American Institute of Mathematical Sciences

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

Interest rates risk-premium and shape of the yield curve

Srdjan Stojanovic 1,

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.
Keywords: optimal pricing., yield curve, risk premium, Interest rate.
Mathematics Subject Classification: Primary: 60H10, 60G40, 93E2.
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.  Google Scholar

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities,, J. Political Econ., 81 (1973), 637.  doi: 10.1086/260062.  Google Scholar

[3]

D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice,, Springer, (2001).  doi: 10.1007/978-3-662-04553-4.  Google Scholar

[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.  Google Scholar

[5]

A. Friedman, Stochastic Differential Equations,, Vol 1 & 2, (1975).   Google Scholar

[6]

S. D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs,, Review of Futures Markets, 8 (1989), 222.   Google Scholar

[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.  Google Scholar

[8]

L. Jiang, Mathematical Modeling and Methods of Option Pricing,, World Scientific Publishing, (2005).  doi: 10.1142/5855.  Google Scholar

[9]

J. Kallsen, Utility-based derivative pricing in incomplete markets,, Mathematical Finance-Bachelier Congress 2000, (2000), 313.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. C. Merton, Theory of rational option pricing,, Bell Journal of Economics and Management Science, 4 (1973), 141.  doi: 10.2307/3003143.  Google Scholar

[13]

R. C. Merton, Continuous-Time Finance,, Wiley-Blackwell, (1990).   Google Scholar

[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.  Google Scholar

[15]

R. Rouge and N. El Karoui, Pricing via utility maximization and entropy,, Mathematical Finance, 10 (2000), 259.  doi: 10.1111/1467-9965.00093.  Google Scholar

[16]

S. Stojanovic, Computational Financial Mathematics using MATHEMATICA®,, Birkhauser, (2003).  doi: 10.1007/978-1-4612-0043-7.  Google Scholar

[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.  Google Scholar

[18]

S. Stojanovic, Stochastic Volatility & Risk Premium,, Lecture Notes, (2005).   Google Scholar

[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.   Google Scholar

[20]

S. Stojanovic, Advanced Financial Engineering for Interest Rates, Equity, and FX,, Lecture Notes, (2007).   Google Scholar

[21]

S. Stojanovic, Any-utility neutral and indifference pricing and hedging,, Risk and Decision Analysis, 4 (2013), 103.   Google Scholar

[22]

S. Stojanovic, Neutral and Indifference Portfolio Pricing, Hedging and Investing,, Springer, (2011).   Google Scholar

[23]

O. Vasicek, An equilibrium characterization of the term structure,, Journal of Financial Economics, 5 (1977), 177.  doi: 10.1002/9781119186229.ch6.  Google Scholar

show all references

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.  Google Scholar

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities,, J. Political Econ., 81 (1973), 637.  doi: 10.1086/260062.  Google Scholar

[3]

D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice,, Springer, (2001).  doi: 10.1007/978-3-662-04553-4.  Google Scholar

[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.  Google Scholar

[5]

A. Friedman, Stochastic Differential Equations,, Vol 1 & 2, (1975).   Google Scholar

[6]

S. D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs,, Review of Futures Markets, 8 (1989), 222.   Google Scholar

[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.  Google Scholar

[8]

L. Jiang, Mathematical Modeling and Methods of Option Pricing,, World Scientific Publishing, (2005).  doi: 10.1142/5855.  Google Scholar

[9]

J. Kallsen, Utility-based derivative pricing in incomplete markets,, Mathematical Finance-Bachelier Congress 2000, (2000), 313.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. C. Merton, Theory of rational option pricing,, Bell Journal of Economics and Management Science, 4 (1973), 141.  doi: 10.2307/3003143.  Google Scholar

[13]

R. C. Merton, Continuous-Time Finance,, Wiley-Blackwell, (1990).   Google Scholar

[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.  Google Scholar

[15]

R. Rouge and N. El Karoui, Pricing via utility maximization and entropy,, Mathematical Finance, 10 (2000), 259.  doi: 10.1111/1467-9965.00093.  Google Scholar

[16]

S. Stojanovic, Computational Financial Mathematics using MATHEMATICA®,, Birkhauser, (2003).  doi: 10.1007/978-1-4612-0043-7.  Google Scholar

[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.  Google Scholar

[18]

S. Stojanovic, Stochastic Volatility & Risk Premium,, Lecture Notes, (2005).   Google Scholar

[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.   Google Scholar

[20]

S. Stojanovic, Advanced Financial Engineering for Interest Rates, Equity, and FX,, Lecture Notes, (2007).   Google Scholar

[21]

S. Stojanovic, Any-utility neutral and indifference pricing and hedging,, Risk and Decision Analysis, 4 (2013), 103.   Google Scholar

[22]

S. Stojanovic, Neutral and Indifference Portfolio Pricing, Hedging and Investing,, Springer, (2011).   Google Scholar

[23]

O. Vasicek, An equilibrium characterization of the term structure,, Journal of Financial Economics, 5 (1977), 177.  doi: 10.1002/9781119186229.ch6.  Google Scholar

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