# Elucidate Gender Fairness in Singing Voice Transcription

Xiangming Gu  Integrative Sciences and Engineering Programme, NUS Graduate
School, National University of SingaporeSingapore<EMAIL_ADDRESS>,  Wei
Zeng  Integrative Sciences and Engineering Programme, NUS Graduate School,
National University of SingaporeSingapore<EMAIL_ADDRESS>and  Ye Wang
School of Computing, National University of SingaporeSingapore
<EMAIL_ADDRESS>
(2023)

###### Abstract.

It is widely known that males and females typically possess different sound
characteristics when singing, such as timbre and pitch, but it has never been
explored whether these gender-based characteristics lead to a performance
disparity in singing voice transcription (SVT), whose target includes pitch.
Such a disparity could cause fairness issues and severely affect the user
experience of downstream SVT applications. Motivated by this, we first
demonstrate the female superiority of SVT systems, which is observed across
different models and datasets. We find that different pitch distributions,
rather than gender data imbalance, contribute to this disparity. To address
this issue, we propose using an attribute predictor to predict gender labels
and adversarially training the SVT system to enforce the gender-invariance of
acoustic representations. Leveraging the prior knowledge that pitch
distributions may contribute to the gender bias, we propose conditionally
aligning acoustic representations between demographic groups by feeding note
events to the attribute predictor. Empirical experiments on multiple benchmark
SVT datasets show that our method significantly reduces gender bias (up to
more than $50\%$) with negligible degradation of overall SVT performance, on
both in-domain and out-of-domain singing data, thus offering a better
fairness-utility trade-off.

fairness, singing voice transcription, pitch, adversarial learning, bias,
fairness-utility trade-off

††journalyear: 2023††copyright: acmlicensed††conference: Proceedings of the
31st ACM International Conference on Multimedia; October 29-November 3, 2023;
Ottawa, ON, Canada††booktitle: Proceedings of the 31st ACM International
Conference on Multimedia (MM ’23), October 29-November 3, 2023, Ottawa, ON,
Canada††price: 15.00††doi: 10.1145/3581783.3612272††isbn:
979-8-4007-0108-5/23/10††ccs: Applied computing Sound and music
computing††ccs: Information systems Music retrieval††ccs: Information systems
Speech / audio search††ccs: Social and professional topics Gender††ccs:
Networks Network reliability

##  1\. Introduction

Figure 1. Pitch distributions of various singing voice transcription datasets.

Singing is a rich audio signal that consists of information from two aspects:
textual and musical. The textual modality pertains to the lyrics, while the
musical modality encompasses note events. Automatic lyric transcription (ALT)
can be used to retrieve lyrical/textual data, as evidenced by prior literature
(Dabike and Barker, 2019; Demirel et al., 2020, 2021; Zhang et al., 2022b; Gu
et al., 2022; Ou et al., 2022; Gao et al., 2022b, a). On the other hand,
singing voice transcription (SVT) is utilized to retrieve note events, which
include onsets, offsets, and pitches (Mauch et al., 2015; Fu and Su, 2019; Hsu
and Su, 2021; Wang and Jang, 2021; Kum et al., 2022; Wang et al., 2022; Gu et
al., 2023). Transcribed information can enable the development of singing
voice synthesis (Ren et al., 2020; Liu et al., 2022a) and aid in education
(Yang et al., 2022; Molina et al., 2013; Gupta et al., 2017) and therapy (Tam
et al., 2007). It is important to note that males and females have distinct
sound characteristics in their singing voices (Latinus and Taylor, 2012), e.g.
timbre. Additionally, compared to lyrics, the note events tend to be more
subject to explicit gender-related biases, particularly for pitch, as shown in
prior literature (Puts et al., 2007; Simpson, 2009) that males tend to have
lower average pitch than females. This is also consistent with our analysis of
four SVT datasets, namely N20EMv2 (Gu et al., 2023), MIR-ST500 (Wang and Jang,
2021), ISMIR2014 (Molina et al., 2014), and M4Singer (Zhang et al., 2022a). As
shown in Fig. 1, we observe that the pitch range of females is generally
higher than that of males across these four SVT datasets. Besides, the
proportion of males and females for each pitch value is also different.
Consequently, a critical fairness question arises: since SVT systems target
pitches, will the performance of these systems favor one gender over the
other? Before we delve into this question, it is imperative to elucidate why
this question holds substantial significance.

The rapid progress in machine learning techniques has facilitated their
successful integration into various downstream applications, streamlining
decision-making processes and reducing the need for repeated human efforts.
However, the presence of bias in machine learning systems can lead to the
discriminatory treatment of certain groups with sensitive attributes such as
gender, age, and race, resulting in unfair decisions. Moreover, general usage
of biased machine learning systems can reinforce the stereotypes and exclude
certain groups from the opportunities. This phenomenon is not only limited to
traditional decision-making scenarios such as loan applications, hiring, legal
proceedings, and policy-making (Zemel et al., 2013), but also frequently
appears in recent deep learning applications, including visual recognition
(Wang et al., 2020; Sarhan et al., 2020; Tartaglione et al., 2021), natural
language processing (Guo et al., 2022; Ding et al., 2022), speech processing
and recognition (Rajan et al., 2022; Dheram et al., 2022; Liu et al., 2022b;
Gorrostieta et al., 2019; Wagner et al., 2023; Fenu et al., 2020; Chen et al.,
2022a), recommendation systems (Sonboli et al., 2021; Li et al., 2021), and
generative models (Choi et al., 2020; Friedrich et al., 2023). Consequently,
the issue of fairness has gained prominence in the machine learning community
due to its pervasive nature and potential societal consequences. Returning to
our SVT task, the unfairness of systems could directly lead to user
inconvenience and negatively impact their experience. Applying biased SVT
systems to downstream applications could cause more fairness issues. We can
consider an automatic sight-singing exercise (Yang et al., 2022), where an SVT
system could serve as an intermediate stage to transcribe the singing voice
into musical notes. However, if a gender-biased SVT system is utilized, it
could result in an unsatisfying user experience for certain groups, as well as
reinforce gender stereotypes, which would be unfair to individuals of
different genders.

In this study, we elucidate the fairness issue in the field of SVT. Our
investigation demonstrates that SVT systems tend to perform better on females
compared to males, raising questions about their fairness. Then, we assume
that this gender bias is attributable to the differences of singing voices,
especially in terms of pitch distributions, in different demographic groups.
We utilize self-supervised-learning (SSL)-based SVT systems (Gu et al., 2023),
which represent the current state-of-the-art in this field, to develop a bias
mitigation approach. In contrast to previous research, which normally focuses
on the in-domain fairness, we also pay attention to the out-of-domain fairness
as user data is sometimes sampled from out-of-domain distribution. To
implement this, our approach adopts the adversarial learning framework to
“unlearn” gender-related information in the acoustic representations.
Considering the effects of pitch distributions, we propose a note-conditional
attribute predictor to conditionally align the representations between female
and male groups by conditioning on note events. Empirical results from various
SVT datasets confirm the effectiveness of our approach in mitigating gender
bias. Our contributions are summarized as below:

  * •

We provide evidence and analysis of the prevalence and source of gender bias
in SVT systems. To the best of our knowledge, this is the first attempt at
fairness in singing-centric deep learning.

  * •

We first introduce a note-conditioned adversarial learning approach to achieve
fair representation learning in audio modality, resulting in a significant
reduction in the performance gap between the two gender groups on various
benchmark SVT datasets while maintaining a good fairness-utility trade-off.
Our method is effective in reducing biases on both in-domain and out-of-domain
data.

  * •

We demonstrate the superiority of our note-conditioned adversarial learning
approach through comparisons with baseline adversarial learning and domain-
independent training.

##  2\. Related Work

###  2.1. Singing Voice Transcription

Singing voice transcription (SVT) involves various sub-tasks, including pitch
estimation and onset/offset detection. Earlier approaches (Mauch and Dixon,
2014; Mauch et al., 2015; Yang et al., 2017; Nishikimi et al., 2020) typically
relied on statistical models, such as Bayesian models and Hidden Markov Models
(HMM), to predict fundamental frequency (F0) and note segmentation. In
contrast, more recent SVT methods (Fu and Su, 2019; Hsu and Su, 2021; Kum et
al., 2022) have predominantly employed deep learning techniques, such as CNN
and LSTM, and have demonstrated superior performance. Despite the promising
SVT performance achieved by these methods, the intrinsic difficulty of
curating large-scale, high-quality SVT datasets presents an obstacle to
further improvements. To address this challenge, several approaches have been
proposed. Among them, VOCANO (Hsu and Su, 2021) employed the Virtual
Adversarial Training (VAT) (Miyato et al., 2018) to train the note segment
network on both labeled and unlabeled data. In (Kum et al., 2022), pseudo
labels are obtained by quantizing frame-level pitch contours for training on
unlabeled audio data. MusicYOLO (Wang et al., 2022) utilized the object
detection model YOLOX (Ge et al., 2021), which has been trained on the image
domain, to locate notes in the audio spectrogram. Recently, (Gu et al., 2023)
adapted self-supervised learning (SSL) models from the speech domain to the
SVT task, thus alleviating the label insufficiency as well as achieving state-
of-the-art SVT performance.

###  2.2. Fairness and Bias Mitigation

The notion of fairness in machine learning systems is defined as the absence
of any discrimination based on sensitive attributes when making decisions. It
can be categorized into group fairness (Hardt et al., 2016; Zhao et al., 2019)
and individual fairness (Dwork et al., 2012; Kusner et al., 2017). The former
requires that there are no disparities among different demographic groups
(Zhao et al., 2019), while the latter requires that similar individuals
receive similar predictions (Kusner et al., 2017). In our work, we focus on
group fairness, which can be assessed by criteria, including independence,
separation, and sufficiency (Barocas et al., 2019; Shen et al., 2022), along
with metrics such as demographic parity, equalized odds, equal opportunity,
and accuracy parity (Verma and Rubin, 2018; Zhao et al., 2019; Mehrabi et al.,
2021).

Numerous approaches have been proposed to mitigate bias in machine learning
systems. Among them, adversarial learning has emerged as a powerful technique
for removing sensitive attributes in the representations used for prediction.
For instance, (Louppe et al., 2017) repurposed the framework of generative
adversarial networks (GANs) (Goodfellow et al., 2020) to satisfy demographic
parity. (Madras et al., 2018) used an adversary to predict sensitive
attributes from latent representations and a decoder to reconstruct the input
data from the latent representations and predicted attributes in adversarial
learning. Additionally, they proposed different adversarial objectives
according to the target group fairness criteria. (Kim et al., 2019) proposed
using a bias prediction network to minimize mutual information between latent
representations and bias through adversarial learning. In contrast to these
approaches, (Zhao et al., 2019) proposed conditional alignment of latent
representations to strike a better balance between fairness and utility. In
addition to adversarial learning, alternative methods have been proposed to
mitigate bias. (Wang et al., 2020) advocated domain independent training,
where different classifiers are trained for different demographic groups.
(Sarhan et al., 2020; Tartaglione et al., 2021) disentangled the latent
representations for task predictions and sensitive attributes, respectively,
to reduce the influence of the sensitive attributes on the model’s decision-
making process.

