A General Framework of Derivatives Pricing

In this paper, we outline a general framework of derivatives pricing. The framework consists of two modules. The first is a novel simulation and machine learning based calibration module and the second one is a pricing module, which originates from [1] and [2]. Numerical examples show good applicability of the proposed framework. The methodology of calibration utilizes machine learning and simulation methods, combined, to deliver high quality parameter inference results and the pricing module is generic and can be applied to any financial derivatives. The machine learning based pricing methodologies can also generate prices on a future simulation grid, which fa-cilitates XVA computations. Our methodologies can be applied to any pricing problem and the calibration routine is general and useful whenever a parametric model needs to be estimated.

author proposes a deep learning and simulation based approach to calibrate option pricing models. In addition, [4] proposes a similar approach.
In this paper, we propose a novel framework to alleviate the mentioned difficulties in derivatives calibration and pricing. First, we propose a simulation-based calibration method, without the need to use numerical optimization routines to minimize the sum of squares, i.e., the L 2 distance, between the model and the observed prices. The intermediate simulation results can be stored and re-used. Therefore, the proposed methodology is efficient: we only need initial simulation and calibration can be done in a fast manner in an on-going basis.
Second, the calibration method does not need many evaluations of derivative prices, as opposed to a standard optimization routine, which might require thousands of iterations. Third, we leverage the methods proposed in [1] and [2] for the pricing of complex financial derivatives, potentially involving optimal stopping features or other exotic properties such as a breakable swap, where both parties can terminate the contract to the best of their interest (and therefore a stochastic Nash equilibrium has to be found in order for us to obtain the price of this product). Clustering method, as an unsupervised learning method, was first applied to yield nonlinear regression computations in [5] and [2]. In this paper, we apply this approach to calibration of financial derivatives. To the best of our knowledge, our paper is the first to propose such a method. The algorithm is very easy to implement, fast and accurate. Numerical experiments show that it can give an accurate estimate to the speed of mean reversion parameter of a CIR process, which is thought to be very difficult to infer using either bond or bond option data. The calibration methodology has the potential to support joint inference using derivatives from different asset classes using data from both P and Q measures.
The organization of this paper is as follows. Section 2 introduces the main methodologies. Section 3 contains numerical experiments and Section 4 concludes. All the source code can be found in Appendix.

The Methodology
In what follows, we will assume that a pricing model (and equivalently, the model prices) is denoted by , where X is a set of state variables described by the model, ϑ is the model parameters related to X and θ is the parameters related to the financial product. C, defined as the set of control parameters, is related to a numerical method that solves the model. We denote mkt M θ the market observed prices for product θ . We use risk neutral derivatives pricing as an example to illustrate ideas, with the understanding that the method is generic and applies to all the asset pricing problems.

Calibration
The calibration method first simulates N uniform samples of parameter ϑ . , i.e., the specific simulated parameter n ϑ that minimizes the distance between model produced prices and market observed prices. As we expect, when N → ∞ , the estimated parameter * ϑ ϑ → , the true parameter value.
The above methodology works theoretically. However, it is difficult, or often time consuming to implement in practice. The reason is that, even for a European type product, with long time to maturity and no closed-form solution, it might take a long time for the pricer to produce even one sufficiently accurate price, let alone the American products. Often, ϑ is in high dimension, given a portfolio of financial derivatives, and N is large. This often implies an unreasonably large amount of time needed for the estimation.
An improvement utilizes clustering method on the simulated parameter space Next, let us focus on cluster * k Θ . As long as * k K Θ ≥ , i.e., the number of elements in * k Θ is no less than K, we can repeat the above operations, until we find a k such that

Pricing
We use the pricing of financial derivatives as an example to illustrate ideas. Under no arbitrage framework and some sufficient condition, the present value of all marketed cash flows is martingales under a so-called risk-neutral measure, or Q measure. In general, derivatives pricing follows a reduce-form approach that assumes an underlying price distribution and compute the conditional expected value of the discounted payoff function. The underlying can be modeled by a discrete time-series or a system of stochastic differential equations. The simulated-based numerical methods for the latter case are discussed in [1] and [2]. We refer the readers to those references for more details.

XVA
The proposed methodologies in [1] and [2] enable evaluation in a future simulation grid and this is the foundation for XVA evaluations.

Numerical Experiments
In this paper, we will mainly test the calibration method, with the pricing component already validated in the reference of [1] and [2]. Due to the constraint on the computational budget, we focus on simple products to illustrate ideas. The method we adopt for the pricer is Monte Carlo simulation, which is relatively more time consuming than semi closed-form solutions.

Heston European Equity Option Pricing Model
Assume that under the risk neutral measure, the stock price follows a Heston-type stochastic volatility model, with parameter values described in Table 1 below.
In order to estimate the true values, we generate 75,000 uniform samples of ( )  Figure 1 shows the price fit for case 1. Blue curve is the true price 1 and the orange curve represents the calibrated prices at maturity date across different strikes. We choose 250 clusters for the estimation. Table 2 contains the results.

CIR Bond Pricing Model
In this section, we study a zero-coupon bond pricing problem, where the short rate process follows a Cox-Ingersoll-Ross model. The parametrization of the problem is listed in Table 3 and inference result is in Table 4. We use a whole term structure of bond prices to calibrate the model. For more details, we refer the interested readers to the sample code in the Appendix. The pricing fit is in

Vasicek Bond Option Pricing Model
The results are listed in the Table 5 and Table 6, and Figure 3. Details of this exercise can be found in the source code.

Conclusion
The main contribution of this paper is a general framework of financial asset pricing and calibration, where the calibration module consists of a novel simulation and clustering-based methodology. The simulated numbers and intermediate   pricing results can be re-used and are therefore very efficient. The methodology potentially applies to any problem that requires curve fitting, i.e., minimizing a parametric objective function and obtaining the optimal parameters.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.