Intrinsic Prices of Risk

Author(s)
Truc Le

ABSTRACT

We review the nature of some well-known phenomena such as volatility smiles, convexity adjustments and parallel derivative markets. We propose that the market is incomplete and postulate the existence of intrinsic risks in every contingent claim as a basis for understanding these phenomena. In a continuous time framework, we bring together the notion of intrinsic risk and the theory of change of measures to derive a probability measure, namely risk-subjective measure, for evaluating contingent claims. This paper is a modest attempt to prove that measure of intrinsic risk is a crucial ingredient for explaining these phenomena, and in consequence proposes a new approach to pricing and hedging financial derivatives. By adapting theoretical knowledge to practical applications, we show that our approach is consistent and robust, compared with the standard risk-neutral approach.

We review the nature of some well-known phenomena such as volatility smiles, convexity adjustments and parallel derivative markets. We propose that the market is incomplete and postulate the existence of intrinsic risks in every contingent claim as a basis for understanding these phenomena. In a continuous time framework, we bring together the notion of intrinsic risk and the theory of change of measures to derive a probability measure, namely risk-subjective measure, for evaluating contingent claims. This paper is a modest attempt to prove that measure of intrinsic risk is a crucial ingredient for explaining these phenomena, and in consequence proposes a new approach to pricing and hedging financial derivatives. By adapting theoretical knowledge to practical applications, we show that our approach is consistent and robust, compared with the standard risk-neutral approach.

KEYWORDS

Implied Volatility, Convexity Adjustment, Primary and Parallel Markets, Incomplete Markets, Intrinsic Risk, Risk-Neutral Measure, Risk-Subjective Measure, Fair Valuation, Delta-Hedging

Implied Volatility, Convexity Adjustment, Primary and Parallel Markets, Incomplete Markets, Intrinsic Risk, Risk-Neutral Measure, Risk-Subjective Measure, Fair Valuation, Delta-Hedging

Cite this paper

Le, T. (2014) Intrinsic Prices of Risk.*Journal of Mathematical Finance*, **4**, 318-327. doi: 10.4236/jmf.2014.45029.

Le, T. (2014) Intrinsic Prices of Risk.

References

[1] Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-659.

http://dx.doi.org/10.1086/260062

[2] Cox, J.C. and Ross, C.A. (1976) The Valuation of Options for Alternative Stochastic Processes. Journal of Financial Economics, 3, 145-166.

http://dx.doi.org/10.1016/0304-405X(76)90023-4

[3] Scott, L.O. (1987) Option Pricing When the Variance Changes Randomly: Theory, Estimation and an Application. Journal of Financial and Quantitative Analysis, 22, 419-438.

http://dx.doi.org/10.2307/2330793

[4] Dupire, B. (1994) Pricing with a Smile. Risk, 7, 18-20.

[5] Derman, E. and Kani, I. (1994) Riding on a Smile. Risk, 7, 32-39.

[6] Merton, R. (1976) Option Pricing When Underlying Stock Returns Are Discontinuous. Journal of Financial Economics, 3, 125-144.

http://dx.doi.org/10.1016/0304-405X(76)90022-2

[7] Hull, J. and White, A. (1988) An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility. Advances in Futures and Options Research, 3, 29-61.

[8] Heston, S. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6, 327-343.

http://dx.doi.org/10.1093/rfs/6.2.327

[9] Bates, D. (1996) Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options. Review of Financial Studies, 9, 69-107.

http://dx.doi.org/10.1093/rfs/9.1.69

[10] Dupire, B. (1996) A Unified Theory of Volatility. Working Paper.

[11] Morgan, J.P. (1999) Pricing Exotics under the Smile. Risk, 11, 72-75.

[12] Britten-Jones, M. and Neuberger, A. (2000) Option Prices, Implied Prices Processes, and Stochastic Volatility. Journal of Finance, 55, 839-866.

http://dx.doi.org/10.1111/0022-1082.00228

[13] Blacher, G. (2001) A New Approach for Designing and Calibrating Stochastic Volatility Models for Optimal Delta Vega Hedging of Exotic Options. Conference Presentation at Global Derivatives, Juan-les-Pins.

[14] Hull, J. and White, A. (1990) Pricing Interest Rate Derivative Securities. The Review of Financial Studies, 3, 573-592.

http://dx.doi.org/10.1093/rfs/3.4.573

[15] Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985) A Theory of the Term Structure of Interest Rates. Econometrica, 53, 385-408.

http://dx.doi.org/10.2307/1911242

[16] Black, F., Derman, E. and Toy, W. (1990) A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options. Financial Analysts Journal, 46, 33-39.

http://dx.doi.org/10.2469/faj.v46.n1.33

[17] Black, F. and Karasinski, P. (1991) Bond and Option Pricing When Short Rates Are Lognormal. Financial Analysts Journal, 47, 52-59.

