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July  2021, 17(4): 2097-2118. doi: 10.3934/jimo.2020060

Effect of institutional deleveraging on option valuation problems

Qing-Qing Yang 1, , Wai-Ki Ching 1, , Wan-Hua He 1, and  Na Song 2,,

1. 

Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
2. 

School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, China

* Corresponding author: Na Song

Received  March 2019 Revised  November 2019 Published  March 2020

This paper studies the valuation problem of European call options when the presence of distressed selling may lead to further endogenous volatility and correlation between the stock issuer's asset value and the price of the stock underlying the option, and hence influence the option price. A change of numéraire technique, based on Girsanov Theorem, is applied to derive the analytical pricing formula for the European call option when the price of underlying stock is subject to price pressure triggered by the stock issuer's own distressed selling. Numerical experiments are also provided to study the impacts of distressed selling on the European call option prices.

Keywords: Option Valuation, Fire Sales, Deleverage, Risk-Neutral Pricing, Endogenous Risk.
Mathematics Subject Classification: Primary: 91G20; Secondary: 91B70.
Citation: Qing-Qing Yang, Wai-Ki Ching, Wan-Hua He, Na Song. Effect of institutional deleveraging on option valuation problems. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2097-2118. doi: 10.3934/jimo.2020060
References:
[1]

M. Anton and C. Polk, Connected stocks, The Journal of Finance, 69 (2014), 1099-1127.  doi: 10.1111/jofi.12149.  Google Scholar

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[3]

M. K. Brunnermeier, Deciphering the liquidity and credit crunch 2007-2008, Journal of Economic Perspectives, 23 (2009), 77-100.  doi: 10.1257/jep.23.1.77.  Google Scholar

[4]

M. Carlson, A brief history of the 1987 stock market crash with a discussion of the federal reserve response, in Finance and Economics Discussion Series, Federal Reserve Board, Washington, DC., 2006. Google Scholar

[5]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar

[6]

R. Cont and L. Wagalath, Fire sales forensics: Measuring endogenous risk, Mathematical Finance, 26 (2016), 835-866.  doi: 10.1111/mafi.12071.  Google Scholar

[7]

J. Coval and E. Stafford, Asset fire sales (and purchases) in equity markets, Journal of Financial Economics, 86 (2007), 479-512.  doi: 10.1016/j.jfineco.2006.09.007.  Google Scholar

[8]

D. Duffie and K. Singleton, Modeling term structures of defaultable bonds, The Review of Financial Studies, 12 (1999), 687-720.  doi: 10.1093/rfs/12.4.687.  Google Scholar

[9]

A. Ellul, C. Jotikasthira and T. C. Lundblad, Regulatory pressure and fire sales in the corporate bond market, Journal of Financial Economics, 101 (2011), 596-620.  doi: 10.1016/j.jfineco.2011.03.020.  Google Scholar

[10]

H. Geman, N. El Karoui and J.-C. Rochet, Changes of num'eraire, changes of probability measure and option pricing, Journal of Applied Probability, 32 (1995), 443-458.  doi: 10.2307/3215299.  Google Scholar

[11]

R. Greenwood and D. Thesmar, Stock price fragility, Journal of Financial Economics, 102 (2011), 471-490.  doi: 10.1016/j.jfineco.2011.06.003.  Google Scholar

[12]

T. Hida, J. Potthoff and L. Streit, Dirichlet forms and white noise analysis, Communications in Mathematical Physics, 116 (1988), 235-245.  doi: 10.1007/BF01225257.  Google Scholar

[13]

A. Khandani and A. W. Lo, What happened to the quants in August 2007? Evidence from factors and transactions data, Journal of Financial Markets, 14 (2011), 1-46.  doi: 10.1016/j.finmar.2010.07.005.  Google Scholar

[14]

A. S. Kyle and W. Xiong, Contagion as a wealth effect, The Journal of Finance, 56 (2001), 1401-1440.  doi: 10.1111/0022-1082.00373.  Google Scholar

[15]

R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2.  Google Scholar

[16]

P. Pedler, Occupation times for two state Markov chains, Journal of Applied Probability, 8 (1971), 381-390.  doi: 10.2307/3211908.  Google Scholar

[17]

