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General properties of solutions to inhomogeneous Black-Scholes equations with discontinuous maturity payoffs. (English) Zbl 1341.35167

A terminal value problem of an inhomogeneous Black-Scholes equation with the risk free short rate \(r(t)\), the dividend rate \(q(t)\) of the underlying asset, the volatility \(\sigma(t)\) of the underlying asset’s price process, maturity payoff \(f(x)\), and inhomogeneous term \(g(x)\) is given by
\[ \begin{aligned} &\frac{\partial V}{\partial t}+ \frac{\sigma^2(t)}{2} x^2 \frac{\partial^2 V}{\partial x^2} +(r(t)-q(t))x \frac{\partial V}{\partial x} -r(t)V+g(x)=0, \\ & 0\leq t<T,\,\, 0<x<\infty, \\ & V(x,T)=f(x).\end{aligned} \]
The authors derive a representation of the solution to the considered terminal value problem and obtain the min-max estimates for the solution. Then it is studied the derivatives of the solution with respect to the stock price variable in the case with continuous terminal payoffs and inhomogeneous terms and the solution’s monotonicity and convexity under the conditions of monotonicity and convexity of the functions \(f(x)\) and \(g(x)\). The derivatives with respect to the stock price variable in the case with discontinuous terminal payoffs or inhomogeneous terms, the gradient estimate, and the solutions’ monotonicity are studied. An example of pricing of a defaultable coupon bond is presented.

MSC:

35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
35C15 Integral representations of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
91G20 Derivative securities (option pricing, hedging, etc.)
35B45 A priori estimates in context of PDEs
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References:

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