Package 'FER' reference manual

Title:	Financial Engineering in R
Description:	R implementations of standard financial engineering codes; vanilla option pricing models such as Black-Scholes, Bachelier, CEV, and SABR.
Authors:	Jaehyuk Choi [aut, cre]
Maintainer:	Jaehyuk Choi <[email protected]>
License:	GPL (>= 2)
Version:	0.94
Built:	2025-03-12 04:28:49 UTC
Source:	https://github.com/pyfe/fe-r

Calculate Bachelier model implied volatility

Description

Calculate Bachelier model implied volatility

Usage

BachelierImpvol(
  price,
  strike = forward,
  spot,
  texp = 1,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)
BachelierImpvol(
  price,
  strike = forward,
  spot,
  texp = 1,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`price`	(vector of) option price
`strike`	(vector of) strike price
`spot`	(vector of) spot price
`texp`	(vector of) time to expiry
`intr`	interest rate (domestic interest rate)
`divr`	dividend/convenience yield (foreign interest rate)
`cp`	call/put sign. `1` for call, `-1` for put.
`forward`	forward price. If given, `forward` overrides `spot`
`df`	discount factor. If given, `df` overrides `intr`

Value

Bachelier implied volatility

References

Choi, J., Kim, K., & Kwak, M. (2009). Numerical Approximation of the Implied Volatility Under Arithmetic Brownian Motion. Applied Mathematical Finance, 16(3), 261-268. doi:10.1080/13504860802583436

Examples

spot <- 100
strike <- 100
texp <- 1.2
sigma <- 20
intr <- 0.05
price <- 20
FER::BachelierImpvol(price, strike, spot, texp, intr=intr)

spot <- 100
strike <- 100
texp <- 1.2
sigma <- 20
intr <- 0.05
price <- 20
FER::BachelierImpvol(price, strike, spot, texp, intr=intr)

Calculate Bachelier model option price

Description

Calculate Bachelier model option price

Usage

BachelierPrice(
  strike = forward,
  spot,
  texp = 1,
  sigma,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)
BachelierPrice(
  strike = forward,
  spot,
  texp = 1,
  sigma,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`strike`	(vector of) strike price
`spot`	(vector of) spot price
`texp`	(vector of) time to expiry
`sigma`	(vector of) volatility
`intr`	interest rate (domestic interest rate)
`divr`	dividend/convenience yield (foreign interest rate)
`cp`	call/put sign. `1` for call, `-1` for put.
`forward`	forward price. If given, `forward` overrides `spot`
`df`	discount factor. If given, `df` overrides `intr`

Value

option price

References

Choi, J., Kim, K., & Kwak, M. (2009). Numerical Approximation of the Implied Volatility Under Arithmetic Brownian Motion. Applied Mathematical Finance, 16(3), 261-268. doi:10.1080/13504860802583436

Examples

spot <- 100
strike <- seq(80,125,5)
texp <- 1.2
sigma <- 20
intr <- 0.05
FER::BachelierPrice(strike, spot, texp, sigma, intr=intr)

spot <- 100
strike <- seq(80,125,5)
texp <- 1.2
sigma <- 20
intr <- 0.05
FER::BachelierPrice(strike, spot, texp, sigma, intr=intr)

Calculate Black-Scholes implied volatility

Description

Calculate Black-Scholes implied volatility

Usage

BlackScholesImpvol(
  price,
  strike = forward,
  spot,
  texp = 1,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)
BlackScholesImpvol(
  price,
  strike = forward,
  spot,
  texp = 1,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`price`	(vector of) option price
`strike`	(vector of) strike price
`spot`	(vector of) spot price
`texp`	(vector of) time to expiry
`intr`	interest rate (domestic interest rate)
`divr`	dividend/convenience yield (foreign interest rate)
`cp`	call/put sign. `1` for call, `-1` for put.
`forward`	forward price. If given, `forward` overrides `spot`
`df`	discount factor. If given, `df` overrides `intr`

Value

Black-Scholes implied volatility

References

Giner, G., & Smyth, G. K. (2016). statmod: Probability Calculations for the Inverse Gaussian Distribution. The R Journal, 8(1), 339-351. doi:10.32614/RJ-2016-024

Examples

spot <- 100
strike <- 100
texp <- 1.2
sigma <- 0.2
intr <- 0.05
price <- 20
FER::BlackScholesImpvol(price, strike, spot, texp, intr=intr)

spot <- 100
strike <- 100
texp <- 1.2
sigma <- 0.2
intr <- 0.05
price <- 20
FER::BlackScholesImpvol(price, strike, spot, texp, intr=intr)