B. Sericola, Occupation times in Markov processes, Communications in Statistics. Stochastic Models, 16 (2000), 479-510.  doi: 10.1080/15326340008807601.  Google Scholar

[18]

S. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer Finance. Springer-Verlag, New York, 2004.  Google Scholar

[19]

A. Shleifer and R. W. Vishny, Liquidation values and debt capacity: A market equilibrium approach, The Journal of Finance, 47 (1992), 1343-1366.  doi: 10.1111/j.1540-6261.1992.tb04661.x.  Google Scholar

[20]

A. Shleifer and R. W. Vishny, Fire sales in finance and macroeconomics, Journal of Economic Perspectives, 25 (2011), 29-48.  doi: 10.1257/jep.25.1.29.  Google Scholar

[21]

R. Wiggins, T. Piontek and A. Metrick, The Lehman Brothers Bankruptcy A: Overview. Yale Program on Financial Stability Case Study 2014-3A-V1, SSRN, (2015), 23 pp. doi: 10.2139/ssrn.2588531.  Google Scholar

[22]

Q.-Q. Yang, W.-K. Ching and T.-K. Siu, Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales, Journal of Industrial and Management Optimization, 15 (2019), 293-318.  doi: 10.3934/jimo.2018044.  Google Scholar

show all references

References:
[1]

M. Anton and C. Polk, Connected stocks, The Journal of Finance, 69 (2014), 1099-1127.  doi: 10.1111/jofi.12149.  Google Scholar

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[3]

M. K. Brunnermeier, Deciphering the liquidity and credit crunch 2007-2008, Journal of Economic Perspectives, 23 (2009), 77-100.  doi: 10.1257/jep.23.1.77.  Google Scholar

[4]

M. Carlson, A brief history of the 1987 stock market crash with a discussion of the federal reserve response, in Finance and Economics Discussion Series, Federal Reserve Board, Washington, DC., 2006. Google Scholar

[5]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar

[6]

R. Cont and L. Wagalath, Fire sales forensics: Measuring endogenous risk, Mathematical Finance, 26 (2016), 835-866.  doi: 10.1111/mafi.12071.  Google Scholar

[7]

J. Coval and E. Stafford, Asset fire sales (and purchases) in equity markets, Journal of Financial Economics, 86 (2007), 479-512.  doi: 10.1016/j.jfineco.2006.09.007.  Google Scholar

[8]

D. Duffie and K. Singleton, Modeling term structures of defaultable bonds, The Review of Financial Studies, 12 (1999), 687-720.  doi: 10.1093/rfs/12.4.687.  Google Scholar

[9]

A. Ellul, C. Jotikasthira and T. C. Lundblad, Regulatory pressure and fire sales in the corporate bond market, Journal of Financial Economics, 101 (2011), 596-620.  doi: 10.1016/j.jfineco.2011.03.020.  Google Scholar

[10]

H. Geman, N. El Karoui and J.-C. Rochet, Changes of num'eraire, changes of probability measure and option pricing, Journal of Applied Probability, 32 (1995), 443-458.  doi: 10.2307/3215299.  Google Scholar

[11]

R. Greenwood and D. Thesmar, Stock price fragility, Journal of Financial Economics, 102 (2011), 471-490.  doi: 10.1016/j.jfineco.2011.06.003.  Google Scholar

[12]

T. Hida, J. Potthoff and L. Streit, Dirichlet forms and white noise analysis, Communications in Mathematical Physics, 116 (1988), 235-245.  doi: 10.1007/BF01225257.  Google Scholar

[13]

A. Khandani and A. W. Lo, What happened to the quants in August 2007? Evidence from factors and transactions data, Journal of Financial Markets, 14 (2011), 1-46.  doi: 10.1016/j.finmar.2010.07.005.  Google Scholar

[14]

A. S. Kyle and W. Xiong, Contagion as a wealth effect, The Journal of Finance, 56 (2001), 1401-1440.  doi: 10.1111/0022-1082.00373.  Google Scholar

[15]

R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2.  Google Scholar

[16]

P. Pedler, Occupation times for two state Markov chains, Journal of Applied Probability, 8 (1971), 381-390.  doi: 10.2307/3211908.  Google Scholar

[17]

