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Calibration and Simulation Engine for Local Volatility Models

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Calibration and Simulation Engine for Local Volatility Models
-------------------------------------------------------------

This repository contains calibration and simulation routines for deterministic/stochastic local volatility models with deterministic/stochastic interest rates, implemented in Python.

The manuscript that gives the details of the calibration algorithms can be accessed at https://arxiv.org/abs/2009.14764. This repository provides a sample implementation of these algorithms.

The repository contains implementation of calibration of local volaility surface for EUR/USD currency pair though the example is easily adapted to other currency pairs with appropriately calibrated G1pp parameters and discount curves.

The repository contains notebooks detailing:
- The calibration procedure 
- Visualization of local volatility surface
- Repricing with the calibrated local volality surface
- Comparing repriced vs market implied volatility

For three different cases:
- Local volatility surface with 2 (Domestic and Foreign) stochastic interest rates (LV2SR)
- Stochastic Local volatility with 2 (Domestic and Foreign) deterministic interest rates (SLV2DR)
- Stochastic Local volatility with 2 (Domestic and Foreign) stochastic interest rates (SLV2SR)

# Files:
- **marketdata_JSON_asof_04_30_2020/**: Market data in JSON format containing EUR and USD discount curves. Risk-Reversal and Butterfly calibrated market prices and implied volatility for EUR/USD.
  - **EURUSD_Heston.json**: Heston parameterization of Stochastic Local Volatility for EUR/USD.

- **lib/**: library and utilities for reading and interpolating market data as well as calibration routines.
  - **bsanalytic.py**: Black-Scholes analytic formulas for relevant market instruments
  - **calibrator.py**: Main utilities for calibrating local volatility surface. Classes include local volatility calibration under Call surface or Total implied variance (TIV) formulation
  - **fxivolinterpolator.py**: Utilities for interpolating time-discretized and strike-discretized market data for implied volatility.
  - **surfaces.py**: Classes and utilities for constructing Call surface and TIV surface needed for calibration.
  - **interpolator.py**: Classes and utilities for interpolating constructed TIV and Call surface needed for calibration.

- **sim_lib/StochasticSim_Multiprocessing.py**: Numpy based local volality and stochastic local volatility simulator. This simulates assets, interest rates, and volatility under the given local or stochastic local volatility and interest rates.
  - **Assets'** time-evolution described as GBM.
  - **Stochastic Interest rates** parametrized and described as Hull-White/G1pp processes
  - **Stochastic Local Volatility** described by Heston Model.

- **LV_2SIR/FX_LocalVol_Calibration.ipynb**: Notebook demonstrating calibration of LV_2SIR and Repricing under Local Volatility.

- **/SLV_2DIR/**:
  - **Calibrate_SLV_2DIR.ipynb**: Notebook demonstrating calibration of Leverage surface of the Stochastic Local volatility under deterministic rates.
  - **Reprice_SLV_2DIR.ipynb**: Reprice under the calibrated stochastic local volatility model under determistic rates.

- **/SLV_2SIR/**:
  - **Calibrate_SLV_2DIR.ipynb**: Notebook demonstrating calibration of Leverage surface of the Stochastic Local volatility under stochastic rates.
  - **Reprice_SLV_2DIR.ipynb**: Reprice under the calibrated stochastic local volatility model under stochastic rates.

## Usage:
Clone the repository. With an installation of Jupyter with Python kernel >=3.6, run notebooks in folders **/LV_2SIR, /SLV_2DIR and /SLV_2SIR** for the Local Volatility with stochastic rates, Stochastic Local volatlity with deterministic rates and Stochastic Local volatility with Stochastic rates respectively.

# Overview of the model and Summary of Results

## The model (LV2SR)
This corresponding model for the underlying FX process with 2 (domestic and foreign) rate is assumed to be:

where the domestic and foreign rates are parametrized using the G1pp model. The domestic rate evolves in the domestic risk neutral measure as

whereas the foreign short rate evolves in foreign risk neutral measure,

## Calibration of local volatility surface (LV2SR)
The local volality surface or state dependent diffusion coefficient <img src="https://render.githubusercontent.com/render/math?math=\sigma(S_t, t)"> is calibrated in the domestic **T-Forward measure**. The procedure is as follows:

- The calibration is performed in a time slice-by-slice basis.
- The underling FX model and the domestic and foreign rates are first simulated in the T-Fwd measure with the local volatility until current time slice.
- The expectation <img src="https://render.githubusercontent.com/render/math?math=\mathbf{E}^{\mathbb{Q}^{\text{T}}}\left[(K r_T^d - S_T r_T^f) \mathbb{1}_{S_T > K}\right]"> is gathered from the T-Fwd simulation of the interest rates and underliers.
- The following extension to Dupire's formula for stochastic rates is used to compute the local volatility at the current time slice.

