On the utility maximization of the discrepancy between a perceived and market implied risk neutral distribution. (English) Zbl 1507.91205
Summary: A method is developed to determine the portfolio that maximizes the expected utility of an agent that trades the difference between a perceived future price distribution of an asset and the associated market implied risk neutral density. Exact results to construct and price such a portfolio are presented under the assumption that the underlying asset price evolves according to a geometric Brownian motion. Integer programming optimization techniques are applied to the general case where one first calibrates the asset price risk neutral density directly from option market data using Gatheral’s SVI parameterization. Several numerical examples approximating the optimal payoff function with liquid securities are given.
References:
| [1] | Agrawal, A.; Verschueren, R.; Diamond, S.; Boyd, S., A rewriting system for convex optimization problems, Journal of Control and Decision, 5, 1, 42-60 (2018) |
| [2] | Aït-Sahalia, Y.; Lo, A., Nonparametric estimation of state-price densities, Journal of Finance, 53, 2, 499-547 (1998) |
| [3] | Angelelli, E.; Mansini, R.; Speranza, M. G., Kernel search: A general heuristic for the multi-dimensional knapsack problem, Computers & Operations Research, 37, 11, 2017-2026 (2010) · Zbl 1188.90207 |
| [4] | Angelelli, E.; Mansini, R.; Speranza, M. G., Kernel search: A new heuristic framework for portfolio selection, Computational Optimization and Applications, 51, 1, 345-361 (2012) · Zbl 1246.91119 |
| [5] | Arrow, K. J., The role of securities in the optimal allocation of risk-bearing, The Review of Economic Studies, 31, 2, 91-96 (1964) |
| [6] | Bernoulli, D., Exposition of a new theory on the measurement of risk, Econometrica, 22, 1, 23-36 (1954) · Zbl 0055.12004 |
| [7] | Bossu, S.; Carr, P.; Papanicolaou, A., A functional analysis approach to the static replication of european options, Quantitative Finance, 21, 4, 637-655 (2021) · Zbl 1477.91052 |
| [8] | Breeden, D. T.; Litzenberger, R. H., Prices of state-contingent claims implicit in option prices, Journal of business, 621-651 (1978) |
| [9] | Breiman, L., Optimal gambling systems for favorable games (1961) · Zbl 0109.36803 |
| [10] | Carr, P.; Ellis, K.; Gupta, V., Static hedging of exotic options, Journal of Finance, 53, 3, 1165-1190 (1998) |
| [11] | Carr, P.; Lee, R., Robust replication of volatiliy derivatives, NYU Mathematics in Finance Working Paper Series, 1-49 (2008) |
| [12] | Carr, P., Lee, R., & Lorig, M. (2017). Robust replication of barrier-style claims on price and volatility. Papers 1508.00632, arXiv.org, revised May 2017, 1-25. |
| [13] | Carr, P.; Madan, D., Towards a theory of volatility trading, Risk Management and Hedging: Part 3, 458-476 (2001) · Zbl 0990.91037 |
| [14] | Debreu, G., Theory of value: An axiomatic analysis of economic equilibrium (1959), Yale University Press · Zbl 0193.20205 |
| [15] | Diamond, S.; Boyd, S., CVXPY: A python-embedded modeling language for convex optimization, Journal of Machine Learning Research, 17, 83, 1-5 (2016) · Zbl 1360.90008 |
| [16] | Dupire, B., Pricing with a smile, Risk, 7, 1, 18-20 (1994) |
| [17] | Ferhati, T., Robust calibration for SVI model arbitrage free, Available at SSRN 3543766 (2020) |
| [18] | Ferhati, T., SVI model free wings, HAL Archives, hal-02517572 (2020) |
| [19] | Gatheral, J., A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives, Presentation at Global Derivatives & Risk Management, Madrid, 0 (2004) |
| [20] | Gatheral, J., The volatility surface: A practitioner’s guide (2011), John Wiley and Sons |
| [21] | Gatheral, J.; Jacquier, A., Arbitrage-free SVI volatility surfaces, Quantitative Finance, 14, 1, 59-71 (2014) · Zbl 1308.91187 |
| [22] | Green, R. C.; Jarrow, R. A., Spanning and completeness in markets with contingent claims, Journal of Economic Theory, 41, 1, 202-210 (1987) · Zbl 0621.90015 |
| [23] | Guastaroba, G.; Savelsbergh, M.; Speranza, M. G., Adaptive kernel search: A heuristic for solving mixed integer linear programs, European Journal of Operational Research, 263, 3, 789-804 (2017) · Zbl 1380.90290 |
| [24] | Guastaroba, G.; Speranza, M. G., Kernel search: An application to the index tracking problem, European Journal of Operational Research, 217, 1, 54-68 (2012) · Zbl 1244.91109 |
| [25] | Gurobi Optimization, L. (2021). Gurobi optimizer reference manual. http://www.gurobi.com. |
| [26] | Jackwerth, J.; Rubinstein, M., Recovering probability distributions from option prices, Journal of Finance, 51, 5, 1611-1631 (1998) |
| [27] | Kelly, J., A new interpretation of the information rate, The Bell System Technical Journal, 35, 917-926 (1956) |
| [28] | Kramkov, D.; Schachermayer, W., The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Annals of Applied Probability, 904-950 (1999) · Zbl 0967.91017 |
| [29] | Leung, T.; Lorig, M., Optimal static quadratic hedging, Quantitative Finance, 16, 9, 1341-1355 (2016) · Zbl 1400.91599 |
| [30] | Merton, R. C., Optimum consumption and portfolio rules in a continuous-time model, Stochastic optimization models in finance, 621-661 (1975), Elsevier |
| [31] | Nachman, D. C., Spanning and completeness with options, The Review of Financial Studies, 1, 3, 311-328 (1988) |
| [32] | Navratil, R.; Taylor, S.; Vecer, J., On equity market inefficiency during the COVID-19 pandemic, International Review of Financial Analysis, 77, 101820 (2021) |
| [33] | Richard, M.; Vecer, J., Efficiency testing of prediction markets: Martingale approach, likelihood ratio and Bayes factor analysis, Risks, 9, 2, 31 (2021) |
| [34] | Tavin, B., Implied distribution as a function of the volatility smile, Bankers Markets and Investors, 119, 31-42 (2011) |
| [35] | Vavasis, S. A., Quadratic programming is in NP, Information Processing Letters, 36, 2, 73-77 (1990) · Zbl 0719.90052 |
| [36] | Vecer, J., Optimal distributional trading gain: State price density equilibrium and Bayesian statistics, Available at SSRN (2020) |
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