SPX Index Option Pricing Model Calibration

The goal of this project is to calibrate options pricing models to a recent set of market data, including dividend rates, jumps, stochastic volatility and interest rates, and use the result to price American options with a Monte Carlo approach.

Models Being Calibrated

• Bates (Jumps + Stochastic Volatility)
  • The code also includes a second version of the Bates model
    with a constant jump in volatility coinciding with a jump in spot
• Heston Stochastic Volatility
• Merton Jump Diffusion
• CIR Interest Rate Model
• American Option using Monte Carlo

Market Data

We are working with S&P Index options for the following maturities: 6/21/2024 (10 options), 9/20/2024 (10 options), 12/20/2024 (10 options), 3/21/2025 (9 options), 6/20/2025 (6 options), and 12/19/2025 (3 options). Prices are as of 04/12/2024. We will use the closing price for that day ($5,123.41) as our index price in the models.

1. Calibrate an overal dividend yield

$$\begin{aligned} E^2 = \sum_{i=1}^{N} (S e^{-yT_{i}}- K_{i}D(T_{i}) - C_{i} + P_{i})^{2} \end{aligned}$$

where 𝑆, K_{i}, T_{i}, 𝑟_{i}, 𝐶_{i}, 𝑃_{i} are, respectively, current SPX index price, strike price of the call-put pair for that strike price, expiration date, interest rate to that expiration date, call and put price. Minimize E over all values of y.

We used Newton's method to find:

$$\begin{aligned} \frac{\partial E^{2}}{\partial y} = 0 \end{aligned}$$

The partial derivative is:

$$\begin{aligned} \frac{\partial E^{2}}{\partial y} = -2 \sum_{i=1}^{N} (S e^{-yT_{i}} - K_{i}D(T_{i}) - C_{i} + P_{i}) \cdot S T_{i} e^{-yT_{i}} \end{aligned}$$

2. Error Calculation for Calibration of Options Models

We calculate the least squares error of calibration for each model:

$$\begin{aligned} E = \sqrt{\sum_{i=1}^{N} ((C_{i}^{*} - C_{i})^{2}+(P_{i}^{*} - P_{i})^{2})} \end{aligned}$$

The stars mean “theoretical” value, obtained with the calibrated parameters.

Errors for each model

Model	Error
Bates	26.669
Bates (with constant jump	26.687
Heston	42.802
Merton	28.822

3. Calibrate the Bates model

$$\begin{aligned} \frac{dS}{S} = (r - y - \lambda (e^{\theta+\frac {1}{2} \beta^{2}} -1)) dt + \sqrt {v} dW_{1} + (e^{Y} - 1) dN \end{aligned}$$

$$\begin{aligned} dv = \kappa(\theta - v) dt + \alpha \sqrt {v} dW_{2} \end{aligned}$$

$$\begin{aligned} Y \sim \mathcal{N}(\theta, \beta^2) \end{aligned}$$

$$\begin{aligned} Q(dN = 1) = \lambda dt \end{aligned}$$

$$\begin{aligned} Q(dN = 0) = 1 - \lambda dt \end{aligned}$$

$$\begin{aligned} dW_{1} \cdot dW_{2} = \rho dt \end{aligned}$$

Calibrated Parameters

Parameter	Value
kappa_v	16.24
theta_v	0.01
sigma_v	0.094
rho	-0.43
v0	0.02
lambda	0.211
mu	-0.312
delta	0.192

Bates Results

4. Calibrate the Heston Model

$$\begin{aligned} \frac{dS}{S} = (r - y) dt + \sqrt {v} dW_{1} \end{aligned}$$

$$\begin{aligned} dv = \kappa(\theta - v) dt + \alpha \sqrt {v} dW_{2} \end{aligned}$$

Just like with Bates, we will calculate the least squares error for Heston ($E_{Heston}$).

Calibrated Parameter

Parameter	Value
kappa_v	16.32
theta_v	0.01
sigma_v	0.09
rho	-0.44
v0	0.02

Heston Results

5. Calibrate the Merton Model

$$\begin{aligned} \sum_{n=0}^{\infty} \frac{e^{-\tilde{\lambda} T}(\tilde{\lambda} T)^{n}}{n!} (S_{0} e^{-yT}N(d_{1,n}) - Ke^{-r_{n}T}N(d_{2,n})) \end{aligned}$$

$$\begin{aligned} d_{1,n} = \frac{ln(\frac{S_{0}}{K}) + (r_{n} - y + \frac{1}{2} \sigma_{n}^{2})T}{\sigma_{n} \sqrt{T}} \end{aligned}$$

$$\begin{aligned} d_{2,n} = d_{1,n} - \sigma_{n} \sqrt{T} \end{aligned}$$

Calibrated Parameter

Parameter	Value
sigma	0.112
lambda	0.155
mu	-0.382
delta	0.16

Merton Results

6. Calibrate the CIR Interest Rate Model

$$\begin{aligned} dr = \kappa_{r}(\theta_{r} - r)dt + \alpha_{r}rdW_{r} \end{aligned}$$

CIR Results

7. Compute Value of American Option Using Monte Carlo

We put everything together for a 3-factor model

$$\begin{aligned} \frac{dS}{S} = (r - y - \lambda (e^{\theta+\frac {1}{2} \beta^{2}} -1)) dt + \sqrt {v} dW_{1} + (e^{Y} - 1) dN \end{aligned}$$

$$\begin{aligned} dv = \kappa(\theta - v) dt + \alpha \sqrt {v} dW_{2} \end{aligned}$$

$$\begin{aligned} Y \sim \mathcal{N}(\theta, \beta^2) \end{aligned}$$

$$\begin{aligned} Q(dN = 1) = \lambda dt \end{aligned}$$

$$\begin{aligned} Q(dN = 0) = 1 - \lambda dt \end{aligned}$$

$$\begin{aligned} dW_{1} \cdot dW_{2} = \rho dt \end{aligned}$$

$$\begin{aligned} dr = \kappa_{r}(\theta_{r} - r)dt + \alpha_{r}rdW_{r} \end{aligned}$$

$$\begin{aligned} dW_{r} \cdot dW_{1} = dW_{r} \cdot dW_{2} = 0 \end{aligned}$$