasets

(b) Difference to training on raw GenImage dataset

Figure 5: Cross-generator perfomance for ResNet50 detectors trained on
compression and size constrained GenImage training subset.

Fig. 5(a) shows the cross-generator performance of detectors trained with this
size constraint, while Fig. 5(b) highlights the difference compared to
detectors trained on the unconstrained, raw dataset. Note that for the
evaluation, images stored as PNG files are compressed using a JPEG quality
factor of 96, ensuring the model is not exposed to uncompressed images, which
it had not encountered during training. The results demonstrate an overall
improvement in cross-generator generalization, with a maximum increase of
41.29 percentage points and an average increase of 11.06 percentage points,
achieving state-of-the-art results in GenImage cross-generator performance. We
note a slight decrease in performance for subsets containing generated images
of the same size as those used in training. This observation supports our
hypothesis, since the original evaluation likely overestimated the
generalization ability for generators of the same size: For instance,
detectors trained on the stable-diffusion subset performed exceptionally well
on the Wukong subset, which shares the same size bias. Mitigating the size
bias leads to a decline in generalization from stable-diffusion to Wukong of
approximately 7 percentage points.

###  4.3 Detector Ablation

To ensure that the constraints not only improve the performance of ResNet50
detectors, we likewise examined our constrained training for a transformer
based detector Swin-T [29]. Table 4 summarizes the average scores: Note that
we outperform the current stat-of-the-art, GenDet [10], which only reports
results of 81.6% for training on SD1.4. When validating our Swin-T detectors,
we achieve an average accuracy of 85.83%. Specifically, the detector trained
on the SD1.4 subset reaches an accuracy of 86.8%. We provide the detailed
cross-generator matrix of Swin-T in Appendix C. The overall improvement in
cross-generator performance is especially impressing when considering that the
selection of training data reduced the number of samples by more than 75%.
Nevertheless, the key insight of our work is not merely enhanced performance
and robustness but rather showing that the named biases significantly affect
the detectors leading to a misjudgment when evaluating.

Table 4: Average cross generator performance for ResNet50 and Swin-T given in accuracy (in %) when trained on raw GenImage subsets and our constrained subsets. Training Subset | ResNet50 | Swin-T
---|---|---
| Classic | Ours | Diff | Classic | Ours | Diff
SD1.5 | 72.16 | 83.90 | +11.74 | 74.14 | 85.90 | +11.76
SD1.4 | 71.27 | 83.39 | +12.12 | 74.93 | 86.80 | +11.87
Wukong | 71.61 | 80.93 | +9.32 | 73.20 | 84.80 | +11.60
Total | 71.68 | 82.74 | +11.06 | 74.09 | 85.83 | +11.74

##  5 Discussion and Conclusion

Our findings demonstrate that datasets used for AI-generated image detection
exhibit biases in compression artifacts and image dimensions. These biases
hinder models from learning the core task of detecting generation specific
artifacts and result in misaligned evaluations. By imposing constraints on the
training dataset to mitigate these biases, we observed a significant shift in
the evaluation of ResNet50 and Swin-T detectors, yielding substantially
improved robustness and generalization. Since we only inspected these baseline
methods, it would be interesting for future works to analyze if we can improve
the performance of detection methods specifically designed for AI-generated
image detection even more or whether they expose as only being able to perform
well on biased datasets. While computational and time limitations prevented
our evaluation of the DIRE method yet, it remains a promising candidate for
further investigation.

Nevertheless, it’s important to highlight that even with these constraints,
datasets may contain other undesirable biases. A notable issue is the
disparity in the source of natural images used for training generative models
compared to those used for detector models, as shown in Table 1. For instance,
in the GenImage dataset, Stable Diffusion generators are trained on images
from LAION, whereas detector models use images from ImageNet. Despite efforts
to align image content through text prompts corresponding to ImageNet classes,
there is a potential risk that detectors might learn to distinguish between
the styles of LAION and ImageNet images. Even more concerning, some generative
models like Stable Diffusion actively fingerprint the generated content [30],
leading to a undesirable distinction from the natural images of the detector
that is clearly not transferable to other generative models.

To create truly unbiased datasets, we propose that the detector models should
be trained on the same natural images as their corresponding generative model.
Additionally, natural and generated images should have near equal
distributions in both compression and image dimensions.

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## Appendix

Appendix A shows detailed insights into size (Fig. 7)and compression (Fig. 6)
distribution in ImageNet images, which are used in GenImage. Appendix B shows
the precision and recall corresponding to the experiment in section 3.1 to
show that compression leads to classifying an image as natural. Appendix C
provides the detailed cross-generator matrix for the Swin-T detector when
trained on the constrained dataset and it’s difference to training on the raw
GenImage dataset.

## Appendix A

Figure 6: Distribution of JPEG quality factor in ImageNet with logarithmic
scale. (Graphic from [27]) Figure 7: Distribution of image size in ImageNet
per interval.

## Appendix B

Figure 8: Precision (in %) corresponding to Fig. 2. This demonstrates that the
classification of natural images is not affected by the compression. Figure 9:
Recall (in %) corresponding to Fig. 2. This demonstrates that compressed AI-
generated images are likely classified as natural

## Appendix C

(a) Accuracy (in %) for constrained training in 4.2. on full validation
datasets

(b) Difference to training on raw GenImage dataset

Figure 10: Cross-generator perfomance for SWIN-T detectors trained on
compression and size constrained GenImage training subset.


# Risk premium and rough volatility

Ofelia Bonesini  Department of Mathematics, Imperial College London
<EMAIL_ADDRESS>,  Antoine Jacquier  Department of Mathematics,
Imperial College London and the Alan Turing Institute
<EMAIL_ADDRESS>and  Aitor Muguruza  Department of Mathematics,
Imperial College London and Kaiju Capital Management  aitor.muguruza-
<EMAIL_ADDRESS>
###### Abstract.

One the one hand, rough volatility has been shown to provide a consistent
framework to capture the properties of stock price dynamics both under the
historical measure and for pricing purposes. On the other hand, market price
of volatility risk is a well-studied object in Financial Economics, and
empirical estimates show it to be stochastic rather than deterministic.
Starting from a rough volatility model under the historical measure, we take
up this challenge and provide an analysis of the impact of such a non-
deterministic risk for pricing purposes.

###### Key words and phrases:

risk premium, fractional Brownian motion, rough volatility

###### 2010 Mathematics Subject Classification:

60F17, 60F05, 60G15, 60G22, 91G20, 91G60, 91B25

OB and AJ are supported by the EPSRC grant EP/T032146/1. For the purpose of
open access, the author(s) has applied a Creative Commons Attribution (CC BY)
licence (where permitted by UKRI, ‘Open Government Licence’ or ‘Creative
Commons Attribution No-derivatives (CC BY-ND) licence’ may be stated instead)
to any Author Accepted Manuscript version arising’

## Introduction

Rough volatility is a recent paradigm proposed by Gatheral, Jaisson and
Rosenbaum [10], which has attracted the attention of many academics and
practitioners thanks to its numerous attractive properties. Despite some
debate about whether volatility should be rough [12, 13, 17, 16, 23], this
class of models provides a general framework to analyse both time series of
the instantaneous volatility (under the historical measure $\mathbb{P}$) and
prices of financial derivatives (under the pricing measure $\mathbb{Q}$).
Starting from a rough version of the Bergomi model [5] under $\mathbb{P}$,
Bayer, Friz and Gatheral [3] showed that a deterministic market price of risk
preserved its structure under $\mathbb{Q}$ (somehow akin to the Heston model
[14] specification).

However, the Financial Economics literature has long shown that this market
price of risk, monitoring the transition from $\mathbb{P}$ to $\mathbb{Q}$ via
Girsanov’s transform, is not constant nor deterministic but instead
stochastic. Its estimation has been the source of long academic discussions,
outside the scope of the present paper though, and we refer the interested
reader to [1, 6, 7, 8, 19, 21] for some useful pointers. This of course has
serious practicals implications for risk management, and [11, 24] are
fascinating sources of information. We focus here on this particular bridge
between $\mathbb{P}$ and $\mathbb{Q}$ and show, not surprisingly, that the
required stochasticity of the market price of risk unfortunately breaks the
structure of the rough Bergomi model under $\mathbb{Q}$. However, we link the
Hölder regularity of the volatility process (lower in this class of rough
models) with that of the change of measure, and design several specifications
making the model tractable under $\mathbb{Q}$. While the rough Bergomi model
tracks well the behaviour of the historical volatility, it is less powerful
for option prices, especially when considering VIX smiles (which are more or
less flat under this model). Our new setup allows for more flexibility there,
while preserving the $\mathbb{P}$-tractability of the model.

Section 1 provides the technical setup and analysis of the market price of
risk, while the design of useful continuous-time rough stochastic volatility
models with non-deterministic market prices of risk is detailed in Section 2.
Finally, in Section 3, we perform an empirical analysis, estimating risk
premia from historical options data.

