# American Institue of Mathematical Sciences

2016, 21(5): 1603-1615. doi: 10.3934/dcdsb.2016013

## Interest rates risk-premium and shape of the yield curve

 1 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, United States

Received  April 2015 Revised  December 2015 Published  April 2016

We apply the general theory of pricing in incomplete markets, due to the author, on the problem of pricing bonds for the Hull-White stochastic interest rate model. As pricing in incomplete markets involves more market parameters than the classical theory, and as the derived risk premium is time-dependent, the proposed methodology might offer a better way for replicating different shapes of the empirically observed yield curves. For example, the so-called humped yield curve can be obtained from a normal yield curve by only increasing the investors risk aversion.
Citation: Srdjan Stojanovic. Interest rates risk-premium and shape of the yield curve. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1603-1615. doi: 10.3934/dcdsb.2016013
##### References:
 [1] D. Becherer, Utility-indifference hedging and valuation via reaction-diffusion systems,, Proc. R. Soc. Lond. A, 460 (2004), 27. doi: 10.1098/rspa.2003.1234. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities,, J. Political Econ., 81 (1973), 637. doi: 10.1086/260062. [3] D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice,, Springer, (2001). doi: 10.1007/978-3-662-04553-4. [4] M. Davis, V. G. Panas and T. Zariphopoulou, European option pricing with transaction costs,, SIAM Journal on Control and Optimization, 31 (1993), 470. doi: 10.1137/0331022. [5] A. Friedman, Stochastic Differential Equations,, Vol 1 & 2, (1975). [6] S. D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs,, Review of Futures Markets, 8 (1989), 222. [7] J. Hull and A. White, Pricing interest-rate derivative securities,, The Review of Financial Studies, 3 (1990), 573. doi: 10.1093/rfs/3.4.573. [8] L. Jiang, Mathematical Modeling and Methods of Option Pricing,, World Scientific Publishing, (2005). doi: 10.1142/5855. [9] J. Kallsen, Utility-based derivative pricing in incomplete markets,, Mathematical Finance-Bachelier Congress 2000, (2000), 313. [10] Z. Kang and S. Stojanovic, Interest rate risk premium and equity valuation,, Journal of Systems Science and Complexity, 23 (2010), 484. doi: 10.1007/s11424-010-0142-y. [11] G. Liang and L. Jiang, A modified structural model for credit risk,, IMA Journal of Management Mathematics, 23 (2012), 147. doi: 10.1093/imaman/dpr004. [12] R. C. Merton, Theory of rational option pricing,, Bell Journal of Economics and Management Science, 4 (1973), 141. doi: 10.2307/3003143. [13] R. C. Merton, Continuous-Time Finance,, Wiley-Blackwell, (1990). [14] M. Musiela and T. Zariphopoulou, An example of indifference prices under exponential preferences,, Finance and Stochastics, 8 (2004), 229. doi: 10.1007/s00780-003-0112-5. [15] R. Rouge and N. El Karoui, Pricing via utility maximization and entropy,, Mathematical Finance, 10 (2000), 259. doi: 10.1111/1467-9965.00093. [16] S. Stojanovic, Computational Financial Mathematics using MATHEMATICA®,, Birkhauser, (2003). doi: 10.1007/978-1-4612-0043-7. [17] S. Stojanovic, Risk premium and fair option prices under stochastic volatility: The HARA solution,, C. R. Acad. Sci. Paris Ser. I, 340 (2005), 551. doi: 10.1016/j.crma.2004.11.002. [18] S. Stojanovic, Stochastic Volatility & Risk Premium,, Lecture Notes, (2005). [19] S. Stojanovic, Pricing and hedging of multi type contracts under multidimensional risks in incomplete markets modeled by general Itô SDE systems,, Asia Pacific Financial Markets, 13 (2006), 345. [20] S. Stojanovic, Advanced Financial Engineering for Interest Rates, Equity, and FX,, Lecture Notes, (2007). [21] S. Stojanovic, Any-utility neutral and indifference pricing and hedging,, Risk and Decision Analysis, 4 (2013), 103. [22] S. Stojanovic, Neutral and Indifference Portfolio Pricing, Hedging and Investing,, Springer, (2011). [23] O. Vasicek, An equilibrium characterization of the term structure,, Journal of Financial Economics, 5 (1977), 177. doi: 10.1002/9781119186229.ch6.

