After Andreas Binder gave insight to you that pricing financial instruments using trees can end with disaster I will show you in a multi-part blog that numerical methods for partial differential equations (PDEs) give you reliable tools for fast and stable pricing. How to choose the appropriate technique and what pitfalls you have to take care of will be discussed in detail.
To apply solution techniques for PDEs like Finite Difference or Finite Element methods you have to develop the corresponding PDE from your stochastic differential equation (SDE). In the first part of the blog series I will show this procedur by means of the well known Black Scholes model.
For the derivation we will use the definition of Brownian processes and the Lemma of Ito:
For the derivation we will use the definition of Brownian processes and the Lemma of Ito:
Brownian Processes:
Starting with infinitesimal increments dWt a whole family of processes describing their dynamics by Brownian motion can be defined:
Here μ and σ are deterministic functions. Equations like the one shown above are called stochastic differential equations.
Ito-Lemma:
Let f(x,t) be a twice continuously differentiable function and Xt a stochastic process like defined before. Then the lemma of Ito states
The Black Scholes Model:
In 1965 P. Samuelson modified the seminal work of Bachelier by assuming the geometric Brownian motion (lognormal distribution instead of normal distribution) for the dynamics of the price of a share. Assuming that St is the price of a share at time t the SDE for the Black Scholes model reads like
The constants μ and σ are the expected return rate and the volatility. It is evident that this process is part of the family of stochastic processes described before.
We denote C(S,t) as the price of a European call at time t for an asset with price process S=St. Furthermore πt denotes the value of a portfolio at time t including such a call and a number Δ of short positions in the traded asset its self to hedge the variability caused by the call.
A change in the value of the portfolio in an infinitesimal time interval is therefore
After applying Ito's Lemma and doing some straightforward calculations
we see that we can eliminate the stochastic component (represented by dS) when our portfolio consists of
short positions in the underlying asset at time t. As the resulting portfolio is risk less we can use the No-Arbitrage-Condition to state
Therefore for each S>0
This is the famous Black-Scholes equation. The properties of this equation will be discussed in the next part of our blog.
Let f(x,t) be a twice continuously differentiable function and Xt a stochastic process like defined before. Then the lemma of Ito states
The Black Scholes Model:
In 1965 P. Samuelson modified the seminal work of Bachelier by assuming the geometric Brownian motion (lognormal distribution instead of normal distribution) for the dynamics of the price of a share. Assuming that St is the price of a share at time t the SDE for the Black Scholes model reads like
The constants μ and σ are the expected return rate and the volatility. It is evident that this process is part of the family of stochastic processes described before.
We denote C(S,t) as the price of a European call at time t for an asset with price process S=St. Furthermore πt denotes the value of a portfolio at time t including such a call and a number Δ of short positions in the traded asset its self to hedge the variability caused by the call.
A change in the value of the portfolio in an infinitesimal time interval is therefore
After applying Ito's Lemma and doing some straightforward calculations
we see that we can eliminate the stochastic component (represented by dS) when our portfolio consists of
short positions in the underlying asset at time t. As the resulting portfolio is risk less we can use the No-Arbitrage-Condition to state
Therefore for each S>0
This is the famous Black-Scholes equation. The properties of this equation will be discussed in the next part of our blog.
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