# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020060

## Effect of institutional deleveraging on option valuation problems

 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.

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, doi: 10.3934/jimo.2020060
##### References:

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##### References:
Variation of European Call Option with Respect to Distressed Selling Impact
Variation of Delta of a European Call Option with Respect to Distressed Selling Impact
Variation of Vega of a European Call Option with Respect to Distressed Selling Impact
Variation of Gamma of a European Call Option with Respect Distressed Selling Impact
Variation of European Call Option Price with Respect to Distressed Selling Impact. $f = \frac{\log{x}}{\eta}$
Variation of European Call Option Price with Respect to Distressed Selling Impact. $f = \frac{1-e^x}{\eta}$
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($\Gamma$) $\frac{ n(d_+)}{S(t)\bar{\sigma}_t\sqrt{T-t}}$
 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($\Gamma$) $\frac{ n(d_+)}{S(t)\bar{\sigma}_t\sqrt{T-t}}$
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 $b=0.0243$
 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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