XLOptionPricer Documentation

Installation & Setup

Getting Started

XLOptionPricer is compatible with Excel for PC, Mac, and the Web. To install it, go to the "Home" tab in the Excel ribon, click on "Add-ins", and select "More Add-ins" at the bottom of the dialogue. Search for "XLOptionPricer" in the resulting window, and install it.

Some employers/schools block the installation of all add-ins from Microsoft. In those situations, you can still use XLOptionPricer using Excel online. Launch Excel online in an incognito window, and create a free account, separate from your employer/school account. You will then be able to install XLOptionPricer there and use it in Excel online.

Excel Functions

This section details the Excel functions that the add-in makes available.

Black-Scholes Option Pricing

xop.BlackScholes ▼
Calculate the price of an option using the Black-Scholes pricing model
Parameters
🖇️ Array PC
Are you pricing a put option or a call option
1: Call option
2: Put option
🖇️ Array S
The current price of the underlying asset
🖇️ Array K
The strike price of the option
🖇️ Array T
The time to expiration, in years
🖇️ Array rf
The annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
The annualized dividend yield of the underlying asset, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array IV
The implied volatility of the underlying asset's returns (the standard deviation of the underlying asset's returns)
Returns
Returns a single number - the price of the option.
xop.Black ▼
Calculate the price of European options of forwards/futures using the method in Black (1976)
Parameters
🖇️ Array PC
Are you pricing a put option or a call option
1: Call option
2: Put option
🖇️ Array F
The forward (or futures) price of the underlying
🖇️ Array K
The strike price of the option
🖇️ Array T
The time to maturity, in years
🖇️ Array rf
The continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array IV
The Black implied volatility of the underlying asset's returns in annual units
Returns
Returns a single number - the price of the option.

Black-Scholes Option Implied Volatility

xop.BlackScholesIV ▼
Calculates the implied volatility of an option using the Black-Scholes pricing model
Parameters
🖇️ Array PC
Are you pricing a put option or a call option
1: Call option
2: Put option
🖇️ Array S
The current price of the underlying asset
🖇️ Array K
The strike price of the option
🖇️ Array T
The time to expiration, in years
🖇️ Array rf
The annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
The annualized dividend yield of the underlying asset, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array Price
The observed price of the option in the market
vol_low optional
The lowest implied volatily to look at when finding the implied volatility (default: 1e-9)
vol_high optional
The highest implied volatily to look at when finding the implied volatility (default: 5)
Returns
Returns a single number - the implied volatility of the option.
xop.BlackIV ▼
Calculates the implied volatility in the Black model from option prices
Parameters
🖇️ Array PC
Put option or call option
1: Call option
2: Put option
🖇️ Array F
Forward (or futures) price of the underlying
🖇️ Array K
The strike price of the option
🖇️ Array T
The time to expiration, in years
🖇️ Array rf
The annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array Price
The observed option market price
vol_low optional
The lowest implied volatily to look at when finding the implied volatility (default: 1e-9)
vol_high optional
The highest implied volatily to look at when finding the implied volatility (default: 5)
Returns
Returns a single number - the implied volatility of the option.

VIX and VIX term structure

xop.VIX_analytical ▼
Model-implied VIX index as a function of maturity
Parameters
V0
Initial variance level
🖇️ Array T
Maturities in years corresponding to the VIX futures/options
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
sigmaz
The volatility of the deterministic normal jump component under Q
parameters optional
The risk-neutral model parameters corresponding to the model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
Returns
Returns a single number - the model-implied VIX index.
xop.VRP_Analytical ▼
Model-implied VIX index as a function of maturity
Parameters
V0
Initial variance level
🖇️ Array T
Maturities in years
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
muz
Drift of deterministic normal jump component under P
sigmaz
The volatility of the deterministic normal jump component under P
sigmazQ
The volatility of the deterministic normal jump component under Q
parametersP
Physical-measure model parameters (P). Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
parametersQ optional
Risk-neutral model parameters (Q). Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
Returns
Returns a matrix with as many rows as elements in T; the first column contains the expected integrated variance under P, and the second under Q.

