Application of Stochastic Differential Equation in Insurance Portfolio Construction Involving Leverage Function and Elasticity of Debt and Equity

Abstract

Leverage effect specifies the functional relationship between stock returns and volatility. As stock price declines, volatility tends to rise. Thus, the variability in market prices of a company's stock has pervasive effect when measuring the level of leverage in the capital structure. The determination of portfolio value of an investor using the continuous time second order stochastic differential equation has major consequence on leverage function. Usually, structural stochastic value of leveraged firms treats company's portfolio as equity whose underlying instrument is the company's asset. In this paper, the objectives are to theoretically: (i) measure the value of insurance company's portfolio by second order stochastic differential equation, (ii) apply Ito's rule to obtain a value on its leverage function and, (iii) obtain the analytical correspondence between equity and volatility in a leveraged company through infinitesimal calculus. The stochastic second order differential equation of portfolio value under two arguments results in equilibrium position which provides the traded price of the derivative, furthermore the linear combination of first order derivative of volatilities with respect to equity and debt is vanishingly zero based on the underlying elasticity of stock volatilities. The resultant effect is that elasticity = of debt and equity cancel out and -15E, SOSE, 51