###  2.3. Adversarial Learning for Invariance

In addition to the adversarial learning framework used in the fairness
community, our work also shares similarities with unsupervised domain
adaptation, which aims to learn domain-invariant representations. (Ganin and
Lempitsky, 2015) proposed a domain classifier to learn the representations
that are invariant to the domain shift, with an adversary implemented by a
gradient reverse layer. (Tzeng et al., 2017) used a discriminator to align the
distributions of source domain and target domain in the representation space,
following the training of GANs (Goodfellow et al., 2020). Similar to (Zhao et
al., 2019), (Zhao et al., 2017; Long et al., 2018) proposed a conditional
discriminator that takes into account the multimodal nature of feature
distributions. One key difference compared to the fairness scenario is that
labels in the target domain are not available in domain adaptation. Therefore,
the conditions in (Zhao et al., 2017; Long et al., 2018) are the label
predictions, rather than the ground-truth labels used in (Zhao et al., 2019).

##  3\. Preliminary for SVT

We recap the formulation and solutions of singing voice transcription (SVT),
which is defined according to the framework introduced in (Wang and Jang,
2021; Gu et al., 2023). The input to an SVT system is the waveform $\bm{x}$
while the output $\bm{y}=[(o_{1},f_{1},p_{1}),...,(o_{n},f_{n},p_{n}),...,$
$(o_{N},f_{N},p_{N})]$ is a sequence of note events, where $o_{n},f_{n},p_{n}$
represent the onset/offset/pitch of each note, respectively. Consequently, the
task of SVT can be regarded as a sequence-to-sequence problem. Since it is
challenging to supervise the entire model using the ground truth note events
directly, frame-level labels $\bm{O},\bm{S},\bm{V},\bm{P}$ are constructed to
mark the onset/silence/octave class/pitch class of each frame. Specifically,
$\bm{O},\bm{S}$ comprise binary classes, whereas $\bm{V}$ and $\bm{P}$ have
multiple classes. The number of categories of pitch classes is fixed to 12,
while the number of categories of octaves are chosen based on the pitch range.
For example, the octave class of C4 is 4 and the pitch class is C. We add an
additional octave/pitch class to represent the pitch of silence. The loss
function is formulated to minimize the empirical risk as follows:

(1)
$\displaystyle\mathcal{L}_{\text{SVT}}(\hat{\bm{y}},\bm{y})=\frac{1}{T}\sum_{t=1}^{T}[$
$\displaystyle
l_{\text{BCE}}(\hat{O}_{t},O_{t})+l_{\text{BCE}}(\hat{S}_{t},S_{t})+$
$\displaystyle
l_{\text{CE}}(\hat{V}_{t},V_{t})+l_{\text{CE}}(\hat{P}_{t},P_{t})],$

where $T$ refers to the number of frames,
$\hat{O}_{t},\hat{S}_{t},\hat{V}_{t},\hat{P}_{t}$ denote the frame-level
predictions while $\hat{\bm{y}}$ represents note-level predictions. After
post-processing, $\hat{O}_{t},\hat{S}_{t},\hat{V}_{t},\hat{P}_{t}$ are
transformed to $\hat{\bm{y}}$. Interested readers can refer to (Wang and Jang,
2021; Gu et al., 2023) for more detailed information. An SVT system consists
of an acoustic encoder and a note predictor. The note predictor is a simple
linear layer, while the design of acoustic encoder can vary. For instance,
(Wang and Jang, 2021) employed EfficientNet (Tan and Le, 2019), while (Gu et
al., 2023) adapted self-supervised-learning models, e.g. wav2vec 2.0 as the
acoustic encoder. To evaluate SVT systems, f1-scores of COnPOff (correct
onset, pitch, and offset), COnP (correct onset and pitch), and COn (correct
onset) are commonly used. These metrics were proposed in (Molina et al., 2014)
and have since been widely adopted. The first two metrics are related to the
accuracy of pitch estimation. We use the default tolerances as implemented in
the python package mir_eval (Raffel et al., 2014) for the evaluation of SVT
systems in this work.

Table 1. COnPOff/COnP/COn F1-score (%) of state-of-the-art SVT systems in (Gu et al., 2023) on multiple datasets. We using bold face to highlight the gender group with better performance and red bold face to mark the results with large bias. Dataset | Model | COnPOff (%) | COnP (%) | COn (%)
---|---|---|---|---
total $\uparrow$ | female $\uparrow$ | male $\uparrow$ | gap | total $\uparrow$ | female $\uparrow$ | male $\uparrow$ | gap | total $\uparrow$ | female $\uparrow$ | male $\uparrow$ | gap
MIR-ST500 | model1 | 52.39 | 53.11 | 51.47 | -1.64 | 70.73 | 72.54 | 68.42 | -4.12 | 78.32 | 79.36 | 77.00 | -2.36
model2 | 34.55 | 35.29 | 33.60 | -2.55 | 51.64 | 52.76 | 50.21 | -2.55 | 71.33 | 72.07 | 70.39 | -1.68
model3 | 52.84 | 54.02 | 51.33 | -2.69 | 70.00 | 71.17 | 67.85 | -3.32 | 78.05 | 78.84 | 77.05 | -1.79
N20EMv2 | model1 | 55.20 | 60.96 | 51.55 | -9.41 | 72.03 | 79.76 | 67.11 | -12.65 | 88.51 | 90.63 | 87.16 | -3.47
model2 | 68.62 | 74.82 | 64.68 | -10.14 | 75.69 | 81.27 | 72.14 | -9.12 | 92.83 | 94.33 | 91.88 | -2.44
model3 | 73.06 | 78.34 | 69.69 | -8.65 | 79.56 | 84.38 | 76.49 | -7.89 | 93.66 | 94.96 | 92.83 | -2.13
ISMIR2014 | model1 | 52.58 | 61.22 | 45.28 | -15.94 | 67.75 | 74.90 | 61.70 | -13.20 | 92.13 | 91.93 | 92.30 | +0.37
model2 | 57.35 | 62.42 | 53.06 | -9.36 | 72.15 | 79.35 | 66.06 | -13.29 | 91.53 | 92.25 | 90.92 | -1.33
model3 | 59.95 | 65.61 | 55.16 | -10.45 | 73.85 | 80.55 | 68.19 | -12.36 | 92.80 | 93.18 | 92.49 | -0.69
M4Singer | wav2vec2 | 53.66 | 57.27 | 49.93 | -7.34 | 61.95 | 66.36 | 57.38 | -8.98 | 82.60 | 81.66 | 83.58 | +1.91
Hubert | 55.11 | 58.39 | 51.73 | -6.66 | 64.17 | 68.10 | 60.10 | -8.00 | 82.13 | 81.53 | 82.76 | +1.23
wavLM | 57.06 | 60.33 | 53.68 | -6.66 | 65.40 | 68.87 | 61.82 | -7.05 | 82.73 | 82.29 | 83.19 | +0.90
data2vec | 53.98 | 57.45 | 50.40 | -7.05 | 63.04 | 67.09 | 58.86 | -8.23 | 82.30 | 82.13 | 82.48 | +0.35

##  4\. Fairness Analysis for SVT

###  4.1. Female Superiority in SVT Performance: Evidence from Multiple
Datasets

We evaluate the performance of state-of-the-art singing voice transcription
(SVT) systems from (Gu et al., 2023) on three benchmark datasets, including
MIR-ST500 (Wang and Jang, 2021), N20EMv2 (Gu et al., 2023), and ISMIR2014
(Molina et al., 2014). The results for two gender groups are presented in
Table 1, where “model1” is trained on the MIR-ST500 training split, “model2”
on the N20EMv2 training split, and “model3” on both training sets. These
models were built based on wav2vec 2.0 (Baevski et al., 2020). The ISMIR2014
dataset already has the gender label for each song. The gender labels for the
N20EMv2 and MIR-ST500 can be obtained by directly listening to the audio
recordings. For double confirmation, we also check the video modality provided
in N20EMv2 and the original Youtube links in MIR-ST500. Across all datasets,
we observe that the SVT performances of three models on females consistently
outperform that on males in terms of COnPOff and COnP f1-scores, which are the
metrics related to pitch estimation. However, the COn performance does not
demonstrate the consistent female superiority. We compute the performance gap
as the difference between male and female metrics, i.e.,
$\text{metric}_{\text{gap}}=\text{metric}_{\text{male}}-\text{metric}_{\text{female}}$.
We find that the performance gaps between gender groups are significant for
pitch-related metrics on the N20EMv2 and ISMIR2014 datasets. Even though the
performance gap is comparably small for MIR-ST500, female superiority in SVT
performance remains valid.

Apart from the datasets evaluated in (Gu et al., 2023), we conduct experiments
on a recent Mandarin singing dataset called M4Singer (Zhang et al., 2022a).
This dataset comprises data from 20 singers across four main voice types
(soprano, alto, tenor, and bass). Since M4Singer lacks official training-test
splits, we manually partition the data, selecting data from two male and two
female singers representing the above four voice types for the test split and
the data from the remaining singers for the training split. To ensure a fair
test split, we make sure that the total duration of selected female data in
the test split is almost equal to that of male data. We follow the training
configuration in (Gu et al., 2023) to train our SVT system from scratch using
wav2vec 2.0, making minimal modifications to achieve high SVT performance. We
add an additional octave category to the classifier to account for the larger
pitch range of M4Singer. The M4Singer dataset already provides the gender
labels, similar to ISMIR2014. The results in Table 1 show that the SVT
performance of females is still better than that of males, with a significant
performance gap.

Figure 2. (a) SVT performance of EfficientNet on ISMIR2014. (b) SVT
performance of model3 on ISMIR2014. (c) SVT performance of EfficientNet on
MIR-ST500. (d) Pitch distributions of male/female/child groups on ISMIR2014.

###  4.2. Female Superiority in SVT Performance: Evidence from Multiple Models

The models evaluated in Sec. 4.1 are all based on wav2vec 2.0 (Baevski et al.,
2020). However, it is important to investigate whether the observed female
superiority is limited to this particular model choice. To explore this
question, we replace the wav2vec 2.0 trained on M4Singer with other self-
supervised-learning (SSL) models, e.g. Hubert (Hsu et al., 2021), wavLM (Chen
et al., 2022b), and data2vec (Baevski et al., 2022). Although these models
have different SSL objectives and slightly different model architectures, we
find that the SVT performance of the female group is still better than the
male group with significant margins, as shown in Table 2. To strengthen our
findings, we further evaluate the performance of EfficientNet-based SVT system
in (Wang and Jang, 2021). As presented in Fig. 2 (a) and (c), we note that on
the MIR-ST500 and ISMIR2014 datasets, the SVT performance of female group
still outperforms that of the male group in terms of COnPOff and COnP
f1-scores, which is consistent with our earlier conclusion.

