Option Pricing in the Heston Stochastic Volatility Model: An Empirical Evaluation

Master project by Patrick Altmeyer, Jacob Daniel Grapendal, Makar Pravosud, and Gand Derry Quintana ’18

Editor’s note: This post is part of a series showcasing Barcelona GSE master projects by students in the Class of 2018. The project is a required component of every master program.


Authors:

Patrick Altmeyer, Jacob Daniel Grapendal, Makar Pravosud, and Gand Derry Quintana

Master’s Program:

Finance

Paper Abstract:

There exists a substantial body of literature concerned with the calibration of the Heston model for pricing financial derivatives under stochastic volatility, many of which rely on computationally expensive algorithms. Our paper evaluates a calibration method of the Heston model proposed by Alòs, De Santiago, and Vives (2015), which can be used to price derivatives with little computational effort. The calibration method is innovative in the sense that it considers only the three most critical regions of the implied volatility surface. The regions where the underlying option is, firstly, at-the-money, secondly, close to maturity and lastly, far away from maturity. Although their procedure is parsimonious and very easy to implement, they calibrate a model whose empirical applicability is contested.

The main contribution of our paper is the evaluation of their model in an extensive numerical exercise as well as an application to real data. Collecting empirical option data has been one of the main challenges with respect to this work, since historical data on financial derivatives is not accessible to the public. Faced with this issue we have written a script that allowed us to automatically scrape option data at a high frequency over just a couple of weeks. Thus, we build our own extensive data base. Also, we have made the data and code available on https://griipen.shinyapps.io/bgse/ and https://github.com/HitKnit/BGSE2018/tree/HitKnit-optionscraping, respectively.

In terms of our results, we find that whilst the calibration method has solid theoretical foundations and produces satisfactory estimation results within the theoretical Heston universe. However, it fails in practice. Specifically, for the numerical exercise we find that out of all simulations the maximum average error across the entire volatility surface is 0.999 percent while the mean error across simulations is only 0.481 percent. In sharp contrast to that, absolute percentage errors for our empirical data are on the order of 30-40 percent in many cases. In the following figure, we present our findings for intra-daily data from May 16, 2018. The left column shows empirical implied volatilities for a European call option on Facebook Inc. (FB) stocks. From top to bottom volatilities are shown for the opening, lunch and closing sessions. The central column shows the fitted volatility surfaces while the right column shows absolute percentage differences between empirical and estimated values. The finding that errors are particularly high for at-the-money options with short times to maturity is robust across the entire data sample.

Conclusions and key results:

In light of these results, we conclude that inherent limitations of the Heston Model disqualify the calibration for practical use. Nonetheless, we believe that similarly simple calibration methods as the one examined here should be used in combination with more sophisticated option pricing models.

References:

Alòs, Elisa, Rafael De Santiago, and Josep Vives. 2015. “Calibration of Stochastic Volatility Models via Second-Order Approximation: The Heston Case.” International Journal of Theoretical and Applied Finance 18 (06). World Scientific: 1550036.

Download the full paper [pdf]


More about the Finance Program at the Barcelona Graduate School of Economics

Economics articles by BGSE alumni at CaixaBank Research

Ricard Murillo, Marta Guasch, and Mar Domènech in front of Caixabank. Photo by Marta Guasch.

We’ve just come across some articles written by several Barcelona GSE Alumni who are now Research Assistants and Economists at Caixabank Research in Barcelona. New articles are published each month on a range of topics.

Below is a list of all the alumni we found listed as article contributors, as well as their most recent publications in English (click each author to view his or her full list of articles in English, Catalan, and Spanish).

If you’re an alum and you’re also writing about Economics, let us know where we can find your stuff!

Gerard Arqué (Master’s in Macroeconomic Policy and Financial Markets ’09)

The (r)evolution in the regulatory and supervisory framework resulting from the crisis

Mar Domènech (Master’s in International Trade, Finance, and Development ’17)

Registered workers affiliated to Social Security: situation and outlook across sectors

Active labour market policies: a results-based evaluation

Equal opportunities: levelling the playing field for everyone

Cristina Farràs (Master’s in Macroeconomic Policy and Financial Markets ’17)

The financial situation of Millennial households in the US and Spain: will they catch up with previous generations?

