Financial Engineering

Stochastic Numerical Methods in Financial Mathematics

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	$\{80,100,120\}$

Monte Carlo Techniques for Stock Pricing

Simulating the stock price of Microsoft for the upcoming 250 trading days MC techniques were used to forecast the prices (100 trajectories were simulated):

$$\text{PriceToday}=\text{PriceYesterday} \times e^{\underbrace{\mu -\frac{\sigma^2}{2}}_{\text{drift}} + \underbrace{\sigma \mathbf{Z}(\text{Rand[0,1]})}_{\text{volatility}}}.$$

Automated Trading Strategies

Overview

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:

$\text{RSI}= 100 - \frac{100}{1 + \frac{\overline{\text{Gain}}}{\overline{\text{Loss}}}}.$

Requirements

• python-binance
• TA-Lib
• numpy
• websocket_client

References

[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