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A Tractable Framework for Option Pricing with Dynamic Market Maker Inventory and Wealth

Published online by Cambridge University Press:  19 July 2019

Mathieu FournierOpen the ORCID record for Mathieu Fournier[Opens in a new window] and
Kris Jacobs
Mathieu Fournier*
Affiliation:
Fournier, mathieu.fournier@hec.ca, HEC Montréal
Kris Jacobs
Affiliation:
Jacobs, kjacobs@bauer.uh.edu, University of Houston
*
Fournier (corresponding author), mathieu.fournier@hec.ca
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Abstract

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We develop a tractable dynamic model of an index option market maker with limited capital. We solve for the variance risk premium and option prices as a function of the asset dynamics and market maker option holdings and wealth. The market maker absorbs end users’ positive demand and requires a more negative variance risk premium when she incurs losses. We estimate the model using returns, options, and inventory and find that it performs well, especially during the financial crisis. The restrictions imposed by nested existing reduced-form stochastic-volatility models are strongly rejected in favor of the model with a market maker.

Information

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 55 , Issue 4 , June 2020, pp. 1117 - 1162
Copyright
Copyright © Michael G. Foster School of Business, University of Washington 2019

Footnotes

For helpful comments we thank an anonymous referee, Daniel Andrei, David Bates, Peter Christoffersen, Julien Cujean, Ian Dew-Becker, Christian Dorion, Redouane Elkamhi, Wayne Ferson, Bruno Feunou, Jean-Sébastien Fontaine, Pascal François, Christian Gouriéroux, Amit Goyal, Ruslan Goyenko, Michael Hasler, Alexandre Jeanneret, Bryan Kelly, Aytek Malkhozov, Tom McCurdy, Dmitriy Muravyev, Chayawat Ornthanalai, Stylianos Perrakis, Norman Schüerhoff, Masahiro Watanabe, Jason Wei, and Alan White; seminar participants at the Bank of Canada, HEC Lausanne/Swiss Finance Institute, HEC Montréal, and Rotman; conference participants at the American Finance Association (AFA) meeting in San Francisco, the Northern Finance Association (NFA) meeting in Ottawa, Canadian Derivatives Institute (CDI), and IFM2; and market makers at the Chicago Board Options Exchange (CBOE). A previous version of this article was circulated as “Inventory Risk, Market Maker Wealth, and the Variance Risk Premium: Theory and Evidence.”

