Calibrate Pricing Model

Calibrate option pricing model in the Live Editor

Since R2022a

Description

The Calibrate Pricing Model task lets you interactively calibrate an equity, FX, or commodity option pricing model using market data. The task automatically generates MATLAB^® code for your live script.

Using this task, you can:

Select data.
Select a model.
Edit parameter constraints.
Specify an optimization solver and options.
Display the results in a volatility surface plot.

For general information about Live Editor tasks, see Add Interactive Tasks to a Live Script.

Set parameters and solver for model calibration using Calibrate Pricing Model live task

Open the Task

To add the Calibrate Pricing Model task to a live script in the MATLAB Editor:

On the Live Editor tab, select Task > Calibrate Pricing Model.
In a code block in the script, type a relevant keyword, such as calibrate. Select Calibrate Pricing Model from the suggested command completions.

Examples

Calibrate Option Pricing Model Using Heston Model

Parameters

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`Select data` — Data for model
matrix for Price, Strike, Maturity | object for Discount Curve | string for Option Type | positive numeric for Spot Price

Select data enables you to select the data that you want to use to calibrate your pricing model:

Price — M-by-N numeric matrix for prices
Strike — M-by-N numeric matrix or M-by-1 column vector for strikes and Strike must be a nonnegative value.
Discount Curve — Rate curve (ratecurve) for the zero curve
Option Type — Put or call option type
Maturity — Option maturity dates
Spot Price — Current underlying asset price

`Select model` — Specify model type
object for Heston, Bates, or Merton

Select model enables you to select the model type that you want to use to calibrate pricing:

Heston — The Heston model is an extension of the Black-Scholes model, where the volatility (square root of variance) is no longer assumed to be constant, and the variance now follows a stochastic (CIR) process. This option allows modeling the implied volatility smiles observed in the market.
The stochastic differential equation is

$\begin{array}{l} d S_{t} = (r - q) S_{t} d t + \sqrt{v_{t}} S_{t} d W_{t} \\ d v_{t} = κ (θ - v_{t}) d t + σ_{v} \sqrt{v_{t}} d W_{t}^{v} \\ E [d W_{t} d W_{t}^{v}] = p d t \end{array}$
where
- r is the continuous risk-free rate.
- q is the continuous dividend yield.
- S_t is the asset price at time t.
- v_t is the asset price variance at time t
- v₀ is the initial variance of the asset price at t = 0 for (v₀ > 0).
- θ is the long-term variance level for (θ > 0).
- κ is the mean reversion speed for the variance for (κ > 0).
- σ_v is the volatility of the variance for (σ_v > 0).
- p is the correlation between the Wiener processes W_t and W^v_t for (-1 ≤ p ≤ 1).
Bates — The Bates model extends the Heston model by including stochastic volatility and (similar to Merton) jump diffusion parameters in the modeling of sudden asset price movements.
The stochastic differential equation is

$\begin{array}{l} d S_{t} = (r - q - λ_{p} μ_{J}) S_{t} d t + \sqrt{v_{t}} S_{t} d W_{t} + J S_{t} d P_{t} \\ d v_{t} = κ (θ - v_{t}) d t + σ_{v} \sqrt{v_{t}} d W_{t} \\ E [d W_{t} d W_{t}^{v}] = p d t \\ prob(d P_{t} = 1) = λ_{p} d t \end{array}$
where
- r is the continuous risk-free rate.
- q is the continuous dividend yield.
- S_t is the asset price at time t.
- v_t is the asset price variance at time t.
- J is the random percentage jump size conditional on the jump occurring, where ln(1+J) is normally distributed with mean $\ln (1 + μ_{J}) - \frac{δ^{2}}{2}$ and the standard deviation δ, and (1+J) has a lognormal distribution:
  
  $\frac{1}{(1 + J) δ \sqrt{2 π}} \exp {{\frac{- [\ln (1 + J) - (\ln (1 + μ_{J}) - \frac{δ^{2}}{2}]}{2 δ^{2}}}^{2}}$
  where
  - v₀ is the initial variance of the asset price at t = 0 (v₀> 0).
  - θ is the long-term variance level for (θ > 0).
  - κ is the mean reversion speed for (κ > 0).
  - σ_v is the volatility of variance for (σ_v > 0).
  - p is the correlation between the Wiener processes W_t and $W_{t}^{v}$ for (-1 ≤ p ≤ 1).
  - μ_J is the mean of J for (μ_J > -1).
  - $λ_{p}$ is the annual frequency (intensity) of Poisson process P_t for ( $λ_{p}$ ≥ 0).
  - δ is the standard deviation of ln(1+J) for (δ ≥ 0).
Merton — The Merton jump diffusion model extends the Black-Scholes model by using the Poisson process to include jump diffusion parameters in the modeling of sudden asset price movements (both up and down).
The stochastic differential equation is

