# Wiener Process

## Rate of return

Let $\{ S_t \}_{t \geq 0}$ be the stock price.
E: $\frac{S_{t + \Delta t} - S_t}{S_t} = \mu \delta t + \text{`noise'}$
'noise' has zero expectation
Consider $\{W_t\}_{t \geq 0}$, the Wiener Process

1. $W_0 = 0$
2. Independent Increment, $\forall t_1 < t_2 \leq t_3 < t_4, W_4 - W_3 \perp W_2 - W_1$
3. $W_{t+h} - W_t \sim \Nu(0, h)$

Are $W_t$ and $W_s$ independent? No!

$$ \begin{align*}
\operatorname{Cov}(W_t, W_s) &= E\{(W_t - E[W_t])(W_s - E[W_s])\} \\
&= E\{(W_t - E[W_t - W_0])(W_s - E[W_s - W_0])\} \\
&= E\{(W_t - 0)(W_s - 0)\} \\
&= E\{W_t W_s \} \\
\text{Suppose } t > s, &= E\{(W_t + W_s - W_s)(W_s)\} \\
&= E\{(W_t - W_s)W_s + W^2_s\} \\
&= E[(W_t - W_s)W_s] + E(W_s^2) \\
&= E[W_t - W_s]E[W_s] + E[W^2_s] \\
&= 0 + s
\end{align*}
$$

Therefore, $\operatorname{Cov}(W_t, W_s) = \min(t, s)$

## Naive Model

$$\begin{align*}
\frac{S_{t + \Delta t} - S_t}{S_t} &= \mu \Delta t + \sigma(W_{t + \Delta t} - W_t) \\
S_{t + \Delta t} &= S_t + S_t[\mu \Delta t + \sigma \Delta W_t] \: (\text{Euler's Method}) \\
S_{t + 2\Delta t} &= S_{t + \Delta t} + S_{t + \Delta t}[\mu \Delta t + \sigma \Delta W_{t + \Delta t}]
\end{align*}
$$

## Weakness

The simulation can goes to negative, which is not possible in real world.
There is sample error/estimate error/simulation error.

### Improvment **Idea**

Shorten the time period for simulation because the Uncertainty of Wiener Process is depend on the time spend. It is the one way to reduce the error.

## Stochastic Differential Equation

$\lim_{\Delta t \rightarrow 0} \frac{S_{t + \Delta t} - S_t}{S_t}$ doesn't exist.
$$\begin{align*}
\frac{S_{t + \Delta t} - S_t}{S_t} &= \mu \Delta t + \sigma(W_{t + \Delta t} - W_t) \\
d S_t &= S_t [\mu dt + \sigma dW_t] \\
\int^t_0 dS_t &= \int^t_0 S_t [\mu dt + \sigma dW_t] \\
S_t - S_t &= \int^t_0 S_t \mu dt + \int ^t_0 \sigma S_t dW_t
\end{align*}
$$

## More General Model

### Black-Scholes Model

$$ \begin{align*}
\frac{dS_t}{S_t} &= \mu dt + \sigma dW_t
\end{align*}
$$

### Ito's process

$$
dX_t = \mu(t, X_t) dt + \sigma(t, X_t)dW_t \\
$$

#### Black-Scholes Model in Ito's Form

$$
dS_t = (\mu \cdot S_t) dt + (\sigma \cdot S_t)dW_t
$$

### Brownian Motion

$$
dX_t = \mu dt + \sigma dW_t \\
X_t - X_0 = \int^t_0 \mu dt + \int^t_0 \sigma dW_t \\
X_t = X_0 + \mu t + \sigma (W_t - W_0) \\
X_t = X_0 + \mu t + \sigma W_t \\
X_t \sim \Nu(X_0 + \mu t, \sigma^2 t)
$$
The bound is known, so not very desireable.

