- The paper introduces a deep signature approach that reformulates asset dynamics using rough stochastic differential equations, enabling tractable non-Markovian option pricing.
- It details both linear and nonlinear signature representations, with signature truncation ensuring computational feasibility and rigorous convergence proofs.
- Numerical simulations demonstrate superior accuracy and robustness of the method compared to traditional Markovian-based pricing models.
Option Pricing under Non-Markovian Stochastic Volatility Models: A Deep Signature Approach
Introduction
In the context of mathematical finance, option pricing represents a key problem frequently solved by evaluating expectations over stochastic processes. Traditional methods assuming Markovian stochastic volatility models encounter significant difficulties when extended to non-Markovian settings where asset prices depend not only on the current state but also on historical paths. This paper proposes an innovative approach combining rough path theory and signature transforms to address this challenge. By reformulating the asset dynamics as rough stochastic differential equations (RSDEs), the method exploits signature representations to convert rough paths into more tractable classical stochastic differential equations (SDEs).
Methodology
The authors introduce a rough signature-based framework to model non-Markovian stochastic volatility. Specifically, the method involves transforming the original non-Markovian dynamics into an RSDE:
dXt=f(t,Xt)vtdWt+g(t,Xt)vtdBt,
where X and v represent the asset price and volatility, respectively, and W and B are independent Brownian motions. The core challenge lies in representing the volatility process vt, which is conditioned on the entire path history.
The paper utilizes the concept of signature transforms, which capture path-dependent information via time-extended Brownian motion signatures. This representation transforms the RSDE into a classical SDE framework, which can be analyzed using conventional tools. By truncating infinite signature representations to finite levels, the approach becomes computationally feasible, with convergence rigorously proved for both linear and nonlinear representations.
Implementation Details
Linear Signature Representation
For linear representations, the asset dynamics are described using infinite linear combinations of the path signature coefficients. Proposition redefines the RSDE with signature truncation, yielding a classical SDE accommodating traditional mathematical finance tools like the Feynman-Kac theorem for pricing.
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class LinearSignatureModel:
def __init__(self, coefficients, truncation_level):
self.coefficients = coefficients
self.truncation_level = truncation_level
def evaluate(self, paths):
truncated_signature = self.compute_signature(paths, self.truncation_level)
return np.dot(self.coefficients, truncated_signature)
def compute_signature(self, paths, level):
# Implementation of signature computation up to the specified level |
Nonlinear Signature Representation
For more complex volatility models, the paper introduces a deep nonlinear signature approach using neural networks. Unlike linear methods, this approach learns non-linear dependencies from data, capturing more intricate path dynamics.
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from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import LSTM, Dense
def build_deep_signature_model(input_shape):
model = Sequential()
model.add(LSTM(128, input_shape=input_shape, return_sequences=True))
model.add(LSTM(128))
model.add(Dense(64, activation='relu'))
model.add(Dense(1))
return model
model = build_deep_signature_model((None, input_dim))
model.compile(optimizer='adam', loss='mse') |
Numerical Results
The method's efficacy is demonstrated via numerical simulations of European option prices under various stochastic volatility models, including both Markovian (e.g., OU and mGBM models) and non-Markovian (e.g., rough Heston and rBergomi) settings. The deep nonlinear signature method notably excels in capturing complex volatility structures, reflected in accurate option pricing.
- Markovian Models: Linear signature methods successfully recover Markovian dynamics, achieving effective convergence at reduced computational costs.
- Non-Markovian Models: The deep nonlinear signature model offers superior accuracy, significantly outperforming linear techniques in scenarios where direct analytic solutions are unattainable.
Conclusion
This paper's deep signature approach represents a significant advancement in pricing options under non-Markovian volatility models, extending classical PDE tools to previously intractable settings. By bridging traditional stochastic calculus and modern rough path theory, it provides a versatile, tractable framework applicable to a wide range of financial models. Future research may expand on adaptive signature truncation strategies and explore real-time applications in evolving market conditions.