[18] Heath, D., Jarrow, R. and Morton, A. (1992) Bond Pricing and the Term Structure of Interest Rates. Econometrica, 61, 77-105.

[19] Sandmann, K. and Sondermann, D. (1994) On the Stability of Log-Normal Interest Rate Models and the Pricing of Eurodollar Futures. Discussion Paper, University of Bonn, Bonn.

[20] Harrison, J.M. and Kreps, D.M. (1979) Martingales and Arbitrage in Multiperiod Securities Markets. Journal of Economic Theory, 20, 381-408.

http://dx.doi.org/10.1016/0022-0531(79)90043-7

[21] Harrison, J.M. and Pliska, S.R. (1981) Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Processes and Their Applications, 11, 215-260.

http://dx.doi.org/10.1016/0304-4149(81)90026-0

[22] Harrison, J.M. and Pliska, S.R. (1983) A Stochastic Calculus Model of Continuous Trading: Complete Markets. Stochastic Processes and Their Applications, 15, 313-316.

http://dx.doi.org/10.1016/0304-4149(83)90038-8

[23] Delbaen, F. and Schachermayer, W. (2004) What Is a Free Lunch? Notices of the AMS, 51, 526-528.

[24] Back, K. (1993) Asymmetric Information and Options. Review of Financial Studies, 6, 435-472.

http://dx.doi.org/10.1093/rfs/6.3.435

[25] Follmer, 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, Gordon and Breach, London/ New York, 38.

[26] Karatzas, I. and Shreve, S.E. (1998) Methods of Mathematical Finance. Springer-Verlag, New York.

http://dx.doi.org/10.1007/b98840

[27] Klebaner, F.C. (2012) Introduction to Stochastic Calculus with Applications. 3rd Edition, 2nd Edition, Imperial College Press.

[28] Jackwerth, J.C. (2000) Recovering Risk Aversion from Option Prices and Realized Returns. The Review of Financial Studies, 13, 433-451.

http://dx.doi.org/10.1093/rfs/13.2.433

[29] Schonbucher, P.J. (1999) A Market Model for Stochastic Implied Volatility. Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 357, 2071-2092.

[30] Cont, R., Fonseca, J.D. and Durrleman, V. (2002) Stochastic Models of Implied Volatility Surfaces. Economic Notes by Banca Monte dei Paschi di Siena SpA, 31, 361-377.

[31] Le, T. (2005) Stochastic Market Volatility Models. Applied Financial Economics Letters, 1, 177-188.

http://dx.doi.org/10.1080/17446540500101986

[32] Carr, P. and Wu, L. (2010) A New Simple Approach for Constructing Implied Volatility Surfaces. Preprint, SSRN.

[33] Rebonato, R. (2004) The Perfect Hedger and the Fox: Volatility and Correlation in the Pricing of FX, Equity and Interest-Rate Options. 2nd Edition, Wiley, Hoboken.

[34] El Karoui, N., Jeanblanc-Picque, M. and Shreve, S. (1998) Robustness of the Black and Scholes Formula. Mathematical Finance, 8, 93-126.

http://dx.doi.org/10.1111/1467-9965.00047

[35] Linetsky, V. (2006) Pricing Equity Derivatives Subject to Bankruptcy. Mathematical Finance, 16, 255-282.

http://dx.doi.org/10.1111/j.1467-9965.2006.00271.x

[36] Keynes, J.M. (1964) The General Theory of Employment, Interest and Money. Harcourt Brace Jovanovich, New York.

[37] Mehra, R. and Prescott, E.C. (1985) The Equity Premium Puzzle. Journal of Monetary Economics, 15, 145-161.

[1] Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-659.

http://dx.doi.org/10.1086/260062

[2] Cox, J.C. and Ross, C.A. (1976) The Valuation of Options for Alternative Stochastic Processes. Journal of Financial Economics, 3, 145-166.

http://dx.doi.org/10.1016/0304-405X(76)90023-4

[3] Scott, L.O. (1987) Option Pricing When the Variance Changes Randomly: Theory, Estimation and an Application. Journal of Financial and Quantitative Analysis, 22, 419-438.

http://dx.doi.org/10.2307/2330793

[4] Dupire, B. (1994) Pricing with a Smile. Risk, 7, 18-20.

[5] Derman, E. and Kani, I. (1994) Riding on a Smile. Risk, 7, 32-39.

[6] Merton, R. (1976) Option Pricing When Underlying Stock Returns Are Discontinuous. Journal of Financial Economics, 3, 125-144.

http://dx.doi.org/10.1016/0304-405X(76)90022-2

[7] Hull, J. and White, A. (1988) An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility. Advances in Futures and Options Research, 3, 29-61.

[8] Heston, S. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6, 327-343.

http://dx.doi.org/10.1093/rfs/6.2.327

[9] Bates, D. (1996) Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options. Review of Financial Studies, 9, 69-107.

http://dx.doi.org/10.1093/rfs/9.1.69

[10] Dupire, B. (1996) A Unified Theory of Volatility. Working Paper.