Measures to improve equality of opportunities

Marta Guasch (Master’s in International Trade, Finance, and Development ’17)
and Adrià Morron (Master’s in Economics ’12)

Jay Gatsby’s American Dream: between inequality and social mobility

Ricard Murillo (Master’s in International Trade, Finance, and Development ’17)

Inflation will gradually recover in the euro area

Millenials and politics: mind the gap!

The sensitivity of inflation to the euro’s appreciation

Ariadna Vidal Martínez (Master’s in Finance ’12)

Situation and outlook for consumer financing


Source: Caixabank Research

Can misguided monetary policy explain the European housing bubble?

Patrick Altmeyer (Finance student ’18) who has an interest in monetary policy, shares his work on whether misguided monetary policy can explain the European housing bubble.


Property prices surged throughout Europe in the early 2000s before collapsing during the crisis and causing tremendous welfare losses. This dissertation uses Structural Vector Autoregression (SVAR) to analyse the role of house prices within the monetary transmission mechanism in Europe over the past decades in order to understand whether lax interest rate policy had caused the bubble. Quarterly observations of inflation, output, consumption, real estate prices and mortgage variables for eight European countries were used. Sample periods vary by model specification but generally four decades.

Impulse response functions for the baseline SVAR suggest that real estate prices did indeed respond positively to dovish monetary policy and thereby amplified conventional effects on consumer spending. However, the interpretation of these preliminary results is complicated by explosive house price dynamics during the early 2000s. The linear vector autoregressions fail to fully capture these non-linear elements of the time series. A statistical test developed by Homm and Breitung (2012) is therefore used to identify bubble periods in the various countries analysed. Explosive house price dynamics are found in all countries but Germany as shown in Figure 1.

Figure 1: House price trends in European countries. Shaded areas indicate bubble periods.

Information about house price bubbles is subsequently used to augment the baseline SVAR in various ways. Consequently, the measured effect of a decrease in interest rates on house prices remains positive, but to a lesser extent. Overall, evidence found here suggests that interest rate policy alone was not responsible for the European housing bubble. Rather, it appears that the boom could be better explained by joint effects of loose monetary policy, financial liberalisation and associated mortgage market innovations. Note, for example, that total securitisation activity measured in terms of the number of euro-denominated asset-backed securities outstanding increased six fold from 2000 until the credit bubble burst in mid 2007. Unsurprisingly, many have drawn a connection between monetary policy and securitisation commonly arguing that the latter amplified the conventional credit effects of the former. Information about mortgage rates and lending activity is used as a proxy for mortgage securitisation and added to the SVAR in the final section of the empirical part. Indeed, these variables are found to have high explanatory power with respect to house price trends in most countries as evident in Figure 2, which plots forecast error variance decompositions for each country under the preferred model specification.

Figure 2: Forecast error variance decompositions.

The paper therefore concludes that stricter interest rates more closely aligned with policy rules could not have entirely avoided the bubbles, hence this approach is not recommended for the future. Putting more focus on asset price stability and thereby departing from the policy rate’s traditional role of smoothing consumption and consumer prices would be too complicated and is therefore not advisable, either. In light of the finding that financial innovations have greatly contributed to bubbles, policy makers should continue current efforts on imposing stricter regulation through macroprudential measures.

The full article can be read here.

Partial Adjustment in Policy Functions of Structural Models of Capital Structure

Editor’s note: This post is part of a series showcasing Barcelona GSE master projects by students in the Class of 2017. The project is a required component of every master program.


Authors:

Mattia Bongini

Master’s Program:

Finance

Paper Abstract:

We present a tradeoff model of capital structure to investigate the sources of adjustment costs and study how firms’ financing decisions determine partial adjustment toward target leverage ratios. The presence of market imperfections, like taxes and collateral constraints, is shown to play a decisive role in the behavior of the policy function of capital and leverage. By means of a contraction argument, we are able to show the existence of a target leverage towards which optimal leverage converges with a speed of adjustment that depends on a firm marginal productivity of capital. Our predictions are consistent with the empirical literature regarding both the magnitude of the speed of adjustment and the relationship between leverage ratios and the business cycle.