Calculate Black-Scholes option price

Description

Calculate Black-Scholes option price

Usage

BlackScholesPrice(
  strike = forward,
  spot,
  texp = 1,
  sigma,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)
BlackScholesPrice(
  strike = forward,
  spot,
  texp = 1,
  sigma,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`strike`	(vector of) strike price
`spot`	(vector of) spot price
`texp`	(vector of) time to expiry
`sigma`	(vector of) volatility
`intr`	interest rate (domestic interest rate)
`divr`	dividend/convenience yield (foreign interest rate)
`cp`	call/put sign. `1` for call, `-1` for put.
`forward`	forward price. If given, `forward` overrides `spot`
`df`	discount factor. If given, `df` overrides `intr`

Value

option price

References

Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654. doi:10.1086/260062

Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3(1), 167-179. doi:10.1016/0304-405X(76)90024-6

https://en.wikipedia.org/wiki/Black-Scholes_model

Examples

spot <- 100
strike <- seq(80,125,5)
texp <- 1.2
sigma <- 0.2
intr <- 0.05
FER::BlackScholesPrice(strike, spot, texp, sigma, intr=intr)

spot <- 100
strike <- seq(80,125,5)
texp <- 1.2
sigma <- 0.2
intr <- 0.05
FER::BlackScholesPrice(strike, spot, texp, sigma, intr=intr)

Calculate the mass at zero under the CEV model

Description

Calculate the mass at zero under the CEV model

Usage

CevMassZero(
  spot,
  texp = 1,
  sigma,
  beta = 0.5,
  intr = 0,
  divr = 0,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)
CevMassZero(
  spot,
  texp = 1,
  sigma,
  beta = 0.5,
  intr = 0,
  divr = 0,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`spot`	(vector of) spot price
`texp`	(vector of) time to expiry
`sigma`	(vector of) volatility
`beta`	beta
`intr`	interest rate
`divr`	dividend rate
`forward`	forward price. If given, `forward` overrides `spot`
`df`	discount factor. If given, `df` overrides `intr`

Value

mass at zero

Examples

spot <- 100
texp <- 1.2
beta <- 0.5
sigma <- 2
FER::CevMassZero(spot, texp, sigma, beta)

spot <- 100
texp <- 1.2
beta <- 0.5
sigma <- 2
FER::CevMassZero(spot, texp, sigma, beta)

Calculate the constant elasticity of variance (CEV) model option price

Description

Calculate the constant elasticity of variance (CEV) model option price

Usage

CevPrice(
  strike = forward,
  spot,
  texp = 1,
  sigma,
  beta = 0.5,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)
CevPrice(
  strike = forward,
  spot,
  texp = 1,
  sigma,
  beta = 0.5,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`strike`	(vector of) strike price
`spot`	(vector of) spot price
`texp`	(vector of) time to expiry
`sigma`	(vector of) volatility
`beta`	elasticity parameter
`intr`	interest rate (domestic interest rate)
`divr`	dividend/convenience yield (foreign interest rate)
`cp`	call/put sign. `1` for call, `-1` for put.
`forward`	forward price. If given, `forward` overrides `spot`
`df`	discount factor. If given, `df` overrides `intr`

Value

option price

References

Schroder, M. (1989). Computing the constant elasticity of variance option pricing formula. Journal of Finance, 44(1), 211-219. doi:10.1111/j.1540-6261.1989.tb02414.x

Examples

spot <- 100
strike <- seq(80,125,5)
texp <- 1.2
beta <- 0.5
sigma <- 2
FER::CevPrice(strike, spot, texp, sigma, beta)

spot <- 100
strike <- seq(80,125,5)
texp <- 1.2
beta <- 0.5
sigma <- 2
FER::CevPrice(strike, spot, texp, sigma, beta)

Calculate the option price under the NSVh model with lambda=1 (Choi et al. 2019)

Description

Calculate the option price under the NSVh model with lambda=1 (Choi et al. 2019)

