BASE_THEORY.md

Foundations: SDEs, Path Signatures and the Dual Problem

This document outlines the core mathematical and financial theories that form the foundation of the DeepQuant library. It covers the models for asset price simulation and the feature representation of price paths.

1. The Probabilistic Setup and American Options

The theoretical framework for modern quantitative finance is built upon a formal probabilistic setup. We begin by defining a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where:

• $\Omega$ is the sample space of all possible outcomes.
• $\mathcal{F}$ is a $\sigma$-algebra representing the set of all possible events.
• $\mathbb{P}$ is the probability measure.
• $\{\mathcal{F}_t\}_{t \in [0,T]}$ is the filtration, an increasing family of $\sigma$-algebras where $\mathcal{F}_t$ represents the information available up to time $t$.

All stochastic processes, such as the asset price $S_t$, are assumed to be adapted to this filtration.

The pricing of an American option within this framework is an optimal stopping problem. The key challenge is to find the optimal time to exercise to maximize the expected payoff.

Definition: The price of an American option ($V_0$) is the supremum, or least upper bound, of the expected discounted payoff over the set of all possible stopping times adapted to the filtration ${\mathcal{F}_t}$. This means we're hunting for the absolute best possible moment to exercise, using only the information available at the time!

Its value is given by the formula:

$$V_0 = \sup_{\tau \in \mathcal{T}_{[0,T]}} \mathbb{E}\left[e^{-r\tau} \text{Payoff}(S_\tau)\right]$$

where $\tau$ is a stopping time with respect to ${\mathcal{F}_t}$.

To solve this, the DeepQuant library uses a primal-dual approach. This method computes a mathematically rigorous price interval, trapping the true price between a lower and an upper bound. The difference, the duality gap, serves as a built-in measure of accuracy.

2. Modeling Financial Markets with SDEs

Stochastic Differential Equations (SDEs) are the mathematical language used to describe the random evolution of asset prices and their volatility. The library's adaptive framework selects the most appropriate SDE model based on market conditions, which are quantified by the Hurst parameter (H).

Mathematical Formulation of an SDE

Definition: A Stochastic Differential Equation (SDE) is a differential equation in which at least one term is a stochastic process, resulting in a solution that is also a stochastic process. It's a mathematical recipe for describing something that has both a predictable trend and a dose of pure randomness!

The rigorous mathematical formulation of an SDE is expressed as an integral equation:

$$X_t = X_0 + \int_0^t a(s, X_s)ds + \int_0^t b(s, X_s)dW_s$$

This states that the value of the process at time $t$ is the sum of its initial value ($X_0$), a standard integral for the deterministic trend (drift), and an Itô integral for the cumulative random fluctuations (diffusion).

For convenience, the SDE is often written in its simpler, equivalent differential form:

$$dX_t = a(t, X_t)dt + b(t, X_t)dW_t$$

This shorthand clearly separates the deterministic drift ($a(t, X_t)dt$) from the random diffusion ($b(t, X_t)dW_t$).

The Heston Model (Classical Volatility, $H \ge 0.5$)

For markets exhibiting classical random-walk behavior, the library employs the Heston model. It is defined by a system of two SDEs:

1. Asset Price ($S_t$):

$$dS_t = r S_t dt + \sqrt{V_t} S_t dW^s_t$$

2. Variance ($V_t$):

$$dV_t = \kappa(\theta - V_t)dt + \xi\sqrt{V_t}dW^v_t$$

where:

• $s_0$: The initial stock price.
• $v_0$: The initial variance.
• $\kappa$ (kappa): The rate of mean reversion, which controls how quickly the variance process returns to its long-term average.
• $\theta$ (theta): The long-term mean of the variance.
• $\xi$ (xi): The volatility of variance, or "vol of vol," which determines the volatility of the variance process itself.
• $\rho$ (rho): The correlation between the asset's random process ($W^s_t$) and the variance's random process ($W^v_t$).
• $r$: The risk-free interest rate.

