Abstract
We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. First, we apply our method to price options on non-traded assets for which there is a traded asset that is correlated to the non-traded asset. Our main contribution to this particular problem is to show that our seller/buyer prices are the upper/lower good deal bounds of Cochrane and Saá-Requejo (J Polit Econ 108:79–119, 2000) and of Björk and Slinko (Rev Finance 10:221–260, 2006) and to determine the analytical properties of these prices. Second, we apply our method to price options in the presence of stochastic volatility. Our main contribution to this problem is to show that the instantaneous Sharpe ratio, an integral ingredient in our methodology, is the negative of the market price of volatility risk, as defined in Fouque et al. (Derivatives in financial markets with stochastic volatility. Cambridge University Press, 2000).
Similar content being viewed by others
References
Barles G., Biton S., Bourgoing M. and Ley O. (2003). Uniqueness results for quasilinear parabolic equations through viscosity solutions’ methods. Calc Var 18: 159–179
Björk T. and Slinko I. (2006). Towards a general theory of good deal bounds. Rev Financ 10: 221–260
Cheridito, P., Kupper, M.: Time-consistency of indifference prices and monetary utility functions, preprint. Princeton University (2006). Available at http://www.princeton.edu/∼dito/papers/timf.pdf
Cochrane J. and Saá-Requejo J. (2000). Beyond arbitrage: Good deal asset price bounds in incomplete markets. J Polit Econ 108: 79–119
Crandall M.G., Ishii H. and Lions P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull Am Math Soc 27(1): 1–67
Davis, M.: Optimal hedging with basis risk, working paper. London: Imperial College (2000). Available at http://www.ma.ic.ac.uk/~mdavis/docs/basisrisk.pdf
Föllmer H. and Schweizer M. (1991). Hedging of contingent claims under incomplete information. In: Davis, M.H.A. and Elliott, R.J. (eds) Applied Stochastic Analysis. Stochastics Monographs, vol. 5., pp 389–414. Gordon and Breach, London
Forsyth, P., Labahn, G.: Numerical methods for controlled Hamilton–Jacobi–Bellman PDEs in finance, preprint. Waterloo: University of Waterloo (2006). Available at http://www.cs.uwaterloo.ca/paforsyt/hjb.pdf
Fouque J.P., Papanicolaou G. and Sircar R. (2000). Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, New York
Friedman A. (1975). Stochastic Differential Equations and Applications 1. Academic, New York
Gerber, H.U.: Introduction to mathematical risk theory. Huebner Foundation Monograph 8 Wharton School of the University of Pennsylvania, published by Richard D. Irwin, Homewood (1979)
Gerber H.U. and Shiu E.S.W. (1994). Option pricing by Esscher transforms (with discussions). Trans Soc Actuar 46: 99–191
Ilhan, A., Jonsson, M., Sircar, R.: Portfolio optimization with derivatives and indifference pricing. In: Carmona, R. (ed.) Volume on Indifference Pricing. Princeton: Princeton University Press (2004, to appear). Available at http://www.princeton.edu/∼sircar/
Karatzas I. and Shreve S.E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York
Leung, T., Sircar, R.: Accounting for risk aversion, vesting, job termination risk and multiple exercises in valuation of employee stock options, preprint. Princeton: Princeton University (2006). Available at http://www.princeton.edu/∼sircar/Public/ARTICLES/ESO_utility.pdf
Musiela M. and Zariphopoulou T. (2004). An example of indifference prices under exponential preferences. Financ Stoch 8: 229–239
Royden H.L. (1968). Real Analysis. Macmillan, New York
Schachermayer, W.: Introduction to the mathematics of financial markets. In: Lecture Notes in Mathematics, vol. 1816, pp. 111–177. Heidelberg: Springer (2000)
Schweizer M. (2001). From actuarial to financial valuation principles. Insur Math Econ 28: 31–47
Sircar R. and Zariphopoulou T. (2005). Bounds & asymptotic approximations for utility prices when volatility is random. SIAM J Control Optim 43(4): 1328–1353
Walter W. (1970). Differential and Integral Inequalities. Springer, New York
Windcliff H., Wang J., Forsyth P.A. and Vetzal K.R. (2007). Hedging with a correlated asset: Solution of a nonlinear pricing PDE. J Comput Appl Math 200: 86–115
Young, V.R.: Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio. Insurance: Math Econ (2007, to appear). Available at http://arxiv.org/abs/0705.1297v1
Zariphopoulou T. (2001). Stochastic control methods in asset pricing. In: Kannan, D. and Lakshmikantham, V. (eds) Handbook of Stochastic Analysis and Applications., pp 102–145. Marcel Dekker, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bayraktar, E., Young, V. Pricing options in incomplete equity markets via the instantaneous Sharpe ratio. Ann Finance 4, 399–429 (2008). https://doi.org/10.1007/s10436-007-0084-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10436-007-0084-0
Keywords
- Pricing derivative securities
- Incomplete markets
- Sharpe ratio
- Correlated assets
- Stochastic volatility
- Non-linear partial differential equations
- Good deal bounds