###  4.3. Possible Reasons for Female Superiority

From Sec. 4.1 and Sec. 4.2, we conclude that female superiority in SVT
performance is valid across different datasets and model choices. To explore
possible reasons behind this phenomenon, we first examine the statistics of
the SVT datasets to investigate whether the singing datasets typically possess
the property of gender data imbalance. As presented in Table 2, we include the
statistics of two gender groups in the training splits of MIR-ST500, N20EMv2,
and M4Singer (ISMIR2014 is only used for evaluation). While MIR-ST500 has a
larger proportion of female data, N20EMv2 and M4Singer have larger proportions
of male data. Nevertheless, all the SVT systems trained on these datasets
favored females, indicating that the source of bias is not merely the data
imbalance. We hypothesize that the gender bias in SVT performance is
attributed to the differences of sound characteristics across different
demographic groups. Specifically, we assume that pitch distributions make
substantial contributions to the female superiority. As shown in Fig. 1, we
find that (1) the female group generally has higher pitch range than the male
group; (2) the proportion of male and female labels for each specific pitch is
different.

Table 2. Demographic statistics of SVT training splits. Dataset | Songs Num | Duration (h)
---|---|---
total | female | male | total | female | male
MIR-ST500 | 400 | 221 | 179 | 27.62 | 15.13 | 12.49
N20EMv2 | 123 |  52 |  71 |  6.44 |  2.74 |  3.70
M4Singer | 521 | 246 | 275 | 22.71 | 10.27 | 12.44

To further support our assumption, we perform an additional evaluation on the
ISMIR2014 dataset, which includes the child group. Typically, children have
different inherent properties in sound voices compared to female adults and
male adults (Moore, 1991; Whiteside and Hodgson, 2000). For instance, the
pitch distribution of the child group is different from both the female group
and male group, as present in Fig. 2 (d). Consequently, we find that both the
EfficientNet-based SVT system and wav2vec 2.0-based SVT system (model3)
demonstrate performance disparity among the three groups. As shown in Fig. 2
(a) and (b), the SVT performance of the child group significantly outperforms
that of the male group while is close to that of the female group. To
interpret this, the pitch distribution difference between the male group and
the child group is large while the difference between the female group and the
child group is comparatively subtle. As there are no existing SVT annotations
for child training data, we narrow down the scope of this work to gender
fairness and leave the discussion on age fairness to future work. Similarly,
as present in Table 1, the performance gap on the MIR-ST500 dataset is not as
large as the other three datasets. We assume the reason is that the singing
recordings in MIR-ST500 were performed by professional singers, resulting in
smaller pitch distribution difference between two gender groups.

Figure 3. Framework of note-conditioned adversarial learning for singing voice
transcription.

##  5\. Bias Mitigation for SVT

###  5.1. Problem Formulation and Basic Framework

Suppose the training samples are drawn from the domain
$\mathcal{D}_{\mathcal{S}}=\\{(\bm{x}^{(n)},\bm{y}^{(n)},A^{(n)})\\}_{n=1}^{N_{\mathcal{S}}}$,
while the test data are sampled from the domain
$\mathcal{D}_{\mathcal{T}}=\\{(\bm{x}^{(n)},\bm{y}^{(n)},A^{(n)})\\}_{n=1}^{N_{\mathcal{T}}}$.
Here $\bm{x}^{(n)}$ represents the raw waveform of the singing audio,
$\bm{y}^{(n)}$ represents the ground-truth note events, and $A^{(n)}$ is the
sensitive attribute. In our study, $A=0$ represents female singers, while
$A=1$ represents male singers. As explained in Sec. 3, the basic SVT system
consists of two primary components: an acoustic encoder $\theta$ and a note
predictor $\phi$. We select self-supervised-learning (SSL) models as our
acoustic encoder due to their state-the-of-art performance in SVT tasks (Gu et
al., 2023).

Fig. 3 illustrates our proposed bias mitigation framework for the SVT task.
The input singing waveform $\bm{x}$ is first processed by the acoustic
encoder, which consists of a CNN and a transformer (Vaswani et al., 2017). We
refer readers to original papers of self-supervised-learning (SSL) models,
e.g. wav2vec 2.0 (Baevski et al., 2020), Hubert (Hsu et al., 2021), wavLM
(Chen et al., 2022b), and data2vec (Baevski et al., 2022) for details on these
models, since they are not the focus of this paper. These models are first
pre-trained on unlabeled speech data and then adapted to our singing data
using a linear probing and then full fine-tuning approach (Gu et al., 2023).
After the acoustic encoder $\theta$, we obtain the acoustic features
$\bm{z}\in\mathcal{R}^{T\times D}$, where $T$ is the number of frames and $D$
is the number of feature dimensions. The note predictor $\phi$ is
parameterized by a linear layer, and accepts the acoustic features $\bm{z}$ to
predict the frame-level labels:
$\hat{O}_{t},\hat{S}_{t},\hat{V}_{t},\hat{P}_{t}=\phi(\bm{z}_{t})$. The loss
function for SVT task in domain $\mathcal{D}_{\mathcal{S}}$ is:

(2)
$\mathcal{L}_{y}=\mathbb{E}_{(\bm{x},\bm{y})\sim\mathcal{D}_{\mathcal{S}}}\left[\mathcal{L}_{\text{SVT}}(\hat{\bm{y}},\bm{y})\right],\,\,\hat{\bm{y}}=\phi(\bm{z})=\phi\circ\theta(\bm{x}),$

where $\mathcal{L}_{\text{SVT}}$ is defined in Eq. 1. To evaluate the SVT
performance, we compute the f1-scores of the COnPOff, COnP, and COn in domain
$\mathcal{D}_{\mathcal{T}}$. We refer to these metrics as utility metrics:

(3)
$U=\mathbb{E}_{(\bm{x},\bm{y})\sim\mathcal{D}_{\mathcal{T}}}\left[\text{metric}(\hat{\bm{y}},\bm{y})\right],$

where metric can be the f1-score of COnPOff or COnP. As our objective is to
mitigate gender bias in SVT systems, we propose the following fairness
metrics, which focus on the performance disparity between two demographic
groups:

(4)
$F=\mathbb{E}_{(\bm{x},\bm{y},A=1)\sim\mathcal{D}_{\mathcal{T}}}\left[\text{metric}(\hat{\bm{y}},\bm{y})\right]-\mathbb{E}_{(\bm{x},\bm{y},A=0)\sim\mathcal{D}_{\mathcal{T}}}\left[\text{metric}(\hat{\bm{y}},\bm{y})\right],$

This definition is similar to the accuracy parity in (Zhao et al., 2019):
$P(\hat{\bm{y}}\neq\bm{y}|A=0)=P(\hat{\bm{y}}\neq\bm{y}|A=1)$ only except that
we replace accuracy with f1-score as the evaluation metric for parity. To
achieve bias mitigation, we formulate the problem as a bundle of optimization
objectives:

(5) $\left\\{\begin{array}[]{l}\max_{\theta,\phi}\min\\{F(\theta,\phi),0\\}\\\
\max_{\theta,\phi}U(\theta,\phi)\end{array}\right.$

###  5.2. Adversarial Learning for Fairness

The optimization objectives in Eq. 5 cannot be directly optimized due to the
unavailability of the test data $\mathcal{D}_{\mathcal{T}}$ during the
training of $\theta,\phi$. To mitigate the gender bias, we propose an
adversarial learning framework to learn fair acoustic representations by
assuming that the acoustic encoder cannot discriminate between two gender
groups for SVT task. To achieve this, we employ an attribute predictor $\psi$
that predicts the labels of sensitive attributes, such as gender. The goal is
to eliminate the gender information in the acoustic features $\bm{z}$ while
preserving the information necessary for predicting the note events. The
binary cross-entropy between gender predictions and ground truth gender labels
serves as the learning objective of the attribute predictor:

(6)
$\mathcal{L}_{A}=\mathbb{E}_{(\bm{x},s)\sim\mathcal{D}_{\mathcal{S}}}\left[\frac{1}{T}\sum_{t=1}^{T}l_{\text{BCE}}(\hat{A}_{t},A)\right],\,\,\hat{A}=\psi(\bm{z})=\psi\circ\theta(\bm{x}).$

To perform frame-level gender classification independently, each frame of
acoustic features, denoted as $\bm{z}_{t}$, is annotated with the same
attribute label $A$. We then average the frame-level classification loss to
obtain the song-level or utterance-level loss. We assume that the temporal
model structure in SSL models has sufficiently learned the gender
representations and thus do not require a temporal attribute predictor. Our
preliminary experiments show no empirical gains by including a temporal
attribute predictor.

From a distribution alignment perspective, our objective is to achieve gender-
invariant acoustic features, which requires that the distribution
$P(\bm{z}|A=0)$ and the distribution $P(\bm{z}|A=1)$ be similar. The loss
function in Eq. 6 serves as a proxy measure for the distance between the two
distributions, and thus, the attribute predictor $\psi$ must be trained to be
powerful enough to distinguish between $P(\bm{z}|A=0)$ and $P(\bm{z}|A=1)$.
Additionally, the acoustic encoder $\theta$ must be trained to deceive $\psi$.
Therefore, the loss function for the acoustic encoder is formulated as
$\mathcal{L}_{y}-\lambda\mathcal{L}_{A}$, where $\lambda$ is a hyper-parameter
that balances the two loss terms. Theoretically, this can be implemented by a
gradient reverse layer (GRL) (Ganin and Lempitsky, 2015). From the perspective
of fairness criteria, the adversarial learning approach enforces that
$\bm{z}\perp A$. Since $\hat{\bm{y}}=\phi(\bm{z})$, $\hat{\bm{y}}\perp A$ can
be further enforced, which is the independence criteria (Barocas et al.,
2019).

###  5.3. Note-conditioned Adversarial Learning

We have observed that there is a difference in pitch distributions between
males and females, which may contribute to the performance disparity.
Motivated by this, we propose the conditional distribution alignment, which
enforces that $P(\bm{z}|\bm{y},A=0)=P(\bm{z}|\bm{y},A=1)$. By incorporating
note labels as an extra input to the attribute predictor $\psi$, we can
eliminate the conditional dependencies between the acoustic features and the
gender labels. Since the note labels are available for both demographic
groups, we propose two variants for the design of condition. In variant 1, the
ground-truth notes $\bm{y}$ are transformed into frame-level labels
$\bm{O},\bm{S},\bm{V},\bm{P}$, which are then fed into the attribute
predictor. For variant 2, the logits of frame-level predictions
$\hat{O}_{t},\hat{S}_{t},\hat{V}_{t},\hat{P}_{t}$ are used instead.