The Rough Bergomi Model (Rough Volatility, $H < 0.5$)

To capture the jagged, "rough" nature of volatility observed in real financial data, the library uses the modern rough Bergomi model. Its core component is the Volterra process:

1. Asset Price ($S_t$):

$$dS_t = r S_t dt + \sqrt{V_t} S_t dB_t$$

2. Variance ($V_t$):

$$V_t = V_0 \exp\left(\eta Y_t - \frac{1}{2}\eta^2 t^{2H}\right) \text{ where } Y_t = \int_0^t (t-s)^{H-1/2} , dW^v_s$$

where:

• $s_0$: The initial stock price.
• $v_0$: The initial forward variance.
• $H$ (H): The Hurst parameter, which must be in the range (0, 0.5) to model rough volatility.
• $\eta$ (eta): The volatility of volatility parameter.
• $\rho$ (rho): The correlation between the volatility and price processes.
• $r$: The risk-free interest rate.

3. Path Signatures: A Modern Feature Representation

Definition: The path signature is a collection of iterated integrals of a path, providing a graded, hierarchical summary of its geometric properties. It's the ultimate path summary, turning a wiggly line into a precise list of its essential features!

For a path $X: [0, T] \to \mathbb{R}^d$, its signature is the infinite sequence:

$$S(X)_{0,T} = \left(1, S^1(X), S^2(X), \dots \right)$$

In practice, the library uses the truncated signature up to a specified order. This provides a powerful, finite-dimensional feature vector that serves as the primary input for the library's deep learning solvers.

4. The SDE-to-Signature Transformation

The connection between a Stochastic Differential Equation (SDE) and a path signature is a formal mathematical mapping. The SDE generates a sample path, and the signature function transforms that path into a unique, structured feature vector. This document details the formal mathematics of that transformation.

Step 1: Path Construction from a Discrete Simulation

The process begins with the output of an SDE simulation: a discrete time series of points $(t_0, X_0), (t_1, X_1), \dots, (t_N, X_N)$, where $X_t \in \mathbb{R}^d$.

To prepare this for the signature calculation, two mathematical steps are taken:

1. The path is augmented with the time dimension to create a new path $Y_t \in \mathbb{R}^{d+1}$, where the mathematical definition is:

$$Y_t = (t, X_t)$$

3. A continuous, piecewise linear path is constructed by connecting the points. For any time $s$ within a simulation interval $[t_i, t_{i+1}]$, the path is formally defined by the linear interpolation formula:

$$Y_s = Y_{t_i} + \frac{s - t_i}{t_{i+1} - t_i}(Y_{t_{i+1}} - Y_{t_i})$$

Step 2: Signature Calculation

The signature of the constructed path $Y_s$ is a sequence of terms, where the $M$-th term, $S^M(Y)$, is formally defined as the iterated integral of the path up to order M:

$$ S^M(Y) = \int_{0< t_1 < \dots < t_M < T} dY_{t_1} \otimes \dots \otimes dY_{t_M} $$

While this is the general form, for a piecewise linear path, it can be calculated exactly. To simplify the notation, we first define the increment of the $i$-th linear segment as the vector $\Delta_i$:

$$\Delta_i = Y_{t_{i+1}} - Y_{t_i}$$

The First-Order Term: Displacement

The first-order term, $S^1(Y)$, resolves to the path's total displacement. For a piecewise linear path, this is the sum of the increments:

$$S^1(Y) = Y_T - Y_0 = \sum_{i=0}^{N-1} \Delta_i$$

This vector contains the total time elapsed and the total change in the asset's value.

A Brief Connection: Deriving the Second-Order Term

The second-order term, $S^2(Y)$, captures the path's area-like properties. The change in this term over a single segment is found to be:

$$\Delta S^2_i = S^1(Y)_{0, t_i} \otimes \Delta_i + \frac{1}{2} \Delta_i^{\otimes 2}$$

The total second-order term is the sum of these changes, $S^2(Y) = \sum_i \Delta S^2_i$.

The General Formula

This pattern generalizes to any level M. The $M$-th term of the signature is given by the exact recursive formula:

$$S^M(Y) = \sum_{i=0}^{N-1} \sum_{k=1}^{M} \frac{1}{k!} S^{M-k}(Y)_{0, t_i} \otimes \Delta_i^{\otimes k}$$

Step 3: The Resulting Feature Vector

The final, usable signature is the truncated signature, a finite vector containing all computed terms up to level M. The mathematical representation of this final vector is the concatenation of the terms:

$$S^M(Y) = \text{concat}\left(S^0(Y), S^1(Y), S^2(Y), \dots, S^M(Y)\right)$$

This formal process provides a deterministic and unique mapping from any given SDE sample path to a rich feature vector.

