Category: Quants

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:

1. Martingale: E(S_{t+k}|F_t)=E(S_t|F_t) for k>1.
2. Quadratic Variation: \sum_{i=1}^n (S_i-S_{i-1})^2 \to t
3. 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,

See the HTML file for full detals.

Key Takeaways

• Property of \{W_t\}:

1. $ W_t – W_s \sim N(0, t-s) $
2. $(W_t – W_{t-1})$ and $W_{t-i} – W_{t-i+1}$ are uncorrelated. So, $\int_0^t dW_u = \sum^t dW_u = W_t$

• For,

$$ dS_t = \mu S_t \ dt +\sigma S_t \ dW_t $$

Why the Geometric Brownian Motion of \{S_t\} is designed in that form?

The answer might be,

$$ dS_t /S_t = \mu \ dt +\sigma \ dW_t $$

$$\int_0^t dS_t /S_t = \int_0^t \mu \ dt + \int_0^t \sigma \ dW_t $$

$$ log(S_t) = \mu \ t +\sigma \ W_t $$

Taking the first difference is similar to differentiation. (d(log(S_T)) = log(S_t/S_t-1) = log(1+r_t) \ approx r_t).

$$ r_t = \mu + \sigma \ \Delta W_t $$

The return, \{r_t\}, is equal to a mean, \mu, plus a stochastic term. That is a random walk.

• We apply a transformation, if S_t follows a Geometric Brownian Motion, then f(S_t) follows another,

In calculating d f(S_t), we would get, (by Taylor Expansion)

$$ df(S_t) = \bigg( \frac{\partial f}{\partial t} + \frac{\partial f}{\partial S_t}\mu S_t +\frac{1}{2}\frac{\partial^2 f}{\partial S_t^2}\sigma^2 S_t^2 \bigg)dt + \frac{\partial f}{\partial S_t}\sigma S_t \ dW_t $$

• A Special Form of f(\cdot) is f(S) = log(S),

$$ d\ log(S_t) = \bigg( \mu – \frac{1}{2}\sigma^2 \bigg)dt + \sigma \ dW_t $$

Integrate the above equation from time 0 to time t, then we would get,

$$ log(S_t) = log(S_0) + (\mu – \frac{1}{2}\sigma^2)t + \sigma W_t $$

Codes are shown in the HTML file.

Markov Property

The Markov Property states that the expected value of the random variable \(S_i\) conditional upon all of the past events only depends on the previous value \(S_{i−1}\). The implication of the Markov Property is that the

Martingale Property

best expectation of future price, \((S_{t+k}\), is the current price, \(S_t\). The current price contains all the information until now.

$$ \mathbb{E}(S_{t+k}|F_t)=S_t, \text{ for k>=0} $$

That is called the Martingale Property.

Quadratic Variation

The Quadratic Variation is defined as,

$$ \sum_{i=1}^k (S_i – S_{i-1})^2 $$

In the coin toss case, each movement must be either “1” or “-1”, so \(S_i – S_{i-1} = \pm 1\). Thus, \((S_i – S_{i-1})^2 =1\) for all “i”. Therefore, the Quadratic Variation equals “k”.

A Realisation

Here below is the Code Realisation of a Brownian motion – Chapter 5.6 Paul Wilmott Introduces Quantitative Finance.