##  1\. Rough volatility models and change of measure

Rough volatility models are a natural extension of classical stochastic
volatility models. Starting from such a model under the historical measure
$\mathbb{P}$, we characterise below its dynamics under equivalent martingale
measures $\mathbb{Q}\sim\mathbb{P}$, which then, by the fundamental theorem of
asset pricing, allows for arbitrage-free option pricing. Following for example
[4, 10] we consider a rather general class of (rough) stochastic volatility
model under $\mathbb{P}$, where the stock price process admits the following
dynamics:

(1.1)
$\left\\{\begin{array}[]{rl}\displaystyle\frac{\mathrm{d}S_{t}}{S_{t}}&=\displaystyle\mu_{t}\mathrm{d}t+\sqrt{v_{t}}\,\mathrm{d}W^{\mathbb{P}}_{t},\\\
v_{t}&=\displaystyle\psi(t,Y_{t}),\\\
Y_{t}&=\displaystyle\int_{0}^{t}k(t,s)\mathrm{d}Z^{\mathbb{P}}_{s},\end{array}\right.$

starting from $S_{0}>0$, over a fixed time interval $\mathbb{T}:=[0,T]$, for
$T>0$. Here
$\boldsymbol{W}^{\mathbb{P}}=(W^{\mathbb{P}},W^{\mathbb{P},\perp})$ is a two-
dimensional standard Brownian motion defined on a given filtered probability
space $(\Omega,\mathcal{F},\mathbb{P})$, with
$\mathcal{F}=\mathcal{F}^{W^{\mathbb{P}}}\vee\mathcal{F}^{W^{\mathbb{P},\perp}}$,
and $Z^{\mathbb{P}}:=\rho W^{\mathbb{P}}+\overline{\rho}W^{\mathbb{P},\perp}$,
with $\rho\in[-1,1]$ and $\overline{\rho}:=\sqrt{1-\rho^{2}}$. We further
introduce the set $\mathbb{F}_{b}$ of $\mathbb{P}$-bounded and
$(\mathcal{F}_{t})_{t\in\mathbb{T}}$-progressively measurable process and
recall the Doléans-Dade stochastic exponential of a square integrable process
$X$:

$\mathcal{E}(X)_{t}:=\exp{\left(X_{t}-\frac{1}{2}\langle
X\rangle_{t}\right)},\qquad t\in\mathbb{T}.$

We finally consider the following set of assumptions, in place for the rest of
the paper:

######  Assumption 1.1.

  1. (i)

The function $\psi:\mathbb{T}\times\mathbb{R}\to(0,\infty)$ is continuous,
bounded, and bounded away from the origin on $\mathbb{T}\times(-\infty,a]$ for
each $a>0$;

  2. (ii)

For each $t\in\mathbb{T}$, $\mathbb{E}\left[v_{t}^{-1}\right]$ is finite;

  3. (iii)

The process $\mu$ belongs to $\mathbb{F}_{b}$;

  4. (iv)

Given an interest rate process $r:=(r_{t})_{t\in\mathbb{T}}\in\mathbb{F}_{b}$,
there exist a sequence $(\widehat{\mathbb{P}}_{n})_{n\in\mathbb{N}}$ and a
process $\gamma:=(\gamma_{t})_{t\in\mathbb{T}}$ in $\mathbb{F}_{b}$ and
bounded $\widehat{\mathbb{P}}_{n}$-almost surely, for each $n\in\mathbb{N}$,
such that

$\sup_{t\in\mathbb{T}}\left\\{\rho\int_{0}^{t}k(t,u)\chi_{u}\mathrm{d}u\right\\}\leq
0,\qquad\widehat{\mathbb{P}}_{n}\text{-almost surely},\text{ for all
}n\in\mathbb{N},$

where we introduce the Sharpe ratio
$\chi_{u}:=\frac{r_{u}-\mu_{u}}{\sqrt{v_{u}}}$ and

(1.2)
$\frac{\mathrm{d}\widehat{\mathbb{P}}_{n}}{\mathrm{d}\mathbb{P}}\bigg{|}_{\mathcal{F}_{t}}:=\mathcal{E}\left(\int_{0}^{\cdot}\chi_{u}\mathrm{d}W^{\mathbb{P}}_{u}+\int_{0}^{\cdot}\gamma_{u}\mathrm{d}W^{\mathbb{P},{\perp}}_{u}\right)_{t\wedge\tau_{n}}\quad\text{and}\quad\tau_{n}:=\inf\\{t\geq
0,\;Y_{t}=n\\};$

  5. (v)

The correlation is negative: $\rho\leq 0$;

  6. (vi)

For each $t\in\mathbb{T}$, the kernel $k(t,\cdot)$ is null on
$\mathbb{T}\setminus[0,t]$ and
$\int_{0}^{\cdot}k(\cdot,s)\mathrm{d}Z^{\mathbb{P}}_{s}$ is a well-defined
Gaussian process.

######  Remark 1.2.

Condition (vi) may be replaced in terms of conditions on the kernel, for
example $k(t,\cdot)\in L^{2}([0,t])$ for each $t\in\mathbb{T}$. In light of
(v), the first constraint in (iv) may be rewritten as
$\inf_{t\in\mathbb{T}}\left\\{\int_{0}^{t}k(t,u)\chi_{u}\mathrm{d}u\right\\}\geq
0$, $\widehat{\mathbb{P}}_{n}\text{-almost surely}$, for all $n\in\mathbb{N}$.

The following examples are common choices of such kernels:

######  Example 1.3.

The Gamma kernel, common in the $\mathcal{BSS}$ literature pioneered by
Barndorff-Nielsen and Schmiegel [2], is given by

(1.3) $k(t,s)=(t-s)^{H-\frac{1}{2}}\mathrm{e}^{-\beta(t-s)}1{1}_{\\{t\geq
s\\}},\qquad\text{with }H\in(0,1),\quad\beta\geq 0.$

In order to state the main result, define the Radon-Nikodym derivative, for
each $t\in\mathbb{T}$,

(1.4)
$\mathcal{D}^{\gamma}_{t}:=\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}\bigg{|}_{\mathcal{F}_{t}^{\mathbb{P}}}=\mathcal{E}\left(\int_{0}^{\cdot}\chi_{u}\mathrm{d}W^{\mathbb{P}}_{u}+\int_{0}^{\cdot}\gamma_{u}\mathrm{d}W^{\mathbb{P},{\perp}}_{u}\right)_{t},$

so that, from (1.2) above,
$\mathcal{D}^{\gamma}_{t\wedge\tau_{n}}=\frac{\mathrm{d}\widehat{\mathbb{P}}_{n}}{\mathrm{d}\mathbb{P}}\bigg{|}_{\mathcal{F}_{t}}$.

######  Proposition 1.4.

For any $\gamma$ satisfying Assumption 1.1(iv), the process
$\displaystyle\int_{0}^{\cdot}\chi_{u}\mathrm{d}W^{\mathbb{P}}_{u}+\int_{0}^{\cdot}\gamma_{u}\mathrm{d}W^{\mathbb{P},{\perp}}_{u}$
is a $\mathbb{P}$-local martingale on $\mathbb{T}$.

###### Proof.

Recall that the sum of two local martingale (with respect to the same
filtration) is a local martingale whose sequence of stopping times is given by
the minimum of the sequences of stopping times for the terms in the sum. The
boundedness of $\gamma$ implies that
$\int_{0}^{\cdot}\gamma_{s}\mathrm{d}W^{\mathbb{P},{\perp}}_{s}$ is a true
$\mathbb{P}$-martingale. Regarding
$\int_{0}^{\cdot}\chi_{u}\mathrm{d}W^{\mathbb{P}}_{u}$, the integrand is
$\mathcal{F}_{s}$-measurable and locally bounded with respect to the sequence
of stopping times $(\tau_{n})_{n\in\mathbb{N}}$ in (1.2) thanks to Assumption
1.1(i)-(iii)-(iv). Exploiting Assumption 1.1(ii)-(iii)-(iv), then
$\mathbb{E}^{\mathbb{P}}[\chi_{t}^{2}]$ is finite for all $t\in\mathbb{T}$.
Thus, the first term in the sum is a local martingale as well, and so is the
whole sum. ∎

Proposition 1.4 justifies the use of a Doléans-Dade stochastic exponential in
the definition of $\mathcal{D}^{\gamma}$. Furthermore, we obtain the following
result.

######  Theorem 1.5.

Under Assumption 1.1, for any $\gamma$ as in Assumption 1.1(iv), then

  1. (I)

the Radon-Nikodym derivative process $\mathcal{D}^{\gamma}$ in (1.4) is a true
$\mathbb{Q}$-martingale;

  2. (II)

under the (arbitrage-free) equivalent risk-neutral martingale measure
$\mathbb{Q}$,

(1.5)
$\left\\{\begin{array}[]{rl}\displaystyle\frac{\mathrm{d}S_{t}}{S_{t}}&=\displaystyle
r_{t}\mathrm{d}t+\sqrt{v_{t}}\,\mathrm{d}W^{\mathbb{Q}}_{t},\\\
v_{t}&=\displaystyle\psi\left(t,\widehat{Y}_{t}+\int_{0}^{t}k(t,s)\lambda_{s}\mathrm{d}s\right),\\\
\widehat{Y_{t}}&=\displaystyle\int_{0}^{t}k(t,s)\mathrm{d}Z^{\mathbb{Q}}_{s},\end{array}\right.$

with $S_{0},v_{0}>0$, and $\lambda$ is the market price of volatility risk
defined by

(1.6) $\lambda_{t}:=\rho\chi_{t}+\overline{\rho}\;\gamma_{t},$

and where $W^{\mathbb{Q}}$ and $Z^{\mathbb{Q}}$ are $\mathbb{Q}$-Brownian
motions defined as

(1.7) $W^{\mathbb{Q}}_{t}:=\displaystyle
W^{\mathbb{P}}_{t}+\int_{0}^{t}\chi_{u}\mathrm{d}u,\qquad\text{and}\qquad
Z^{\mathbb{Q}}_{t}:=Z_{t}^{\mathbb{P}}+\int_{0}^{t}\lambda_{u}\mathrm{d}u;$

  3. (III)

the discounted stock price $\widetilde{S}_{t}:=\frac{S_{t}}{B_{t}}$ with
$\mathrm{d}B_{t}=r_{t}B_{t}\mathrm{d}t$, $B_{0}=1$, is a true
$\mathbb{Q}$-martingale.

