Preface
The authors would like to dedicate this paper to Peter Carr, with whom they have had several discussions in the past about swaps, including variance, volatility, covariance, and correlation swaps.
The second author’s first experience with swaps was in Vancouver in 2002 on a 5-day Industrial Problem Solving Workshop (IPSW) organized by PIMS. The problem was proposed by RBC Financial Group (see (
RBC Financial Group Team 2002)), and it concerned the pricing of swaps involving the so-called pseudo-statistics, namely, the pseudo-variance, pseudo-covariance, pseudo-volatility, and pseudo-correlation. The team consisted of nine graduate students, with whom we solved the problem and prepared a report (see (
Badescu et al. 2002)). I would like to thank them all for a very productive collaboration during this time. I would also like to mention that the idea to use the change in time method for solving swap pricing problems (see (
Swishchuk 2004)) actually appeared to the second author during this workshop.
Peter was always humorous regarding any topic during our lighthearted conversations, and there is no exception for the latter one. I remember when I told him that we had some results on pricing pseudo-variance, pseudo-covariance, pseudo-volatility, and pseudo-correlation swaps (discretely defined as volatility derivatives), which we obtained during this IPSW in Vancouver in 2002, together with nine graduate students and prepared a paper, he immediately sent me an email saying “Please, send me your pseudo-paper!”.
1. Introduction
The simplest measure of a stock’s risk or uncertainty is its volatility. The volatility, which we denote as , is the annualized standard deviation of the stock’s returns during the period of interest, where the subscript R denotes the observed or “realized” volatility. Volatility swaps, or realized volatility forward contracts, are the easiest way to trade volatility because they provide pure exposure to volatility (and only to volatility). Therefore, volatility swaps are forward contracts on future realized stock volatility, and variance swaps are similar contracts on variance, the square of the future volatility. Both these instruments provide an easy way for investors to gain exposure to the future level of volatility.
Even though volatility is commonly talked about amongst options market participants, variance and volatility squared have more fundamental significance. A pricing covariance swap, from a theoretical point of view, is similar to a pricing variance swap. A covariance swap is a covariance forward contact of two underlying assets.
The following is in reference to the work of (
Carr and Lee 2009). In the earlier part of the 1990s, one can find the first evidence of volatility derivatives being traded in the OTC market. Both variance and volatility derivatives were traded sporadically between 1993 and 1998. The first contracts to enjoy any liquidity were variance swaps. Due to historically high implied volatility in 1998, one can see the emergence of variance swaps in that year. The so-called covariance and correlation swaps have become available as well (
Brockhaus and Long 2000). Some papers advocate for the creation of volatility indices and other financial products, whose payoff is tied to these indices, including the VIX index. We would also like to mention that (
Gastineau 1977) and (
Galai 1979) introduced option indices similar to stock indices. Futures and option contracts based on realized volatility indices were proposed by (
Brenner and Galai 1989). (
Whaley 1993) introduced derivative contracts written on the VIX index, while (
Fleming et al. 1995) described the construction of the original VIX index. More information about the development of a volatility indices can be found in (
Carr and Lee 2009).
Extending the Black Scholes model for stochastic volatility, some papers proposed a parametric process. Specifically, (
Grünbichler and Longstaff 1996) valued options and futures considering a continuous time GARCH process for variance, while
Brenner and Galai (
1993,
1996) created an evaluation model for options on volatility via a binomial process. Some papers proposed an alternative non-parametric approach even before swaps on variance had been introduced. In this direction, the first paper was a working paper by Neuberger (
Neuberger 1990), and it was published in 1994 (
Neuberger 1994). Neuberger assumed that the price of stocks is continuous over time and that the limit of the sum of squared returns should exist, in place of assuming a specific stochastic process. Independently of Neuberger, (
Dupire 1993) developed the same argument. Building on a prior working paper by (
Dupire 1996), (
Carr and Madan 2001) completed the task of developing a robust replicating strategy for continuously monitored variance swap. The next main breakthrough in the robust pricing of volatility derivatives occurred in path-breaking work by (
Dupire 1996) and by (
Derman et al. 1997). Moments swaps were studied in (
Schoutens 2005). Modelling and pricing of variance swaps for stochastic volatilities with delay and for multi-factor stochastic volatilities with delay were considered in (
Swishchuk 2005) and (
Swishchuk 2006), respectively. Extensive overview on volatility derivatives may be found in (
Carr and Lee 2009).