Return moments

xop.LR_moment ▼
Moments of the log-return of the underlying
Parameters
V0
Initial variance level
T
Investment horizon in years
mu
Drift of log-returns under the physical measure P
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
muz
The drift of the deterministic normal jump component under P
sigmaz
The volatility of the deterministic normal jump component under P
parameters optional
The risk-neutral model parameters corresponding to the model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.HPR_moment ▼
Moments of the simple holding-period return of the underlying
Parameters
V0
Initial variance level
T
Investment horizon in years
mu
Drift of log-returns under the physical measure P
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
muz
The drift of the deterministic normal jump component under P
sigmaz
The volatility of the deterministic normal jump component under P
parameters optional
The risk-neutral model parameters corresponding to the model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.LR_pdf ▼
Probability density function of the log returns
Parameters
🖇️ Array RT
Vector of log-return values r at which to evaluate the density
V0
Initial variance level
T
Investment horizon in years
mu
Drift of log-returns under the physical measure P
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
muz
The drift of the deterministic normal jump component under P
sigmaz
The volatility of the deterministic normal jump component under P
parameters optional
Physical-measure model parameters. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.HPR_pdf ▼
Probability density function of the simple holding-period return
Parameters
🖇️ Array RT
Vector of log-return values r at which to evaluate the density
V0
Initial variance level
T
Investment horizon in years
mu
Drift of log-returns under the physical measure P
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
muz
The drift of the deterministic normal jump component under P
sigmaz
The volatility of the deterministic normal jump component under P
parameters optional
Physical-measure model parameters. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.S_pdf ▼
Probability density function of the terminal price S_T
Parameters
🖇️ Array ST
Grid of terminal prices S_T > 0 at which to evaluate the density
S0
Current underlying spot price
V0
Initial variance level
T
Investment horizon in years
mu
Drift of log-returns under the physical measure P
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
muz
The drift of the deterministic normal jump component under P
sigmaz
The volatility of the deterministic normal jump component under P
parameters optional
Physical-measure model parameters. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.

Prices and IV

xop.OptionPrice ▼
Fourier-based European option pricing under an affine model
Parameters
🖇️ Array S0
Current underlying spot price
🖇️ Array V0
Current variance level
🖇️ Array PC
Are you pricing a put option or a call option
1: Call option
2: Put option
🖇️ Array K
Strike price(s)
🖇️ Array T
Time to maturity in years
🖇️ Array r
Continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
Continuously compounded annualized dividend yield, expressed as a decimal (eg: 0.02 for 2%)
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
🖇️ Array sigmaz
The volatility of the deterministic normal jump component under Q
parameters optional
Risk-neutral model parameters corresponding to model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.OptionIV ▼
Model-implied Black volatility for at-the-money call options
Parameters
🖇️ Array S0
Current underlying spot price
🖇️ Array V0
Current variance level
🖇️ Array K
Strike price(s)
🖇️ Array T
Time to maturity in years
🖇️ Array r
Continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
Continuously compounded annualized dividend yield, expressed as a decimal (eg: 0.02 for 2%)
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
🖇️ Array sigmaz
The volatility of the deterministic normal jump component under Q
parameters optional
Risk-neutral model parameters corresponding to model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.

Greeks

xop.OptionDelta ▼
Spot delta of European options under an affine model
Parameters
🖇️ Array S0
Current underlying spot price
🖇️ Array V0
Current variance level
🖇️ Array PC
Are you pricing a put option or a call option
1: Call option
2: Put option
🖇️ Array K
Strike price(s)
🖇️ Array T
Time to maturity in years
🖇️ Array r
Continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
Continuously compounded annualized dividend yield, expressed as a decimal (eg: 0.02 for 2%)
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
🖇️ Array sigmaz
The volatility of the deterministic normal jump component under Q
parameters optional
Risk-neutral model parameters corresponding to model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.OptionGamma ▼
Spot gamma of European options under an affine model
Parameters
🖇️ Array S0
Current underlying spot price
🖇️ Array V0
Current variance level
🖇️ Array K
Strike price(s)
🖇️ Array T
Time to maturity in years
🖇️ Array r
Continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
Continuously compounded annualized dividend yield, expressed as a decimal (eg: 0.02 for 2%)
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
🖇️ Array sigmaz
The volatility of the deterministic normal jump component under Q
parameters optional
Risk-neutral model parameters corresponding to model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.OptionVega ▼
Vega of European options with respect to initial variance V0
Parameters
🖇️ Array S0
Current underlying spot price
🖇️ Array V0
Current variance level
🖇️ Array K
Strike price(s)
🖇️ Array T
Time to maturity in years
🖇️ Array r
Continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
Continuously compounded annualized dividend yield, expressed as a decimal (eg: 0.02 for 2%)
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
🖇️ Array sigmaz
The volatility of the deterministic normal jump component under Q
parameters optional
Risk-neutral model parameters corresponding to model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.OptionVanna ▼
Vanna of European options under an affine model
Parameters
🖇️ Array S0
Current underlying spot price
🖇️ Array V0
Current variance level
🖇️ Array K
Strike price(s)
🖇️ Array T
Time to maturity in years
🖇️ Array r
Continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
Continuously compounded annualized dividend yield, expressed as a decimal (eg: 0.02 for 2%)
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
🖇️ Array sigmaz
The volatility of the deterministic normal jump component under Q
parameters optional
Risk-neutral model parameters corresponding to model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.OptionTheta ▼
Theta of European options under an affine model
Parameters
🖇️ Array S0
Current underlying spot price
🖇️ Array V0
Current variance level
🖇️ Array PC
Are you pricing a put option or a call option
1: Call option
2: Put option
🖇️ Array K
Strike price(s)
🖇️ Array T
Time to maturity in years
🖇️ Array r
Continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
Continuously compounded annualized dividend yield, expressed as a decimal (eg: 0.02 for 2%)
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
🖇️ Array sigmaz
The volatility of the deterministic normal jump component under Q
parameters optional
Risk-neutral model parameters corresponding to model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.