From the perspective of fairness criteria, the variant 1 enforces $\bm{z}\perp
A|\bm{y}$ while the variant 2 enforces $\bm{z}\perp A|\hat{\bm{y}}$.
Therefore, the former can enforce $\hat{\bm{y}}\perp A|\bm{y}$, which is the
separation criteria (Barocas et al., 2019). The formulation of variant 2 is
motivated by the perspective of distribution alignment. The prediction
$\hat{\bm{y}}$ contains prior knowledge about the classifier $\phi$ and
similarities among different pitches (the predicted probabilities of other
pitch values besides the true pitch value are not zeros, similar to dark
knowledge in knowledge distillation (Hinton et al., 2015)), while the ground
truth $\bm{y}$ is one hot and has no information about other pitches.
Considering that the pitch distributions of two gender groups are different,
the proportion of male and female labels for each specific pitch value is also
different. When conducting the conditional alignment for a specific pitch, the
similarities contained in the pitch logits can assist in aligning other
pitches, resulting in an improved and more efficient conditional alignment by
the attribute predictor.

The framework of note-conditioned adversarial learning for SVT is depicted in
Fig. 3, and its variant 2 is presented in Alg. 1. The algorithm of variant 1
can be similarly derived. To train the SVT system, we follow the linear-
probing and full-finetuning strategy proposed in (Gu et al., 2023). During the
linear probing stage, only the label predictor $\phi$ and attribute predictor
$\psi$ are updated. This stage serves as a warm-up for these two models and
preserves the pre-trained features of SSL models. Then in the full-finetuning
stage, all three models $\theta,\phi,\psi$ are updated using different
learning rates. To parameterize the note-conditioned attribute predictor, we
first use two linear layers to embed each frame of acoustic features $\bm{z}$
and notes $\bm{y}$ (or $\hat{\bm{y}}$), respectively. Then the two embeddings
are concatenated and passed through two linear layers with ReLU activations.
We note that by modifying the loss function in Eq. 6, our bias mitigation
framework can be readily applied to other types of sensitive attributes, such
as $A$ is a multi-class attribute, or a continuous attribute, or a vector that
encompasses multiple attributes.

###  5.4. Fairness-Utility Trade-off

Our task involves an inherent trade-off between fairness $F$ and utility $U$,
as noted in previous literature (Kleinberg et al., 2016). This is due to the
fact that when the gender information is removed from the acoustic features,
the representations used for the SVT task may be affected. Optimizing
$\mathcal{L}_{y}-\lambda\mathcal{L}_{A}$ of the acoustic encoder $\theta$
poses a challenge as the the two loss terms have conflicting natures.
Empirical experiments demonstrate that in some cases, improvements in the
fairness metric $F$ come at a cost of reduced utility metric $U$. Our goal is
to improve the fairness metric $U$ without significantly degrading the utility
metric $U$, starting from the initial point $(F_{0},U_{0})$ without mitigating
bias. To achieve this, we introduce a tolerance hyper-parameter $\delta$ for
$U$. We aim to increase the value of $F$ as much as possible within the range
of $U>U_{0}-\delta$. Meanwhile, sacrificing utility beyond this range is not
acceptable for real-world applications. This trade-off criterion facilitates
the model selection. Typically, we set $\delta$ as $2\%$ or $5\%$ for
f1-scores of COnPOff and COnP.

Algorithm 1 Note-conditioned adversarial learning for SVT

Acoustic encoder $\theta^{(0)}$ pre-trained on speech data under SSL
objective, randomly initialized label predictor $\phi^{(0)}$ and attribute
predictor $\psi^{(0)}$, learning rates $\eta_{1},\eta_{2},\eta_{3}$ for
$\theta,\phi,\psi$, training steps $K_{1},K_{2}$ for linear probing and full
finetuning.

for $k=1$ to $K_{1}+K_{2}$ do

$\bm{z}=\theta^{(k-1)}(\bm{x})$, $\hat{\bm{y}}=\phi^{(k-1)}(\bm{z})$

$\hat{A}_{t}=\psi^{(k-1)}(\bm{z}\text{.detach}(),\hat{\bm{y}}\text{.detach}())$,
compute $\mathcal{L}_{A}$ in Eq. 6

$\psi^{(k)}=\psi^{(k-1)}-\eta_{3}\frac{\partial\mathcal{L}_{A}}{\partial\psi^{(k-1)}}$
$\triangleright$ Update $\psi$

$\hat{A}_{t}=\psi^{(k)}(\bm{z},\hat{\bm{y}})$, compute
$\mathcal{L}_{y},\mathcal{L}_{A}$ in Eq. 2 and 6

$\phi^{(k)}=\phi^{(k-1)}-\eta_{2}\frac{\partial\mathcal{L}_{y}}{\partial\phi^{(k-1)}}$
$\triangleright$ Update $\phi$

if $k\leq K_{1}$ then $\triangleright$ Update $\theta$

$\theta^{(k)}=\theta^{(k-1)}$

else

$\theta^{(k)}=\theta^{(k-1)}-\eta_{1}(\frac{\partial\mathcal{L}_{y}}{\partial\theta^{(k-1)}}-\lambda\frac{\partial\mathcal{L}_{A}}{\partial\theta^{(k-1)}})$

end if

end for

Table 3. Bias mitigation performance of note-conditioned adversarial learning. Train set | Test set | Method | COnPOff (%) | COnP (%)
---|---|---|---|---
Utility ($U$) $\uparrow$ | Fairness ($F$) $\uparrow$ | Utility ($U$) $\uparrow$ | Fairness ($F$) $\uparrow$
M4Singer | M4singer | ERM | 53.66 | \- 7.34 | 61.95 | \- 8.98
Ours | 52.48 (-1.18) | \- 3.61 (+3.73) | 60.67 (-1.28) | \- 4.21 (+4.77)
MIR-ST500 | N20EMv2 | ERM | 55.20 | \- 9.41 | 72.03 | -12.65
Ours | 53.29 (-1.91) | \- 4.14 (+5.27) | 72.71 (+0.68) | -10.82 (+1.83)
ISMIR2014 | ERM | 52.58 | -15.94 | 67.75 | -13.20
Ours | 48.18 (-4.40) | \- 8.29 (+7.65) | 65.40 (-2.35) | \- 9.08 (+4.12)
| N20EMv2 | ERM | 73.06 | \- 8.65 | 79.56 | \- 7.89
MIR-ST500 | Ours | 72.43 (-0.63) | \- 5.78 (+2.87) | 78.47 (-1.09) | \- 6.82 (+1.07)
N20EMv2 | ISMIR2014 | ERM | 59.95 | -10.45 | 73.85 | -12.36
| Ours | 59.57 (-0.38) | \- 7.49 (+2.96) | 73.33 (-0.52) | \- 7.75 (+4.61)

##  6\. Empirical Experiments

###  6.1. Bias Mitigation Performance

To evaluate in-domain fairness, we adopt the proposed note-condi- tioned
adversarial learning method on the M4Singer dataset. The SVT system is based
on wav2vec 2.0 (Baevski et al., 2020) and trained on the training split of
M4Singer and evaluated on its test split. We set the learning rates for
acoustic encoder and label predictor to be fixed at $\eta_{1}=\eta_{2}=3\times
10^{-4}$. To further evaluate out-of-domain fairness, we conduct experiments
on model1 and model3, as displayed in Table 1. We set the learning rates to
fixed values of $\eta_{1}=5\times 10^{-5},\eta_{2}=3\times 10^{-4}$ following
(Gu et al., 2023). During the bias mitigation, we select the learning rate for
the attribute predictor $\eta_{3}$ from the set $\\{0.1,0.01,0.001,0.0001\\}$
and the balancing term $\lambda$ from the set $\\{0.2,0.5,1.0,2.0\\}$. We
report the best results we can achieve. This hyper-parameter selection also
applies to the baselines we compare with in the following Sec. 6.2. We find
that further increasing the learning rate $\eta_{3}$ or the balancing term
$\lambda$ results in severe degradation in utility, even though the gender
bias seems to be eliminated, as elaborated in Sec. 6.3. Additionally, when
$\eta_{1}$ is large, we find that the best results are achieved when
$\eta_{3}$ is also large. Given our framework is based on adversarial
learning, aiming to achieve equilibrium between the acoustic encoder and the
attribute predictor, a larger learning rate for the encoder necessitates a
corresponding increase in the learning rate of the attribute predictor to
attain equilibrium111We conducted our experiments using the open-sourced repo:
https://github.com/guxm2021/SVT_SpeechBrain..

Table 3 presents a comparison between the SVT systems trained with our note-
conditioned adversarial learning (variant 2) and those trained using empirical
risk minimization (ERM) in Eq. 2 without bias mitigation. Our experiments on
the M4Singer dataset reveal that the fairness metrics improve by over 50% for
COnPOff and COnP, respectively. At the same time, the utility metrics drop
only by 1.18% and 1.28% for COnPOff and COnP, respectively. Apart from the in-
domain fairness results, we evaluate the out-of-domain fairness results on
model1 and model3 (mentioned in Sec. 4.1). Our experiments demonstrate that
applying the bias mitigation method on model1 significantly improves its
fairness. In particular, the performance disparity decreases by $50\%$ on
N20EMv2 and ISMIR2014, in terms of COnPOff. These results validate the
effectiveness of our bias mitigation method on out-of-domain data.
Furthermore, we achieve fairer SVT performance on both in-domain and out-of-
domain scenarios with minor total performance degradation when using the
state-of-the-art performing SVT system (model3).