Conclusions:

In this work we showed how financial and economic frictions are able to generate a partial adjustment dynamics in leverage policy functions. In the model we studied, the key factors of this phenomenon are collateral constraints (which strike a balance between tax benefits of debt and distress costs) and firm productivity of capital. The latter, in particular, determines the speed of adjustment towards the (state-dependent) target leverage ratio.

Our model fits well several stylized facts of leverage dynamics established by the empirical literature: an example is given by the magnitude of the speed of adjustment, which falls into the confidence intervals estimated by several authors. Another one, is the countercyclical behavior of leverage dynamics with respect to the business cycle, which is due to the fact that in recessions it is easier for the collateral constraint to be binding.

Future work should first address the translation of the hypotheses of Theorem 5.4 on the Lagrange multiplier into assumptions on the components of the model (the production function and the various market frictions). The next step would then be to extend the model to a full general equilibrium model to study thoroughly the effects of preference and monetary shocks on leverage dynamics. Pairing consumers’ utility maximization with firms’ financing problem would also allow to study the interaction between expected returns and partial adjustment: in such framework, the collateral constraint should probably be replaced by several credit rating inequalities determining both firm specific discount rates and target leverage ratios.

Gender Differentials in Returns to Education in Developing Countries

Editor’s note: This post is part of a series showcasing Barcelona GSE master projects by students in the Class of 2017. The project is a required component of every master program.


Authors:

Ignatius Barnardt, Golschan Khun Jush, Thies Wollesen, Samuel Hayden and Eva Sotosek

Master’s Program:

Economics and Finance

Paper Abstract:

We investigate a possible gender gap in returns to education using data from the World Bank’ STEP program for seven developing and emerging countries. We control for cognitive skills, non-cognitive skills and parental education – previously unobserved due to unavailability of data – to investigate how this heterogeneity is playing a role in estimating the gender differential in educational returns. We also model selection using the Heckman two-step estimation procedure to examine whether selection may be driving this phenomenon. Our findings suggest that gender gaps in returns to education are not as prominent in the countries in our sample as previously suggested. We also find that controlling for unobserved heterogeneity on the one hand, and selection on the other, has different effects in different countries, highlighting the importance of understanding individual countries’ labour markets in detail before drawing conclusions regarding the existence of a gender gap in returns to education.

Conclusions:

This paper explores gender gaps in returns to education for seven developing and emerging countries. First, we investigate the existence of such a gap in a standard Mincerian framework. We find a significant returns gap in only two countries, namely Ukraine and Ghana, while the estimates for the other countries are centred relatively tightly around statistically insignificant point estimates close to zero. Using quantile regressions to dig deeper does not materially affect our findings, although it does allow us to specify that the returns gaps estimated for Ghana and Ukraine are significant at two out of three quartiles of the wage distribution, and that in Vietnam there is a small but significant returns gap at the upper two quartiles of the distribution. These findings are important in providing context for the existing literature, showing that returns premiums in favour of females are not universally prevalent in developing countries for urban wage workers. This suggests that where large, significant returns gaps have been found in the literature, this seems to be driven to a large extent by other segments of the labour market.

Second, we use our novel dataset to analyse the extent to which controlling for previously unobserved heterogeneity, namely cognitive skills, personality traits and family background, affect OLS estimates of the returns gap. We find that controlling for these STEP variables does not materially affect our baseline estimates for Bolivia, Colombia, Georgia, Kenya and Vietnam (where the estimated gap remains insignificant and close to zero), or for Ukraine, where the estimated gap is of similar magnitude and remains significant. Only in Ghana we find that adding the STEP controls has a material effect, reducing the point estimate of the gap substantially and rendering it insignificant. The results of the quantile regressions qualify this finding somewhat, showing that controlling for the STEP variables does make a difference for estimates of the gap at certain quantiles of the distribution in Ukraine and Vietnam. Overall, our finding regarding the importance of these sources of previously unobserved heterogeneity is cautiously negative: although they do appear to make a small difference for the level estimates and have an important effect in Ghana, they do not appear to be universal sources of endogeneity in estimating the returns gap for urban wage workers.

