Abstract
Typical financial pricing models in continuous time are built upon hypothesized stochastic processes to assess the models’ dynamics over time and the underlying economy. A common and convenient method to describe the evolution of the modeled state variables is to specify their probabilistic behavior via Itô processes. Here, the basic building block is the Brownian motion which we examine in this chapter accompanied by the necessary stochastic calculus to set up appropriate stochastic differential equations. The final objective is to come close to finding the real data generating process behind the modeled capital market. The second part of this chapter introduces the fundamental no-arbitrage principle saying there should be no free lunch in efficient capital markets. This essential asset pricing argument can be employed in continuous time using the partial-differential-equation(PDE)and the equivalent-martingale-method(EMM)-approach which we elaborate on two classical cases of asset pricing. Thereafter, we are equipped with the financial modeling framework for building and implementing financial pricing models for valuation and hedging purposes.
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References
The earliest modeling is by Bachelier (1900), who used this type of stochastic processes to describe stock price movements. To cover jumps in the dynamics of a security, one can add a Poisson process component; see, for example, Merton (1976).
See, for example, Karlin and Taylor (1975, p. 376) and also for further standard distributions Abramowitz and Stegun (1965) and Stuart and Ord (1994).
See, for example, Black and Scholes (1973), Duffle (2001), Dybvig and Ross (1987), Ingersoll (1987), and Varian (1987).
In case of further interest in a detailed analysis of the differences between arbitrage-and equilibrium-oriented pricing see, for example, Schöbel (1995b).
For a comparative overview of different hypothesized distributions in financial modeling see, for example, McDonald (1996); among the first noticing empirical inconsistencies with normally distributed stock returns were Fama (1965) and Mandelbrot (1963).
In chapter 2 we examine the estimation principles underlying our applications in parts II, III, and IV.
For a broad variety of applications in equity, fixed-income, and foreign-exchange markets see, for example, Musiela and Rutkowski (1997).
The theory is essentially due to Harrison and Pliska (1981), building on the work in discrete time by Harrison and Kreps (1979); see also Stricker (1984) and Delbaen and Schachermayer (1994).
Named after the contributions of Feynman (1948) and Kac (1949). See theorem 15. 1.
For a complete background of this theorem see, for example, Duffle (2001) and Musiela and Rutkowski (1997).
This results in discounted asset prices being martingales with respect to Q; see Harrison and Pliska (1981) and Geman, Karoui, and Rochet (1995). However, in pricing term structure derivatives the forward martingale measure is used as in section 11. 2.
For the choice of the money market account as numéraire see, for example, Artzner (1997).
For an overview of the ‘incomplete’ models before Black and Scholes (1973), the various model extensions, and for empirical tests see, for example, Smith (1976).
For an analysis of volatility being a meaningful variation parameter see, for example, Bates (1996).
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© 2004 Springer-Verlag Berlin Heidelberg
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Kellerhals, B.P. (2004). Financial Modeling. In: Asset Pricing. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24697-8_1
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DOI: https://doi.org/10.1007/978-3-540-24697-8_1
Publisher Name: Springer, Berlin, Heidelberg
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