The Heston model is amongst the most popular stochastic models for the pricing of volatility swaps, but new approaches and modifications are continually being suggested. In (
Salvi and Swishchuk 2014), the Heston model is used along with a new probabilistic approach to study volatility swaps, while a variance drift-adjusted version of the Heston model is presented in (
Swishchuk and Vadori 2014). Models and processes that incorporate jumps are also increasingly popular for modeling the fluctuations of financial markets. Amongst these jump processes is the Barndorff-Nielsen and Shephard (BN-S) model, which was used in (
Benth et al. 2007) to analyze swaps written on powers of realized volatility. The arbitrage-free pricing of variance and volatility swaps under the BN-S model is examined in (
Habtemicael and SenGupta 2016b), while covariance swaps under the BN-S model are looked at in (
Habtemicael and Sengupta 2016a). (
Issaka and SenGupta 2017) further examine the BN-S model by calculating the bounds of the arbitrage-free variance swap price. Derivative asset analysis and pricing of European style options under the BN-S model is found in (
Sengupta 2016).
The main problem in pricing variance, volatility, and correlation swaps is how to determine the evolution of the stochastic processes for the underlying assets and their volatilities. Thus, sometimes it is simpler to consider the pricing of swaps by so-called pseudo-statistics, namely, the pseudo-variance, -covariance, -volatility, and -correlation (
Badescu et al. 2002;
RBC Financial Group Team 2002).
The main motivation of this paper is to consider the pricing of swaps based on pseudo-statistics, instead of stochastic models, and to compare this approach with the most popular stochastic volatility model in the Cox–Ingresoll–Ross (CIR) model. Within this paper, we will demonstrate how to value different types of swaps (variance, volatility, covariance, and correlation swaps) using pseudo-statistics (pseudo-variance, pseudo-volatility, pseudo-correlation, and pseudo-covariance). As a result, we will arrive at a method for pricing swaps that does not rely on any stochastic models for a stochastic stock price or stochastic volatility, and instead relies on data/statistics. A data/statistics-based approach to swap pricing is very different from stochastic volatility models such as the Cox–Ingresoll–Ross (CIR) model, which, in comparison, follows a stochastic differential equation (see (
Swishchuk 2004)). Although there are many other stochastic models that provide an approach to calculating the price of swaps, we will use the CIR model for comparison within this paper, due to the popularity of the CIR model. Therefore, in this paper, we will compare the CIR model approach to pricing swaps to the pseudo-statistic approach to pricing swaps, in order to compare a stochastic model to the data/statistics-based approach to swap pricing that is developed within this paper.
In
Section 2, we present the analytical closed-form formulas of pseudo-variance, pseudo-volatility, pseudo-correlation, and pseudo-covariance.
Section 3 defines a pseudo-swap and presents the associated equations. In
Section 4, four data sets that are to be used in the numerical examples that follow are defined.
Section 5 defines the logarithmic return of a stock price, while in
Section 6, we calculate the expected sample variance and expected sample volatility. The pseudo-variance, pseudo-volatility, pseudo-covariance, and pseudo-correlation are calculated in
Section 7,
Section 8,
Section 10 and
Section 11, respectively. In
Section 9, we calculate the expected sample covariance of the two stock data sets presented in
Section 4.
Section 12 outlines the data sets and purpose of the comparison between the pseudo-statistic approach and the approach based on the Cox–Ingresoll–Ross (CIR) model. The results of the numerical comparison between the variance and volatility swap payoffs when using the CIR model and the pseudo-statistic approach are shown in
Section 13 and
Section 14, respectively. The paper is concluded in
Section 15.
Within this paper, we will make use of statistical analysis, martingales, and stochastic calculus to provide analytical closed-form formulas of the pricing of swaps and calculate numerical values using these formulas. In addition, we have also used data analysis methods in the selection and preparation of the data sets that are presented in
Section 4.
7. Realized Pseudo-Volatility Square and Pseudo-Variance Swap Payoff
In order to calculate the realized pseudo-volatility square and the payoff of a pseudo-variance swap, we must first define the following parameters:
- –
Position Taken:
- –
Converting Parameter:
- –
Strike Price: .