Third, we examine the importance of controlling for selection in estimating the returns differential using the Heckman two-step procedure, dropping Kenya from our sample due to missing data. Here we find that after controlling for selection, our point estimates of the returns gap remain insignificant in Ghana, Georgia and Vietnam, albeit with a relatively high point estimate in Georgia. Similarly, our estimate of the returns gap in Ukraine does not change considerably and remains significant. In contrast, we obtain higher and significant point estimates of the returns gap in Bolivia and Colombia. As explained above, this somewhat counterintuitive result is due to positive selection of females into employment in Bolivia and Colombia, and the positive relationship between education levels and probability of employment. Interestingly, in the two countries where selection appears to be important, we found earlier that controlling for the STEP variables did not have an observable effect. Our findings therefore suggest that it is likely to be important to control for selection when estimating returns gaps in developing countries, even if only to exclude the possibility of selection bias. In addition, our approach suggests that selection is likely to operate through channels other than cognitive or non-cognitive abilities, or parental background.

Taken together, our findings show that, at least for urban wage workers in the countries in our sample, a returns premium for females may not be as prevalent as previously suggested. We also find that controlling for potential sources of endogeneity, such as unobserved heterogeneity and selection, substantially changes the estimates of the gender returns gap in three out of seven of the countries in our sample. This highlights the importance of considering these channels to avoid the risk of biased estimation. This paper therefore represents a starting point for more detailed research, which could zoom in on the existence and drivers of returns differentials in individual countries, and overcome some of the limitations of this paper by extending it to rural areas and using samples with a larger number of clusters. These findings are also relevant to policy makers, since they demonstrate the importance of understanding the characteristics and dynamics of each country’s individual labour market prior to making policy proposals.

How can time series help in delimiting the relevant market.

Delimiting the relevant market is a key concept for the analysis of mergers and acquisitions. The theoretical framework introduced by the SNNIP test helps to understand the conditions needed to do it. Nevertheless, there exist so many methods and the scientific community does not coincide in what of them is better to use. In this article based on previous work[1], some methods grounded on time series are presented.

In general, the concept of relevant market is associated with arbitrage. In this sense, two regions belong to the same market when arbitrage is possible. Therefore, it is possible to check whether the prices of these areas hold a pattern of convergence. As exposed mainly in Haldrup (2003)[2] , we can differentiate two types of convergence:

Absolute convergence: it appears when there is perfect arbitrage with no transportation costs, then the stationary price difference between regions is zero. It can be expressed as:

limitRelative convergence: it is analogous to the previous concept but, in this case, transportation cost does not completely disappear. It can be expressed as:

limitTherefore, absolute convergence is a specific case of relative convergence for the case of α=0, which is mainly that transportation costs are equal to zero.

There are several methods used to analyse time series of prices. They are useful to define the relevant market. There are two main dimensions: defining the market of substitute goods and delimiting the area where a company is competing.

CORRELATION

Correlation is one of the most common methods used to analyse prices. In this sense, Stigler & Sherwin (1985)[3] proposed to do it with series transformed in logarithms to avoid problems arising from divergences in variance.

Ideally, two prices of goods or regions inside the same market should have high correlation in both logarithms and its first derivative (that works as an approximation of the growth rate).

This method presents many problems. Firstly, high correlations can be produced because of a spurious relation (Granger & Newbold, 1975)[4]. Moreover, Bishop & Walker (1996)[5] argue that highly volatile exchange rates can distort the results. Nevertheless, Haldrup (2003)[2] argue the since 90s exchange rates have a stable structure and, therefore, the analysis is not injured.

COINTEGRATION

Cointegration can be determined by the procedure defined by Engle & Granger (1987)[6]. In a more general insight, if time series are integrated of order 1, it is possible to use the Johansen’s test (1991)[7]. In this sense, Alexander & Wyeth (1994)[8] argue that a common market can be defined with only one cointegration relationship. In contrast, Haldrup (2003)[2] argues that the single market is determined with k-1, the maximum, cointegration relationships. Cointegration cannot be applied when one of the series is not integrated, that is, it is stationary.

Given that cointegration relationships can be understood as a log-run equilibrium, it is possible to define best response functions to find results corresponding to price-based models, as Bertrand’s Oligopoly.