Table 4. More bias mitigation results. Train set | Test set | Method | $\eta_{3}$ | $\lambda$ | COnPOff (%) | COnP (%)
---|---|---|---|---|---|---
Utility ($U$) $\uparrow$ | Fairness ($F$) $\uparrow$ | Utility ($U$) $\uparrow$ | Fairness ($F$) $\uparrow$
M4Singer | M4singer | ERM | - | - | 53.66 | \- 7.34 | 61.95 | \- 8.98
AL | 1.0 | 3.0 | 42.56 (-11.10) | \+ 4.65 | 49.60 (-12.35) | \+ 6.35
NCAL | 0.1 | 5.0 | 49.97 (\- 3.69) | \+ 0.39 | 57.66 (\- 4.29) | \+ 1.57
NCAL | 0.1 | 3.0 | 49.87 (\- 3.79) | \+ 1.29 | 58.53 (\- 3.42) | \+ 0.60
MIR-ST500 | N20EMv2 | ERM | - | - | 55.20 | \- 9.41 | 72.03 | -12.65
NCAL | 0.0001 | 5.0 | 45.78 (\- 9.42) | \+ 2.45 | 69.54 (\- 2.49) | \- 3.77
ISMIR2014 | ERM | - | - | 52.58 | -15.94 | 67.75 | -13.20
NCAL | 0.0001 | 5.0 | 42.24 (-10.34) | \+ 3.01 | 62.44 (\- 5.31) | \+ 1.22
| N20EMv2 | ERM | - | - | 73.06 | \- 8.65 | 79.56 | \- 7.89
MIR-ST500 | NACL | 0.001 | 5.0 | 72.28 (\- 0.78) | \- 7.78 | 78.70 (\- 0.86) | \- 8.23
N20EMv2 | ISMIR2014 | ERM | - | - | 59.95 | -10.45 | 73.85 | -12.36
| NACL | 0.001 | 5.0 | 50.95 (\- 9.00) | \+ 2.49 | 65.87 (\- 7.98) | \+ 0.64

###  6.2. Comparisons with Baselines

We compare our note-conditioned adversarial learning framework with two
baseline methods: adversarial learning (AL), and domain independent training
proposed in (Wang et al., 2020). We denote our method as “NCAL (variant 1)”
and “NCAL (variant 2)”. For AL, we keep the same configuration as our NCAL,
except that we do not feed any condition into the attribute predictor. For
domain independent training, we compare our method with two variants:
calibrated inference and miscalibrated inference, which differ in whether the
gender labels are used during the inference. These two inferences are
abbreviated as “ DInD (w/ calibr.)” and “DInD (w/o calibr.)”, respectively. We
refer readers to (Wang et al., 2020) for more technical details. We evaluate
the comparisons on model3 and report the results on the N20EMv2 and ISMIR2014
datasets.

Figure 4. Comparisons among different bias mitigation methods of Fairness-
Utility trade-off on N20EMv2 and ISMIR2014.

We present the Fairness-Utility trade-off of various bias mitigation methods
in Fig. 4. According to the optimization bundle in Eq. 5, an upper-right point
signifies a better trade-off compared to a lower-left point. Firstly, we
observe that our two variants of NACL, “NCAL (variant 1)” and “NCAL (variant
2)”, perform the best in the most cases in terms of fairness-utility trade-
off. These two variants perform similarly on N20EMv2. On the ISMIR2014
dataset, NACL variant 2 performs better for COnPOff while variant 1 exhibits
superiority for COnP. We then note that NCAL consistently outperforms the
baseline AL. The performance of AL is similar to NACL only in terms of COnP on
N20EMv2. However, in other cases, AL shows lower utility and less fairness.
With respect to DInD, it shows better fairness than NCAL only in terms of COnP
on N20EMv2 with calibration. However, its utility is more severely degraded
than NCAL. In other cases, NACL consistently outperforms DInD. Although
adversarial learning approaches are prone to instability during training
compared to non-adversrial learning approaches, such as DInD, our proposed
NCAL offers a better fairness-utility trade-off. Moreover, DInD cannot be
easily applied to the scenarios where continuous sensitive attributes or
multiple sensitive attributes are considered.

###  6.3. Further Empirical Analysis

In our experiments, we observe that setting a larger learning rate $\eta_{3}$
or the balancing term $\lambda$ leads to near-perfect gender fairness but a
drastic degradation in utility. As presented in Table 4, on M4Singer, our NCAL
method can achieve almost gender performance equality when $\lambda$ is set to
3.0 or 5.0, but at the cost of more utility degradation than the results in
Table 3. However, the baseline AL method fails to achieve such equality. We
only observe a better performance for the male group with much more utility
deterioration compared to NCAL when $\eta_{3}=1.0$ and $\lambda=3.0$. This
trend is consistently observed on model1 and model3 using NCAL for bias
mitigation. We hypothesize that increasing either $\eta_{3}$ or $\lambda$
enhances the discrimination ability of the attribute predictor, causing the
models to focus more on the fairness while neglecting the utility. As
explained in Sec. 5.4, such models may not be suitable for real-world
applications where both fairness and utility are important.

The results in Table 3 and 4 can also validate our assumptions behind the
performance disparity in SVT. Our baseline AL aims to ensure that the learnt
acoustic representations contains as little gender information as possible. In
this way, the effects of sound characteristics, which are related to gender,
can be implicitly mitigated. The improvements in terms of fairness metrics
brought by baseline AL provide evidence about our assumption that gender bias
in SVT performance is attributed to the differences of sound characterises
across different demographic groups. Additionally, when conditioning on the
pitch information, our SVT systems could achieve nearly perfect fairness, as
presented in Table 4, which further demonstrates the substantial contributions
of pitch distribution difference to the performance disparity.

##  7\. Discussion and Future Work

In addition to the primary focus on group fairness in this work, max-min
fairness (Pham et al., 2023) is also raised in certain cases. Max-min fairness
aims to minimize the worse-case error rates, offering an alternative
perspective on fairness evaluation. Our motivation stems from the objective of
enhancing user experience and mitigating potential discriminatory treatment
when utilizing SVT systems and their downstream applications. Consequently,
our main target is to strive for performance equalization across different
demographic groups. Therefore, there is less discussion on max-min fairness.
Despite this, we still observed improved performance for male data in most
cases after applying our bias mitigation approach.

In this work, we formulate our solution from the perspective of fairness. We
think the perspective of signal processing may also be beneficial to further
interpret our findings. By incorporating signal processing approaches, our
adversarial learning framework may further enhance SVT performance in terms of
both fairness and utility. We identify this as an avenue for future
exploration. Furthermore, we think our approach could be extended to consider
other sensitive attributes, such as age, race, language. Beyond demographic
groups, we also recognize that different instruments generally exhibit
distinct pitch distributions and timbre characteristics. Hence, our approach
holds potential applicability in the domain of automatic music transcription
(Gardner et al., 2022), wherein musical notes are inferred from audio signals
produced by diverse instruments.

##  8\. Conclusion

This work represents the first attempt of fairness topic within the singing-
centric deep learning community. We presented evidence that the performance of
singing voice transcription (SVT) on female data surpasses that of male data,
irrespective of the models or datasets employed. Our findings suggested that
this performance disparity is attributed to the inherent differences between
male and female singing voices, especially in pitch distribution. Given the
significance of this fairness issue, we proposed a note-conditioned
adversarial learning approach to mitigate gender bias in SVT. Specifically,
our approach leveraged an attribute predictor to learn gender-invariant
acoustic representations. By conditioning on the note events, we further
achieved conditional alignment between acoustic features of different groups.
Our results demonstrated the effectiveness of our bias mitigation method, as
it significantly improves fairness metrics while maintaining utility metrics
across both in-domain and out-of-domain data.

###### Acknowledgements.

We would like to thank anonymous reviewers for their valuable suggestions. We
also appreciated Nicholas Wong’s writing advice and Xudong Shen’s comments on
fairness. This project is funded in part by a research grant MOESOL-2021-0017
from the Ministry of Education in Singapore.

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agsm

# On “A General Framework for Pricing Asian Options Under Markov Processes"

Zhenyu Cui 111Financial Engineering division, School of Systems and
Enterprises, Stevens Institute of Technology. Email<EMAIL_ADDRESS>Chihoon
Lee222School of Business, Stevens Institute of Technology. Email:
<EMAIL_ADDRESS>Yanchu Liu333Department of Finance, Lingnan (University)
College, Sun Yat-Sen University. Email<EMAIL_ADDRESS>
###### Abstract

[CSK15] [Cai, N., Y. Song, S. Kou (2015) A general framework for pricing Asian
options under Markov processes. Oper. Res. 63(3): 540-554] made a breakthrough
by proposing a general framework for pricing both discretely and continuously
monitored Asian options under one-dimensional Markov processes. In this note,
under the setting of continuous-time Markov chain (CTMC), we explicitly carry
out the inverse $\mathcal{Z}-$transform and the inverse Laplace transform
respectively for the discretely and the continuously monitored cases. The
resulting explicit single Laplace transforms improve their Theorem 2, p.543,
and numerical studies demonstrate the gain in efficiency.

Key words:  Asian option; Markov process; Continuous-Time Markov Chain;
Laplace transform

##  1 Introduction

Asian options are popular path-dependent options actively traded in the
financial markets, yet their valuation is challenging and has attracted a
significant amount of interest in the literature ([CK12], [CLS14], [F99],
[L04]). A breakthrough was achieved in [CSK15], where the authors proposed a
general framework for pricing both discretely and continuously monitored Asian
options under one-dimensional Markov processes through a novel and elegant
functional equation approach. They employed results from [MP13], a weak
approximation scheme from the continuous-time Markov chain (CTMC) to the
Markov process, and explicitly solved the functional equations in the CTMC
case. The resulting approximation is shown to yield an efficient valuation of
Asian option prices. The analytical solutions provided in their paper are
given as a $\mathcal{Z}-$Laplace transform for the discrete case, and a two-
dimensional Laplace transform for the continuous case (Theorem 2, p.543 of
[CSK15]).

In this note, under the setting of CTMC, we explicitly carry out the inverse
$\mathcal{Z}-$transform and the inverse Laplace transform, and hence obtain
explicit single Laplace transforms for both discretely and continuously
monitored Asian options. This improves the double transforms in Theorem 2 of
[CSK15]. As a result, the theoretical complexity and computational efficiency
are improved accordingly.

##  2 Main Result

Given the filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geqslant 0},\mathbb{P})$, and
assume that we work under the risk-neutral measure $\mathbb{P}$. Denote $r$
the risk-free interest rate, and assume that the dividend rate is $0$.

One can construct a suitable CTMC to approximate a general one-dimensional
Markov process (the stock price); see [MP13] or p.544 of [CSK15]. Thus in the
following, we shall restrict our discussions to CTMC.

We consider a non-negative CTMC, $\\{X_{t}\\}_{t\geqslant 0}$ with finite
state space $\\{x_{1},\ldots,x_{N}\\}$, whose transition probability matrix is
${\bf P}(t)=(p_{ij}(t))_{N\times N}$, where
$p_{ij}(t)=\mathbb{P}(X_{t+u}=x_{j}\mid X_{u}=x_{i}),1\leqslant i,j\leqslant
N,t,u\geqslant 0$. Its transition rate matrix is ${\bf G}=(q_{ij})_{N\times
N}$, where $q_{ij}=p_{ij}^{\prime}(0),$ $1\leqslant i,j\leqslant N$. Define
${\bf x}=(x_{1},\ldots,x_{N})^{T}$, and let ${\bf I}$ denote the identity
matrix. We use ${\bf 1}$ to denote an $N\times 1$ column vector with all
entries equal to $1$. Let ${\bf D}=(d_{ij})_{N\times N}$ be a diagonal matrix
with $d_{jj}=x_{j}$, $j=1,\ldots,N$. We use $\mathds{E}^{x}[\,\cdot\,]$ to
denote the expectation conditional on $X_{0}=x$.

