Quantitative Research for Master's Thesis & Career as Quantiative Trader

Logic and Methodology

The objective of this study is to estimate, at any given time \( t \), the probability that the asset price \( S_T \) at a future horizon \( T \) exceeds a predefined strike \( K_t \), which in this context corresponds to the hourly open price. The forecasted probability is denoted as:

\[ p_t = \mathbb{P}(S_T > K_t \mid \mathcal{F}_t) \]

where \( \mathcal{F}_t \) represents all information available up to time \( t \). In a cash-or-nothing binary option, the payoff at maturity is given by:

\[ \Pi_T = \begin{cases} 1, & \text{if } S_T > K_t, \\ 0, & \text{otherwise.} \end{cases} \]

Under the risk-neutral measure \( \mathbb{Q} \), the theoretical value of this contract at time \( t \) is:

\[ V_t = e^{-r(T-t)} \mathbb{E}_{\mathbb{Q}}[\Pi_T \mid \mathcal{F}_t] = e^{-r(T-t)} \mathbb{Q}(S_T > K_t). \]

In cryptocurrency markets, where the risk-free rate \( r \) is approximately zero, this simplifies to:

\[ V_t = \mathbb{Q}(S_T > K_t). \]

Hence, the market quote itself represents the implied probability that the event \( S_T > K_t \) occurs. The machine learning model aims to construct an empirical estimator \( \hat{p}_t \) that approximates this risk-neutral probability and identifies systematic deviations \( \hat{p}_t - V_t \), which serve as actionable trading signals.

Stochastic Foundation

The next stage of this research focuses on extending the stochastic framework for option pricing by exploring models that capture both volatility dynamics and market jumps. The starting point is the Heston model, which incorporates stochastic variance as an additional source of uncertainty. The model is defined as follows:

\[ \begin{cases} dS_t = \mu S_t \, dt + \sqrt{v_t} S_t \, dW_t^{(1)}, \\[6pt] dv_t = \kappa(\theta - v_t) \, dt + \xi \sqrt{v_t} \, dW_t^{(2)}, \end{cases} \]

where \( S_t \) is the asset price, \( v_t \) the instantaneous variance, \( \mu \) the drift, \( \kappa \) the mean-reversion rate, \( \theta \) the long-term variance level, \( \xi \) the volatility of variance, and \( W_t^{(1)}, W_t^{(2)} \) are Brownian motions with correlation \( \rho \). The inclusion of stochastic variance allows the model to reproduce volatility clustering and leverage effects commonly observed in financial markets.

The Heston framework serves as a foundation for constructing more complex hybrid models that can integrate neural network estimation or data-driven calibration. In particular, machine learning techniques can be applied to estimate parameters such as \( \kappa \), \( \theta \), and \( \xi \) dynamically from observed data, enabling adaptive modeling of volatility surfaces and option price distributions in real time.

Future extensions will consider jump components of the form:

\[ dS_t = \mu S_t \, dt + \sqrt{v_t} S_t \, dW_t^{(1)} + S_{t^-}(J - 1) \, dN_t, \]

where \( N_t \) is a Poisson process with intensity \( \lambda \) and \( J \) represents the random proportional jump size. The integration of jump terms with stochastic variance (often referred to as the Bates model) will allow for richer representations of empirical return distributions, including heavy tails and volatility spikes.

These continuous-time formulations provide a mathematically consistent foundation for option valuation and can be linked to machine learning frameworks through parameter inference and state estimation. Neural networks trained on simulated or real market data can be employed to approximate the conditional expectations implied by these stochastic models, bridging analytical finance and data-driven prediction.

Feature Engineering

For each time step, a vector of input features \( x_t \in \mathbb{R}^d \) is generated from real-time market microstructure variables:

\[ x_t = [\Delta P_t, \; \Delta V_t, \; \text{spread}_t, \; \text{OBI}_t, \; \sigma_t, \; \lambda_t, \; \kappa_t, \ldots] \]

where \( \Delta P_t \) and \( \Delta V_t \) denote normalized price and volume changes, \( \text{spread}_t \) is the relative bid–ask spread, \( \text{OBI}_t \) the order book imbalance, \( \sigma_t \) the estimated local volatility, \( \lambda_t \) the empirical jump frequency, and \( \kappa_t \) the average jump size. These features encode both short-term dynamics and structural risk components that affect binary option pricing.

Neural Network Integration

The feature sequences \( \{x_{t-L+1}, \ldots, x_t\} \) are processed by a Temporal Convolutional Network (TCN). The network employs dilated causal convolutions to capture temporal dependencies across multiple scales while preserving the chronological order of observations. The output layer produces a vector of predicted probabilities:

\[ \hat{y}_t = \begin{bmatrix} P(\Delta_{t,5} > 0) \\ P(\Delta_{t,10} > 0) \\ P(\Delta_{t,20} > 0) \\ P(\Delta_{t,30} > 0) \\ P(\Delta_{t,T} > 0) \end{bmatrix}, \]

where \( \Delta_{t,h} = \text{close}_{t+h} - \text{strike}_t \) measures the difference between the future close price and the strike. Each component corresponds to the estimated probability that the asset ends above the strike at horizon \( h \). These probabilities can then be compared against market-implied values to generate long or short trading signals.

References

Shreve, S. E. (2004). Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. New York: Springer Finance Textbooks. Retrieved from https://link.springer.com/book/10.1007/978-0-387-22527-2

Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer Finance Textbooks. Retrieved from https://link.springer.com/book/10.1007/978-0-387-40105-8