<img src="https://render.githubusercontent.com/render/math?math=\huge{\sigma_{\text{LV (stochastic rates)}}^2 = \frac{\frac{\partial C_{\text{BS}}}{\partial T}- P^d(0, T) \mathbb{E}^{Q^{\text{T}}}\left[(K r_T^d - S_T r_T^f) \mathbb{1}_{S_T > K}\right]}{\frac{\partial C_{\text{BS}}}{\partial w} \left[1 - \frac{y}{w} \frac{\partial w}{\partial y} %2B \frac{1}{2} \frac{\partial^2 w}{\partial y^2} %2B \frac{1}{4} \left(\frac{\partial w}{\partial y}\right)^2\left(-\frac{1}{4}- \frac{1}{w} %2B \frac{y^2}{w^2}\right)\right]}}">

- The procedure is repeated for all time slices starting from 0 to maturity.

where: 
<img src="https://render.githubusercontent.com/render/math?math=T">: Time to maturity

<img src="https://render.githubusercontent.com/render/math?math=C_{\text{BS}}"> : The Black-Scholes call price

<img src="https://render.githubusercontent.com/render/math?math=w=\sigma_{\text{BS}}T^2"> : Total implied variance

<img src="https://render.githubusercontent.com/render/math?math=S_T">: Value of Underlier at time.

<img src="https://render.githubusercontent.com/render/math?math=K">: Strike

<img src="https://render.githubusercontent.com/render/math?math=y=\log\left(\frac{S_T}{K}\right)"> : Log-moneyness

Local volatility surface obtained via different number of monte-carlo paths.
![LV_2SR_Convergence](https://user-images.githubusercontent.com/12563351/141022625-c281469d-dd3e-4bf9-94cb-c31cd48ddac7.png)

Call Option price and implied volatility Recovery:
![LV_2SR_maturity_diff_call_and_ivol](https://user-images.githubusercontent.com/12563351/141033827-4e3c9b81-1911-4a50-9c1e-77d73a61bc62.png)

## Stochastic Local Volatility with 2 Deterministic Rates (SLV2DR)
## The model.

The model of stochastic local volatility is modeled as a CIR(Cox-Ingersoll-Ross) process.

where: 
<img src="https://render.githubusercontent.com/render/math?math=\Large{S_t}">: Underlier's value at time

<img src="https://render.githubusercontent.com/render/math?math=\Large{r^d_t, r^f_t}"> : The corresponding domestic and foreign rates (deterministic)

<img src="https://render.githubusercontent.com/render/math?math=\Large{L(S_t, t)}"> : State dependent leverage function

<img src="https://render.githubusercontent.com/render/math?math=\Large{U_t}">: The variance process

<img src="https://render.githubusercontent.com/render/math?math=\Large{\kappa_t, \theta_t, \xi_t}">: The corresponding mean reversion, long-term variance and vol-of-vol parameters of the Heston process.

<img src="https://render.githubusercontent.com/render/math?math=\Large{dW_t^{U\text{(DRN)}}, dW_t^{S\text{(DRN)}}}"> : Browninan drivers of the underlier and variance processes in Domestic Risk neutral measure

## Calibration of the Leverage Surface

The leverage function is related to the local volatility calibrated with deterministic rates via the Dupire's formula and the expectation of variance process as:

- The conditional expectation <img src="https://render.githubusercontent.com/render/math?math=\mathbf{E}^{\mathbb{Q}^{\text{(DRN)}}}\left[U_t \mid S_t=x \right]"> can be computed by binning the underlier values at time from the simulation sample paths as a function of <img src="https://render.githubusercontent.com/render/math?math=S_t, t">.
- Alternatively, the expectation can also obtained by regressing on the risk-factor, (here <img src="https://render.githubusercontent.com/render/math?math=S_t">) as described in the paper.
- The value of <img src="https://render.githubusercontent.com/render/math?math=\sigma_{\text{LV}}(x, t)^2"> is computed from deterministic Dupire's formula.
- Finally the value of leverage function <img src="https://render.githubusercontent.com/render/math?math=L(x, t)^2"> on the grid is obtained by dividing <img src="https://render.githubusercontent.com/render/math?math=\sigma_{\text{LV}}(x, t)^2"> by <img src="https://render.githubusercontent.com/render/math?math=\mathbf{E}^{\mathbb{Q}^{\text{(DRN)}}}\left[U_t \mid S_t=x \right]">