Consider the following payoff functions studied in equation (1), p.541 of
[CSK15]:

$\displaystyle v_{c}(t,k;x)$
$\displaystyle=\mathds{E}^{x}[(A_{t}-k)^{+}],\quad\quad\quad
v_{d}(n,k;x)=\mathds{E}^{x}[(B_{n}-k)^{+}],$

where $A_{t}:=\int_{0}^{t}X_{u}du$ and
$B_{n}:=\sum\limits_{i=0}^{n}X_{t_{i}}$. Denoting $T$ the maturity and $K$ the
strike price, the price of the continuously monitored Asian call option
$V_{c}(T,K;x)$ at time $0$ is given by $(e^{-rT}/T)v_{c}(T,TK;x)$. Similarly,
the price of the discretely monitored Asian call option $V_{d}(n,K;x)$ at time
$0$ is given by $(e^{-rT}/(n+1))v_{d}(n,(n+1)K;x)$. Henceforth, $\Delta=T/n$.
The following is our main result, which improves Theorem 2, p.543 of [CSK15].
Define

$v_{c}(n,k;{\bf x}):=(v_{c}(n,k;x_{1}),\ldots,v_{c}(n,k;x_{N}))^{T},\quad
v_{d}(n,k;{\bf x}):=(v_{d}(n,k;x_{1}),\ldots,v_{d}(n,k;x_{N}))^{T}.$

######  Proposition 2.1.

(Single Laplace transforms for fixed strike Asian options)

(i)(Discretely monitored Asian options)

Let $g_{d}(n,\theta;{\bf x}):=\int_{0}^{\infty}e^{-\theta k}v_{d}(n,k;{\bf
x})dk$, then for any complex $\theta$ such that $Re(\theta)>0$, we have

$\displaystyle g_{d}(n,\theta;{\bf x})$
$\displaystyle=\frac{1}{\theta^{2}}(e^{-\theta{\bf D}}{\bf
P}(\Delta))^{n}e^{-\theta{\bf D}}{\bf 1}-\frac{1}{\theta^{2}}{\bf
1}+\frac{{\bf x}}{\theta}\frac{1-e^{(n+1)r\Delta}}{1-e^{r\Delta}}.$ (1)

(ii) (Continuously monitored Asian options)

Let $g_{c}(t,\theta;{\bf x}):=\int_{0}^{\infty}e^{-\theta k}v_{c}(t,k;{\bf
x})dk$, then for any complex $\theta$ such that $Re(\theta)>0$, we have

$\displaystyle g_{c}(t,\theta;{\bf x})$
$\displaystyle=\frac{1}{\theta^{2}}e^{({\bf G}-\theta{\bf D})t}{\bf
1}-\frac{1}{\theta^{2}}{\bf 1}+\frac{{\bf x}}{r\theta}(e^{rt}-1).$ (2)

Proof of Proposition 2.1.

(i) Let $L_{d}(z,\theta;{\bf
x}):=\sum\limits_{n=0}^{\infty}z^{n}\int_{0}^{\infty}e^{-\theta
k}v_{d}(n,k;{\bf x})dk=\sum\limits_{n=0}^{\infty}z^{n}g_{d}(n,\theta;{\bf
x})$. From Proposition 1(i) and Theorem 2(i) of [CSK15], we have

$\displaystyle L_{d}(z,\theta;{\bf x})$
$\displaystyle=\frac{1}{\theta^{2}}(e^{\theta{\bf D}}-z{\bf
P}(\Delta))^{-1}{\bf 1}-\frac{1}{\theta^{2}(1-z)}{\bf 1}+\frac{{\bf
x}}{\theta(1-z)(1-ze^{r\Delta})}.$ (3)

From definition, we observe that $g_{d}(n,\theta;{\bf x})$ can be treated as
the coefficient of $z^{n}$ in the power series expansion of
$L_{d}(z,\theta;{\bf x})$ with respect to the transform variable $z$. This
motivates us to expand the right hand side of (3) into a power series of $z$.

Similar as in the proof of Theorem 2 in [CSK15], we can show that
$e^{\theta{\bf D}}-z{\bf P}(\Delta)$ is strictly diagonally dominant, and by
the Lévy-Desplanques theorem (Corollary $5.6.17$ of [HJ85]), we have that
$e^{\theta{\bf D}}-z{\bf P}(\Delta)$ is invertible. From Corollary $5.6.16$ of
[HJ85], if there is a matrix norm $||\cdot||$ (without loss of generality, we
can take the maximum norm, i.e., $||{\bf A}||=\max\\{|a_{ij}|\\})$ such that
$||{\bf I}-{\bf A}||<1$, then we have ${\bf A}^{-1}=\sum_{k=0}^{\infty}({\bf
I}-{\bf A})^{k}$.

In the following, we assume that the transform variable satisfies
$|z|<\min\\{1,e^{-r\Delta},1/||(e^{\theta{\bf D}})^{-1}{\bf P}(\Delta)||\\}$,
so that the power series expansions with respect to $z$ are well-defined. We
have

$\displaystyle(e^{\theta{\bf D}}-z{\bf P}(\Delta))^{-1}{\bf 1}=(e^{\theta{\bf
D}}({\bf I}-z(e^{\theta{\bf D}})^{-1}{\bf P}(\Delta)))^{-1}{\bf 1}$
$\displaystyle=({\bf I}-z(e^{\theta{\bf D}})^{-1}{\bf
P}(\Delta))^{-1}(e^{\theta{\bf D}})^{-1}{\bf 1}$ $\displaystyle=\left({\bf
I}+z(e^{\theta{\bf D}})^{-1}{\bf P}(\Delta)+z^{2}((e^{\theta{\bf D}})^{-1}{\bf
P}(\Delta))^{2}+\cdots+z^{n}((e^{\theta{\bf D}})^{-1}{\bf
P}(\Delta))^{n}+\cdots\right)(e^{\theta{\bf D}})^{-1}{\bf 1}$
$\displaystyle=(e^{\theta{\bf D}})^{-1}{\bf 1}+z(e^{\theta{\bf D}})^{-1}{\bf
P}(\Delta)(e^{\theta{\bf D}})^{-1}{\bf 1}+\cdots+z^{n}((e^{\theta{\bf
D}})^{-1}{\bf P}(\Delta))^{n}(e^{\theta{\bf D}})^{-1}{\bf 1}+\cdots$ (4)

The coefficient of $z^{n}$ in (4) is $((e^{\theta{\bf D}})^{-1}{\bf
P}(\Delta))^{n}(e^{\theta{\bf D}})^{-1}{\bf 1}$. Expand the remaining parts in
(3) as

$\displaystyle-\frac{1}{\theta^{2}(1-z)}{\bf 1}$
$\displaystyle=-\frac{1}{\theta^{2}}{\bf 1}\sum\limits_{i=0}^{\infty}z^{i},$
(5)

and

$\displaystyle\frac{{\bf x}}{\theta(1-z)(1-ze^{r\Delta})}$
$\displaystyle=\frac{{\bf
x}}{\theta}\sum\limits_{i=0}^{\infty}z^{i}\times\sum\limits_{j=0}^{\infty}z^{j}e^{jr\Delta}.$
(6)

It is clear that the coefficient of $z^{n}$ in (5) is
$-\frac{1}{\theta^{2}}{\bf 1}$, and the coefficient of $z^{n}$ in (6) is
$\frac{{\bf x}}{\theta}\frac{1-e^{(n+1)r\Delta}}{1-e^{r\Delta}}$.

Thus the inverse $\mathcal{Z}-$transform of $L_{d}(z,\theta;x)$ is given by

$\displaystyle g_{d}(n,\theta;{\bf x})$
$\displaystyle=\mathcal{Z}^{-1}\left(L_{d}(z,\theta;x)\right)=\frac{1}{\theta^{2}}((e^{\theta{\bf
D}})^{-1}{\bf P}(\Delta))^{n}(e^{\theta{\bf D}})^{-1}{\bf
1}-\frac{1}{\theta^{2}}{\bf 1}+\frac{{\bf
x}}{\theta}\frac{1-e^{(n+1)r\Delta}}{1-e^{r\Delta}},$ (7)

and, together with $(e^{\theta{\bf D}})^{-1}=e^{-\theta{\bf D}}$, it completes
the proof of part (i).

(ii) Let $L_{c}(\mu,\theta;{\bf x}):=\int_{0}^{\infty}e^{-\mu
t}\int_{0}^{\infty}e^{-\theta k}v_{c}(t,k;{\bf
x})dkdt=\int_{0}^{\infty}e^{-\mu t}g_{c}(t,\theta;{\bf x})dt$. From
Proposition 1(ii) and Theorem 2(ii) of [CSK15], we have

$\displaystyle L_{c}(\mu,\theta;{\bf x})$
$\displaystyle=\frac{1}{\theta^{2}}m(\mu,\theta;{\bf
x})-\frac{1}{\theta^{2}\mu}{\bf 1}+\frac{{\bf x}}{\theta\mu(\mu-r)},$ (8)

where $m(\mu,\theta;{\bf x}):=(\theta{\bf D}+\mu{\bf I}-{\bf G})^{-1}{\bf 1}$.

It can be shown that $\theta{\bf D}+\mu{\bf I}-{\bf G}$ is strictly diagonally
dominant, and thus invertible by the Lévy-Desplanques theorem. We assume
$|\mu|>\max\\{||{\bf G}-\theta{\bf D}||,0\\}$, so that the following power
series expansions with respect to $\mu$ are well-defined. We obtain

$\displaystyle(\theta{\bf D}+\mu{\bf I}-{\bf G})^{-1}{\bf 1}$
$\displaystyle=\left(\mu\left({\bf I}-\left(-\frac{\theta}{\mu}{\bf
D}+\frac{1}{\mu}{\bf G}\right)\right)\right)^{-1}{\bf 1}$
$\displaystyle=\frac{1}{\mu}\left({\bf I}-\frac{{\bf G}-\theta{\bf
D}}{\mu}\right)^{-1}{\bf 1}$ $\displaystyle=\frac{1}{\mu}\left({\bf
I}+\frac{{\bf G}-\theta{\bf D}}{\mu}+\left(\frac{{\bf G}-\theta{\bf
D}}{\mu}\right)^{2}+\cdots+\left(\frac{{\bf G}-\theta{\bf
D}}{\mu}\right)^{n}+\cdots\right){\bf 1}$ $\displaystyle=\frac{{\bf
1}}{\mu}+\frac{({\bf G}-\theta{\bf D}){\bf 1}}{\mu^{2}}+\frac{({\bf
G}-\theta{\bf D})^{2}{\bf 1}}{\mu^{3}}+\cdots+\frac{({\bf G}-\theta{\bf
D})^{n}{\bf 1}}{\mu^{n+1}}+\cdots.$ (9)