References

Adrian, T.; Etula, E.; and Muir, T.. “Financial Intermediaries and the Cross-Section of Asset Returns.” Journal of Finance, 69 (2014), 2557–2596.CrossRefGoogle Scholar
Aït-Sahalia, Y.; Karaman, M.; and Mancini, L.. “The Term Structure of Variance Swaps, Risk Premia and the Expectations Hypothesis.” Working Paper, Swiss Finance Institute (2018).Google Scholar
Aït-Sahalia, Y., and Kimmel, R.. “Maximum Likelihood Estimation of Stochastic Volatility Models.” Journal of Financial Economics, 83 (2007), 413–452.CrossRefGoogle Scholar
Amihud, Y., and Mendelson, H.. “Dealership Market: Market Making with Inventory.” Journal of Financial Economics, 8 (1980), 31–53.CrossRefGoogle Scholar
Andersen, T. G.; Fusari, N.; and Todorov, V.. “The Risk Premia Embedded in Index Options.” Journal of Financial Economics, 117 (2015), 558–584.CrossRefGoogle Scholar
Ang, A.; Hodrick, R.; Xing, Y.; and Zhang, X.. “The Cross-Section of Volatility and Expected Returns.” Journal of Finance, 61 (2006), 259–299.CrossRefGoogle Scholar
Bakshi, G.; Cao, C.; and Chen, Z.. “Empirical Performance of Alternative Option Pricing Models.” Journal of Finance, 52 (1997), 2003–2049.CrossRefGoogle Scholar
Bakshi, G., and Kapadia, N.. “Delta-Hedged Gains and the Negative Market Volatility Risk Premium.” Review of Financial Studies, 16 (2003), 527–566.CrossRefGoogle Scholar
Barras, L., and Malkhozov, A.. “Does Variance Risk Have Two Prices? Evidence from the Equity and Option Markets.” Journal of Financial Economics, 121 (2016), 79–92.CrossRefGoogle Scholar
Bates, D.“Post-’87 Crash Fears in the S&P 500 Futures Option Market.” Journal of Econometrics, 94 (2000), 181–238.CrossRefGoogle Scholar
Bates, D.“Empirical Option Pricing: A Retrospection.” Journal of Econometrics, 116 (2003), 387–404.CrossRefGoogle Scholar
Black, F., and Scholes, M.. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81 (1973), 637–654.CrossRefGoogle Scholar
Bollen, N. P., and Whaley, R. E.. “Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?” Journal of Finance, 59 (2004), 711–753.CrossRefGoogle Scholar
Bollerslev, T.; Tauchen, G.; and Zhou, H.. “Expected Stock Returns and Variance Risk Premia.” Review of Financial Studies, 22 (2009), 4463–4492.CrossRefGoogle Scholar
Boyer, B., and Vorkink, K.. “Stock Options as Lotteries.” Journal of Finance, 69 (2014), 1485–1527.CrossRefGoogle Scholar
Britten-Jones, M., and Neuberger, A.. “Option Prices, Implied Price Processes, and Stochastic Volatility.” Journal of Finance, 55 (2000), 839–866.CrossRefGoogle Scholar
Broadie, M.; Chernov, M.; and Johannes, M.. “Model Specification and Risk Premiums: Evidence from Futures Options.” Journal of Finance, 62 (2007), 1453–1490.CrossRefGoogle Scholar
Broadie, M.; Chernov, M.; and Johannes, M.. “Understanding Index Option Returns.” Review of Financial Studies, 22 (2009), 4493–4529.CrossRefGoogle Scholar
Brunnermeier, M., and Pedersen, L.. “Market Liquidity and Funding Liquidity.” Review of Financial Studies, 22 (2009), 2201–2238.CrossRefGoogle Scholar
Bruno, V., and Shin, H. S.. “Capital Flows and the Risk-Taking Channel of Monetary Policy.” Journal of Monetary Economics, 71 (2015), 119–132.CrossRefGoogle Scholar
Buraschi, A.; Trojani, F.; and Vedolin, A.. “When Uncertainty Blows in the Orchard: Comovement and Equilibrium Volatility Risk Premia.” Journal of Finance, 69 (2014), 101–137.CrossRefGoogle Scholar
Carr, P., and Wu, L.. “Stochastic Skew in Currency Options.” Journal of Financial Economics, 86 (2007), 213–247.CrossRefGoogle Scholar
Carr, P., and Wu, L.. “Variance Risk Premiums.” Review of Financial Studies, 22 (2009), 1311–1341.CrossRefGoogle Scholar
Chen, H.; Joslin, S.; and Ni, S.. “Demand for Crash Insurance, Intermediary Constraints, and Risk Premia in Financial Markets.” Review of Financial Studies, 32 (2019), 228–265.CrossRefGoogle Scholar
Cheng, I. H.“The VIX Premium.” Review of Financial Studies, 32 (2019), 180–227.CrossRefGoogle Scholar
Christoffersen, P.; Jacobs, K.; and Mimouni, K.. “Volatility Dynamics for the S&P 500: Evidence from Realized Volatility, Daily Returns, and Option Prices.” Review of Financial Studies, 23 (2010), 3141–3189.CrossRefGoogle Scholar
Comerton-Forde, C.; Hendershott, T.; Jones, C.; Moulton, P.; and Seasholes, M.. “Time Variation in Liquidity: The Role of Market Maker Inventories and Revenues.” Journal of Finance, 65 (2010), 295–331.CrossRefGoogle Scholar
Coval, J., and Shumway, T.. “Expected Option Returns.” Journal of Finance, 56 (2001), 983–1009.CrossRefGoogle Scholar
Cox, J. C., and Huang, C. F.. “Optimal Consumption and Portfolio Policies When Asset Prices Follow a Diffusion Process.” Journal of Economic Theory, 49 (1989), 33–83.CrossRefGoogle Scholar
Cremers, M.; Halling, M.; and Weinbaum, D.. “Aggregate Jump and Volatility Risk in the Cross-Section of Stock Returns.” Journal of Finance, 70 (2015), 577–614.CrossRefGoogle Scholar
Dew-Becker, I.; Giglio, S.; Le, A.; and Rodriguez, M.. “The Price of Variance Risk.” Journal of Financial Economics, 123 (2017), 225–250.CrossRefGoogle Scholar
Driessen, J.; Maenhout, P.; and Vilkov, G.. “The Price of Correlation Risk: Evidence from Equity Options.” Journal of Finance, 64 (2009), 1377–1406.CrossRefGoogle Scholar