###### Proof.

To satisfy the no-arbitrage conditions, the change of measure for
$W^{\mathbb{P}}$ is constrained by the martingale restriction on the
discounted spot dynamics, while the Brownian motion $Z^{\mathbb{P}}$ gives
freedom to the model and makes the market incomplete by the free choice of the
process $\gamma$. Consequently, the change of measure from $\mathbb{P}$ to
$\mathbb{Q}$ and the corresponding Radon-Nikodym derivative directly follow
from Girsanov’s Theorem via (1.4), provided that $\mathcal{D}^{\gamma}_{t}\in
L^{1}$ and $\mathcal{D}^{\gamma}$ is a true martingale. Thus, once we have
shown (I), then (II) automatically follows. By Proposition 1.4,
$\mathcal{D}^{\gamma}_{t}\in L^{1}$ and, being a non-negative local
martingale, it is a supermartingale, and a true martingale on $\mathbb{T}$ if
and only if $\mathbb{E}\left[\mathcal{D}^{\gamma}_{T}\right]=1$. To prove
this, we closely follow [9, Proof of Theorem 1.1] with some modifications, and
recall the stopping time $\tau_{n}:=\inf\\{t\geq 0,\;Y_{t}=n\\}$ from (1.2).
For any $s\in\mathbb{T}$, the random function
$f(x):=\frac{r_{s}-\mu_{s}}{\sqrt{\psi(s,x)}}$ is $\mathbb{P}$-bounded on
$(-\infty,a]$ for any $a>0$ since $r$ and $\mu$ are $\mathbb{P}$-bounded, and
$\psi(s,\cdot)$ bounded away from zero on intervals of the form $(-\infty,a]$,
by Assumption 1.1(iii)-(iv)-(i), respectively. Then, again by Proposition 1.4,

(1.8)
$1=\mathbb{E}\left[\mathcal{D}^{\gamma}_{T\wedge\tau_{n}}\right]=\mathbb{E}\left[\mathcal{D}^{\gamma}_{T}1{1}_{\\{T<\tau_{n}\\}}\right]+\mathbb{E}\left[\mathcal{D}^{\gamma}_{\tau_{n}}1{1}_{\\{\tau_{n}\leq
T\\}}\right].$

The first term in (1.8) converges to
$\mathbb{E}\left[\mathcal{D}^{\gamma}_{T}\right]$ as $n$ tends to infinity,
yielding

$1-\mathbb{E}\left[\mathcal{D}^{\gamma}_{T}\right]=\lim_{n\uparrow\infty}\mathbb{E}\left[\mathcal{D}^{\gamma}_{\tau_{n}}1{1}_{\\{\tau_{n}\leq
T\\}}\right].$

Girsanov’s theorem implies
$\mathbb{E}\left[\mathcal{D}^{\gamma}_{\tau_{n}}1{1}_{\\{\tau_{n}\leq
T\\}}\right]=\widehat{\mathbb{P}}_{n}(\tau_{n}\leq T)$, where
$\widehat{\mathbb{P}}_{n}$ is defined such that

$\widehat{W}^{n}_{t}=\displaystyle
W^{\mathbb{P}}_{t}-\int_{0}^{t\wedge\tau_{n}}\chi_{u}\mathrm{d}u$

is a $\widehat{\mathbb{P}}_{n}$-Brownian motion. Then, under
$\widehat{\mathbb{P}}_{n}$, the process $Y$ becomes

$Y_{t}=\widehat{Y}_{t}^{n}+\int_{0}^{t\wedge\tau_{n}}k(t,s)\Big{(}\rho\chi_{u}+\overline{\rho}\;\gamma_{u}\Big{)}\mathrm{d}u,$

where
$\displaystyle\widehat{Y}^{n}_{t}:=\int_{0}^{t}k(t,s)\mathrm{d}\widehat{Z}^{n}_{s}$
and

$\widehat{Z}^{n}_{t}=Z^{\mathbb{P}}_{t}-\int_{0}^{t\wedge\tau_{n}}\Big{(}\rho\chi_{u}+\overline{\rho}\;\gamma_{u}\Big{)}\mathrm{d}u,$

for $t\geq 0$, where $\widehat{Z}^{n}$ is a
$\widehat{\mathbb{P}}_{n}$-Brownian motion. Furthermore, by Assumption
1.1(iv), there exists $M_{\gamma}>0$ such that $|\gamma_{t}|<M_{\gamma}$
$\widehat{\mathbb{P}}_{n}$-almost surely, and

$\displaystyle\widehat{\mathbb{P}}_{n}\left(\sup_{t\in\mathbb{T}}Y_{t}\geq
n\right)$
$\displaystyle=\widehat{\mathbb{P}}_{n}\left(\sup_{t\in\mathbb{T}}\left\\{\widehat{Y}_{t}^{n}+\int_{0}^{t\wedge\tau_{n}}k(t,u)\Big{(}\rho\chi_{u}+\overline{\rho}\;\gamma_{u}\Big{)}\mathrm{d}s\right\\}\geq
n\right)$
$\displaystyle\leq\widehat{\mathbb{P}}_{n}\left(\sup_{t\in\mathbb{T}}\widehat{Y}_{t}^{n}+\sup_{t\in\mathbb{T}}\left\\{\rho\int_{0}^{t\wedge\tau_{n}}k(t,u)\chi_{u}\mathrm{d}u\right\\}+\sup_{t\in\mathbb{T}}\left\\{\overline{\rho}\int_{0}^{t\wedge\tau_{n}}k(t,u)\gamma_{u}\mathrm{d}u\right\\}\geq
n\right)$
$\displaystyle\leq\widehat{\mathbb{P}}_{n}\left(\sup_{t\in\mathbb{T}}\widehat{Y}_{t}^{n}+\sup_{t\in\mathbb{T}}\left\\{\rho\int_{0}^{t\wedge\tau_{n}}k(t,u)\chi_{u}\mathrm{d}u\right\\}\geq
n-|\overline{\rho}|K_{\mathbb{T}}M_{\gamma}\right),$

where $K_{\mathbb{T}}:=\sup_{t\in\mathbb{T}}\int_{0}^{t}k(t,s)\mathrm{d}s$.
Then, an application of Assumption 1.1(iv), we obtain

(1.9) $\widehat{\mathbb{P}}_{n}\left(\sup_{t\in\mathbb{T}}Y_{t}\geq
n\right)\leq\widehat{\mathbb{P}}_{n}\left(\sup_{t\in\mathbb{T}}\widehat{Y}_{t}^{n}\geq
n-|\overline{\rho}|K_{\mathbb{T}}M_{\gamma}\right).$

Inequality (1.9), in turn, implies $\widehat{\mathbb{P}}_{n}(\tau_{n}\leq
T)\leq\widehat{\mathbb{P}}_{n}(\widehat{\tau}_{n}\leq T)$, for
$\widehat{\tau}_{n}:=\inf\\{t\geq
0,\;\widehat{Y}_{t}^{n}=n-|\overline{\rho}|K_{\mathbb{T}}M_{\gamma}\\}$.
Finally, since $\widehat{Z}^{n}$ is a $\widehat{\mathbb{P}}_{n}$-Brownian
motion, we obtain

$\lim_{n\uparrow\infty}\widehat{\mathbb{P}}_{n}(\tau_{n}\leq
T)\leq\lim_{n\uparrow\infty}\widehat{\mathbb{P}}_{n}(\widehat{\tau}_{n}\leq
T)=\lim_{n\uparrow\infty}\mathbb{P}\left(\sup_{t\in\mathbb{T}}Y_{t}\geq
n-|\overline{\rho}|K_{\mathbb{T}}M_{\gamma}\right)=0,$

and it follows that
$\displaystyle\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}$ is indeed a
true martingale and note that
$\displaystyle\lim_{n\uparrow\infty}\widehat{\mathbb{P}}_{n}=\mathbb{Q}$. In
the sense that the relation (1.7) holds between the $\mathbb{P}$ and
$\mathbb{Q}$ Brownian motions. This concludes the proof of (I) and, as stated
in the beginning, the results in (II) then readily follow by direct
application of the change of measure theorem.