Usage

Nsvh1Choi2019(
  strike = forward,
  spot,
  texp = 1,
  sigma,
  vov = 0,
  rho = 0,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)
Nsvh1Choi2019(
  strike = forward,
  spot,
  texp = 1,
  sigma,
  vov = 0,
  rho = 0,
  intr = 0,
  divr = 0,
  cp = 1L,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`strike`	(vector of) strike price
`spot`	(vector of) spot price
`texp`	(vector of) time to expiry
`sigma`	(vector of) volatility
`vov`	(vector of) vol-of-vol
`rho`	(vector of) correlation
`intr`	interest rate
`divr`	dividend rate
`cp`	call/put sign. `1` (default) for call price, `-1` for put price, `NULL` for Bachelier volatility
`forward`	forward price. If given, `forward` overrides `spot`
`df`	discount factor. If given, `df` overrides `intr`

Value

BS volatility or option price based on cp

References

Choi, J., Liu, C., & Seo, B. K. (2019). Hyperbolic normal stochastic volatility model. Journal of Futures Markets, 39(2), 186–204. doi:10.1002/fut.21967

Examples


spot <- 100
strike <- seq(80,125,5)
texp <- 1.2
sigma <- 20
vov <- 0.2
rho <- -0.5
strike <- seq(0.1, 2, 0.1)

FER::Nsvh1Choi2019(strike, spot, texp, sigma, vov, rho)

spot <- 100
strike <- seq(80,125,5)
texp <- 1.2
sigma <- 20
vov <- 0.2
rho <- -0.5
strike <- seq(0.1, 2, 0.1)

FER::Nsvh1Choi2019(strike, spot, texp, sigma, vov, rho)

Calculate the equivalent BS volatility (Hagan et al. 2002) for the Stochatic-Alpha-Beta-Rho (SABR) model

Description

Calculate the equivalent BS volatility (Hagan et al. 2002) for the Stochatic-Alpha-Beta-Rho (SABR) model

Usage

SabrHagan2002(
  strike = forward,
  spot,
  texp = 1,
  sigma,
  vov = 0,
  rho = 0,
  beta = 1,
  intr = 0,
  divr = 0,
  cp = NULL,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)
SabrHagan2002(
  strike = forward,
  spot,
  texp = 1,
  sigma,
  vov = 0,
  rho = 0,
  beta = 1,
  intr = 0,
  divr = 0,
  cp = NULL,
  forward = spot * exp(-divr * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`strike`	(vector of) strike price
`spot`	(vector of) spot price
`texp`	(vector of) time to expiry
`sigma`	(vector of) volatility
`vov`	(vector of) vol-of-vol
`rho`	(vector of) correlation
`beta`	(vector of) beta
`intr`	interest rate (domestic interest rate)
`divr`	convenience rate (foreign interest rate)
`cp`	call/put sign. `NULL` for BS vol (default), `1` for call price, `-1` for put price.
`forward`	forward price. If given, `forward` overrides `spot`
`df`	discount factor. If given, `df` overrides `intr`

Value

BS volatility or option price based on cp

References

Hagan, P. S., Kumar, D., Lesniewski, A. S., & Woodward, D. E. (2002). Managing Smile Risk. Wilmott, September, 84-108.

Examples

sigma <- 0.25
vov <- 0.3
rho <- -0.8
beta <- 0.3
texp <- 10
strike <- seq(0.1, 2, 0.1)
FER::SabrHagan2002(strike, 1, texp, sigma, vov, rho, beta)
FER::SabrHagan2002(strike, 1, texp, sigma, vov, rho, beta, cp=1)

sigma <- 0.25
vov <- 0.3
rho <- -0.8
beta <- 0.3
texp <- 10
strike <- seq(0.1, 2, 0.1)
FER::SabrHagan2002(strike, 1, texp, sigma, vov, rho, beta)
FER::SabrHagan2002(strike, 1, texp, sigma, vov, rho, beta, cp=1)

Spread option under the Bachelier model

Description

The payout of the spread option is max(S1_T - S2_T - K, 0) where S1_T and S2_T are the prices at expiry T of assets 1 and 2 respectively and K is the strike price.

