A Stochastic Second Order Partial Differential Equation of Portfolio Construction Involving Leverage Effect in continuous Time Actuarial Finance

Abstract

Leverage effect specifies the functional relationship between stock returns and volatility. As stock price declines,
volatility tends to rise. This is due to the effect that change in market valuation of a company’s equity has on the degree
of leverage function in the capital structure. The determination of portfolio value of an investor using the continuous
time second order stochastic differential equation has a 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 (1) measure the value of a company’s portfolio by second order
stochastic differential equation, (2) apply Ito’s rule to obtain the value of its leverage function and then (3) determine
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 first order derivative of volatilities
with respect to equity and debt is vanishingly zero based on the underlying elasticity of stock volatilities. The resulting
implication is that elasticity of debt and equity cancel out.