We now prove (III): the discounted price $\widetilde{S}=S/B$ is a true
martingale for $\rho\leq 0$. A straightforward application of Itô’s formula
yields under $\mathbb{Q}$

(1.10)
$\left\\{\begin{array}[]{rl}\displaystyle\frac{\mathrm{d}\widetilde{S}_{t}}{\widetilde{S}_{t}}&=\sqrt{v_{t}}\,\mathrm{d}W^{\mathbb{Q}}_{t},\\\
v_{t}&=\displaystyle\psi\left(t,\widehat{Y}_{t}+\int_{0}^{t}k(t,s)\lambda_{s}\mathrm{d}s\right),\\\
\widehat{Y_{t}}&=\displaystyle\int_{0}^{t}k(t,s)\mathrm{d}Z^{\mathbb{Q}}_{s}\end{array}\right.$

Define the stopping time $\iota_{n}:=\inf\\{t\geq 0,\;\widehat{Y}_{t}=n\\}$.
For any $t\in\mathbb{T}$, the random function
$g(x):=\psi\left(t,x+\int_{0}^{t}k(t,s)\lambda_{s}\mathrm{d}s\right)$ is
bounded $\mathbb{Q}$-almost surely on $(-\infty,a]$ by Assumption
1.1(i)-(iii)-(iv), with $\lambda$ in (1.6), and by boundedness of $\gamma$, so
that, since $\widetilde{S}$ is a $\mathbb{Q}$-local martingale:

(1.11)
$\displaystyle\widetilde{S}_{0}=\mathbb{E}^{\mathbb{Q}}[\widetilde{S}_{T\wedge\iota_{n}}]=\mathbb{E}^{\mathbb{Q}}[\widetilde{S}_{T}1{1}_{\\{T<\iota_{n}\\}}]+\mathbb{E}^{\mathbb{Q}}[\widetilde{S}_{\iota_{n}}1{1}_{\\{T>\iota_{n}\\}}].$

The first term converges to $\mathbb{E}^{\mathbb{Q}}[\widetilde{S}_{T}]$ as
$n$ tends to infinity, hence

(1.12)
$\displaystyle\widetilde{S}_{0}-\mathbb{E}^{\mathbb{Q}}[\widetilde{S}_{T}]=\lim_{n\uparrow\infty}\mathbb{E}^{\mathbb{Q}}[\widetilde{S}_{\iota_{n}}1{1}_{\\{T>\iota_{n}\\}}].$

Girsanov’s Theorem further gives
$\mathbb{E}^{\mathbb{Q}}[\widetilde{S}_{T}1{1}_{\\{T>\iota_{n}\\}}]=\widetilde{S}_{0}\widetilde{\mathbb{P}}_{n}(T>\iota_{n})$,
where $\widetilde{\mathbb{P}}_{n}$ is such that

(1.13)
$\displaystyle\widetilde{W}^{n}_{t}=W^{\mathbb{Q}}_{t}-\int_{0}^{t\wedge\iota_{n}}v_{s}\mathrm{d}s$

is a $\widetilde{\mathbb{P}}_{n}$-Brownian motion. Note that, for
$t<\iota_{n}$,
$\widehat{Y}_{t}=\widetilde{Y}_{t}+\rho\int_{0}^{t}k(t,s)v_{s}\mathrm{d}s$,
where $\widetilde{Y}_{t}=\int_{0}^{t}k(t,s)\mathrm{d}\widetilde{Z}^{n}_{s}$
and
$\widetilde{Z}^{n}_{t}:=Z^{\mathbb{Q}}_{t}-\rho\int_{0}^{t}k(t,s)v_{s}\mathrm{d}s$
is also a $\widetilde{\mathbb{P}}_{n}$-Brownian motion. We conclude that, if
$\rho\leq 0$, then $\widehat{Y}_{t}\geq\widetilde{Y}_{t}$ and

$\lim_{n\uparrow\infty}\widetilde{\mathbb{P}}_{n}(\iota_{n}\leq
T)\leq\lim_{n\uparrow\infty}\widetilde{\mathbb{P}}_{n}(\widetilde{\iota}_{n}\leq
T)=\lim_{n\uparrow\infty}\mathbb{P}\left(\sup_{t\in[0,T]}Y_{t}\leq
n\right)=0,$

where $\widetilde{\iota}_{n}:=\inf\\{t\geq 0,\;\widetilde{Y}_{t}=n\\}$, and
hence $\widetilde{S}$ is a true martingale. ∎

### Discussion

Under Assumption 1.1, consider $\rho\leq 0$, some valid function $\psi$ and
kernel $k$, and the constant values $\gamma_{s}=\overline{\gamma}$ and
$\overline{\mu}=\mu_{s}\leq r_{s}=\overline{r}$ for
$\overline{\gamma},\overline{\mu},\overline{r}\in\mathbb{R}$ ensuring
Assumption 1.1(iv)., so that Theorem 1.5 applies. In this scenario, a
sufficient condition for the change of measure to be well defined is that the
physical drift must be smaller than the risk-free rate.

###  1.1. Characterisation of rough volatility models via Generalised
Fractional Operators

It is natural to represent rough volatility models in terms of fractional
operators. In this section we present the Generalised Fractional Operators
(GFO) and a representation result for rough volatility models in terms of GFO.
To do so, let us first present the GFO introduced in [15, Definition 1.1]
defined as follows.

######  Definition 1.6.

For any $\beta\in(0,1)$, $\alpha\in(-\beta,1-\beta)$ and
$h\in\mathcal{C}^{1}_{b}((0,\infty))$ such that $h^{\prime}(\cdot)\leq 0$, the
GFO associated to the kernel $k(x):=x^{\alpha}h(x)$ applied to $f\in
C^{\beta}(\mathbb{R})$ is defined as

(1.14)
$(\mathcal{G}^{\alpha}f)(t):=\left\\{\begin{array}[]{ll}\displaystyle\int_{0}^{t}(f(s)-f(0))\frac{\mathrm{d}}{\mathrm{d}t}k(t-s)\mathrm{d}s,&\text{if
}\alpha\in[0,1-\beta),\\\
\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{0}^{t}(f(s)-f(0))k(t-s)\mathrm{d}s,&\text{if
}\alpha\in(-\beta,0).\end{array}\right.$

To simplify future notations, we let $H_{\pm}:=H\pm\frac{1}{2}$ for
$H\in(0,\frac{1}{2})$. We now introduce a specific setup that will drive the
rest of our computations: consider the power-law kernel

(1.15) $\mathtt{k}_{\alpha}(u):=u^{\alpha}1{1}_{\\{u\geq 0\\}},$

as well as the set

(1.16) $\Lambda_{\beta,H}:=\Big{\\{}\lambda\in\mathcal{C}^{\beta}\text{ for
some $\beta\in(0,1]$ such that $H_{-}\in(-\beta,0)$ and
$\lambda_{0}=0$}\Big{\\}}.$

To this particular power-law kernel, the GFO (from Definition 1.6, since
$H_{-}\in(-\frac{1}{2},0)$) reads

$(\mathcal{G}^{H_{-}}f)(t)=\frac{\mathrm{d}}{\mathrm{d}t}\int_{0}^{t}(f(s)-f(0))\mathtt{k}_{H_{-}}(t-s)\mathrm{d}s.$

Denote further

$\mathtt{K}(t,s):=\int_{0}^{t}\mathtt{k}_{H_{-}}(u-s)\mathrm{d}u=\frac{\mathtt{k}_{H_{+}}(t-s)}{H_{+}},$

so that the corresponding GFO is precisely
$\frac{1}{H_{+}}\mathcal{G}^{H_{+}}$. To streamline notations and emphasise
nice symmetries, we introduce the notations

(1.17)
$\mathfrak{G}^{-}:=\mathcal{G}^{H_{-}}\qquad\text{and}\qquad\mathfrak{G}^{+}:=\frac{1}{H_{+}}\mathcal{G}^{H_{+}}.$

From the properties of GFO [15, Proposition 1.2], then
$\mathfrak{G}^{+}\lambda\in\mathcal{C}^{\beta+H_{+}}$ as soon as
$\lambda\in\Lambda_{\beta,H}$.

######  Corollary 1.7 (GFO representation of rough volatility).

With the kernel $\mathtt{k}_{H_{-}}$ in (1.15) and
$\lambda\in\Lambda_{\beta,H}$, the system (1.5) under the risk-neutral measure
$\mathbb{Q}$ can be rewritten as

$\left\\{\begin{array}[]{rl}\displaystyle\frac{\mathrm{d}S_{t}}{S_{t}}&=\displaystyle
r_{t}\mathrm{d}t+\sqrt{v_{t}}\mathrm{d}W^{\mathbb{Q}}_{t},\\\
v_{t}&=\displaystyle\psi\Big{(}t,(\mathfrak{G}^{-}Z^{\mathbb{Q}})(t)+(\mathfrak{G}^{+}\lambda)(t)\Big{)}.\end{array}\right.$

###### Proof.

The fact that
$\displaystyle\int_{0}^{\cdot}\mathtt{k}_{H_{-}}(\cdot-s)\mathrm{d}Z^{\mathbb{Q}}_{s}=\mathfrak{G}^{-}Z^{\mathbb{Q}}$
is straightforward by the properties of GFO in [15, Proposition 1.4].
Furthermore, for any $\lambda\in\Lambda_{\beta,H}$ and any $t\in\mathbb{T}$,

$\int_{0}^{t}\mathtt{k}_{H_{-}}(t-s)\lambda_{s}\mathrm{d}s=\int_{0}^{t}\frac{\mathrm{d}}{\mathrm{d}t}\mathtt{K}(t,s)(\lambda_{s}-\lambda_{0})\mathrm{d}s=\left(\mathfrak{G}^{+}\lambda\right)(t).$

∎

Note that since $\mathfrak{G}^{+}\lambda\in\mathcal{C}^{\beta+H_{+}}$, then
the risk premium has sample paths with Hölder regularity greater than
$\frac{1}{2}$, regardless of the value of $H$.