FORNI’S TEST

Since cointegration procedure is based on unit roots tests, Forni (2004)[9] defined a way of determining the long-run equilibrium in a more flexible way. This test tries to analyse the stationarity of the logarithm of the ratios of both price series. It is possible to run different unit root test.

  NULL HYPOTHESIS
ADF They do not belong to the same market  (non-stationarity)
ADF-GLS They do not belong to the same market  (non-stationarity)
KPSS Both goods or regions belong to the same market (stationarity)

Figure 1 shows the time-series of the logarithm of the ratio of the price of two different goods. It is an example of relative convergence. Even with some outliers it is possible to see how the ratio fluctuates around an equilibrium. In this case, the test allows to conclude that the series is stationary. We could conclude that with the evidence extracted from this procedure, both goods are part of the same relevant market.

Figure 1: Ratio of two prices seeming to be in the same marketestacionario Source: Own elaboration in previous work [1]

Figure 2 shows the same time-series but for different goods. It shows an unclear pattern of co-movement between prices. Not only prices seem not be related but also, they seem to move away. In this case, the series is not stationary and thus, according with this test, we could conclude that both goods do not belong to the same market.

Figure 2: Ratio of two prices seeming not to be in the same market.
no-est   Source: Own elaboration in previous work[1]

From my point of view, for this purpose, unit root tests can be applied either with or without trend and intercept in the auxiliary regression. Initially, to test whether two goods or regions belong to the same market the trend is not relevant, since they should have a constant long-run equilibrium. In the case that the series were not stationary, repeating the test with trend would be interesting. It could explain if there exists a pattern of divergence between goods or regions. The intercept can be understood as the α coefficient exposed above. If it were zero and the test concluded stationarity, it could be a case of absolute convergence.

GRANGER CAUSALITY

Granger causality is based on the analysis of VAR models. In an easy approach, with VAR models we try to estimate the price of one good or area in function of the lags of the other price and its own lags. Granger causality analyses the null of all coefficient of the other price are zero. If the null is rejected, one price causes the other and they seem to belong to the same market.

It is possible to carry out the regressions in both ways, the first one for estimating a price and the second one for estimating the other. There could be causality in both ways but it is not a necessary requirement to conclude that there exists a causality relationship between them.

Prices displayed as the ratio of Figure 1, showed a two-way causality relationship. However, prices of Figure 2 did not show any causality relationship.

CONCLUSIONS

There are many methods to analyse if some regions or goods belong to the same relevant market. Apart from the ones exposed above, other price-based ways can be used as VEC models or PCA, and other non-price-based methods as the shock analysis or the Elzinga & Hogarty Test (1973)[10].

In general, different procedures do not use to issue contradictory answers, but they are not self-explanatory by themselves. They need to be complemented with each other to bring back the most accurate conclusion.

REFERENCES

[1] See García García, Alberto (2016). El mercado relevante: técnicas económicas y econométricas para la delimitación. Trabajo Fin de Grado. Universidad de Oviedo.

[2] Haldrup, N. (2003). “Empirical Analysis of Price Data in the Delineation of the Relevant Geographical Market in Competition Analysis. University of Aarhus, Economic Working Paper .

[3] Stigler, G. J., & Sherwin, R. A. (1985). The Extent of the Market. Journal of Law and Economics, Vol. 28, No. 3, 555-585.

[4] Granger, C. W., & Newbold, P. (1974). Spurious Regressions in Econometrics. Journal of Econometrics, 2;, 111-20.

[5] Bishop, S., & Walker, M. (1996). “Price correlation analysis: still a useful tool for relevant market definition. Lexecon.

[6] Engle, R. F. & Granger, C.W. (1987). Co-Integration and Error Correction: Representation, Estimation and Testing. Econometrica, 55(2), 251-76.

[7] Johansen, S. (1988). Statistical Analysis of Cointegration Vectors. Journal of Economic Dynamics and Control, 231-254.

[8] Alexander, Carol and Wyeth, John (1994) Cointegration and market integration: an application to the Indonesian rice market. The Journal of Development Studies, 30 (2). pp. 303-334. ISSN 0022-0388

[9] Forni, M. (2004). Using Stationarity Test in Antitrust Market Definition. American Law and Economic Review, 441-64.

