Abstract
Trading on a stock market is obviously a discrete process, as it consists of single transactions performed at distinct times. There are, however, so many transactions in such a high frequency that it is for many applications better to model them in a time-continuous setting, i.e., to assume that they take place at all times. In this chapter we will provide a short introduction to time-continuous models. An important difference will be that prices are exogenously given. In particular, we will derive the famous Black-Scholes model for asset pricing as it has been introduced by Fischer Black and Myron Scholes (J Pol Econ 81(3):637–654, 1973) and by Robert C. Merton (Econometrica 41(5):867–887, 1973). In 1997, the Nobel prize has been awarded for this work. The importance of this model can not be overemphasized, as the Royal Swedish Academy put it:
Their innovative work … has provided us with completely new ways of dealing with financial risk, both in theory and in practice. Their method has contributed substantially to the rapid growth of markets for derivatives in the last two decades.
In fact, their formula is probably the most used formula on banks and stock exchanges worldwide even today. In this chapter we will derive this celebrated formula, but we will not reduce all to one equation. There is a whole theory behind this result and we will see how this theory can be used to solve many more problems. We will also see what limitations the classical theory still has and sketch some ways how to overcome them.
\(\varPi \acute{a}\nu \tau a\ \chi \omega \rho \epsilon \tilde{\iota }\ \kappa a\grave{\iota }\ o\!\mathop{\upsilon }\limits ^{,}\!\delta \grave{\epsilon }\nu \ \mu \acute{\epsilon }\nu \epsilon \iota.\) (All is flux and nothing stays still.) Heraklite, as quoted by Platon.
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Notes
- 1.
With this innocent looking assumption we have excluded many derivatives that are path dependent, i.e., their value does not only depend on S and t, but also on the previous prices of the underlying asset. We will come back to this later when we talk about numerical methods for the computation of asset prices.
- 2.
An ansatz is a specific functional form which we assume the solution to have in order to compute it. This assumption is a posteriori justified if we obtain a solution that is indeed of the assumed form. In the appendix more ideas on how to solve PDEs and further references are given.
- 3.
More precisely, we study a probability space \((\varOmega,\mathcal{F},p)\), where \(\mathcal{F}\) is a \(\sigma\)-algebra and p is a probability measure on Ω with respect to \(\mathcal{F}\), see Sect. A.4 for details.
- 4.
This and the following definitions can also be adapted to discrete-time problems.
- 5.
We have seen such a trading strategy already in Sect. 8.3: the delta hedge strategy.
- 6.
Restrictive laws seem to hinder their success in the US.
- 7.
The need for studying other processes will be discussed in Sect. 8.8.
- 8.
Compare also [CV02].
- 9.
It is of course possible that u 1 is time-independent and that both utilities are in effect the same, however, in reality this is unlikely to be the case: consider an investment problem over one year without earnings, then the consumption in that year induces probably a smaller utility than the saved money at the end of that year does, simply because the latter has to be used in all the subsequent years still to come.
- 10.
More precisely: let X(t) denote the process of the value of the portfolio defined by the investment strategy (i.e., X depends in particular on S, c and \(\theta\)), then the expected value of the intertemporal consumption, \(\int _{0}^{T}\min \{0,u_{1}(t,c(t))\}\,\mathrm{d}t\), has to be larger than \(-\infty \) and the expected value of the final wealth, \(\min \{0,u_{2}(X(T))\}\) has to be larger than \(-\infty \), as well.
- 11.
For mathematical terminology compare Appendix A.
- 12.
Up to now we have only considered the case where the risk-free rate and the volatility were constant and borrowing and investing in the risk-free asset had the same fixed interest rate. Moreover, we have not considered dividends. All of these extensions are not essential to understand this theorem, but are stated for completeness. These extensions include as special case particular the setting we have previously used where r = b, δ = 0 and \(\sigma\) and r being constant in time. For details see [KS98].
- 13.
For technical definition, e.g. predictable process, see [Duf96]
- 14.
Originally, NIG distributions have been used in physics, more precisely in the modeling of turbulence and sand grain distributions. Only years later they made it into finance.
- 15.
Excess kurtosis refers to the amount of kurtosis that exceeds that of the normal distribution.
- 16.
Other data gives very similar results. For illustration we concentrate on one particular case. See [RSW] for details.
- 17.
A vector of random variables is heteroskedastic if the random variables have different variances.
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Hens, T., Rieger, M.O. (2016). Time-Continuous Model. In: Financial Economics. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49688-6_8
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