##  2\. Modelling the risk premium process: A practical approach

In practice, the process $\lambda$ is directly modelled without resorting to a
change of measure starting from $\gamma$. We now consider different modelling
choices for the risk premium $\lambda$ and analyse some of its properties. In
spite of the formal derivation of Theorem 1.5, a numerical treatment of the
integral $\int_{0}^{t}\bullet\;\mathrm{d}s$ is rather intricate. To overcome
this issue, Bayer, Friz and Gatheral [3] elegantly came up with the forward
variance form of rough volatility in the spirit of Bergomi [5]. We shall
restrict ourselves to this functional form (defined in the following
proposition) for the reminder of the section. Consider (1.5) with
$\psi(t,x)=\xi_{0}(t)\mathrm{e}^{\nu x}$, with
$\xi_{0}(t):=\mathbb{E}[v_{t}|\mathcal{F}_{0}]$ and $\nu>0$. Then the risk-
neutral dynamics in forward variance form read

(2.1)
$\left\\{\begin{array}[]{rl}\displaystyle\frac{\mathrm{d}S_{t}}{S_{t}}&=\displaystyle
r_{t}\mathrm{d}t+\sqrt{v_{t}}\,\mathrm{d}W^{\mathbb{Q}}_{t},\\\
v_{t}&=\displaystyle\xi_{0}(t)\exp\left(\nu\left(\int_{0}^{t}\mathtt{k}_{H_{-}}(t-s)\mathrm{d}Z^{\mathbb{Q}}_{s}+\int_{0}^{t}\mathtt{k}_{H_{-}}(t-s)\lambda_{s}\mathrm{d}s\right)\right).\end{array}\right.$

In the remaining of this section, the process $X^{\mathbb{Q}}$ will denote a
$\mathbb{Q}$-Brownian motion possibly correlated with $W^{\mathbb{Q}}$ and
$Z^{\mathbb{Q}}$.

###  2.1. Risk premium driven by Itô diffusion

Generalised Fractional Operators provide a natural framework to model risk
premium processes driven by diffusions. The statement below shows the details
of such a construction. Recall that the Beta function is defined as
$\mathfrak{B}(x,y):=\int_{0}^{1}s^{x-1}(1-s)^{y-1}\mathrm{d}s$, for $x,y>0$.

######  Proposition 2.1.

For $H\in(0,\frac{1}{2})$ and $\alpha\in\left(-\frac{1}{2},0\right)$, consider
$\lambda=\mathfrak{b}\mathcal{G}^{\alpha}Y^{\mathbb{Q}}\in\mathcal{C}^{\alpha+\frac{1}{2}}$,
where $\mathfrak{b}:=\mathfrak{B}(H_{+},\alpha+1)^{-1}$ and

$Y^{\mathbb{Q}}_{t}=\int_{0}^{t}b(s,Y^{\mathbb{Q}}_{s})\mathrm{d}s+\int_{0}^{t}\sigma(s,Y^{\mathbb{Q}}_{s})\mathrm{d}X^{\mathbb{Q}}_{s},$

where $b(\cdot)$ and $\sigma(\cdot,\cdot)$ satisfy Yamada-Watanabe conditions
[20, Section 5.2, Proposition 2.13] for pathwise uniqueness ensuring that a
weak solution exists. Then
$\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}}\in\mathcal{C}^{H+\alpha+1}$ and

(2.2)
$v_{t}=\xi_{0}(t)\exp\Big{\\{}\nu\left((\mathfrak{G}^{-}Z^{\mathbb{Q}})(t)+(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}})(t)\right)\Big{\\}}.$

Furthermore, if $Y^{\mathbb{Q}}=X^{\mathbb{Q}}$ and $\mathrm{d}\langle
Y^{\mathbb{Q}},Z^{\mathbb{Q}}\rangle_{t}=\rho\,\mathrm{d}t$ with $\rho\leq 0$,
then

(2.3) $\displaystyle\mathbb{E}^{\mathbb{Q}}\left[v_{t}|\mathcal{F}_{s}\right]$
$\displaystyle=\xi_{0}(t)\exp\Big{\\{}\nu\left[\left(\mathfrak{G}^{-}Z^{\mathbb{Q}}\right)(s,t)+\left(\mathcal{G}^{\alpha+H_{+}}X^{\mathbb{Q}}\right)(s,t)\right]\Big{\\}}$
(2.4)
$\displaystyle\times\exp\left\\{\frac{\nu^{2}}{2}\left(\frac{\mathtt{k}_{2H}(t-s)}{2H}+\frac{\mathtt{k}_{2(H+1)}(t-s)}{2H_{+}^{2}(H+1)}+\rho\frac{\mathtt{k}_{2H_{+}}(t-s)}{H_{+}^{2}}\right)\right\\},$

where
$\left(\mathcal{G}^{H_{-}}Z^{\mathbb{Q}}\right)(s,t):=\int_{0}^{s}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}$
for $0\leq s\leq t$, and similarly for
$\left(\mathcal{G}^{\alpha+H_{+}}X^{\mathbb{Q}}\right)(s,t)$.

###### Proof.

We first prove (2.2), which follows from [15, Proposition 1.2] and the
identities highlighted above in Corollary 1.7. Indeed, in view of Corollary
1.7 we only need to show that

$\int_{0}^{t}\mathtt{k}_{H_{-}}(t-s)\lambda_{s}\mathrm{d}s=(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}})(t).$

Replacing the expression for $\lambda$ in the integral and using stochastic
Fubini theorem, we obtain

$\displaystyle\int_{0}^{t}\mathtt{k}_{H_{-}}(t-s)\lambda_{s}\mathrm{d}s=\mathfrak{b}\int_{0}^{t}\mathtt{k}_{H_{-}}(t-s)(\mathcal{G}^{\alpha}Y^{\mathbb{Q}})(s)\mathrm{d}s$
$\displaystyle=\mathfrak{b}\int_{0}^{t}\mathtt{k}_{H_{-}}(t-s)\int_{0}^{s}\mathtt{k}_{\alpha}(s-u)\mathrm{d}Y^{\mathbb{Q}}_{u}\,\mathrm{d}s$
$\displaystyle=\mathfrak{b}\int_{0}^{t}\int_{u}^{t}\mathtt{k}_{H_{-}}(t-s)\mathtt{k}_{\alpha}(s-u)\mathrm{d}s\,\mathrm{d}Y^{\mathbb{Q}}_{u}.$

Now, direct computations for the inner integral yields

$\int_{u}^{t}\mathtt{k}_{H_{-}}(t-s)\mathtt{k}_{\alpha}(s-u)\mathrm{d}s=\mathtt{k}_{\alpha+H_{+}}(t-u)\int_{0}^{1}(1-s)^{H_{-}}s^{\alpha}\mathrm{d}s=\mathfrak{B}(\alpha+1,H_{+})\mathtt{k}_{\alpha+H_{+}}(t-u).$

Therefore

$\int_{0}^{t}\mathtt{k}_{H_{-}}(t-s)\lambda_{s}\mathrm{d}s=\mathfrak{b}\int_{0}^{t}\mathfrak{B}(\alpha+1,H_{+})\mathtt{k}_{\alpha+H_{+}}(t-u)\mathrm{d}Y^{\mathbb{Q}}_{u}=\int_{0}^{t}\mathtt{k}_{\alpha+H_{+}}(t-u)\mathrm{d}Y^{\mathbb{Q}}_{u}=(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}})(t).$

We now move to the proof of the identity (2.3). Exploiting the representation
of $v_{t}$ in this specific case and the measurability and independence
properties of the Brownian increments,