Expected return moments

xop.OptionHPReturnMoment ▼
Expected option holding-period simple return over a sub-horizon.
Parameters
🖇️ Array S0
Underlying spotprice at time 0
🖇️ Array V0
Initial variance level at time 0
🖇️ Array PC
Call option (1) or put option (2)?
1: Call option
2: Put option
🖇️ Array K
Strike price(s)
🖇️ Array T
Time to maturity in years
🖇️ Array S
Start of the holding period in years, with S <= T
🖇️ Array r
Continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
Continuously compounded annualized dividend yield, expressed as a decimal (eg: 0.02 for 2%)
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
🖇️ Array muzP
The drift of the deterministic normal jump component under P.
🖇️ Array sigmazP
Volatility of the deterministic normal jump component under P.
🖇️ Array sigmazQ
Volatility of the deterministic normal jump component under Q.
parametersP
Physical-measure model parameters. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
parametersQ optional
Risk-neutral model parameters corresponding to model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.OptionReturnMoment ▼
Moments of the option holding-period return using affine pricing formulas and payoff transforms.
Parameters
🖇️ Array S0
Underlying spotprice at time 0
🖇️ Array V0
Initial variance level at time 0
🖇️ Array PC
Call option (1) or put option (2)?
1: Call option
2: Put option
🖇️ Array K
Strike price(s)
🖇️ Array T
Time to maturity in years
🖇️ Array r
Continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
Continuously compounded annualized dividend yield, expressed as a decimal (eg: 0.02 for 2%)
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
🖇️ Array muzP
The drift of the deterministic normal jump component under P.
🖇️ Array sigmazP
Volatility of the deterministic normal jump component under P.
🖇️ Array sigmazQ
Volatility of the deterministic normal jump component under Q.
parametersP
Physical-measure model parameters. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
parametersQ optional
Risk-neutral model parameters corresponding to model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
xop.OptionReturnMoment_2 ▼
Alternative computation of option holding-period return moments via raw payoff moments
Parameters
🖇️ Array S0
Underlying spotprice at time 0
🖇️ Array V0
Initial variance level at time 0
🖇️ Array PC
Call option (1) or put option (2)?
1: Call option
2: Put option
🖇️ Array K
Strike price(s)
🖇️ Array T
Time to maturity in years
🖇️ Array r
Continuously compounded annualized risk-free rate, expressed as a decimal (eg: 0.02 for 2%)
🖇️ Array q
Continuously compounded annualized dividend yield, expressed as a decimal (eg: 0.02 for 2%)
model
The model identifier
SVJ: SVJ model
SV: SV model
Merton: Merton model
DPS: DPS model
BS: Black-Scholes model
🖇️ Array muzP
The drift of the deterministic normal jump component under P.
🖇️ Array sigmazP
Volatility of the deterministic normal jump component under P.
🖇️ Array sigmazQ
Volatility of the deterministic normal jump component under Q.
parametersP
Physical-measure model parameters. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
parametersQ optional
Risk-neutral model parameters corresponding to model. Depending on the model selected, provide the following parameters in the following order:
  • SVJ model: kappa (mean-reversion speed of he variance process V_t), theta (the long-run mean level of variance to which V_t reverts), sigma (volatility of volatility; diffusion volatility of the variance process), rho (correlation between the Brownian shocks to returns and variance), lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns, per jump), sigma_y (standard deviation of the jump size in log-returns)
  • SV model:kappa (mean-reversion speed of the variance process), theta (long-run mean level of variance), sigma (volatility of volatility of the variance process), rho (correlation between return and variance Brownian shocks)
  • Merton model: lambda_y (Poisson arrival intensity of jumps in log-returns), mu_y (mean jump size in log-returns), sigma_y (standard deviation of the jump size in log-returns)
  • DPS model: kV (mean-reversion speed of the variance process V_t), tV (long-run mean level of variance; the variance 'target' v-bar), sV (volatility of volatility; diffusion volatility of the variance process), p (correlation between Brownian schoks to log-price and variance), ly (jump intensity for price-only (y-only) jumps), lv (jump intensity for variance only (v-only) jumps), lc (jump intensity for common price-variance jumps), muy (mean jump size in log-price for price-only jumps), sy (standard deviation of jump size in log-price for price-only jumps), muv (mean jump size in variance for variance-only jumps), mucy (mean log-price component of common jumps), scy (standard deviation of the log-price component of common jumps), mucv (mean variance component of common jumps), pJ (parameter controlling the dependence/coupling between price and variance in common jumps; appears as the cross term in the common-jump transform)
  • Black-Scholes model: no parameters required; do not pass parameters to the function.
Results