Stochastic Calculus for Finance II

The problem of the timing of an investment decision under partial information is analyzed in a framework where the firm is ambiguity averse. The analysis yields the description of a robust decision rule for an investment in a finite life project in presence of a stochastic instantaneous return. It is demonstrated that ambiguity aversion may accelerate investment in the short run. Ex post validation of the determined investment policy treats the impact of ambiguity aversion on the proper way of discounting of the profit flow resulting from the project and the fair price of risk associated with ambiguity aversion.

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“…To do so, we refer to the concept of the market price of risk known from stochastic finance (see [31], Chap. 15 or [32], Chap. 5).…”

Section: Endogenous Price Of Riskmentioning

confidence: 99%

The Worst Case for Real Options

Trojanowska

Kort

2010

J Optim Theory Appl

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“…Let P (S, y, z, t) be the price of a down-and-out binary option. According to the Feynman-Kac theorem [18], the pricing system for such an option under the multi-scale SV model can be derived as…”

Section: Solution Processmentioning

confidence: 99%

Pricing credit default swaps under a multi-scale stochastic volatility model

Chen

2017

Physica A: Statistical Mechanics and its Applications

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In this paper, we consider the pricing of credit default swaps (CDSs) with the reference asset driven by a geometric Brownian motion with a multi-scale stochastic volatility (SV), which is a two-factor volatility process with one factor controlling the fast time scale and the other representing the slow time scale. A key feature of the current methodology is to establish an equivalence relationship between the CDS and the down-and-out binary option through the discussion of "no default" probability, while balancing the two SV processes with the perturbation method. An approximate but closed-form pricing formula for the CDS contract is finally obtained, whose accuracy is in the order of θ (ϵ+δ+ϵδ) AbstractIn this paper, we consider the pricing of credit default swaps (CDSs) with the reference asset driven by a geometric Brownian motion with a multi-scale stochastic volatility (SV), which is a two-factor volatility process with one factor controlling the fast time scale and the other representing the slow time scale. A key feature of the current methodology is to establish an equivalence relationship between the CDS and the down-and-out binary option through the discussion of "no default" probability, while balancing the two SV processes with the perturbation method. An approximate but closed-form pricing formula for the CDS contract is finally obtained, whose accuracy is in the order of O(ϵ + δ + √ ϵδ).

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“…These models lack the precise foundations of Newton's, Maxwell's, or Schrödinger's equations. Instead, there is a growing role for nonlinear stochastic PDEs [112,146,42] with a wide range of applications, from financial models and data assimilation in atmospheric sciences to material sciences and biological models; see [118,185,116,121,154,169,166] and the references therein. Moreover, more often than not, realistic models from social and biological sciences do not allow separation of scales; instead, one is forced to study nonlinear PDEs across scales.…”

Section: Mathematical Models and Modern Mathematical Toolsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

142

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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Stochastic Calculus for Finance II

Cited by 886 publications

References 0 publications

The Worst Case for Real Options

The Worst Case for Real Options

Pricing credit default swaps under a multi-scale stochastic volatility model

A review of numerical methods for nonlinear partial differential equations

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