B. Sericola, Occupation times in Markov processes, Communications in Statistics. Stochastic Models, 16 (2000), 479-510.  doi: 10.1080/15326340008807601.  Google Scholar

[18]

S. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer Finance. Springer-Verlag, New York, 2004.  Google Scholar

[19]

A. Shleifer and R. W. Vishny, Liquidation values and debt capacity: A market equilibrium approach, The Journal of Finance, 47 (1992), 1343-1366.  doi: 10.1111/j.1540-6261.1992.tb04661.x.  Google Scholar

[20]

A. Shleifer and R. W. Vishny, Fire sales in finance and macroeconomics, Journal of Economic Perspectives, 25 (2011), 29-48.  doi: 10.1257/jep.25.1.29.  Google Scholar

[21]

R. Wiggins, T. Piontek and A. Metrick, The Lehman Brothers Bankruptcy A: Overview. Yale Program on Financial Stability Case Study 2014-3A-V1, SSRN, (2015), 23 pp. doi: 10.2139/ssrn.2588531.  Google Scholar

[22]

Q.-Q. Yang, W.-K. Ching and T.-K. Siu, Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales, Journal of Industrial and Management Optimization, 15 (2019), 293-318.  doi: 10.3934/jimo.2018044.  Google Scholar

Figure 1.  Variation of European Call Option with Respect to Distressed Selling Impact
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Figure 2.  Variation of Delta of a European Call Option with Respect to Distressed Selling Impact
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Figure 3.  Variation of Vega of a European Call Option with Respect to Distressed Selling Impact
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Figure 4.  Variation of Gamma of a European Call Option with Respect Distressed Selling Impact
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Figure 5.  Variation of European Call Option Price with Respect to Distressed Selling Impact. $ f = \frac{\log{x}}{\eta} $
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Figure 6.  Variation of European Call Option Price with Respect to Distressed Selling Impact. $ f = \frac{1-e^x}{\eta} $
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Table 1.  Greeks
Option Price($C(t)$) $S(t)\mathcal{N}(d_+(t))-KB(t,T)\mathcal N(d_-(t))$
Delta($\Delta$) $\mathcal N(d_+(t))$
Vega($\mathcal V$) $S(t) n(d_+(t))\sqrt{T-t}$
Gamma($\Gamma$) $\frac{ n(d_+)}{S(t)\bar{\sigma}_t\sqrt{T-t}}$
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Option Price($C(t)$) $S(t)\mathcal{N}(d_+(t))-KB(t,T)\mathcal N(d_-(t))$
Delta($\Delta$) $\mathcal N(d_+(t))$
Vega($\mathcal V$) $S(t) n(d_+(t))\sqrt{T-t}$
Gamma($\Gamma$) $\frac{ n(d_+)}{S(t)\bar{\sigma}_t\sqrt{T-t}}$
Table 2.  Preference parameters
Parameters Values Parameters Values
Market depth $ L=10 $ MLR $ \eta=1 $
Volatility $ \sigma_S=0.2 $ Volatility $ \sigma_X=0.1 $
Volatility $ \sigma_r=0.15 $ Time to maturity $ T-t=1 $
Initial price $ S_0=40 $ Strike price $ K=40 $
Initial price $ X_0=100 $ Initial price $ B(t,T)=0.05 $
Correlation $ \rho=0.7 $ Time steps $ N=100 $
Correlation $ \rho_{1r}=0.5 $ Correlation $ \rho_{2r}=0.6 $
Mean-reverting speed $ a=100 $ Long-term interest rate $ b=0.0243 $
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Parameters Values Parameters Values
Market depth $ L=10 $ MLR $ \eta=1 $
Volatility $ \sigma_S=0.2 $ Volatility $ \sigma_X=0.1 $
Volatility $ \sigma_r=0.15 $ Time to maturity $ T-t=1 $
Initial price $ S_0=40 $ Strike price $ K=40 $
Initial price $ X_0=100 $ Initial price $ B(t,T)=0.05 $
Correlation $ \rho=0.7 $ Time steps $ N=100 $
Correlation $ \rho_{1r}=0.5 $ Correlation $ \rho_{2r}=0.6 $
Mean-reverting speed $ a=100 $ Long-term interest rate $ b=0.0243 $
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