The leverage function calibrated with deterministic rates with different number of monte-carlo calibration paths.
![SLV_2DR_Convergence](https://user-images.githubusercontent.com/12563351/141051960-bcf55031-9cd1-419f-87b6-87a48d1588e2.png)

The repriced call function at maturity and the corresponding implied vol recovered within +-2 Monte Carlo errors.
![SLV_2DR_maturity_diff_call_and_ivol](https://user-images.githubusercontent.com/12563351/141052061-d024c83f-deb1-4b1e-bc81-331684bdaa20.png)

## Stochastic Local Volatility with 2 Stochastic Rates (SLV2SR)
## The model.
The model which is a mixture of stochastic local volatility with stochastic rates can be written as:

The underlier modeled as CIR dynamics: 
<img src="https://render.githubusercontent.com/render/math?math=\Large{dS_t = \left[r^d_t - r^f_t\right] S_t dt %2B L(S_t, t)\sqrt{U_t} S_t dW_t^{S\text{(DRN)}}}"> 
<img src="https://render.githubusercontent.com/render/math?math=\Large{dU_t = \kappa_t (\theta_t - U_t) dt %2B \xi_t \sqrt{U_t} dW_t^{U\text{(DRN)}}}">

Domestic rates modeled as a G1pp process: 
<img src="https://render.githubusercontent.com/render/math?math=\Large{r^d_t = x^d_t %2B \phi^d_t}"> 
<img src="https://render.githubusercontent.com/render/math?math=\Large{dx^d_t = -a^d_t x^d_t dt %2B \sigma^d_t dW_t^{d\text{(DRN)}}}">

Foreign rates modeled as a second G1pp process: 
<img src="https://render.githubusercontent.com/render/math?math=\Large{r^f_t = x^f_t %2B \phi^f_t}"> 
<img src="https://render.githubusercontent.com/render/math?math=\Large{dx^f_t = \left[-a^f_t x^f_t - \rho_{Sf} \sigma^f_t L(S_t, t) \sqrt{U_t}\right] dt %2B \sigma^f_t dW_t^{f\text{(DRN)}}}">

## Calibration of the leverage surface

The calibrated leverage function is related to the local volatility (LV2SR) and the variance process by:
<img src="https://render.githubusercontent.com/render/math?math=\Large{\sigma_{\text{LV}}(x, t)^2 = L(x, t)^2 \mathbf{E}^{\mathbb{Q}^{\text{T}}} \left[U_t \mid S_t=x \right]}">

- The conditional expectation <img src="https://render.githubusercontent.com/render/math?math=\mathbf{E}^{\mathbb{Q}^{\text{T}}}\left[U_t \mid S_t=x \right]"> can be computed by binning the underlier values at time from the simulation sample paths as a function of <img src="https://render.githubusercontent.com/render/math?math=S_t, t"> in **T-Fwd measure**.
- Alternatively, the expectation can also obtained by regressing on the risk-factor, (here <img src="https://render.githubusercontent.com/render/math?math=S_t">) as described in the paper.
- The value of <img src="https://render.githubusercontent.com/render/math?math=\sigma_{\text{LV}}(x, t)^2"> is computed as in the LV2SIR method.
- Finally the value of leverage function <img src="https://render.githubusercontent.com/render/math?math=L(x, t)^2"> on the grid is obtained by dividing <img src="https://render.githubusercontent.com/render/math?math=\sigma_{\text{LV}}(x, t)^2"> by <img src="https://render.githubusercontent.com/render/math?math=\mathbf{E}^{\mathbb{Q}^{\text{(DRN)}}}\left[U_t \mid S_t=x \right]">

The repriced Call option prices and the implied volatility recovered with the repriced options fall within +- 2 MC errors.
![SLV_2SR_maturity_diff_call_and_ivol](https://user-images.githubusercontent.com/12563351/141054532-cb876819-1bf7-4729-b88e-6bdfc7113c50.png)

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