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Contents

Prerequisites

The following environment configuration setup would guarantee reproducibility of the obtained results.

  • The latest release of the Ubuntu Linux distribution was used, i.e. Ubuntu 24.04 LTS. In fact, any Linux distribution should also be sufficient.
  • Docker Engine installation (docker engine).

Docker

The linux bash script run.sh is responsible for building the docker image specified in the Dockerfile and running the Jupyter application.

> ./run.sh

UbuntuCommands

Then simply copy the produced token value from the terminal and open your web browser and navigate to http://localhost:8080.

Make sure to paste the token value to the corresponding authentication field to log in.

LocalhostConnect

Pricing

Fourier Method

European Put Option

The payoff function of a European Put Option is the following.

$$f(x) = (K - e^x)^+$$

The fourier transform corresponding to the put option is equivallent to the following expression.

$$\hat{f}(u) = \int_{-\infty}^{\infty} e^{iux}(K - e^x)^+ dx = \frac{K^{1+iu} - K}{1+iu} - \frac{K^{iu} - 1}{iu}$$

Model Free Bounds

IBM Optimization CPLEX

Use IBM Decision Optimization CPLEX Modeling for Python

Let's use the DOcplex Python library to write the mathematical model in Python.

Create the model

All objects of the model belong to one model instance.

Define the decision variables.

  • The continuous variable z represents the dual variable corresponding to the probability measure in the primal problem.
  • The continuous matrix variable y represents dual variable corresponding to the call prices in the primal problem.

Express the objective

We want to find the sub-replicating portfolio (interpretation of dual problem).

European Basket Call Option

Under a number of assumptions listed in this paper here, the model-free lower bound of the basket call is obtained by solving the following optimization problem.

$$\begin{aligned}[t] \inf_{\pi} \ \mathbb{E}_{\pi}\left [ \left ( \mathbf{w \cdot S}(T) - K \right )^+ \right ]&, \\ \\\ \mathbb{E}_{\pi}\left [ \left ( S_i(T) - K_{ij} \right )^+ \right ]&=c_{ij}, \quad i\in \{1, \dots,n\}, \quad j\in \{1,\dots,m\}. \end{aligned}$$

The dual problem is given as follows.

$$\begin{aligned}[t] \sup_{z \in \mathbb{R}, \ y \in \mathbb{R}^{n \times m}} \ &z+ \sum \limits _{i=1}^n \sum \limits_{j=1}^m c_{ij} y_{ij}, \\ \\\ &z+ \sum \limits _{i=1}^n \sum \limits_{j=1}^m \left( s_i - K_{ij} \right)^+y_{ij} \le \left( \mathbf{w \cdot s} - K \right)^+, \quad \forall \ \mathbf{s} \in \mathbb{R}^n_{\ge 0}. \end{aligned}$$

It is enough to check the constraints for each

$$s \in \mathcal{C}_0$$

not for all

$$s \in \mathbb{R}^n_{+}$$

where

$$\mathcal{C}_0=\mathcal{C}_1\cup \mathcal{C}_2$$ $$\mathcal{C}_1=\left \{s\in \mathbb{R}^n_+ \mid \forall i, s_i=0 \text{ or } K_{ij} \text{ for some }j\right\}$$ $$\mathcal{C}_2=\bigcup\limits_{k=1}^n\mathfrak{B}_k$$ $$\mathfrak{B}_k=\left\{s\in \mathbb{R}^n_+ \mid \forall i \not= k, s_i=0 \text{ or } K_{ij} \text{ for some }j, \ s_k=w_k^{-1}\Big( K- \sum\limits_{i=1,i\not=k}^n w_is_i \Big)\ge 0\right\}.$$

The resulting linear program can be written as follows.

$$\begin{aligned}[t] \max_{z \in \mathbb{R}, \ y \in \mathbb{R}^{n \times m}} \ &z+ \sum \limits _{i=1}^n \sum \limits_{j=1}^m c_{ij} y_{ij}, \\ \\\ &z+ \sum \limits _{i=1}^n \sum \limits_{j=1}^m \left( s_i - K_{ij} \right)^+y_{ij} \le \left( \mathbf{w \cdot s} - K \right)^+, \quad \forall \ \mathbf{s} \in \mathfrak{C}_0, \\\ &\sum \limits_{j=1}^m y_{ij} \le w_i, \quad \forall \ i \in \{1, \dots,n\}. \end{aligned}$$

Separation Problem

$$\begin{align*} &\textbf{Check if } z+ \sum \limits _{i=1}^n \sum \limits_{j=1}^m \left( s_i - K_{ij} \right)^+y_{ij} \le \left( \mathbf{w \cdot s} - K \right)^+, \quad \forall \ \mathbf{s} \in \mathfrak{C}_0, \\\ &\textbf{or find } \quad \mathbf{s^*} \in \mathfrak{C}_0 \ \textbf{ such that the inequality is violated.} \\\ \end{align*}$$

Numerical Results

For the option strikes of the calls for each asset, the starting value is 100 and more calls are added with step size $\text{step} \mathrel{+}=1$ as m increases.

The algorithm is dependent on the choice of the initial set of constraints $\mathcal{C}$.

Assume that C3 is defined as follows.

$$\mathcal{C}_3=\left \{(s_1,\dots,s_n) \mid \text{ for some } i, s_i=K_{i0} \text{ or } K_{im} \text{ and for some }p, s_j=K_{jp} \ \forall \ j\not=i\right\}$$

Volatilities and weight vectors in the numerical tests.

$n$ Values
3 $\boldsymbol{\sigma}$ 1.0 1.6 2.0
$\mathbf{w}$ 0.3 0.35 0.35
4 $\boldsymbol{\sigma}$ 0.3 0.3 1.8 1.2
$\mathbf{w}$ 0.1 0.2 0.3 0.4
5 $\boldsymbol{\sigma}$ 0.3 0.4 0.8 1.8 1.9
$\mathbf{w}$ 0.2 0.2 0.2 0.2 0.2
6 $\boldsymbol{\sigma}$ 0.3 0.5 1.3 1.5 1.9 2.1
$\mathbf{w}$ 0.1 0.1 0.1 0.1 0.3 0.3
8 $\boldsymbol{\sigma}$ 0.1 0.2 1.5 1.6 1.7 1.8 1.9 2.0
$\mathbf{w}$ 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3

Monte Carlo

European Call Option

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 ${80,100,120}$

Geometric Asian Option

Paoyff functions of arithmetic

$$\left( \frac{1}{n} \sum_{i=1}^{n} S_{t_i} - K \right)^+$$

and geometric Asian Options

$$\left( \left( \prod_{i=1}^{n} S_{t_i} \right) ^{1/n} - K \right)^+.$$

MSFT Stock Price

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

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

MSFT Stock Price Forecast

Trading Strategies

Trading Bot

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:

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

References

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Some applications in Financial Mathematics.

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numerical-methods-implementation computational-finance financial-engineering pricing-derivatives heston-model stochastic-methods

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