5. The Primal-Dual Framework in Detail

The Primal Problem: Finding the Lower Bound

The primal problem directly addresses the optimal stopping problem. Its theoretical foundation is the Bellman principle of dynamic programming. In a discrete-time setting, the value of the option $V_t$ at time $t$ is the greater of its intrinsic value (if exercised) or its continuation value (if held):

$$V_t = \max\left(\text{Payoff}(S_t), C_t\right)$$

The continuation value ($C_t$) is the central quantity. It is formally defined as the discounted expected value of the option at the next time step, conditional on all information available up to the current time, $\mathcal{F}_t$:

$$C_t = \mathbb{E}\left[e^{-r\Delta t} V_{t+\Delta t} | \mathcal{F}_t\right]$$

To solve for $V_0$, one must work backward from maturity ($T$), where $V_T = \text{Payoff}(S_T)$. At each preceding step, the key theoretical challenge is the evaluation of this conditional expectation to determine the optimal exercise decision.

The Lower Bound Formulation

The backward induction process yields an estimated continuation value $\hat{C}_t$ for each time step $t$. This allows for the definition of a near-optimal stopping time, $\hat{\tau}$, for each simulated path:

$$\hat{\tau} = \inf { t \in {0, \dots, T} \mid \text{Payoff}(S_t) \ge \hat{C}_t }$$

where $\hat{C}_T$ is defined as the intrinsic payoff at maturity. This rule states that the option should be exercised at the first time its immediate payoff value is greater than or equal to the estimated value of holding it.

The resulting lower bound, $V_{\text{lower}}$, is the expected discounted payoff achieved by applying this exercise strategy across all paths:

$$V_{\text{lower}} = \mathbb{E}\left[e^{-r\hat{\tau}} \text{Payoff}(S_{\hat{\tau}})\right]$$

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The Dual Problem: Finding the Upper Bound

The dual problem provides a powerful method for finding a provable upper bound on the option's price. The formulation is derived from the principle that subtracting a "fair game"—a martingale process that starts at zero—from the option's payoff does not change its true value, but allows for the construction of a clear upper bound. ⚖️

The true value of an American option, $V_0$, is the expected discounted payoff received by exercising at the optimal time, $\tau^*$. If we define the discounted payoff process as $P_t = e^{-rt}\text{Payoff}(S_t)$, then $V_0 = \mathbb{E}[P_{\tau^*}]$. Based on the Optional Stopping Theorem, we can subtract any suitable martingale $M_t$ (with $M_0=0$) without changing the expected value, which gives the identity $V_0 = \mathbb{E}[P_{\tau^*} - M_{\tau^*}]$.

The key insight comes from recognizing that for any single path, the value of the process $(P_t - M_t)$ at the specific exercise time $\tau^*$ can never be greater than the maximum value that process achieves over the option's entire lifetime. This simple fact, $P_{\tau^*} - M_{\tau^*} \le \sup_{t \in [0,T]} (P_t - M_t)$, must also hold when we take the expectation over all paths.

Combining these steps leads directly to the martingale duality formula:

$$V_0 \le \mathbb{E}\left[\sup_{t \in [0,T]} \left(e^{-rt}\text{Payoff}(S_t) - M_t\right)\right]$$

Since this relationship holds for any valid martingale, the goal of the dual solver is to find the specific martingale that makes this upper bound as tight (as low) as possible. Therefore, the upper bound is the infimum over the space of all valid martingales ($\mathcal{M}$):

$$V_{\text{upper}} = \inf_{M \in \mathcal{M}} \mathbb{E}\left[\sup_{t \in [0,T]} \left(e^{-rt}\text{Payoff}(S_t) - M_t\right)\right]$$

To make this computationally feasible, the search is restricted to a rich family of martingales constructed as stochastic integrals. A general martingale process is defined as:

$$M_t = \int_0^t \alpha_s dW_s$$

where $\alpha_s$ is a predictable process known as the martingale integrand. The problem of finding the optimal martingale $M$ is thus transformed into the more concrete problem of finding the optimal integrand process $\alpha_s$.

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The Duality Gap

The final outputs of the primal and dual formulations provide the price interval $[V_{\text{lower}}, V_{\text{upper}}]$. The difference between these two values is the duality gap. A small duality gap is a strong theoretical indicator of an accurate and reliable price, as it means the lower and upper bounds have converged to a narrow range.