$\displaystyle\mathbb{E}^{\mathbb{Q}}\left[v_{t}|\mathcal{F}_{s}\right]$
$\displaystyle=\xi_{0}(t)\mathbb{E}^{\mathbb{Q}}_{s}\left[\exp\left\\{\nu\left[\left(\mathcal{G}^{H_{-}}Z^{\mathbb{Q}}\right)(t)+\left(\mathcal{G}^{\alpha+H_{+}}X^{\mathbb{Q}}\right)(t)\right]\right\\}\right]$
$\displaystyle=\xi_{0}(t)\mathbb{E}^{\mathbb{Q}}_{s}\left[\exp\left\\{\nu\left[\int_{0}^{t}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}+\int_{0}^{t}\mathtt{k}_{\alpha+H_{+}}(t-u)\mathrm{d}X^{\mathbb{Q}}_{u}\right]\right\\}\right]$
$\displaystyle=\xi_{0}(t)\exp\left\\{\nu\left[\int_{0}^{s}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}+\int_{0}^{s}\mathtt{k}_{\alpha+H_{+}}(t-u)\mathrm{d}X^{\mathbb{Q}}_{u}\right]\right\\}$
$\displaystyle\qquad\times\mathbb{E}^{\mathbb{Q}}_{s}\left[\exp\left\\{\nu\left[\int_{s}^{t}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}+\int_{s}^{t}\mathtt{k}_{\alpha+H_{+}}(t-u)\mathrm{d}X^{\mathbb{Q}}_{u}\right]\right\\}\right]$
$\displaystyle=\xi_{0}(t)\exp\left\\{\nu\left[\int_{0}^{s}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}+\int_{0}^{s}\mathtt{k}_{\alpha+H_{+}}(t-u)\mathrm{d}X^{\mathbb{Q}}_{u}\right]\right\\}$
$\displaystyle\qquad\times\exp\left\\{\frac{\nu^{2}}{2}\left(\int_{s}^{t}\mathtt{k}_{2H_{-}}(t-u)\mathrm{d}u+\int_{s}^{t}\frac{\mathtt{k}_{2H+1}(t-u)}{H_{+}^{2}}\mathrm{d}u+\rho\int_{s}^{t}\frac{\mathtt{k}_{2H}(t-u)}{H_{+}}\mathrm{d}u\right)\right\\}$
$\displaystyle=\xi_{0}(t)\exp\left\\{\nu\left[\int_{0}^{s}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}+\int_{0}^{s}\mathtt{k}_{\alpha+H_{+}}(t-u)\mathrm{d}X^{\mathbb{Q}}_{u}\right]\right\\}$
$\displaystyle\qquad\times\exp\left\\{\frac{\nu^{2}}{2}\left(\frac{\mathtt{k}_{2H}(t-s)}{2H}+\frac{\mathtt{k}_{2(H+1)}(t-s)}{2H_{+}^{2}(H+1)}+\rho\frac{\mathtt{k}_{2H_{+}}(t-s)}{H_{+}^{2}}\right)\right\\}.$

Thus we only have to show that

$\left(\mathcal{G}^{H_{-}}Z^{\mathbb{Q}}\right)(s,t)=\int_{0}^{s}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}\qquad\text{and}\qquad\left(\mathcal{G}^{\alpha+H_{+}}X^{\mathbb{Q}}\right)(s,t)=\int_{0}^{s}\mathtt{k}_{\alpha+H_{+}}(t-u)\mathrm{d}X^{\mathbb{Q}}_{u}.$

We prove the first identity, the second being analogous. It is a
straightforward consequence of the definitions and the properties of Brownian
increments:

$\displaystyle\left(\mathcal{G}^{H_{-}}Z^{\mathbb{Q}}\right)(s,t):=\mathbb{E}^{\mathbb{Q}}_{s}\left[\int_{0}^{t}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}\right]$
$\displaystyle=\int_{0}^{s}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}+\mathbb{E}^{\mathbb{Q}}_{s}\left[\int_{s}^{t}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}\right]$
$\displaystyle=\int_{0}^{s}\mathtt{k}_{H_{-}}(t-u)\mathrm{d}Z^{\mathbb{Q}}_{u}.$

∎

######  Remark 2.2.

Since the instantaneous variance in this model is log-Normal, the results in
[18, Proposition 3.1] and numerical methods therein still apply for the VIX
with minimal changes.

###  2.2. A risk premium driven by a CIR process

A second natural choice is to consider the Cox-Ingersoll-Ross (CIR) process

(2.5)
$\displaystyle\mathrm{d}Y^{\mathbb{Q}}_{s}=\kappa(\theta-Y^{\mathbb{Q}}_{s})\mathrm{d}s+\sigma\sqrt{Y^{\mathbb{Q}}_{s}}\
\mathrm{d}X^{\mathbb{Q}}_{s},$

with $\kappa,\theta,\sigma>0$. As tempting as this approach might seem, it is
not trivial at all to compute the basic quantity
$\mathbb{E}^{\mathbb{Q}}[v_{t}]$ here, as the following proposition shows.

######  Proposition 2.3.

Assume that the Brownian motions $Z^{\mathbb{Q}}$ and $X^{\mathbb{Q}}$ are
independent and consider
$\lambda=\mathcal{G}^{\alpha}Y^{\mathbb{Q}}\in\mathcal{C}^{\alpha+\frac{1}{2}}$,
with $Y^{\mathbb{Q}}$ defined in (2.5). Then, for any $s\leq t$,

$\displaystyle\mathbb{E}_{s}^{\mathbb{Q}}[v_{t}]=\xi_{0}(t)$
$\displaystyle\exp\Big{\\{}\nu\left[\left(\mathfrak{G}^{-}Z^{\mathbb{Q}}\right)(s,t)+\left(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}}\right)(s,t)\right]\Big{\\}}$
$\displaystyle\exp\left\\{\frac{\nu^{2}}{2}\int_{s}^{t}\mathtt{k}_{H_{-}}(t-u)^{2}\mathrm{d}u-Y^{\mathbb{Q}}_{s}C(s,T)-A(s,T)\right\\},$

where $A(t,T):=-\kappa\theta\int_{t}^{T}C(u,T)\mathrm{d}u$ and $C$ satisfies
the Riccati equation

$\nu\mathtt{k}_{H_{-}}(T,t)-\partial_{t}C(t,T)+C(t,T)\theta+\frac{\sigma^{2}}{2}C^{2}(t,T)=0,$

for $t\in[0,T)$, with boundary condition $C(T,T)=0$.

###### Proof.

By independence of the driving Brownian motions we have, for any $u\leq t$,

$\displaystyle\mathbb{E}[v_{t}|\mathcal{F}_{u}]$
$\displaystyle=\xi_{0}(t)\exp\left\\{\nu\left(\left(\mathfrak{G}^{-}Z^{\mathbb{Q}}\right)(u,t)+\left(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}}\right)(u,t)\right)\right\\}$
$\displaystyle\qquad\times\mathbb{E}\left[\exp\Big{\\{}\nu\left(\left(\mathfrak{G}^{-}Z^{\mathbb{Q}}\right)(t)-\left(\mathfrak{G}^{-}Z^{\mathbb{Q}}\right)(u,t)\right)\Big{\\}}\right]$
$\displaystyle\qquad\times\mathbb{E}\left[\exp\Big{\\{}\nu\left(\left(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}}\right)(t)-\left(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}}\right)(u,t)\right)\Big{\\}}\right],$

where the first expected value is the MGF of a Gaussian random variable, hence

$\mathbb{E}\left[\exp\Big{\\{}\nu\Big{(}\left(\mathfrak{G}^{-}Z^{\mathbb{Q}}\right)(t)-\left(\mathfrak{G}^{-}Z^{\mathbb{Q}}\right)(u,t)\Big{)}\Big{\\}}\right]=\exp\left\\{\frac{\nu^{2}}{2}\int_{u}^{t}\mathtt{k}_{H_{-}}(t,s)\mathrm{d}s\right\\}.$

We are now interested in computing the second expectation

$\mathbb{E}\left[\exp\Big{\\{}\nu\left(\left(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}}\right)(t)-\left(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}}\right)(u,t)\right)\Big{\\}}\right],$

where

$\left(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}}\right)(t)-\left(\mathcal{G}^{\alpha+H_{+}}Y^{\mathbb{Q}}\right)(u,t)=\int_{u}^{t}\mathtt{k}_{H_{-}}(t,s)Y^{\mathbb{Q}}_{s}\mathrm{d}s.$

This is, in spirit, similar to computing a bond price in the CIR model. To do
so, define

$B(t,T):=\mathbb{E}\left[\exp\left(\nu\int_{t}^{T}\mathtt{k}_{H_{-}}(T,s)Y^{\mathbb{Q}}_{s}\mathrm{d}s\right)\bigg{|}\mathcal{F}_{t}\right],$

where $t\leq T$. We note that $B(\cdot,T)$ is a semimartingale as $T$ is
fixed, therefore applying the conditional version of Feynman-Kac’s formula, we
obtain

(2.6) $\left(\nu
r\mathtt{k}_{H_{-}}(T,t)+\partial_{t}+\kappa(\theta-y)\partial_{r}+\frac{\sigma^{2}}{2}r\partial_{yy}\right)\mathcal{B}(y,t,T)=0.$

where $\mathcal{B}(y,t,T)$ is such that $\mathcal{B}(Y^{\mathbb{Q}}_{t},t,T)$
is a solution to the moment generating function. With an ansatz of the type
$\mathcal{B}(y,t,T)=\exp\\{-yC(t,T)-A(t,T)\\}$, we have at $(y,t,T)$,

$\displaystyle\partial_{t}\mathcal{B}(y,t,T)=-(y\partial_{t}C(t,T)+\partial_{t}A(t,T))\mathcal{B}(y,t,T),$
$\displaystyle\partial_{y}\mathcal{B}(y,t,T)=-C(t,T)\mathcal{B}(y,t,T),\qquad\partial_{yy}\mathcal{B}(y,t,T)=C(t,T)^{2}\mathcal{B}(y,t,T),$

and the PDE (2.6) becomes, with $r=Y^{\mathbb{Q}}_{t}$,

$\left(\nu
Y^{\mathbb{Q}}_{t}\mathtt{k}_{H_{-}}(T,t)-\left(Y^{\mathbb{Q}}\partial_{t}C+\partial_{t}A+\kappa\left(\theta-Y^{\mathbb{Q}}_{t}\right)C\right)+\frac{\sigma^{2}}{2}C^{2}Y^{\mathbb{Q}}_{t}\right)\mathcal{B}(Y^{\mathbb{Q}}_{t},t,T)=0,$

which further simplifies to

$\left(\nu\mathtt{k}_{H_{-}}(T,t)-\partial_{t}C-\kappa
C+\frac{\sigma^{2}}{2}C^{2}\right)Y^{\mathbb{Q}}_{t}\mathcal{B}(Y^{\mathbb{Q}}_{t},t,T)-(\kappa\theta
C+\partial_{t}A)\mathcal{B}(Y^{\mathbb{Q}}_{t},t,T)=0.$

The second term cancels when
$A(t,T)=-\kappa\theta\int_{t}^{T}C(u,T)\mathrm{d}t$, and a Riccati equation
remains:

$\nu\mathtt{k}_{H_{-}}(T,t)-\partial_{t}C(t,T)-\kappa
C(t,T)+\frac{\sigma^{2}}{2}C^{2}(t,T)=0,$

with boundary conditions $A(T,T)=C(T,T)=0$. ∎

Already in the uncorrelated case the computation of
$\mathbb{E}^{\mathbb{Q}}[v_{t}]$ becomes very costly, having to solve a PDE
for each time $t$. In the correlated case there is no hope to obtain any semi-
analytic result since one would need to compute cross terms and there is no
tool coming from Itô’s calculus available in that case.