## Ito's Process

$$
X_t = X_o \int^t_0 \mu(t, X-t)dt + \int^t_0 \sigma(t, X_t)dW_t \\
$$

### Ito's Isometry

Let $y_t = \int^t_0 \sigma(\tau, W_\tau)dW_\tau$

1. $E[y_t] = 0$
2. $Var(y_t) = \int^t_0 E[\sigma^2(\tau, W_\tau)]d\tau$

Proof of Property 1
$$
\tilde{y} = \sum^{n - 1}_{i=0} \sigma(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}] \text{ where } t_{i+1} - t_i = \Delta t \\
E[\tilde{y}] = E\{ \sum^{n - 1}_{i=0} \sigma(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}]\} \\
= E\left\{ E\left[\sum^{n - 1}_{i=0} \sigma(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}] \bigg| W_0, W_1, \cdots, W_{n-1}\right] \right\} \\
= E \left\{ \sum^{n-2}_{i=0} \sigma(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}] + E[\sigma(t_{n-1}, X_{t_{n-1}}) (W_{t_n} - W_{t_{n - 1}})] \bigg| \mathcal{F_{t_{n-1}}}) \right\} \\
= E \left\{ \sum^{n-2}_{i=0} \sigma(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}] + E[0 (W_{t_n} - W_{t_{n - 1}} )]\bigg| \mathcal{F_{t_{n-1}}}) \right\} \\
= E \left\{ \sum^{n-2}_{i=0} \sigma(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}] \bigg| \mathcal{F_{t_{n-1}}}) \right\} \\
= 0 \text{(By Induction)}
$$

Proof of property 2
$$
E[\tilde{y}^2] = E\left\{ \left(\sum^{n - 1}_{i=0} \sigma(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}] \right) \left( \sum^{n - 1}_{j=0} \sigma(t_j, X_{t_j}) [W_{t_{j+1}} - W_{t_j}] \right) \right\} \\
= E\left\{ \sum_{i \neq j} \sigma(t_i, X_{t_i}) \sigma(t_j, X_{t_j}) [W_{t_{i+1}} - W_{t_i}] [W_{t_{j+1}} - W_{t_j}] + \sum^{n-1}_{i=0}\sigma^2(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}]^2 \right\}
$$
Consider $E\left[\sum_{i \neq j} \sigma(t_i, X_{t_i}) \sigma(t_j, X_{t_j}) [W_{t_{i+1}} - W_{t_i}] [W_{t_{j+1}} - W_{t_j}]\right]$
$$
E\left[\sum_{i \neq j} \sigma(t_i, X_{t_i}) \sigma(t_j, X_{t_j}) [W_{t_{i+1}} - W_{t_i}] [W_{t_{j+1}} - W_{t_j}]\right] \\
= \sum_{i \neq j} E\left[\sigma(t_i, X_{t_i}) \sigma(t_j, X_{t_j}) [W_{t_{i+1}} - W_{t_i}] [W_{t_{j+1}} - W_{t_j}]\right]
$$
WLOG $i < j$,
$$
E\left\{ E\left[ \sigma(t_i, X_{t_i}) \sigma(t_j, X_{t_j}) [W_{t_{i+1}} - W_{t_i}] [W_{t_{j+1}} - W_{t_j}] \bigg| \mathcal{F}_{t_j} \right] \right\} \\
= E\left\{ \sigma(t_i, X_{t_i}) \sigma(t_j, X_{t_j}) [W_{t_{i+1}} - W_{t_i}] E\left[ [W_{t_{j+1}} - W_{t_j}] \bigg| \mathcal{F}_{t_j} \right] \right\} \\
= E\left\{ \sigma(t_i, X_{t_i}) \sigma(t_j, X_{t_j}) [W_{t_{i+1}} - W_{t_i}] (0) \right\} \\
= 0
$$

Consider $E\sum^{n-1}_{i=0}\sigma^2(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}]^2$
$$
E\sum^{n-1}_{i=0}\sigma^2(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}]^2 \\
= \sum^{n-1}_{i=0} E\left\{ \sigma^2(t_i, X_{t_i}) [W_{t_{i+1}} - W_{t_i}]^2 \bigg| \mathcal{F}_{t_i} \right\} \\
= \sum^{n-1}_{i=0} E[ \sigma(t_i, X_{t_i})]^2 \Delta t \rightarrow \int^t_0 E[ \sigma^2(t_i, X_{t_i})] dt
$$

