Chapter 4
Stochastic Processes
Abstract After introducing binomial trees as a time-discrete model to value options,
this chapter moves on to discuss how a Wiener process gives rise to the diffusion equation, which is subsequently solved in terms of Green’s functions. A short digression
of the role of Green’s functions in physics follows, before Ito’s lemma is discussed
and used to derive the Fokker-Planck equation for the continuous-time model that
describes the evolution of stocks. Solving the Fokker-Planck equation then leads to
the well-known log-normal distribution. The latter is then used to derive the expectation value of an option’s value, should it be exercised at maturity. Examples from
other uses of expectation values in physics concludes this chapter.
In the previous chapter we ignored the time evolution of the stocks, apart from assuming that the underlying dynamics is governed by an average growth with superimposed fluctuations. In the present and subsequent chapters we relax this constraint
and will deal with the temporal evolution of financial assets; an elaborate system of
guesstimating the unforeseeable. We will not only consider the evolution of stocks,
but also that of other financial derivatives, such as options. Before dealing with quasicontinuous systems below, we first discuss a simple system that helps us understand
the mechanisms. It is based on discretizing time and the method is commonly referred
to as binomial trees.
4.1 Binomial Trees
In the theory of binomial trees, we consider different ways in which the value of a
financial product, for example, an option can move, either up or down and assign
probabilities to the different cases. The suitable initial price is then determined by
the expectation value of the option price at the final time. Note that this will depend
on the strike price K of the option. An option that carries a higher risk, but also the
promise of making a higher profit, will be worth more.
For definiteness, let us consider the pricing of a call option with initial price c for
an underlying asset, which we assume to be a share with initial price S. The tree is
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_4
29
Stochastic Processes
Abstract After introducing binomial trees as a time-discrete model to value options,
this chapter moves on to discuss how a Wiener process gives rise to the diffusion equation, which is subsequently solved in terms of Green’s functions. A short digression
of the role of Green’s functions in physics follows, before Ito’s lemma is discussed
and used to derive the Fokker-Planck equation for the continuous-time model that
describes the evolution of stocks. Solving the Fokker-Planck equation then leads to
the well-known log-normal distribution. The latter is then used to derive the expectation value of an option’s value, should it be exercised at maturity. Examples from
other uses of expectation values in physics concludes this chapter.
In the previous chapter we ignored the time evolution of the stocks, apart from assuming that the underlying dynamics is governed by an average growth with superimposed fluctuations. In the present and subsequent chapters we relax this constraint
and will deal with the temporal evolution of financial assets; an elaborate system of
guesstimating the unforeseeable. We will not only consider the evolution of stocks,
but also that of other financial derivatives, such as options. Before dealing with quasicontinuous systems below, we first discuss a simple system that helps us understand
the mechanisms. It is based on discretizing time and the method is commonly referred
to as binomial trees.
4.1 Binomial Trees
In the theory of binomial trees, we consider different ways in which the value of a
financial product, for example, an option can move, either up or down and assign
probabilities to the different cases. The suitable initial price is then determined by
the expectation value of the option price at the final time. Note that this will depend
on the strike price K of the option. An option that carries a higher risk, but also the
promise of making a higher profit, will be worth more.
For definiteness, let us consider the pricing of a call option with initial price c for
an underlying asset, which we assume to be a share with initial price S. The tree is
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_4
29