[11] Morgan, J.P. (1999) Pricing Exotics under the Smile. Risk, 11, 72-75.

[12] Britten-Jones, M. and Neuberger, A. (2000) Option Prices, Implied Prices Processes, and Stochastic Volatility. Journal of Finance, 55, 839-866.

http://dx.doi.org/10.1111/0022-1082.00228

[13] Blacher, G. (2001) A New Approach for Designing and Calibrating Stochastic Volatility Models for Optimal Delta Vega Hedging of Exotic Options. Conference Presentation at Global Derivatives, Juan-les-Pins.

[14] Hull, J. and White, A. (1990) Pricing Interest Rate Derivative Securities. The Review of Financial Studies, 3, 573-592.

http://dx.doi.org/10.1093/rfs/3.4.573

[15] Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985) A Theory of the Term Structure of Interest Rates. Econometrica, 53, 385-408.

http://dx.doi.org/10.2307/1911242

[16] Black, F., Derman, E. and Toy, W. (1990) A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options. Financial Analysts Journal, 46, 33-39.

http://dx.doi.org/10.2469/faj.v46.n1.33

[17] Black, F. and Karasinski, P. (1991) Bond and Option Pricing When Short Rates Are Lognormal. Financial Analysts Journal, 47, 52-59.

[18] Heath, D., Jarrow, R. and Morton, A. (1992) Bond Pricing and the Term Structure of Interest Rates. Econometrica, 61, 77-105.

[19] Sandmann, K. and Sondermann, D. (1994) On the Stability of Log-Normal Interest Rate Models and the Pricing of Eurodollar Futures. Discussion Paper, University of Bonn, Bonn.

[20] Harrison, J.M. and Kreps, D.M. (1979) Martingales and Arbitrage in Multiperiod Securities Markets. Journal of Economic Theory, 20, 381-408.

http://dx.doi.org/10.1016/0022-0531(79)90043-7

[21] Harrison, J.M. and Pliska, S.R. (1981) Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Processes and Their Applications, 11, 215-260.

http://dx.doi.org/10.1016/0304-4149(81)90026-0

[22] Harrison, J.M. and Pliska, S.R. (1983) A Stochastic Calculus Model of Continuous Trading: Complete Markets. Stochastic Processes and Their Applications, 15, 313-316.

http://dx.doi.org/10.1016/0304-4149(83)90038-8

[23] Delbaen, F. and Schachermayer, W. (2004) What Is a Free Lunch? Notices of the AMS, 51, 526-528.

[24] Back, K. (1993) Asymmetric Information and Options. Review of Financial Studies, 6, 435-472.

http://dx.doi.org/10.1093/rfs/6.3.435

[25] Follmer, 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, Gordon and Breach, London/ New York, 38.

[26] Karatzas, I. and Shreve, S.E. (1998) Methods of Mathematical Finance. Springer-Verlag, New York.

http://dx.doi.org/10.1007/b98840

[27] Klebaner, F.C. (2012) Introduction to Stochastic Calculus with Applications. 3rd Edition, 2nd Edition, Imperial College Press.

[28] Jackwerth, J.C. (2000) Recovering Risk Aversion from Option Prices and Realized Returns. The Review of Financial Studies, 13, 433-451.

http://dx.doi.org/10.1093/rfs/13.2.433

[29] Schonbucher, P.J. (1999) A Market Model for Stochastic Implied Volatility. Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 357, 2071-2092.

[30] Cont, R., Fonseca, J.D. and Durrleman, V. (2002) Stochastic Models of Implied Volatility Surfaces. Economic Notes by Banca Monte dei Paschi di Siena SpA, 31, 361-377.

[31] Le, T. (2005) Stochastic Market Volatility Models. Applied Financial Economics Letters, 1, 177-188.

http://dx.doi.org/10.1080/17446540500101986

[32] Carr, P. and Wu, L. (2010) A New Simple Approach for Constructing Implied Volatility Surfaces. Preprint, SSRN.

[33] Rebonato, R. (2004) The Perfect Hedger and the Fox: Volatility and Correlation in the Pricing of FX, Equity and Interest-Rate Options. 2nd Edition, Wiley, Hoboken.

[34] El Karoui, N., Jeanblanc-Picque, M. and Shreve, S. (1998) Robustness of the Black and Scholes Formula. Mathematical Finance, 8, 93-126.

http://dx.doi.org/10.1111/1467-9965.00047

[35] Linetsky, V. (2006) Pricing Equity Derivatives Subject to Bankruptcy. Mathematical Finance, 16, 255-282.

http://dx.doi.org/10.1111/j.1467-9965.2006.00271.x

[36] Keynes, J.M. (1964) The General Theory of Employment, Interest and Money. Harcourt Brace Jovanovich, New York.

[37] Mehra, R. and Prescott, E.C. (1985) The Equity Premium Puzzle. Journal of Monetary Economics, 15, 145-161.