##  3\. Roughly extracting the Risk Premium from the Market

In this section we consider the risk premium process $\lambda$ to be
deterministic, with the aim of obtaining a formula that links $\mathbb{P}$ and
$\mathbb{Q}$ market observable quantities. The following theorem shows how to
infer the risk premium from the market using forecasts under the physical
measure and Variance Swap prices in the pricing measure.

######  Theorem 3.1.

Consider the rough volatility model (1.1) under $\mathbb{P}$. If
$\psi(t,x)=\xi_{0}(t)\mathrm{e}^{\nu x}$, $\mu_{s}=r_{s}$ for all $s\geq 0$
and $\left(\lambda_{s}\right)_{s\geq 0}\in L^{2}(\mathbb{R})$ is
deterministic, then

(3.1)
$\nu\overline{\rho}\int_{s}^{t}\mathtt{k}_{H_{-}}(t,u)\gamma_{u}\mathrm{d}u=\log\left(\frac{\mathbb{E}^{\mathbb{Q}}[v_{t}|\mathcal{F}_{s}]}{\mathbb{E}^{\mathbb{P}}[v_{t}|\mathcal{F}_{s}]}\right)=\log\left(\frac{\xi_{s}(t)}{\mathbb{E}^{\mathbb{P}}[v_{t}|\mathcal{F}_{s}]}\right).$

###### Proof.

If $\mu=r$ almost surely, the Radon-Nikodym derivative (1.4) in Theorem 1.5
reads

(3.2)
$\displaystyle\mathcal{D}^{\gamma}=\mathcal{E}\left(\int_{0}^{\cdot}\gamma_{s}\mathrm{d}W^{\mathbb{P}\perp}_{s}\right),$

where we recall $\lambda_{s}=\overline{\rho}\gamma_{s}$, and the inverse
Radon-Nikodym derivative is given by

(3.3)
$\displaystyle\reflectbox{$\mathcal{D}$}^{\gamma}:=\frac{1}{\mathcal{D}^{\gamma}}=\mathcal{E}\left(-\int_{0}^{\cdot}\gamma_{s}\mathrm{d}W^{\mathbb{Q}\perp}_{s}\right).$

Then, the conditional change of measure formula yields

(3.4)
$\mathbb{E}^{\mathbb{P}}[v_{t}|\mathcal{F}_{s}]=\frac{\mathbb{E}^{\mathbb{Q}}\left[v_{t}\reflectbox{$\mathcal{D}$}^{\gamma}_{t}|\mathcal{F}_{s}\right]}{\mathbb{E}^{\mathbb{Q}}\left[\reflectbox{$\mathcal{D}$}^{\gamma}_{t}|\mathcal{F}_{s}\right]}.$

On the one hand,
$\mathbb{E}^{\mathbb{Q}}\left[\reflectbox{$\mathcal{D}$}^{\gamma}_{t}|\mathcal{F}_{s}\right]=\mathcal{E}\left(-\int_{0}^{\cdot}\gamma_{u}\mathrm{d}W^{\mathbb{Q}\perp}_{u}\right)_{s}$
by the properties of the stochastic exponential and Gaussian moment generating
functions. On the other hand, since, for $t\in\mathbb{T}$,
$Z^{\mathbb{Q}}_{t}=Z^{\mathbb{P}}_{t}+\int_{0}^{t}\lambda_{s}\mathrm{d}s$ and
$\lambda$ is deterministic, then

(3.5)
$\displaystyle\mathbb{E}^{\mathbb{Q}}\left[v_{t}\reflectbox{$\mathcal{D}$}^{\gamma}_{t}|\mathcal{F}_{s}\right]$
$\displaystyle=\mathbb{E}^{\mathbb{Q}}\left[\left.\exp\left\\{\nu\left(\int_{0}^{t}\mathtt{k}_{H_{-}}(t,s)\mathrm{d}Z^{\mathbb{Q}}_{s}+\int_{0}^{t}\lambda_{s}\mathtt{k}_{H_{-}}(t,s)\mathrm{d}s\right)\right\\}\mathrm{e}^{-\int_{0}^{t}\gamma_{s}\mathrm{d}W^{\mathbb{Q}\perp}_{s}-\frac{1}{2}\int_{0}^{t}\gamma_{s}^{2}\mathrm{d}s}\right|\mathcal{F}_{s}\right]$
(3.6)
$\displaystyle=\mathrm{e}^{\nu\int_{0}^{t}\lambda_{s}\mathtt{k}_{H_{-}}(t,s)\mathrm{d}s-\frac{1}{2}\int_{0}^{t}\gamma_{s}^{2}\mathrm{d}s}\mathbb{E}^{\mathbb{Q}}\left[\left.\exp\left\\{\nu\int_{0}^{t}\mathtt{k}_{H_{-}}(t,s)\mathrm{d}Z^{\mathbb{Q}}_{s}-\int_{0}^{t}\gamma_{s}\mathrm{d}W^{\mathbb{Q}\perp}_{s}\right\\}\right|\mathcal{F}_{s}\right],$

where the second factor in the last term is just the conditional moment
generating function of a Gaussian random variable. Applying Itô’s isometry
then, conditionally on $\mathcal{F}_{s}$, the random variable
$\nu\int_{0}^{t}\mathtt{k}_{H_{-}}(t,s)\mathrm{d}Z^{\mathbb{Q}}_{s}-\int_{0}^{t}\gamma_{s}\mathrm{d}W^{\mathbb{Q}\perp}_{s}$is
distributed as $\mathcal{N}\left(\mu,\sigma^{2}\right)$ with

$\displaystyle\mu$
$\displaystyle:=\nu\int_{0}^{s}\mathtt{k}_{H_{-}}(t,u)\mathrm{d}Z^{\mathbb{Q}}_{u}-\int_{0}^{s}\gamma_{u}\mathrm{d}W^{\mathbb{Q}\perp}_{u},$
$\displaystyle\sigma^{2}$
$\displaystyle:=\nu^{2}\int_{s}^{t}k^{2}(t,u)\mathrm{d}u+\int_{s}^{t}\gamma_{u}^{2}\mathrm{d}u-2\nu\overline{\rho}\int_{s}^{t}\mathtt{k}_{H_{-}}(t,u)\gamma_{u}\mathrm{d}u,$

since $Z^{\mathbb{Q}}=\rho W^{\mathbb{Q}}+\overline{\rho}W^{\mathbb{Q}\perp}$.
Exploiting the identities above and reordering terms,

(3.7)
$\displaystyle\mathbb{E}^{\mathbb{Q}}\left[v_{t}\reflectbox{$\mathcal{D}$}^{\gamma}_{t}|\mathcal{F}_{s}\right]\
$
$\displaystyle=\exp\Bigg{\\{}\nu\int_{0}^{t}\mathtt{k}_{H_{-}}(t,s)\lambda_{s}\mathrm{d}s+\overbrace{\nu\int_{0}^{s}\mathtt{k}_{H_{-}}(t,u)\mathrm{d}Z^{\mathbb{Q}}_{u}-\int_{0}^{s}\gamma_{u}\mathrm{d}W^{\mathbb{Q}\perp}_{u}}^{\mu}$
$\displaystyle\qquad+\frac{1}{2}\Bigg{(}\underbrace{\nu^{2}\int_{s}^{t}k^{2}(t,u)\mathrm{d}u+\int_{s}^{t}\gamma_{u}^{2}\mathrm{d}u-2\nu\overline{\rho}\int_{s}^{t}\mathtt{k}_{H_{-}}(t,u)\gamma_{u}\mathrm{d}u}_{\sigma^{2}}-\int_{0}^{t}\gamma_{s}^{2}\mathrm{d}s\Bigg{)}\Bigg{\\}}$
$\displaystyle=\exp\Bigg{\\{}-\nu\overline{\rho}\int_{s}^{t}\mathtt{k}_{H_{-}}(t,u)\gamma_{u}\mathrm{d}u\Bigg{\\}}\overbrace{\exp\Bigg{\\{}-\int_{0}^{s}\gamma_{u}\mathrm{d}W^{\mathbb{Q}\perp}_{u}-\frac{1}{2}\int_{0}^{s}\gamma_{u}^{2}\mathrm{d}u\Bigg{\\}}}^{\mathbb{E}^{\mathbb{Q}}\left[\reflectbox{$\mathcal{D}$}^{\gamma}_{t}|\mathcal{F}_{s}\right]}$
$\displaystyle\qquad\underbrace{+\exp\Bigg{\\{}\nu\int_{0}^{s}\mathtt{k}_{H_{-}}(t,u)\mathrm{d}Z^{\mathbb{Q}}_{u}+\frac{\nu^{2}}{2}\int_{s}^{t}\mathtt{k}_{H_{-}}^{2}(t,u)\mathrm{d}u+\nu\int_{0}^{t}\mathtt{k}_{H_{-}}(t,s)\lambda_{s}\mathrm{d}s\Bigg{\\}}}_{\mathbb{E}^{\mathbb{Q}}[v_{t}|\mathcal{F}_{s}]},$