[10] Elzinga, K. G., & Hogarty, T. F. (1973). The Problem of Geographic Market Definition in Antimerger Suits. Antitrust Bulletin, 18(1), pp.45-81.

Brexit: BGSE Community Analysis

We want to know what the BGSE community is thinking and reading about the Brexit.

brexit-624x437

We invite all Barcelona GSE students and alumni to share their early reflections on the potential economic consequences of the UK’s recent vote to leave the EU. Did you focus on a related topic in your master project? Are you working at a think tank, central bank, or consulting firm where your projects will be impacted by this decision? Have you seen any articles or links that you found useful for understanding what lies ahead?

Here are a couple of pieces we’ve found to get the discussion going:

After Brexit: What next for the EMU, EU and UK?
(ADEMU webinar)

The BGSE participates in A Dynamic Economic and Monetary Union (ADEMU), a project of the EU Horizon 2020 Program. Last week, ADEMU researchers held a webinar to discuss the Brexit.

Background:

Europe has grown out of its crises when reason and solidarity have prevailed, but it has also been devastated by its crises when fear and nationalism have taken the lead. Brexit, in the aftermath of the euro crisis, brings this dichotomy back to the foreground. Since 2010 there have been important advances in the development of the Economic and Monetary Union (EMU) and flexible forms of participation have allowed other EU countries, reluctant to join the euro, to share the basic principles that define the EU and have a common presence in the interdependent global world.

According to the panelists, Brexit raises 3 crucial questions:

  1. Should the EMU be accelerated to become a centre of gravity within the EU, or slowed down to avoid a centrifugal diaspora? If accelerated, how?
  2. Should an ‘exit’ country be allowed free entry to the single market and other EU public goods without accepting freedom of movement?
  3. Should the EU remain as it is, or increase its capacity to offer common public services (Banking Union, border security, research funding, environment, etc.), or limit its scope of activity to the EU single and integrated market?

Webinar Panel:
– Joaquín Almunia (Former Vice-President of the European Commission, honorary president of the Barcelona GSE)
– Ramon Marimon (European University Institute and UPF – Barcelona GSE; ADEMU)
– Gorgio Monti (European University Institute; ADEMU)
– Morten Ravn (University College London; ADEMU)

Moderator:
Annika Zorn (European University Institute; Florence School of Banking & Finance)


From Brexit to the Future
(Joseph Stiglitz)

Nobel Laureate and Barcelona GSE Scientific Council member Joseph Stiglitz shares some reflections in the wake of the Brexit decision


What are you thoughts on Brexit?

We want to know what the BGSE community is thinking and reading about the Brexit. Please share your ideas, favorite sources for analysis, or observations from economists you respect in the comments below.

Cross ownership and firm performance

Editor’s note: This post is part of a series showcasing Barcelona GSE master projects by students in the Class of 2015. The project is a required component of every master program.


Authors:
Octavi Castells Pera, Jaime López Sastre, and Berenice Ramirez

Master’s Program:
Finance

Paper Abstract:

This paper assesses the impact of cross ownership on firm performance and industry competition through an analysis of shareholder’s networks in Spain using a panel regression model on a sample of non-financial listed companies between the years 2004 and 2012. The results show that there is a positive and significant effect of the number of connections a firm has with other industry rivals through the common ownership mechanism on its markup.

Read the paper or view presentation slides:

What can the risk neutral moments tell us about future returns?

Editor’s note: This post is part of a series showcasing Barcelona GSE master projects by students in the Class of 2015. The project is a required component of every master program.


Authors:
Juan Imbet, Nuria Mata

Master’s Program:
Finance

Paper Abstract:

We test if the first four moments of the risk neutral distribution implicit in options’ prices predict market returns. We estimate the risk
neutral distribution of the S&P 500 over different frequencies using a non parametric polynomial fitting, and test if the first four moments of the distribution predict returns of the S&P 500. Our results suggest that there is no evidence on this predictability power.

Presentation Slides:

Photo Diary: Exams Winter 2015

How masters and PhD students are surviving finals this month…

Staking out a cozy corner in the library

 

It’s all about the snacks

 

Moments of Zen

 

A little help from our friends

 

Have a photo you’d like to share? Email it to thevoice@barcelonagse.eu or mention @barcelonagse on Twitter or Instagram