A financial derivative is a financial contract whose value depends on the price of an underlying asset such as a stock, bond, commodity, or index. Accurate valuation of financial derivatives and their associated sensitivity factors is important for both investment and hedging purposes. However, many complex derivatives exhibit path-dependency and early-exercise features, which means that closed-form solutions rarely exist, and numerical methods must be used.
The issue with numerical methods is that they are often slow. As a result, efforts are being made to improve the efficiency of numerical techniques for valuing financial derivatives. Reference [1] proposed a fast valuation method based on machine learning. It developed a hybrid two-stage valuation framework that applies a machine learning algorithm to highly accurate derivative valuations incorporating full volatility surfaces. The volatility surface is parameterized, and a Gaussian Process Regressor (GPR) is trained to learn the nonlinear mapping from the complete set of pricing inputs directly to the valuation outputs. Once trained, the GPR delivers near-instantaneous valuation results.
The authors provided examples, notably, the valuation of American put options using a Crank-Nicolson finite-difference solver. They pointed out,
In this work, we introduce a ML framework that takes all the relevant risk factors as input, including the parameters modeling the shape of the volatility surface, and generates the price and related Greeks as output directly almost instantly. To illustrate this methodology, we have used idealized volatility surfaces where the volatility surface for any given maturity is described by the 5-parameter SVI parameterization, and the term structure is specified by a single parameter. Within this idealized framework, we then apply this methodology to evaluate two kinds of derivatives products, namely the fair strike Kvar of a variance swap, and the price V and Greeks (∆, Γ, Θ) of an American put. For each of these products, we have prepared a training set and a testing set using valuations obtained by highly accurate numerical models commonly used by derivatives practitioners. The training data are then used to train a GPR to learn the mapping between the input risk factors and the output valuation variables directly, and the performance of the GPR is validated using the testing data where the high accuracy numerical model valuations are used as the ground truth. For the variance swap, a very high precision prediction with an overall 0.5% relative error is achieved. As for the American put, the price V and first order Greeks ∆ and Θ all have accurate predictions with relative error at 1.7%, 3.3% and 3.5% , respectively. However, partly due to the discontinuity of the Gamma Γ profile in the strike dimension, the GPR’s performance of this higher order derivative valuation is notably less accurate. Nonetheless, the key message from this study is that by training ML to directly map the relationships between pricing inputs and valuation outputs, this methodology has reduced the computation time by 3 to 4 orders of magnitude for the American put, offering significant improvement and potential in performing large scale real-time valuations of derivative products with early exercise features.
In summary, the authors developed an efficient method to price complex financial derivatives using a machine learning technique. However, it is noted that GPR’s performance in valuing higher-order greeks is noticeably less accurate. Additionally, the study was conducted using synthetic data, so it would be useful to see the method applied to real-world scenarios.
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References
[1] Lijie Ding, Egang Lu, Kin Cheung, Fast Derivative Valuation from Volatility Surfaces using Machine Learning, arXiv:2505.22957
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