Advanced Trading Strategies

Quant-driven approaches with key mathematical formulas

Momentum Stat-Arb Vol-Arb Gamma Machine Learning

1. Trend Following Strategies

We analyze 10–20 years of price data to uncover persistent market direction using adaptive, non‑linear models such as Genetic Algorithms, SVMs, and ANNs.

Momentum signal (simple): $M_t = P_t - P_{t-L}$
Dual moving average rule: Go long if $\mathrm{SMA}_{\text{fast}}(t) > \mathrm{SMA}_{\text{slow}}(t)$
SVM decision function: $f(\mathbf{x}) = \operatorname{sign}\!\left(\sum_{i=1}^{n} \alpha_i y_i \, K(\mathbf{x}_i, \mathbf{x}) + b\right)$
ANN layer mapping: $\mathbf{a}^{(\ell)} = \sigma\!\left(\mathbf{W}^{(\ell)} \mathbf{a}^{(\ell-1)} + \mathbf{b}^{(\ell)}\right)$

Signal-to-noise via information ratio: $$\mathrm{IR} = \frac{\mathbb{E}[R_p - R_b]}{\sqrt{\operatorname{Var}(R_p - R_b)}}$$

2. Mean Reversion Strategies (Statistical Arbitrage)

We construct market‑neutral long/short spreads from cointegrated instruments and size trades by mean‑reversion speed.

Cointegration (Engle–Granger): For two price series $(X_t, Y_t)$, estimate $Y_t = \beta X_t + u_t$. If $u_t$ is stationary (ADF rejects unit root), the spread $u_t$ is tradeable.

Ornstein–Uhlenbeck (OU) process for spread $X_t$: $$\mathrm{d}X_t = \kappa(\mu - X_t)\,\mathrm{d}t + \sigma\,\mathrm{d}W_t$$

Half‑life of mean reversion: $$t_{1/2} = \frac{\ln 2}{\kappa}$$

Discrete‑time estimation via AR(1): $$\Delta X_t = X_t - X_{t-1} = \phi - \kappa X_{t-1} + \varepsilon_t \;\Rightarrow\; \hat{\kappa} = -\widehat{\text{slope}}\;.$$

z‑score entry/exit: $z_t = \dfrac{X_t - \hat{\mu}}{\hat{\sigma}}$; enter at $|z_t| > z_\text{in}$, exit at $|z_t| < z_\text{out}$.

3. Volatility Arbitrage Strategies

Exploit gaps between implied volatility (IV) and realized volatility (RV) while managing Greeks in a delta‑neutral book.

Realized variance (close‑to‑close): $$\mathrm{RV} = \sum_{t=1}^{T} r_t^2, \quad r_t = \ln\!\left(\frac{P_t}{P_{t-1}}\right)$$

Delta‑neutrality and Greeks: $$\sum_i w_i\,\Delta_i = 0,\qquad \Gamma = \frac{\partial \Delta}{\partial S},\qquad \Theta = \frac{\partial V}{\partial t},\qquad \text{Vega} = \frac{\partial V}{\partial \sigma}$$

First‑order P&L approximation for an options book $V$: $$\mathrm{d}V \approx \Delta\,\mathrm{d}S + \tfrac{1}{2}\,\Gamma\,(\mathrm{d}S)^2 + \text{Vega}\,\mathrm{d}\sigma + \Theta\,\mathrm{d}t$$

Smile exploitation: Short overpriced OTM (high IV) and long relatively underpriced strikes; rebalance to keep $\sum w_i\,\Delta_i \approx 0$.

4. Gamma Scalping Strategies

Harvest convexity from positive gamma while re‑hedging delta as price moves.

With frequent re‑hedging of $\Delta$, expected scalping P&L from convexity over small horizon: $$\mathbb{E}[\mathrm{P\&L}] \approx \tfrac{1}{2}\,\Gamma\,\mathbb{E}[(\Delta S)^2] - \text{(financing/theta costs)}$$

Sign of gamma: Long options $\Rightarrow \Gamma>0$ (benefit from large moves); short options $\Rightarrow \Gamma<0$.
Dynamic hedging rule: After price change $\Delta S$, update hedge by $\Delta h \approx \Gamma\,\Delta S$ to restore $\Delta\!\approx 0$.

5. Machine Learning‑Based Strategies

Supervised DNNs for direction classification and unsupervised clustering for regime discovery.

Binary cross‑entropy (logistic head): $$\mathcal{L}(\theta) = -\big[y\,\log p_\theta(\text{up}|\mathbf{x}) + (1-y)\,\log (1-p_\theta(\text{up}|\mathbf{x}))\big]$$

K‑Means objective: $$\min_{\{\mu_k\}} \sum_{k=1}^{K} \sum_{\mathbf{x}_i \in C_k} \lVert \mathbf{x}_i - \mu_k \rVert^2$$

Regime labels (unsupervised): Apply K‑Means to volatility/return features; map clusters to regimes (e.g., low‑vol trend, high‑vol chop).
Regularization: L2 weight decay $\lambda\lVert\theta\rVert_2^2$ and dropout stabilize noisy financial features.