by using the decomposition of $\sigma^{2}$ as the sum of three terms, and so

(3.8)
$\displaystyle\mathbb{E}^{\mathbb{Q}}\left[v_{t}\reflectbox{$\mathcal{D}$}^{\gamma}_{t}|\mathcal{F}_{s}\right]=\mathbb{E}^{\mathbb{Q}}\left[\reflectbox{$\mathcal{D}$}^{\gamma}_{t}|\mathcal{F}_{s}\right]\mathbb{E}^{\mathbb{Q}}[v_{t}|\mathcal{F}_{s}]\exp\left\\{-\nu\overline{\rho}\int_{s}^{t}\mathtt{k}_{H_{-}}(t,u)\gamma_{u}\mathrm{d}u\right\\}.$

Finally, going back to (3.4) and exploiting the identity in (3.8), the result
follows from

(3.9)
$\displaystyle\mathbb{E}^{\mathbb{P}}[v_{t}|\mathcal{F}_{s}]=\mathbb{E}^{\mathbb{Q}}[v_{t}|\mathcal{F}_{s}]\exp\left\\{-\nu\overline{\rho}\int_{s}^{t}\mathtt{k}_{H_{-}}(t,u)\gamma_{u}\mathrm{d}u\right\\}=\mathbb{E}^{\mathbb{Q}}[v_{t}|\mathcal{F}_{s}]\exp\left\\{-\nu\int_{s}^{t}\mathtt{k}_{H_{-}}(t,u)\lambda_{u}\mathrm{d}u\right\\}.$

∎

###  3.1. Estimating the risk premium in rough Bergomi

In this section we work with the rough Bergomi model under $\mathbb{P}$ and
its $\mathbb{Q}$-version:

$\left\\{\begin{array}[]{rl}\displaystyle\frac{\mathrm{d}S_{t}}{S_{t}}&=\displaystyle\mu_{t}\mathrm{d}t+\sqrt{v_{t}}\,\mathrm{d}W^{\mathbb{P}}_{t},\\\
v_{t}&=\displaystyle\exp\left\\{\nu Z^{H}_{t}\right\\},\\\
\end{array}\right.\quad\text{vs}\quad\left\\{\begin{array}[]{rl}\displaystyle\frac{\mathrm{d}S_{t}}{S_{t}}&=\displaystyle
r_{t}\mathrm{d}t+\sqrt{v_{t}}\,\mathrm{d}W^{\mathbb{Q}}_{t},\\\
v_{t}&=\displaystyle\xi_{0}(t)\exp\left\\{\nu\left(\int_{0}^{t}\mathtt{k}_{H_{-}}(t-s)\mathrm{d}Z^{\mathbb{Q}}_{s}+\int_{0}^{t}\mathtt{k}_{H_{-}}(t-s)\lambda_{s}\mathrm{d}s\right)\right\\},\end{array}\right.$

Assuming $\lambda$ deterministic, Theorem 3.1 gives an explicit procedure to
compute the risk premium. In practice however, we are only able to observe
variance swap quotes in discrete times, and hence it is natural to consider
$\lambda$ piecewise constant.

######  Assumption 3.2.

The deterministic process $\lambda$ admits the following piecewise constant
representation on the time partition $\\{0=T_{0}<T_{1},...,<T_{n}=T\\}$,
namely for $n\in\mathbb{N}$:

(3.10)
$\lambda(t):=\sum_{i=1}^{n}\lambda_{i}1{1}_{\\{t\in[T_{i-1},T_{i})\\}},\quad\lambda_{i}\in\mathbb{R}\text{
for }i=1,\ldots,n.$

Similarly the forward variance $\xi_{0}$ admits the following piecewise
constant representation
$\mathbb{E}_{0}^{\mathbb{Q}}[v_{t}]=\xi_{0}(t):=\sum_{i=1}^{n}\xi_{i}1{1}_{\\{t\in[T_{i-1},T_{i})\\}}$
with $\xi_{i}\in\mathbb{R}$ for $i=1,\ldots,n$, where
$\xi_{i}:=\frac{\mathfrak{V}_{T_{i}}T_{i}-\mathfrak{V}_{T_{i-1}}T_{i-1}}{T_{i}-T_{i-1}}$
and
$\mathfrak{V}_{T}:=\mathbb{E}^{\mathbb{Q}}\left[\frac{1}{T}\int_{0}^{T}v_{s}\mathrm{d}s\right]$
is a market variance swap quote.

We now estimate $\\{\lambda_{1},\cdots,\lambda_{n}\\}$. The dataset consists
of daily Eurostoxx variance swap quotes for maturities $\\{\text{1M, 3M, 6M,
1Y, 2Y}\\}$ (Figures 1 and 2), while Figure 3 shows the daily realised
volatility obtained from Oxford-Man institute data.

Figure 1. Variance Swap volatility daily quotes on SX5E Figure 2. Forward
variances extracted from variance swap quotes on SX5E Figure 3. Annualised
daily realised volatility on SX5E

In order to apply formula (3.1) we need the following ingredients:

(3.11) $\text{Parameters
}H,\nu,\rho,\qquad\mathbb{E}^{\mathbb{P}}[v_{t}|\mathcal{F}_{0}],\qquad\mathbb{E}^{\mathbb{Q}}[v_{t}|\mathcal{F}_{0}].$

So far we have obtained $\mathbb{E}^{\mathbb{Q}}[v_{t}|\mathcal{F}_{0}]$ from
Variance Swap market quotes. The next step is to estimate $(H,\nu,\rho)$ using
historical time-series. Gatheral, Jaisson and Rosenbaum [10], explain how to
estimate $\widehat{H}$ and $\widehat{\nu}$ from daily volatility data (Figure
3), and we follow their approach using a 100-day rolling window (Figure 4) and
refer the reader to the original paper for details.

Figure 4. Estimated $\widehat{H}$ and $\widehat{\nu}$ on SX5E.

In order to estimate the correlation parameter we use the formula

(3.12)
$\mathrm{Corr}\left(Z^{H}_{t}-Z^{H}_{s},\int_{s}^{t}\mathrm{d}W_{s}\right)=\frac{\rho\sqrt{2H}}{H_{+}},$

which allows us to estimate the correlation with the proxy

(3.13)
$\displaystyle\widehat{\rho}=\frac{\widehat{H}+\frac{1}{2}}{\sqrt{2\widehat{H}}}\
\widehat{\mathrm{Corr}}\left(\frac{\log(S_{t_{i}})-\log(S_{t_{i-1}})}{\sqrt{v_{t_{i-1}}}},\log(v_{t_{i}})-\log(v_{t_{i-1}})\right).$

Figure 5 below displays the historical estimates using a estimation window of
100 days.

Figure 5. Daily correlation estimate on SX5E and realised volatility.

To forecast volatility and obtain
$\mathbb{E}^{\mathbb{P}}[v_{t}|\mathcal{F}_{0}]$, we proceed as in [10] and
use the forecasting formula for the fractional Brownian motion due to Nuzman
and Poor [22]:

(3.14)
$Z^{H}_{t+\Delta}|\mathcal{F}_{t}\sim\mathcal{N}\left(\frac{\cos(H\pi)}{\pi}\Delta^{H_{+}}\int_{0}^{t}\frac{Z^{H}_{s}\mathrm{d}s}{(t-s+\Delta)(t-s)^{H_{+}}},\frac{C_{H}\Delta^{2H}}{2H}\right).$

Finally, we orderly estimate $\lambda_{i}$ for each $i=1,\ldots,n$ using
Theorem 3.1 and the piecewise constant assumption (3.10), as

(3.15)
$\displaystyle\sum_{j=1}^{i}\lambda_{j}\int_{T_{j-1}}^{T_{j}}\mathtt{k}_{H_{-}}(t,u)\mathrm{d}u=\frac{1}{\nu(1-\rho^{2})}\log\left(\frac{\xi_{0}(T_{i})}{\mathbb{E}^{\mathbb{P}}[v_{T_{i}}|\mathcal{F}_{0}]}\right).$

Figure 6 shows the historical evolution of the risk premium process.

Figure 6. Daily SX5E risk premia; dashed lines represent means.

######  Remark 3.3.

We would like to emphasise that assessing the best method to estimate (3.11)
is beyond the scope of this paper. However, as highlighted in the
introduction, we stress the importance of Theorem 3.1 towards which this
empirical work provides a first step.

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