Egloff, D.; Leippold, M.; and Wu, L.. “The Term Structure of Variance Swap Rates and Optimal Variance Swap Investments.” Journal of Financial and Quantitative Analysis, 45 (2010), 1279–1310.CrossRefGoogle Scholar
Eraker, B.“Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices.” Journal of Finance, 59 (2004), 1367–1403.CrossRefGoogle Scholar
Eraker, B.; Johannes, M.; and Polson, N.. “The Role of Jumps in Returns and Volatility.” Journal of Finance, 58 (2003), 1269–1300.CrossRefGoogle Scholar
Gârleanu, N.; Pedersen, L. H.; and Poteshman, A.. “Demand-Based Option Pricing.” Review of Financial Studies, 22 (2009), 4259–4299.CrossRefGoogle Scholar
Gourier, E.“Pricing of Idiosyncratic Equity and Variance Risks.” Working Paper, Queen Mary University of London (2016).Google Scholar
Gromb, D., and Vayanos, D.. “Equilibrium and Welfare in Markets with Financially Constrained Arbitrageurs.” Journal of Financial Economics, 66 (2002), 361–407.CrossRefGoogle Scholar
Gruber, P. H.; Tebaldi, C.; and Trojani, F.. “The Price of the Smile and Variance Risk Premia.” Working Paper, Swiss Finance Institute (2015).CrossRefGoogle Scholar
Hansch, O.; Naik, N.; and Viswanathan, S.. “Do Inventories Matter in Dealership Markets? Evidence from the London Stock Exchange.” Journal of Finance, 53 (1998), 1623–1656.CrossRefGoogle Scholar
He, Z.; Kelly, B.; and Manela, A.. “Intermediary Asset Pricing: New Evidence from Many Asset Classes.” Journal of Financial Economics, 126 (2017), 1–35.CrossRefGoogle Scholar
He, Z., and Krishnamurthy, A.. “Intermediary Asset Pricing.” American Economic Review, 103 (2013), 732–770.CrossRefGoogle Scholar
Heston, S.“Closed-Form Solution for Options with Stochastic Volatility, with Application to Bond and Currency Options.” Review of Financial Studies, 6 (1993), 327–343.CrossRefGoogle Scholar
Ho, T., and Stoll, H.. “Optimal Dealer Pricing under Transactions and Return Uncertainty.” Journal of Financial Economics, 9 (1981), 47–73.CrossRefGoogle Scholar
Ho, T., and Stoll, H.. “The Dynamics of Dealer Markets under Competition.” Journal of Finance, 38 (1983), 1053–1074.CrossRefGoogle Scholar
Jones, C. S.“The Dynamics of Stochastic Volatility: Evidence from Underlying and Options Markets.” Journal of Econometrics, 116 (2003), 181–224.CrossRefGoogle Scholar
Jones, C. S.“A Nonlinear Factor Analysis of S&P 500 Index Option Returns.” Journal of Finance, 61 (2006), 2325–2363.CrossRefGoogle Scholar
Karatzas, I.; Lehoczky, J. P.; and Shreve, S.. “Optimal Portfolio and Consumption Decisions for a Small Investor on a Finite Time-Horizon.” SIAM Journal on Control and Optimization, 25 (1987), 1557–1586.CrossRefGoogle Scholar
Leippold, M., and Su, L.. “Collateral Smile.” Journal of Banking and Finance, 58 (2015), 15–28.CrossRefGoogle Scholar
Lewis, A.“Option Valuation under Stochastic Volatility.” Finance Press. Newport Beach, CA, (2000).Google Scholar
Madhavan, A., and Smidt, S.. “An Analysis of Changes in Specialist Inventories and Quotations.” Journal of Finance, 48 (1993), 1595–1628.CrossRefGoogle Scholar
Madhavan, A., and Sofianos, G.. “An Empirical Analysis of NYSE Specialist Trading.” Journal of Financial Economics, 48 (1998), 189–210.CrossRefGoogle Scholar
Malik, S., and Pitt, M.. “Particle Filters for Continuous Likelihood Evaluation and Maximization.” Journal of Econometrics, 165 (2011), 190–209.CrossRefGoogle Scholar
Mildenstein, E., and Schleef, H.. “The Optimal Pricing Policy of a Monopolistic Market Maker in the Equity Market.” Journal of Finance, 38 (1983), 218–231.CrossRefGoogle Scholar
Muravyev, D.“Order Flow and Expected Option Returns.” Journal of Finance, 71 (2016), 673–708.CrossRefGoogle Scholar
Naik, N., and Yadav, P.. “Do Dealer Firms Manage Inventory on a Stock-by-Stock or a Portfolio Basis?” Journal of Financial Economics, 69 (2003), 325–353.CrossRefGoogle Scholar
Pan, J.“The Jump Risk Premia Implicit in Options: Evidence from an Integrated Time Series Study.” Journal of Financial Economics, 63 (2002), 3–50.CrossRefGoogle Scholar
Schürhoff, N., and Ziegler, A.. “Variance Risk, Financial Intermediation, and the Cross-Section of Expected Option Returns.” Working Paper, University of Zurich (2011).Google Scholar
Shleifer, A., and Vishny, R.. “The Limits of Arbitrage.” Journal of Finance, 52 (1997), 35–55.CrossRefGoogle Scholar
Sofianos, G.“Specialist Gross Trading Revenues at the New York Stock Exchange.” Working Paper, New York Stock Exchange (1995).Google Scholar
Todorov, V.“Variance Risk-Premium Dynamics: The Role of Jumps.” Review of Financial Studies, 23 (2010), 345–383.CrossRefGoogle Scholar
Trolle, A., and Schwartz, E.. “Unspanned Stochastic Volatility and the Pricing of Commodity Derivatives.” Review of Financial Studies, 22 (2009), 4423–4461.CrossRefGoogle Scholar
Vilkov, G.“Variance Risk Premium Demystified.” Working Paper, INSEAD (2008).CrossRefGoogle Scholar
Zhang, L.; Mykland, P.; and Aït-Sahalia, Y.. “A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High-Frequency Data.” Journal of the American Statistical Association, 100 (2005), 1394–1411.CrossRefGoogle Scholar
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