$\begin{array}{l} d S_{t} = (r - q - λ_{p} μ_{j}) S_{t} d t + σ S_{t} d W_{t} + J S_{t} d P_{t} \\ prob(d P_{t} = 1) = λ_{p} d t \end{array}$
where
- r is the continuous risk-free rate.
- q is the continuous dividend yield.
- W_t is the Wiener process.
- J is the random percentage jump size conditional on the jump occurring, where ln(1+J) is normally distributed with mean $\ln (1 + μ_{J}) - \frac{δ^{2}}{2}$ and the standard deviation δ, and (1+J) has a lognormal distribution:
  
  $\frac{1}{(1 + J) δ \sqrt{2 π}} \exp {{\frac{- [\ln (1 + J) - (\ln (1 + μ_{J}) - \frac{δ^{2}}{2}]}{2 δ^{2}}}^{2}}$
  where
  - μ_J is the mean of J for (μ_J > -1).
  - δ is the standard deviation of ln(1+J) for (δ≥ 0).
  - ƛ_p is the annual frequency (intensity) of Poisson process P_tfor (ƛ_p ≥ 0).
  - σ is the volatility of the asset price for (σ > 0).

`Edit parameter constraints` — Parameter constraints for model and solver
table

The parameter constraints displayed in table for editing depend on the model type that you specify from Select model.

Heston
- V0 — Initial variance of the underlying asset
- ThetaV — Long-term variance of underlying asset
- Kappa — Mean revision speed for the variance of underlying asset
- SigmaV — Volatility of the variance of underlying asset
- RhoSV — Correlation between Wiener processes for underlying asset and its variance
Bates
- V0 — Initial variance of the underlying asset
- ThetaV — Long-term variance of underlying asset
- Kappa — Mean revision speed for the variance of underlying asset
- SigmaV — Volatility of the variance of underlying asset
- RhoSV — Correlation between Wiener processes for underlying asset and its variance
- MeanJ — Mean of the random percentage jump size
- JumpVol — Standard deviation of log(1+J)
- JumpFreq — Annual frequency of Poisson jump process
Merton
- Volatility — Volatility value for the underlying asset
- MeanJ — Mean of the random percentage jump size
- JumpVol — Standard deviation of log(1+J)
- JumpFreq — Annual frequency of Poisson jump process

In addition, you can edit the table for Lower Bounds (lb) and Upper Bounds (ub).

`Specify optimization solver and options` — Specify solver type and options
drop-down control for Solver, Text Display, and Plot Function | text box for Tolerance | text box for Max Iterations

Specify optimization solver and options enables you to specify the solver and options for optimization and visualizing results:

Solver
- lsqnonlin - Non linear least squares — For information, see lsqnonlin.
- simulannelbnd - Simulated annealing — For information, see simulannealbnd (Global Optimization Toolbox).

Text Display
- Final output
- Each iteration
- No display
For information, see Iterative Display.

Plot Function
- Best value
- Current value
- No plot
For information, see Plot Functions.

Tolerance — Termination tolerance of the objective function lsqnonlin or simulannealbnd (Global Optimization Toolbox). Tolerance must be a positive value.

Max Iterations — Maximum number of iterations allowed. Max Iterations must be a positive numeric value.

`Display results` — Display optimization results
check box to toggle display of Volatility Surface Plot

Select the Volatility Surface Plot check box to display the current optimization results.

Version History

Introduced in R2022a

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R2022b: Support for Maximum Iterations

The Specify optimization solver and options section of the Live Editor task supports an option for Max Iterations.

Calibrate Pricing Model

Description

Open the Task

Examples

Parameters

Select data — Data for model matrix for Price, Strike, Maturity | object for Discount Curve | string for Option Type | positive numeric for Spot Price

Select model — Specify model type object for Heston, Bates, or Merton

Edit parameter constraints — Parameter constraints for model and solver table

Specify optimization solver and options — Specify solver type and options drop-down control for Solver, Text Display, and Plot Function | text box for Tolerance | text box for Max Iterations

Display results — Display optimization results check box to toggle display of Volatility Surface Plot