It is clear from (9) that the inverse Laplace transform of $m(\mu,\theta;{\bf
x})$ with respect to $\mu$ is given by

$\displaystyle\mathcal{L}_{\mu}^{-1}(m(\mu,\theta;{\bf x}))$
$\displaystyle=\sum\limits_{i=0}^{\infty}\frac{({\bf G}-\theta{\bf D})^{i}{\bf
1}}{i!}t^{i}=e^{({\bf G}-\theta{\bf D})t}{\bf 1}.$ (10)

It is easy to identify the inverse Laplace transform with respect to $\mu$ of
the two remaining terms in (8) as

$\displaystyle\mathcal{L}_{\mu}^{-1}\left(-\frac{1}{\theta^{2}\mu}{\bf
1}\right)$ $\displaystyle=-\frac{1}{\theta^{2}}{\bf
1},\quad\quad\quad\mathcal{L}_{\mu}^{-1}\left(\frac{{\bf
x}}{\theta\mu(\mu-r)}\right)=\frac{{\bf x}}{r\theta}(e^{rt}-1).$ (11)

From (10) and (11), we have

$\displaystyle g_{c}(t,\theta;{\bf x})$
$\displaystyle=\mathcal{L}_{\mu}^{-1}\left(L_{c}(\mu,\theta;{\bf
x})\right)=\frac{1}{\theta^{2}}e^{({\bf G}-\theta{\bf D})t}{\bf
1}-\frac{1}{\theta^{2}}{\bf 1}+\frac{{\bf x}}{r\theta}(e^{rt}-1).$ (12)

This completes the proof. $\Box$

######  Remark 2.1.

We note that it is very difficult, if not impossible, to further carry out
Laplace inversion of the function $g_{d}(n,\theta;{\bf x})$ or
$g_{c}(t,\theta;{\bf x})$ and hence obtain closed-form expressions for the
option prices. For $g_{d}(n,\theta;{\bf x})$, we can easily identify the
inverse Laplace transform of $\frac{1}{\theta^{2}}e^{-\theta{\bf D}}$ and
$e^{-\theta{\bf D}}{\bf P}(\Delta)$, respectively, but it is challenging to
obtain a tractable Laplace inversion of $(e^{-\theta{\bf D}}{\bf
P}(\Delta))^{n}$. For $g_{c}(t,\theta;{\bf x})$, the major difficulty for
carrying out further Laplace inversion lies in the matrix exponential
$e^{({\bf G}-\theta{\bf D})t}$, and the fact that ${\bf G}$ and ${\bf D}$ are
in general not commutable. Based on these observations, we argue that the
results in Proposition 2.1 may not be further improved in general.

######  Remark 2.2.

There are two ways of computing the expression given in (1). In the “forward"
way, we first compute the $n$th power of $e^{-\theta{\bf D}}{\bf P}(\Delta)$
and then multiply it by the vector $e^{-\theta{\bf D}}{\bf 1}$. The complexity
would be $O(N^{3}n)$, since the multiplication of two $N-$dimensional square
matrices has cost of $O(N^{3})$. In the “backward" way, we multiply the matrix
$e^{-\theta{\bf D}}{\bf P}(\Delta)$ by the vector $e^{-\theta{\bf D}}{\bf 1}$
first and the result is again a vector. The complexity of this operation is
only $O(N^{2})$, because: (1) $e^{-\theta{\bf D}}$ is a diagonal matrix and
therefore computing $(e^{\theta{\bf D}})^{-1}{\bf P}(\Delta)$ costs only
$O(N)$, and (2) Multiplication of an $N-$dimensional square matrix by a vector
costs $O(N^{2})$. Repeating this procedure $n$ times, we obtain
$((e^{\theta{\bf D}})^{-1}{\bf P}(\Delta))^{n}(e^{\theta{\bf D}})^{-1}{\bf 1}$
with a total cost of $O(N^{2}n)$, which reduces the complexity by $O(N)$ times
compared with the “forward" way. Although in the numerical tests conducted in
the following section, we did not observe significant differences between the
realized running time of these two ways of implementation via Matlab, we
believe that the “backward" way will have some potential benefits in more
computationally intensive settings.

##  3 Numerical Results

We follow the two-step procedure proposed in §5 (p. 544) of [CSK15] to compute
the Asian option prices, except that in the second step we only need to invert
the single Laplace transforms given in (1) and (2). Note that inverting a
single Laplace transform is well studied, and we use the equations (4.6) and
(6.26) in [AW92]. Numerical results are collected under different models,
including the CIR model, the CEV model, the double-exponential jump diffusion
(DEJD) model, the Merton’s jump diffusion (MJD) model, and the Carr-Geman-
Madan-Yor (CGMY) model. Parameter settings for each model involved are exactly
the same as those in §5 of [CSK15]. All numerical experiments in this note are
conducted using Matlab R2014b on a laptop equipped with an Intel Core 2
i7-4510U CPU @ 2.00 GHz 2.60 GHZ and 8 GB of RAM.

The following tables report the Asian option prices computed via our single
Laplace transform formulas, and compare them with certain benchmarks and
results based on the double transform methods, both of which are directly
taken from [CSK15]. Relative errors of our method compared to the benchmarks
are also presented. We also record CPU running times of our method.

We observe similar patterns across the tables: (1) The option prices yielded
by our single Laplace transform inversions are very close to those from the
double transform inversions in [CSK15], while it takes less realized running
time in all cases. (2) For discretely monitored Asian options, the running
time of our method increases with the number of monitoring points, i.e., $n$,
but at a slower rate than that reported in [CSK15].

##  4 Conclusion

In this note, under the setting of CTMC, we obtain explicit single Laplace
transforms for prices of discretely and continuously monitored Asian options.
This improves Theorem 2, p.543 of [CSK15], and numerical studies demonstrate
the gain in efficiency.

## References

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  * [7] [Fusai and Meucci]Fusai and Meucci2008FM08 Fusai, G., and A. Meucci (2008): “Pricing discretely monitored Asian options under the Levy processes,” _J. Banking and Finance_ , 32(10), 2076–2088.
  * [8] [Horn and Johnson]Horn and Johnson1985HJ85 Horn, R., and C. Johnson (1985): “Matrix Analysis,” _Cambridge University Press_.
  * [9] [Linetsky]Linetsky2004L04 Linetsky, V. (2004): “Spectral expansions for Asian (average price) options,” _Oper. Res._ , 52(6), 856–867.
  * [10] [Mijatović and Pistorius]Mijatović and Pistorius2013MP13 Mijatović, A., and M. Pistorius (2013): “Continuously monitored barrier options under Markov processes,” _Math. Finance._ , 23(1), 1–38.
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Table 1: Asian options under the CIR model.

$K$ | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$) | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$)
---|---|---|---|---|---|---|---|---
| $n=12$ | $n=25$
0.90 | 0.21279 | 0.21257 | 0.21300 | 0.10 | 0.21428 | 0.21406 | 0.21449 | 0.10
0.95 | 0.18659 | 0.18638 | 0.18674 | 0.08 | 0.18810 | 0.18789 | 0.18823 | 0.07
1.00 | 0.16282 | 0.16264 | 0.16297 | 0.09 | 0.16432 | 0.16414 | 0.16445 | 0.08
1.05 | 0.14140 | 0.14126 | 0.14158 | 0.13 | 0.14287 | 0.14273 | 0.14303 | 0.11
1.10 | 0.12223 | 0.12213 | 0.12245 | 0.18 | 0.12365 | 0.12355 | 0.12385 | 0.17
| $n=50$ | $n=100$
0.90 | 0.21501 | 0.21406 | 0.21521 | 0.09 | 0.21538 | 0.21515 | 0.21558 | 0.09
0.95 | 0.18883 | 0.18862 | 0.18896 | 0.07 | 0.18920 | 0.18899 | 0.18933 | 0.07
1.00 | 0.16505 | 0.16487 | 0.16517 | 0.07 | 0.16542 | 0.16524 | 0.16554 | 0.07
1.05 | 0.14359 | 0.14344 | 0.14374 | 0.10 | 0.14395 | 0.14381 | 0.14410 | 0.10
1.10 | 0.12434 | 0.12424 | 0.12453 | 0.15 | 0.12470 | 0.12460 | 0.12489 | 0.15
| $n=250$ | $n=+\infty$
0.90 | 0.21560 | 0.21537 | 0.21581 | 0.10 | 0.21575 | 0.21552 | 0.21592 | 0.08
0.95 | 0.18943 | 0.18922 | 0.18956 | 0.07 | 0.18958 | 0.18937 | 0.18976 | 0.09
1.00 | 0.16565 | 0.16547 | 0.16578 | 0.08 | 0.16580 | 0.16562 | 0.16600 | 0.12
1.05 | 0.14418 | 0.14403 | 0.14432 | 0.10 | 0.14433 | 0.14418 | 0.14457 | 0.17
1.10 | 0.12492 | 0.12481 | 0.12510 | 0.14 | 0.12506 | 0.12496 | 0.12534 | 0.22

Note. Pricing Asian options under the CIR model via our single transform CTMC
approximation with $N=50$. The parameter settings are the same as those in
[FMR08] and also in §5.1 of [CSK15]. The columns “Benchmark" are taken from
[FMR08], whose values are computed from their analytical solutions. The
columns “Cai et al." are taken from the Table 3 of [CSK15]. Results based on
our single Laplace transforms are presented in the columns “CTMC." The columns
“Rel. err.($\%$)" document the relative errors of our method compared with the
benchmark values. The realized computing times to output the option price
based on our method are about 0.009, 0.011, 0.013, 0.016, and 0.023 seconds
for $n$ = 12, 25, 50, 100, and 250, respectively, and about 0.031 seconds for
$n=+\infty$ (i.e., the continuously monitored Asian options).