### Vasicek's model

Mean returning Model
a = speed of returning long term equilibrium
b = long term equilibrium
$\sigma$ = volatility

$$
dV_t = a(b - r_t)d_t + \sigma dW_t, r_0 = r \\
r_t = r_0 + \int^t_0 a(b - r_\tau) d\tau + \int^t_0 \sigma dW_\tau \\
E[r_t] = E\{ r_0 + \int^t_0 a(b - r_\tau) d\tau \} + E[\int^t_0 r dW_\tau] \\
E[r_t] = r_0 + E\{\int^t_0 a(b - r_\tau) d\tau \} \\
E[r_t] = r_0 + \int^t_0\{ab - aE[r_\tau] \}d\tau
$$

Let $E[r_t] = f(t)$

$$
\frac{d f(t)}{dt} = 0 + ab - af(t) \\
\frac{d f(t)}{dt} + af(t) = ab \\
$$

Consider, $d(e^{at}f(t)) = f'(t)e^{at} + ae^{at}f(t)$

$$
e^{at}[\frac{d f(t)}{t} + af(t)] = (ab)e^{at} \\
\frac{d(e^{at} f(t))}{dt} = abe^{at} \\
\int d(e^{at} f(t)) = \int abe^{at} dt \\
e^{at} f(t) = be^{at} + C \\
E[r_t] = f(t) = b + Ce^{-at} \\
$$
Solve for C when $t = 0$,
$$
r = f(0) = b + Ce^0 \\
C = (r - b) \\
E[r_t] = f(t) = b + (r - b)e^{-at}
$$

<!-- $E[x] = \int^\infty_{-\infty} x f_x(x) dx$
Recall the Fubini's principle
$\iint\limits_D f(x, y) dx dy = \iint\limits_D f(x,y) dydx$ -->

$$
dX_t = t dt + e^t dW_t
$$
Euler's Scheme: $X_{t+ \Delta t} = X_t + t\Delta t + e^t [W_{t + \Delta t} - W_t]$
Exact SImulation: $X_{t + \Delta t} = X_t + \int^{t + \Delta t}_t \tau d\tau + \int^{t+\Delta t}_t e^\tau dW_\tau$
$=X_t + \frac{(t + \Delta t)^2 - t^2}{2} + \int^{t + \Delta t}_t e^\tau dW_\tau$
Note that, $\int^{t + \Delta t}_t e^\tau dW_\tau$ is SDE. By Ito's Isometry

Let $y = \int^{t + \Delta t}_t e^\tau dW_\tau \sim \mathcal{N}$

### Ito's Lemma

Consider $f(t, X) : [0, \infty) \times \mathbb{R} \rightarrow \mathbb{R} \text{ s.t. } f \in C^{1,2}$ and Ito's process $\{ X_t\}_{t \geq 0}$
$d f(t, X_t) = [\frac{\partial f}{\partial t} + \mu(t, X) \frac{\partial f}{\partial x} +\frac{\sigma^2(t, X)}{2} \frac{\partial^2 f}{\partial X^2}]dt + \sigma(t, X)\frac{\partial f}{\partial X} dW_t$

### Black-Scholes Model

$$
\frac{d S_t}{S_t} = \mu dt + \sigma dW_t \Leftrightarrow dS_t = \mu S_t dt + \sigma S_t dW_t
$$
Euler Scheme: $S_{t+ \Delta t} = S_t + \mu S_t \Delta t + \sigma S_t \sqrt{\Delta t} \epsilon$
Consider: $f(t, S) = \log S$
$$
\frac{\partial f}{\partial t} = 0; \frac{\partial f}{\partial S} = \frac{1}{S} ; \frac{\partial^2 f}{\partial S^2} = \frac{-1}{S^2} \\
d \log S = [0 + \mu S (\frac{1}{S}) + \frac{(\sigma S)^2}{2}(\frac{-1}{S^2})]dt + \sigma S (\frac{1}{S}) dW_t \\
d \log S_t = [\mu - \frac{\sigma^2}{2}]dt + \sigma dW_t \\
\log S_t - \log S_0 = \int^T_0 (\mu - \frac{\sigma^2}{2}) dt + \int^T_0 \sigma dW_t \\
S_T = S_0 \exp\{(\mu - \frac{\sigma^2}{2})T + \sigma(W_T - W_0)\}
$$