Using these parameters, we can then utilize the following pair of equations found in (
Badescu et al. 2002), in order to calculate the realized pseudo-volatility square and pseudo-variance swap payoff:
Realized Pseudo-Volatility Square:
Payoff of Pseudo-Variance Swap:
We will use Equations (29) and (30), along with the defined parameters above, in the following sections, in order to calculate realized pseudo-volatility squares and pseudo-variance swap payoffs using the 6-month data sets introduced in
Section 4.1.
7.1. AAPL: Realized Pseudo-Volatility Square and Pseudo-Variance Swap Payoff
Using the daily closing prices for AAPL in the 6-month period from November 2022 to May 2023, along with Equations (15), (16) and (29), we can calculate the realized pseudo-volatility square of AAPL to be the following:
As previously calculated in
Section 5.1, the mean of
is 0.0021, which is the value we used to calculate the realized pseudo-volatility square above. However, we can examine the relationship between the realized pseudo-volatility square of AAPL and the arithmetic mean of the logarithmic AAPL stock returns by letting
vary around its true value. As a result, we can produce
Figure 9, which is below.
Now, using the realized pseudo-volatility square from November 2022 to May 2023 of the AAPL stock, which we calculated above, we can calculate the payoff of a pseudo-variance swap, whose underlying asset is the variance of the AAPL stock. We will assume the pseudo-variance swap has a maturity of 6 months, and that we are interested in the payoff associated with taking a long position on the swap. Additionally, we will assume that the payoff is calculated using a converting parameter of
$1 per unit of pseudo-statistic, and we will use the expected sample variance of AAPL calculated in
Section 6.1 as the strike price. Therefore, we now have the following parameter values:
- –
Maturity Date:
- –
Position Taken: .
- –
Converting Parameter:
- –
Strike Price: .
Using these parameters, Equation (
16), and the previously calculated realized pseudo-volatility square of AAPL, we can calculate the payoff of a pseudo-variance, swap whose underlying asset is the variance of the AAPL stock, to be the following:
7.2. GOOG: Realized Pseudo-Volatility Square and Pseudo-Variance Swap Payoff
Similarly, we use the 6 months of daily closing prices for GOOG, along with Equations (15), (16) and (29), to calculate the realized pseudo-volatility square of GOOG to be the following:
In
Section 5.1, we found that
is 0.0018; however, we can allow this value to vary in order to examine the relationship between the realized pseudo-volatility square of GOOG and the arithmetic mean of the logarithmic GOOG stock returns. As a result, we can produce
Figure 10, which is included below.
Once again, we will calculate the payoff of a long position on a 6-month pseudo-variance swap with a conversion factor of 1. However, the underlying asset of the variance swap will now be the variance of GOOG, and we will use the expected sample variance of GOOG calculated in
Section 6.2 as the strike price. Therefore, we will now use the following parameter values:
- –
Maturity Date:
- –
Position Taken:
- –
Converting Parameter:
- –
Strike Price: .
Using these parameters, Equation (
30) and the previously calculated realized pseudo-volatility square of GOOG, we can calculate the payoff of a pseudo-variance swap whose underlying asset is the variance of the GOOG stock to be
10. Realized Pseudo-Volatility Cross and Pseudo-Covariance Swap Payoff
In order to calculate the realized pseudo-volatility cross and the payoff of a pseudo-covariance swap, we must first define the following parameters:
- –
Position Taken:
- –
Converting Parameter:
- –
Strike Price: .
Using these parameters allows us to utilize the following pair of equations, found in (
Badescu et al. 2002), to calculate the realized pseudo-volatility cross and pseudo-covariance swap payoff:
Realized Pseudo-Volatility Cross:
Payoff of Pseudo-Covariance Swap:
Using Equations (15), (16) and (34), along with the daily closing prices for AAPL and GOOG in the 6-month period from November 2022 to May 2023, we can calculate the realized pseudo-volatility cross of AAPL and GOOG to be the following:
As previously calculated in
Section 5.1,
; thus, we can allow
to vary around its true value in order to allow us to examine the relationship between the realized pseudo-volatility cross of AAPL and GOOG over 6 months, and the arithmetic mean of the logarithmic AAPL stock returns. As a result, we can produce
Figure 13, which is presented below.
Similarly, we can allow
to vary around its true value of 0.0018 to examine the relationship between the realized pseudo-volatility cross of AAPL and GOOG over 6 months, and the arithmetic mean of the logarithmic GOOG stock returns. As a result, we can produce
Figure 14, which is presented below.