Table 2: Asian options under the CEV model.

| (I) Discretely monitored Asian options | (II) Continuously monitored Asian options
---|---|---
|  under the CEV model ($n=250$) | under the CEV model
$K$ | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$) | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$)
| $\beta=0.25$ | $\beta=0.25$
80 | 21.60167 | 21.60974 | 21.60980 | 0.04 | 21.59408(0.00468) | 21.61076 | 21.61093 | 0.08
90 | 13.15550 | 13.15548 | 13.15551 | 0.00 | 13.15109(0.00425) | 13.15931 | 13.15920 | 0.07
100 | 6.84034 | 6.82619 | 6.82623 | 0.21 | 6.83859(0.00340) | 6.83128 | 6.83146 | 0.10
110 | 3.07180 | 3.05691 | 3.05697 | 0.48 | 3.07333(0.00239) | 3.06138 | 3.06136 | 0.39
120 | 1.22841 | 1.22497 | 1.22502 | 0.28 | 1.23175(0.00154) | 1.22762 | 1.22765 | 0.33
| $\beta=-0.25$ | $\beta=-0.25$
80 | 21.67122 | 21.67979 | 21.67979 | 0.04 | 21.66618(0.00464) | 21.68104 | 21.68112 | 0.07
90 | 13.26903 | 13.26768 | 13.26768 | 0.01 | 13.26741(0.00417) | 13.27147 | 13.27137 | 0.03
100 | 6.84853 | 6.83407 | 6.83409 | 0.21 | 6.85150(0.00327) | 6.83920 | 6.83932 | 0.18
110 | 2.92962 | 2.91597 | 2.91599 | 0.46 | 2.93166(0.00221) | 2.92049 | 2.92050 | 0.38
120 | 1.04072 | 1.04152 | 1.04154 | 0.08 | 1.04453(0.00131) | 1.04429 | 1.04420 | 0.03
| $\beta=-0.5$ | $\beta=-0.5$
80 | 21.71428 | 21.72237 | 21.72238 | 0.04 | 21.71118(0.00465) | 21.72370 | 21.72379 | 0.06
90 | 13.32877 | 13.32675 | 13.32676 | 0.02 | 13.32850(0.00416) | 13.33052 | 13.33044 | 0.01
100 | 6.85365 | 6.83904 | 6.83906 | 0.21 | 6.85984(0.00324) | 6.84420 | 6.84429 | 0.23
110 | 2.86119 | 2.84823 | 2.84824 | 0.45 | 2.86666(0.00215) | 2.85276 | 2.85281 | 0.48
120 | 0.95542 | 0.95803 | 0.95805 | 0.28 | 0.95995(0.00122) | 0.96084 | 0.96070 | 0.08

Note. Pricing Asian options under the CEV model via our single transform CTMC
approximation with $N=50$. All parameter settings are the same as those in
§5.2 of [CSK15]. Part (I) compares our results (i.e., the column “CTMC") with
the asymptotic expansion numerical prices in [CLS14] (i.e., the column
“Benchmark") and results via the double transform method in [CSK15] (i.e., the
column “Cai et al."). These two reference columns are taken from Part (I) of
Table 4 in [CSK15]. It takes about 0.024 seconds on average to output one
price by our method. Part (II) collects results for continuously monitored
Asian options. The benchmark values, taken from Part (II) of Table 4 in
[CSK15], are based on Monte Carlo simulations with numbers in the brackets
indicating standard deviations. The column “Cai et al.", taken from the same
reference, are obtained based on the double transform method in [CSK15]. It
takes about 0.049 seconds on average to output a price via our method.

Table 3: Asian options under the DEJD model.

(I) Discretely monitored Asian options under the DEJD model
---
$n$ | $K$ | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$)
12 | 90 | 12.71236 | 12.70857 | 12.70873 | 0.03
| 100 | 5.01712 | 5.01254 | 5.01263 | 0.09
| 110 | 1.04142 | 1.03988 | 1.03989 | 0.15
50 | 90 | 12.74369 | 12.74016 | 12.74025 | 0.03
| 100 | 5.05809 | 5.05358 | 5.05371 | 0.09
| 110 | 1.06878 | 1.06725 | 1.06725 | 0.14
250 | 90 | 12.75241 | 12.74875 | 12.74881 | 0.03
| 100 | 5.06949 | 5.06491 | 5.06504 | 0.09
| 110 | 1.07646 | 1.07489 | 1.07489 | 0.15
(II) Continuously monitored Asian options under the DEJD model
$K$ | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$) | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$)
| $\sigma=0.05$ | $\sigma=0.1$
90 | 13.47952 | 13.46823 | 13.47752 | 0.01 | 13.55964 | 13.56418 | 13.56389 | 0.03
95 | 9.16588 | 9.18472 | 9.16582 | 0.0007 | 9.41962 | 9.42931 | 9.42470 | 0.05
100 | 5.38761 | 5.37399 | 5.38772 | 0.002 | 5.91537 | 5.91365 | 5.91780 | 0.04
105 | 2.72681 | 2.71628 | 2.72530 | 0.06 | 3.35071 | 3.34830 | 3.35143 | 0.02
110 | 1.28264 | 1.30224 | 1.28198 | 0.05 | 1.74896 | 1.75431 | 1.74943 | 0.03
| $\sigma=0.2$ | $\sigma=0.3$
80 | 14.17380 | 14.17589 | 14.17568 | 0.01 | 15.33688 | 15.33545 | 15.33575 | 0.007
90 | 10.53795 | 10.53824 | 10.53807 | 0.001 | 12.10723 | 12.10414 | 12.10441 | 0.01
100 | 7.48805 | 7.48621 | 7.48648 | 0.02 | 9.35336 | 9.34883 | 9.34914 | 0.02
110 | 5.09001 | 5.08708 | 5.08736 | 0.05 | 7.08059 | 7.07520 | 7.07551 | 0.07
120 | 3.32061 | 3.31802 | 3.31789 | 0.08 | 5.26109 | 5.25561 | 5.25589 | 0.10
| $\sigma=0.4$ | $\sigma=0.5$
80 | 16.81490 | 16.81130 | 16.81958 | 0.03 | 18.46259 | 18.45288 | 18.46148 | 0.006
90 | 13.87995 | 13.87460 | 13.88190 | 0.01 | 15.75006 | 15.73859 | 15.74575 | 0.03
100 | 11.33257 | 11.32581 | 11.33275 | 0.002 | 13.36027 | 13.34737 | 13.35386 | 0.05
110 | 9.16131 | 9.15366 | 9.16048 | 0.009 | 11.27716 | 11.26330 | 11.26950 | 0.07
120 | 7.34063 | 7.33266 | 7.33944 | 0.02 | 9.47826 | 9.46389 | 9.47003 | 0.09

Note. Pricing Asian options under the DEJD model via our single transform CTMC
approximation with $N=50$. All parameter settings are the same as those in
§5.3 of [CSK15]. Part (I) compares our results (i.e., the column “CTMC") with
numerical prices obtained by the recursive algorithm in [FM08] (i.e., the
column “Benchmark") and results via the double transform method in [CSK15]
(i.e., the column “Cai et al."). These two reference columns are taken from
the Part (I) of Table 5 in [CSK15]. The CPU times to output the option prices
based on our method are about 0.015, 0.018, and 0.036 seconds for $n$ = 12,
50, and 250, respectively. Part (II) collects results for continuously
monitored Asian options, where the benchmark values are from [CK12]. The
column “Cai et al.", taken from Part (II) of Table 5 in [CSK15], is based on
the double transform method. It takes about 0.21 seconds to compute one price
via our method for the continuously monitored Asian options.

Table 4: Asian options under the MJD model.

(I) Discretely monitored Asian options under the MJD model
---
$n$ | $K$ | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$)
12 | 90 | 12.71066 | 12.70620 | 12.70636 | 0.03
| 100 | 5.01127 | 5.00539 | 5.00546 | 0.12
| 110 | 1.05162 | 1.04941 | 1.04940 | 0.21
50 | 90 | 12.74093 | 12.73659 | 12.73665 | 0.03
| 100 | 5.05246 | 5.04654 | 5.04667 | 0.11
| 110 | 1.07959 | 1.07736 | 1.07733 | 0.21
250 | 90 | 12.74917 | 12.74485 | 12.74490 | 0.03
| 100 | 5.06381 | 5.05790 | 5.05803 | 0.11
| 110 | 1.08740 | 1.08515 | 1.08512 | 0.21
(II) Continuously monitored Asian options under the MJD model
$K$ | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$)
90 | 12.74857(0.00371) | 12.74705 | 12.74699 | 0.01
100 | 5.05974(0.00399) | 5.05740 | 5.06095 | 0.02
110 | 1.08413(0.00280) | 1.09235 | 1.08712 | 0.28

Note. Pricing Asian options under the MJD model via our single transform CTMC
approximation with $N=50$. All parameter settings are the same as those in
§5.4 of [CSK15]. Part (I) compares our results (i.e., the column “CTMC") with
numerical prices obtained by the recursive algorithm in [FM08] (i.e., the
column “Benchmark") and results via the double transform method in [CSK15]
(i.e., the column “Cai et al."). These two reference columns are taken from
the Part (I) of Table 6 in [CSK15]. The CPU times to output the option price
based on our method are about 0.008, 0.011, and 0.025 seconds for $n$ = 12,
50, and 250, respectively. Part (II) collects results for continuously
monitored Asian options. The benchmark values, taken from Part (II) of Table 6
in [CSK15], are based on Monte Carlo simulations with numbers in the brackets
indicating standard deviations. The column “Cai et al.", taken from the same
reference, is based on the double transform method. It takes about 0.047
seconds to compute one price via our method for the continuously monitored
Asian options.

Table 5: Asian options under the CGMY model.

(I) Discretely monitored Asian options under the CGMY model
---
$n$ | $K$ | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$)
12 | 90 | 12.70625 | 12.70406 | 12.70318 | 0.02
| 100 | 5.03492 | 5.02551 | 5.02612 | 0.17
| 110 | 1.02115 | 1.01464 | 1.01304 | 0.79
50 | 90 | 12.73854 | 12.73745 | 12.73644 | 0.02
| 100 | 5.07570 | 5.06651 | 5.06716 | 0.17
| 110 | 1.04674 | 1.04012 | 1.03854 | 0.78
250 | 90 | 12.74737 | 12.74653 | 12.74549 | 0.01
| 100 | 5.08694 | 5.07783 | 5.07849 | 0.17
| 110 | 1.05389 | 1.04725 | 1.04567 | 0.78
(II) Continuously monitored Asian options under the CGMY model
$K$ | Benchmark | Cai et al. | CTMC |  Rel. err.($\%$)
90 | 12.74788(0.00396) | 12.74689 | 12.74780 | 0.0006
100 | 5.08865(0.00405) | 5.08019 | 5.08138 | 0.14
110 | 1.05810(0.00280) | 1.06028 | 1.05751 | 0.06

Note. Pricing Asian options under the CGMY model via our single transform CTMC
approximation with $N=50$. All parameter settings are the same as those in
§5.6 of [CSK15]. Part (I) compares our results (i.e., the column “CTMC") with
numerical prices obtained by the recursive algorithm in [FM08] (i.e., the
column “Benchmark") and results via the double transform method in [CSK15]
(i.e., the column “Cai et al."). These two reference columns are taken from
the Part (I) of Table 9 in [CSK15]. The CPU times to output the option price
based on our method are about 0.015, 0.018, and 0.025 seconds for $n$ = 12,
50, and 250, respectively. Part (II) collects results for continuously
monitored Asian options. The benchmark values, taken from Part (II) of Table 9
in [CSK15], are based on Monte Carlo simulations with numbers in the brackets
indicating standard deviations. The column “Cai et al.", taken from the same
reference, is based on the double transform method. It takes about 0.092
seconds to compute one price via our method for the continuously monitored
Asian options.