Another example

$$
\frac{d S_t}{S_t} = cos(at) dt + \sigma e^{bt} dW_t \\
f(t, S) = log S \\
d log S_t = [ cos(at) - \frac{\sigma^2 e^{2bt}}{2}]dt + \sigma e^{bt} dW_t \\
\int^T_0 d log S_t = \int^T_0 d[ cos(at) - \frac{\sigma^2 e^{2bt}}{2}]dt + \int^T_0 d\sigma e^{bt} dW_t \\
\log S_t - \log S_0 = \frac{1}{a}sin(at) \bigg|^{t=T}_{t=0} + \int^T_0 \sigma e^{bt} dW_t \\
S_T = S_0 \exp\{ \frac{1}{a} sin(aT) - \frac{\sigma^2}{4b}[e^{2bT} - 1] + \int^T_0 \sigma e^{bt} d W_t\} \\
S_T = S_0 \exp\{ \frac{1}{a}sin(aT) - \frac{\sigma^2}{4b}[e^{2bT} - 1] + \sqrt{\frac{\sigma^2}{2b}[e^{2bT} - 1]} \epsilon\}, \epsilon \sim N(0, 1) \\
\int^T_0 \sigma e^{bt} dW_t \sim \mathcal{N}(0, \int^T_0 \sigma^2 e^{2bt} dt) \\
\int^T_0 \sigma e^{bt} dW_t \sim \mathcal{N}(0, \frac{\sigma^2}{2b}[e^{2bT} -1]) \\
$$

Vasicek model.

$$
dr_t = a(b - r_t)dt + \sigma dW_t \\
dr_t + ar_t dt = ab dt + rdW_t
$$
Consider $f(t,r) = e^{at}r$
$$
\frac{\partial f}{\partial t} = ae^{at} r; \frac{\partial f}{\partial r} = e^{at}; \frac{\partial^2 f}{\partial r^2} = 0 \\
d[e^{at}r] = [ae^{at}r + a(b-r)e^{at} + 0]dt + \sigma e^{at} dW_t \\
d[e^{at}r] = [ abe^{at}]dt + \sigma e^{at} dW_t \\
\int^T_0 d[e^{at}r] = \int^T_0 abe^{at} dt + \int^T_0 \sigma e^{at} dW_t \\
e^{aT}r_T - r_0 = b[e^{aT} - 1] + \int^T_0 \sigma e^{at} W_t \\
r_T = r_0 e^{-aT} + b[1 - e^{-aT}] + \int^T_0 \sigma e^{-a(T-t)} dW_t \\
r_T = r_0 e^{-aT} + b[1 - e^{-aT}] + \sqrt{\frac{\sigma^2}{2a}[1 - e^{-2aT}]} \epsilon, \epsilon \sim N(0,1)\\
\int^T_0 \sigma e^{-a(T-t)} dW_t \sim \mathcal{N}(0, \frac{\sigma^2}{2a}[1 - e^{-2aT}]) \\
var = \int^T_0 \sigma^2 e^{-2a(T-t)}dt \\
= \frac{\sigma^2}{2a}[e^{-2a}(T-t)]^{t = T}_{t = 0}
$$

## Black-Sholes Equation

$$
\frac{d S_t}{S_t} = \mu dt + \sigma dW_t
$$
$f:$ Option pricing function with payoff $F(S_T)$
eg: Call - $F(S_T) = max(S_T - K, 0)$
Put - $F_(S_T) = max(K - S_T, 0)$
log-contract - $F(S_T) = logS_T$