Stochastic Differential Equations (SDEs)

Previously, we introduced Brownian Motion and the Wiener Process as the foundation for modeling asset price paths.However, standard Brownian Motion has a chance of taking negative values, which does not reflect the reality of stock prices that cannot go below 0. Therefore, it is necessary to develop a new model that retains the characteristics of randomness but addresses the issue of negative values. That is Geometric Brownian Motion (GBM).

But before discussing GBM, we need to understand Stochastic Differential Equations (SDE) first.

The necessity of SDE

In the world of randomness, the general rules of calculus cannot be applied directly because Brownian Motion is non-differentiable and exhibits random behavior. Therefore, we need to use stochastic integrals instead.

Definition: Stochastic Integral

Define the non-anticipatory function f(t)f(t)f(t). The stochastic integral of fff with Brownian MotionB(t)B(t)B(t) is:

∫0Tf(t) dB(t)\int_0^T f(t) \, dB(t)∫0T​f(t)dB(t)

The function f(t) must not be able to use future information from B(t). For example, if f(t) is an investment strategy based on the price B(t), using future information would be equivalent to market manipulation.

Write in the form of a derivative

Even though dB(t)dB(t)dB(t) cannot be interpreted as a classical derivative, we often write it in shorthand:

dX(t)=μ(t,X(t)) dt+σ(t,X(t)) dB(t)dX(t) = \mu(t, X(t)) \, dt + \sigma(t, X(t)) \, dB(t)dX(t)=μ(t,X(t))dt+σ(t,X(t))dB(t)

which is a stochastic differential equation (SDE) that describes the changes in X(t)X(t)X(t)

Definition: Stochastic Differential Equation

Let B(t) be a Brownian Motion. If X(t) is a stochastic process that evolves according to:

dX(t)=μ(t,X(t)) dt+σ(t,X(t)) dB(t)dX(t) = \mu(t, X(t)) \, dt + \sigma(t, X(t)) \, dB(t)dX(t)=μ(t,X(t))dt+σ(t,X(t))dB(t)

Where:

• μ(t,X(t))μ(t, X(t))μ(t,X(t)): drift coefficient (non-random component)
• σ(t,X(t))σ(t,X(t)): diffusion coefficient (random component multiplied by dB(t)dB(t)dB(t))

Then X(t)X(t)X(t) is a process called the Ito Process or Ito drift-diffusion process.

SDE is the main tool for modeling the changes of stochastic variables, such as asset prices, by combining:

• Deterministic change (dt term)
• The uncertainty randomness (dB(t) term)
• In the next section, we will see how applying SDE to GBM can create a price model that only has positive values.

Reference: Stochastic Differential Equations

From https://www.quantstart.com/articles/Stochastic-Differential-Equations/