Here is a realisation of the previous blog, A bit Stochastic Calculus.
Implications
- 1. Stochastic Integral is different from Ordinary Integral, so as to differentiation. The Stochastic Differential Equation, S.D.E. we normally use to mimic the movement of a stock is as the following,
$$ d S=\mu S d + \sigma S dW$$
, where dW follows a Brownian Motion.
The Brownian Motion also has the following critical properties:
- Martingale: E(S_{t+k}|F_t)=E(S_t|F_t) for k>1.
- Quadratic Variation: \sum_{i=1}^n (S_i-S_{i-1})^2 \to t
- Normality: An increment of S_{t+dt} is normally distributed, with mean zero and variance t_i – t_{i-1}
- 2. What does Ito’s Lemma tell us?
If the stock price, S_t, follows a stochastic process, then the financial contracts F(S_t), as a function of the underlying stock price, follow another stochastic process.
- 3. Taylor Expansion helps
See notes about Paul Willmott’s book’s paragraph.
- 4. Different Forms of Stochastic Processes could be applied.
Example 1: dS=\mu \ dt + \sigma \ dX
Example 2: dS=\mu S \ dt +\sigma S \ dX
For this, dV = \frac{\partial V}{\partial t}dt+\frac{\partial V}{\partial S}dS+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}dt, which is also called the Geometric Brownian Motion (GBM). If in a specific form of value function V=F(S)=ln(S), then dF = (\mu -\frac{1}{2}\sigma^2)dt + \sigma \ dX. See derivation in attached notes.
Example 3: A mean-reverting random walk. dS=(v-\mu S)dt +\sigma dX
Example 4: Another mean-reverting r.w. dS=(v-\mu S)dt +\sigma S^{1/2}dX
The stochastic term is altered compared with example 3. Now if S ever gets close to zero the randomness decreases,