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##### References:
 [1] D. Becherer, Utility-indifference hedging and valuation via reaction-diffusion systems,, Proc. R. Soc. Lond. A, 460 (2004), 27. doi: 10.1098/rspa.2003.1234. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities,, J. Political Econ., 81 (1973), 637. doi: 10.1086/260062. [3] D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice,, Springer, (2001). doi: 10.1007/978-3-662-04553-4. [4] M. Davis, V. G. Panas and T. Zariphopoulou, European option pricing with transaction costs,, SIAM Journal on Control and Optimization, 31 (1993), 470. doi: 10.1137/0331022. [5] A. Friedman, Stochastic Differential Equations,, Vol 1 & 2, (1975). [6] S. D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs,, Review of Futures Markets, 8 (1989), 222. [7] J. Hull and A. White, Pricing interest-rate derivative securities,, The Review of Financial Studies, 3 (1990), 573. doi: 10.1093/rfs/3.4.573. [8] L. Jiang, Mathematical Modeling and Methods of Option Pricing,, World Scientific Publishing, (2005). doi: 10.1142/5855. [9] J. Kallsen, Utility-based derivative pricing in incomplete markets,, Mathematical Finance-Bachelier Congress 2000, (2000), 313. [10] Z. Kang and S. Stojanovic, Interest rate risk premium and equity valuation,, Journal of Systems Science and Complexity, 23 (2010), 484. doi: 10.1007/s11424-010-0142-y. [11] G. Liang and L. Jiang, A modified structural model for credit risk,, IMA Journal of Management Mathematics, 23 (2012), 147. doi: 10.1093/imaman/dpr004. [12] R. C. Merton, Theory of rational option pricing,, Bell Journal of Economics and Management Science, 4 (1973), 141. doi: 10.2307/3003143. [13] R. C. Merton, Continuous-Time Finance,, Wiley-Blackwell, (1990). [14] M. Musiela and T. Zariphopoulou, An example of indifference prices under exponential preferences,, Finance and Stochastics, 8 (2004), 229. doi: 10.1007/s00780-003-0112-5. [15] R. Rouge and N. El Karoui, Pricing via utility maximization and entropy,, Mathematical Finance, 10 (2000), 259. doi: 10.1111/1467-9965.00093. [16] S. Stojanovic, Computational Financial Mathematics using MATHEMATICA®,, Birkhauser, (2003). doi: 10.1007/978-1-4612-0043-7. [17] S. Stojanovic, Risk premium and fair option prices under stochastic volatility: The HARA solution,, C. R. Acad. Sci. Paris Ser. I, 340 (2005), 551. doi: 10.1016/j.crma.2004.11.002. [18] S. Stojanovic, Stochastic Volatility & Risk Premium,, Lecture Notes, (2005). [19] S. Stojanovic, Pricing and hedging of multi type contracts under multidimensional risks in incomplete markets modeled by general Itô SDE systems,, Asia Pacific Financial Markets, 13 (2006), 345. [20] S. Stojanovic, Advanced Financial Engineering for Interest Rates, Equity, and FX,, Lecture Notes, (2007). [21] S. Stojanovic, Any-utility neutral and indifference pricing and hedging,, Risk and Decision Analysis, 4 (2013), 103. [22] S. Stojanovic, Neutral and Indifference Portfolio Pricing, Hedging and Investing,, Springer, (2011). [23] O. Vasicek, An equilibrium characterization of the term structure,, Journal of Financial Economics, 5 (1977), 177. doi: 10.1002/9781119186229.ch6.
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