TradingPlatform

Ongoing project of creating a trading platform and different alphas to give a user to whole experience.

Mathematics behind one of the strategy

You can read the Md file by copying and pasting the ReadMe.md here: https://stackedit.io/app#

Monte Carlo Strategy

Brownian Motion

The process ${W(t)}{t\geq0}$ is said to be (standard) Brownian Motion if the following are satisfied:

• $W(0)=0$
• For $s,t\geq0$ the random variable $W(s+t)-W(s) \sim N(0,t)$
• Whenever $0\leq t_0\leq t_1\leq....<t_n$ , the quantities $W(t_1)-W(t_0),W(t_2)-W(t_1),....,W(t_n)-W(t{n-1})$ are independent
• $W(t)$ is a continuous function of $t$ with probability $1$

Deriving Ito's Lemma (Time Independent)

Let us suppose that the asset price $S$ satisifies the Stochastic Differential Equation (SDE)

$$ dS = \mu dt + \sigma dW $$

where $\mu (t)$ and $\sigma(t)$ depends on the time-interval we look at (say 10D period) and the $W(s)$ for $s\leq t$ is a Brownian motion i.e the random fluctuation in a stock-price $S$

Now consider a function $f(S(t),t)$ of asset price where $f$ has a continuoous second derivative (i.e $f\in C^2 (0,T)$). For simplicity, let us assume that $f$ is independent of time i.e $f=f(S)$. Then by Taylor's Theorem:

$$ f(S+dS) = f(S) + f'(S)dS + \frac{1}{2}f''(S)(dS)^2+o((dS)^2) $$

Now: $dS = \mu dt + \sigma dW \ (dS)^2 = \mu^2(dt)^2+2\mu\sigma dt*dW + \sigma^2(dW)^2$

But since $dW$ has order $\sqrt{dt}$ then it has order $dt$. So overall we have: $$ (dS)^2 = \sigma^2(dW)^2 + o(dt)$$

After suitable subsitution, we overall get that:

$$ f(S+dS) - f(S) = f'(S)[\mu dt+\sigma dW] + \frac{1}{2}f''(S)\sigma^2(dW)^2 + o(dt) $$ <\p>

Since $W(t)$ is a Brownian Motion $\implies$ $dW=W(t+dt) - W(t)$ is a Brownian Motion with $dW \sim N(0,dt) \ \implies E[(dW)^2]= Var(dW)+0 \ \implies =dt$

So in the limit and replacing $(dW)^2$ by $dt$ , we get:

$$df = \frac{df}{dS}(\mu dt+\sigma dW) + \frac{1}{2}\frac{d^2f}{dS^2}\sigma^2dt$$

Ito's Lemma (Time Independent)

Let $f(S)$ be a continuous twice differentiable and suppose that: $$ dS = \mu dt + \sigma dW$$ Then: $$df = \frac{df}{dS}(\mu dt+\sigma dW) + \frac{1}{2}\frac{d^2f}{dS^2}\sigma^2dt$$ When written out in the Integral form: $$ f(S(T)) - f(S(0)) = \int_{0}^{T} \left(\mu dt+\sigma dW\right) dt + \int_{0}^{T} \sigma\frac{df}{dS} dW$$

Thus we get a relationship between a Stochastic Integral and a Standard Integral with respect to time.

A model for stock price

Consider an asset with price $S(t)$ that evolves according to the SDE $$ dS = \mu Sdt + \sigma SdW$$

Over a period $dt$, the price changes by a deterministic quantity $\mu Sdt$ (representing some underlying deterministic growth) and a random quantity $\sigma SdW$ (where $\sigma$ measures the volatility of the asset).

It is useful to work in terms of $log(S(t))$ so we define: $$ f(S) = log(S) \ f'(S) = \frac{1}{S} \ f''(S) = -\frac{1}{S^2} $$

So after plugging the following in the SDE we get:

$$ df = \left(\mu - \frac{1}{2}\sigma^2\right) + \sigma dW $$

Plugging into Ito's Lemma we get:

$$ log(S(T)) - log(S(0)) = \left(\mu - \frac{1}{2}\sigma^2\right)T + \sigma W(T) \quad (*)$$

We conclude that:

$$ log\left(\frac{S(T)}{S(0)}\right) \sim N\left((\mu - \frac{1}{2}\sigma^2)T,\sigma^2T\right) $$ The above equation can be generalised to give the following:

$$ log\left(\frac{S(t+\Delta t)}{S(t)}\right) \sim N\left((\mu - \frac{1}{2}\sigma^2)\Delta t,\sigma^2\Delta t\right)$$

So we say that $Y = \frac{S(t)}{S(0)}$ is log-normally distributed with: $$ E[Y] = exp\left[{\eta+\frac{1}{2}\sigma^2}t\right] \ $$ where $\eta = (\mu - \frac{1}{2}\sigma^2)t$

Using $(*)$ we can show that:

$$ S(t+\Delta t) = S(t)*exp\left[\left(\mu - \frac{1}{2}\sigma^2\right)\Delta t + \sigma W(t)\right]$$

where $\mu$ is 1+log-return's mean and $\sigma$ is 1+log-return's standard deviation

In my algorithm: $$ \Delta t = 1 \quad \text{since I have daily data} \ S_{t+1} = S_t*exp\left[\left(\mu - \frac{1}{2}\sigma^2\right) + \sigma W_t\right] $$