Using the realized pseudo-volatility cross calculated above, we can calculate the payoff of a pseudo-covariance swap, whose underlying asset is the covariance of AAPL and GOOG stocks over the 6-month period from November 2022 to May 2023. We will assume the pseudo-covariance swap has a maturity of 6 months, and that we are interested in the payoff associated with taking a long position on the swap. Additionally, we will assume that the payoff is calculated using a converting parameter of
$1 per unit of pseudo-statistic, and we will use the expected sample covariance calculated in
Section 9.3 as the strike price. Therefore, we can now write the following parameter values:
- –
Maturity Date:
- –
Position Taken:
- –
Converting Parameter:
- –
Strike Price: .
Using these parameters, Equation (
21), and the previously calculated pseudo-volatility cross of AAPL and GOOG, we can calculate the payoff of a pseudo-covariance swap, whose underlying asset is the covariance of AAPL and GOOG, to be
11. Realized Pseudo-Correlation and Pseudo-Correlation Swap Payoff
In order to calculate the realized pseudo-correlation and the payoff of a pseudo-correlation swap, we must first define the following parameters:
- –
Position Taken:
- –
Converting Parameter:
- –
Strike Price: .
Using these parameters allows us to utilize the following two equations, found in (
Badescu et al. 2002), to calculate the realized pseudo-correlation and pseudo-correlation swap payoff:
Realized Pseudo-Correlation:
Payoff of Pseudo-Correlation Swap:
Using Equations (15), (16) and (36), along with the data sets of daily closing prices for AAPL and GOOG introduced in
Section 4.1, we can calculate the realized pseudo-correlation of AAPL and GOOG to be the following:
Back in
Section 5.1, we found that
; thus, in order to examine the relationship between the realized pseudo-correlation of AAPL and GOOG over 6 months, and the arithmetic mean of the logarithmic AAPL stock returns, we can allow
to vary around its true value. As a result, we can produce
Figure 15 below.
Similarly, we can allow
to vary around its true value of 0.0018 to examine the relationship between the realized pseudo-correlation of AAPL and GOOG over 6 months, and the arithmetic mean of the logarithmic GOOG stock returns. In this case, we can produce
Figure 16 below.
Using the realized pseudo-correlation calculated above, we can calculate the payoff of a pseudo-correlation swap, whose underlying asset is the correlation of AAPL and GOOG stocks over the 6-month period from November 2022 to May 2023. We will assume the pseudo-covariance swap has a maturity of 6 months, and that we are interested in the payoff associated with taking a long position on the swap. Additionally, we will assume that the payoff is calculated using a converting parameter of $1 per unit of pseudo-statistic. For the strike price we will use the realized pseudo-correlation achieved over the last 15 market days before the start of the swap contract on 9 November. Therefore, we can now write the following parameter values:
- –
Maturity Date:
- –
Position Taken:
- –
Converting Parameter:
- –
Strike Price: .
Using these parameters, Equation (
37), and the previously calculated pseudo-correlation of AAPL and GOOG, we can calculate the payoff the payoff of a pseudo-correlation swap, whose underlying asset is the covariance of AAPL and GOOG, to be
12. Comparing the Approach Based on the Cox–Ingresoll–Ross Model to the Realized Pseudo-Statistic Approach
The purpose of this section is to compare the payoffs of volatility and variance swaps, when the realized volatility and variance is calculated using the Cox–Ingresoll–Ross (CIR) model for variance rather than using the realized pseudo-statistic approach. The pseudo-statistic approach to calculating realized volatility and variance is a data/statistic approach that does not rely on a stochastic model for a stochastic stock price or stochastic volatility. Meanwhile, the Cox–Ingresoll–Ross (CIR) model is a stochastic volatility model that follows a stochastic differential equation (see (
Swishchuk 2004)). Moreover, the Cox–Ingresoll–Ross (CIR) model is the most popularly used stochastic model for the calculation of variance, making it ideal for the comparison between a stochastic approach and a data/statistics approach to the pricing of swaps. Therefore, we will use the CIR model to create a comparison between a stochastic approach to pseudo-statistics and a data/statistical approach to pseudo-statistics in the following sections.
