Abstract
We propose a mean-variance framework to analyze the optimal quoting policy of an option market maker. The market maker’s profits come from the bid-ask spreads received over the course of a trading day, while the risk comes from uncertainty in the value of his portfolio, or inventory. Within this framework, we study the impact of liquidity and market incompleteness on the optimal bid and ask prices of the option. First, we consider a market maker in a complete market, where continuous trading in a perfectly liquid underlying stock is allowed. In this setting, the market maker may remove all risk by Delta hedging, and the optimal quotes will depend on the option’s liquidity, but not on the inventory. Second, we model a market maker who may not trade continuously in the underlying stock, but rather sets bid and ask quotes in the option and this illiquid stock. We find that the optimal stock and option quotes depend on the relative liquidity of both instruments as well as on the net Delta of the inventory. Third, we consider an incomplete market with residual risks due to stochastic volatility and large overnight moves in the stock price. In this setting, the optimal quotes depend on the liquidity of the option and on the net Vega and Gamma of the inventory.
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References
Almgren R., Chriss N. (2001) Optimal execution of portfolio transactions. Journal of Risk 3: 5–39
Boyle P., Emanuel D. (1980) Discretely adjusted option hedges. Journal of Financial Economics 8: 259–282
Cont R., da Fonseca J. (2002) Dynamics of implied volatility surfaces. Quantitative Finance 2: 45–60
Davis M.H.A., Panas V.G., Zariphopoulou T. (1993) European option pricing with transaction costs. SIAM Journal on Control and Optimization 31(2): 470–493
Engle, R., & Ferstenberg, R. (2006). Execution risk, Working paper.
Garleanu, N., Pedersen, L. H., & Poteshman, A. M. (2006). Demand based option pricing, Working paper, Wharton School of Business.
Hasbrouck J. (2007) Empirical market microstructure. Oxford University Press, New York
Ho T.S.Y., Macris R.G. (1984) Dealer bid-ask quotes and transaction prices: An empirical study of some AMEX options. Journal of Finance 39(1): 23–45
Ho T.S.Y., Stoll H.R. (1981) Optimal dealer pricing under transactions and return uncertainty. Journal of Financial Economics 9: 47–73
Jameson M., Wilhelm W. (1992) Market making in the options markets and the costs of discrete hedge rebalancing. Journal of Finance 47: 765–779
Minina, V., & Vellekoop, M. (2008). A risk reserve model for hedging in incomplete markets rebalancing, Working paper.
Schönbucher P.J. (1999) A market model for stochastic implied volatility. Philosophical Transactions: Mathematical, Physical and Engineering Sciences 357(1758): 2071–2092
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Stoikov, S., Sağlam, M. Option market making under inventory risk. Rev Deriv Res 12, 55–79 (2009). https://doi.org/10.1007/s11147-009-9036-3
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DOI: https://doi.org/10.1007/s11147-009-9036-3