The Heston Stochastic Volatility Model assumes that the price of an asset is described by the equations:
$S_{t}=rS_{t}dt+\sqrt{V_t} S_t dW_t, \quad S_0=s$
$dV_t= \kappa(\theta-V_t)dt+\eta\sqrt{V_t}d\overline{W_{t}}, \quad V_0 =v.$
If $X_t=log(S_t)$
then by applying the Itô's lemma
the aforementioned equations can be written in the equivallent form:
$dX_t= (r-\frac{V_t}{2})dt+\sqrt{V_t}dW_{t}, \quad X_0 =x$
$dV_t= \kappa(\theta-V_t)dt+\eta\sqrt{V_t}d\overline{W_{t}}, \quad V_0 =v,$
where:
$\overline{W}= \rho W+\sqrt{1-\rho^2}\hat{W}.$
The parameters passed to the model can be found in the following table:
| Parameters | Symbol | Values |
| Mean Reverison | $\kappa$ |
1 |
| Long Run Variance | $\theta$ | 0.09 |
| Current Variance | v | 0.09 |
| Correlation | $\rho$ |
-0.3 |
| Volatility | $\eta$ |
1 |
| Maturity | T | 1 |
| Interest Rate |
r |
0 |
| Strike Prices |
K |
$\{80,100,120\}$ |
Simulating the stock price of Microsoft for the upcoming 250 trading days MC techniques were used to forecast the prices (100 trajectories were simulated):
$$\text{PriceToday}=\text{PriceYesterday} \times e^{\underbrace{\mu -\frac{\sigma^2}{2}}_{\text{drift}} + \underbrace{\sigma \mathbf{Z}(\text{Rand[0,1]})}_{\text{volatility}}}.$$
Basic automated trading bot which implements strategies on real-time price data of the CRYPTO-market. The Relative Strength Index (RSI) measures the magintude of recent price changes to evaluate overbought or oversold conditions in the price of a stock:
$\text{RSI}= 100 - \frac{100}{1 + \frac{\overline{\text{Gain}}}{\overline{\text{Loss}}}}.$
[1] https://www.investopedia.com/terms/h/heston-model.asp
[2] https://www.binance.com
[3] https://finance.yahoo.com/
[4] https://github.com/binance/binance-spot-api-docs