In the following two sections, we will calculate the realized variance and volatility of two stocks over a 6-month period, along with the payoffs of variance and volatility swaps, using the CIR model. Then, we will compare these calculated values to the payoffs of variance and volatility swaps associated with the same stocks over the same time period, which were calculated using the realized pseudo-statistic approach (see (
Badescu et al. 2002)).
For the following report, we have used real-life financial data from Yahoo Finance (
https://ca.finance.yahoo.com/ (accessed on 7 June 2023)) of the publicly traded stocks belonging to
- –
Apple Inc. (AAPL)
- –
Alphabet Inc. Class C (GOOG)
In the sections that follow, we will make use use of four data sets. The first two data sets contain 6 months of daily closing data for AAPL () and GOOG () from 9 November 2022 to 8 May 2023. The last two data sets contain 1 year of daily closing data for AAPL and GOOG from 8 November 2021 to 8 November 2022.
15. Conclusions and Future Work
In accordance with the main motivation of this paper, which was described in
Section 1, in the previous sections of this paper, we have considered the pricing of swaps based on pseudo-statistics, instead of stochastic models, and we have compared this approach with the most popular stochastic volatility model in the Cox–Ingresoll–Ross (CIR) model. Within this paper, we have demonstrated how to price various types of swaps (variance, volatility, covariance, and correlation swaps) using pseudo-statistics (pseudo-variance, pseudo-volatility, pseudo-correlation, and pseudo-covariance). We have also presented analytical closed-form formulas for both the pseudo-statistics and the pricing of swaps based on these pseudo-statistics. As a result, within this paper, we present a method for pricing swaps that does not rely on any stochastic models for a stochastic stock price or stochastic volatility, and instead relies on data/statistics. Consequently, using real-life data along with this data/statistical approach to swap pricing, we were able to calculate numerical values for the value of various swaps based on the volatility, variance, covariance, and correlation of AAPL and GOOG. Some of these results are summarized below in
Table 5.
The data/statistics-based approach to swap pricing we have introduced is very different from stochastic volatility models such as the Cox–Ingresoll–Ross (CIR) model, which, in comparison, follows a stochastic differential equation (see (
Swishchuk 2004)). Therefore, in this paper, we have compared the CIR model approach to pricing swaps to the pseudo-statistic approach to pricing swaps, in order to compare a stochastic model to the data/statistics-based approach to swap pricing, which was developed within this paper. Although there are many other stochastic models that provide an approach to calculating the price of swaps, we have used the CIR model for comparison within this paper, due to the popularity of the CIR model.
After using
Section 12 to define the four data sets that were to be used in the calculations of
Section 13 and
Section 14, along with providing a short description of the approach that was to be used, we were able to use the Cox–Ingresoll–Ross (CIR) model to calculate the realized discretely sampled variance of AAPL and GOOG over a 6-month period of November 2022 to May 2023 in
Section 13.1, the variance swap payoffs in
Section 13.2, the realized discretely sampled volatility in
Section 14.1, and the volatility swap payoffs in
Section 14.2.
Consequently, we were able to compare these values to previously calculated values, which were obtained using the same time period and the same data sets, but with the realized pseudo-statistic approach (see (
Badescu et al. 2002)). Although the values obtained through the realized pseudo-statistic approach were similar to the values obtained through the CIR model, differences between the values did exist, as can be seen in
Table 5 below:
As a result, through the calculations contained within this report, we can conclude that realized volatility and variance, as well as the payoffs associated with volatility and variance swaps, can be calculated with either the CIR model or the realized pseudo-statistic approach, and similar results will be obtained. However, it is clear that even though the results will be similar depending on the approach used, the resulting values will not be identical.
In future work, we plan to further research the comparison between the data/statistical approach to swap pricing and the stochastic approach to swap pricing, beyond the Cox–Ingresoll–Ross (CIR) model. Therefore, future work could include comparing the pseudo-statistic approach to the non-Gaussian Ornstein–Uhlenbeck stochastic volatility model, using the information in (
Benth et al. 2007) as a framework for calculating volatility under the Ornstein–Uhlenbeck model. Other stochastic models that should be compared to the data/statistical approach to calculating volatility include the BN-S stochastic volatility model (see (
Sengupta 2016)), delayed volatility swaps (see (
Swishchuk and Vadori 2014)), and Markov-modulated volatilities (see (
Salvi and Swishchuk 2014)).