Usage

SpreadBachelier(
  strike = 0,
  spot1,
  spot2,
  texp = 1,
  sigma1,
  sigma2,
  corr,
  intr = 0,
  divr1 = 0,
  divr2 = 0,
  cp = 1L,
  forward1 = spot1 * exp(-divr1 * texp)/df,
  forward2 = spot2 * exp(-divr2 * texp)/df,
  df = exp(-intr * texp)
)
SpreadBachelier(
  strike = 0,
  spot1,
  spot2,
  texp = 1,
  sigma1,
  sigma2,
  corr,
  intr = 0,
  divr1 = 0,
  divr2 = 0,
  cp = 1L,
  forward1 = spot1 * exp(-divr1 * texp)/df,
  forward2 = spot2 * exp(-divr2 * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`strike`	(vector of) strike price
`spot1`	(vector of) spot price of asset 1
`spot2`	(vector of) spot price of asset 2
`texp`	(vector of) time to expiry
`sigma1`	(vector of) Bachelier volatility of asset 1
`sigma2`	(vector of) Bachelier volatility of asset 2
`corr`	correlation
`intr`	interest rate
`divr1`	dividend rate of asset 1
`divr2`	dividend rate of asset 2
`cp`	call/put sign. `1` for call, `-1` for put.
`forward1`	forward price of asset 1. If given, overrides `spot1`
`forward2`	forward price of asset 2. If given, overrides `spot2`
`df`	discount factor. If given, `df` overrides `intr`

Value

option price

Examples


FER::SpreadBachelier((-2:2)*10, 100, 120, 1.3, 20, 36, -0.5)

FER::SpreadBachelier((-2:2)*10, 100, 120, 1.3, 20, 36, -0.5)

Spread option pricing method by Bjerksund & Stensland (2014)

Description

The payout of the spread option is max(S1_T - S2_T - K, 0) where S1_T and S2_T are the prices at expiry T of assets 1 and 2 respectively and K is the strike price.

Usage

SpreadBjerksund2014(
  strike = 0,
  spot1,
  spot2,
  texp = 1,
  sigma1,
  sigma2,
  corr,
  intr = 0,
  divr1 = 0,
  divr2 = 0,
  cp = 1L,
  forward1 = spot1 * exp(-divr1 * texp)/df,
  forward2 = spot2 * exp(-divr2 * texp)/df,
  df = exp(-intr * texp)
)
SpreadBjerksund2014(
  strike = 0,
  spot1,
  spot2,
  texp = 1,
  sigma1,
  sigma2,
  corr,
  intr = 0,
  divr1 = 0,
  divr2 = 0,
  cp = 1L,
  forward1 = spot1 * exp(-divr1 * texp)/df,
  forward2 = spot2 * exp(-divr2 * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`strike`	(vector of) strike price
`spot1`	(vector of) spot price of asset 1
`spot2`	(vector of) spot price of asset 2
`texp`	(vector of) time to expiry
`sigma1`	(vector of) volatility of asset 1
`sigma2`	(vector of) volatility of asset 2
`corr`	correlation
`intr`	interest rate
`divr1`	dividend rate of asset 1
`divr2`	dividend rate of asset 2
`cp`	call/put sign. `1` for call, `-1` for put.
`forward1`	forward price of asset 1. If given, overrides `spot1`
`forward2`	forward price of asset 2. If given, overrides `spot2`
`df`	discount factor. If given, `df` overrides `intr`

Value

option price

References

Bjerksund, P., & Stensland, G. (2014). Closed form spread option valuation. Quantitative Finance, 14(10), 1785–1794. doi:10.1080/14697688.2011.617775

Examples


FER::SpreadBjerksund2014((-2:2)*10, 100, 120, 1.3, 0.2, 0.3, -0.5)

FER::SpreadBjerksund2014((-2:2)*10, 100, 120, 1.3, 0.2, 0.3, -0.5)

Kirk's approximation for spread option

Description

The payout of the spread option is max(S1_T - S2_T - K, 0) where S1_T and S2_T are the prices at expiry T of assets 1 and 2 respectively and K is the strike price.

Usage

SpreadKirk(
  strike = 0,
  spot1,
  spot2,
  texp = 1,
  sigma1,
  sigma2,
  corr,
  intr = 0,
  divr1 = 0,
  divr2 = 0,
  cp = 1L,
  forward1 = spot1 * exp(-divr1 * texp)/df,
  forward2 = spot2 * exp(-divr2 * texp)/df,
  df = exp(-intr * texp)
)
SpreadKirk(
  strike = 0,
  spot1,
  spot2,
  texp = 1,
  sigma1,
  sigma2,
  corr,
  intr = 0,
  divr1 = 0,
  divr2 = 0,
  cp = 1L,
  forward1 = spot1 * exp(-divr1 * texp)/df,
  forward2 = spot2 * exp(-divr2 * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`strike`	(vector of) strike price
`spot1`	(vector of) spot price of asset 1
`spot2`	(vector of) spot price of asset 2
`texp`	(vector of) time to expiry
`sigma1`	(vector of) volatility of asset 1
`sigma2`	(vector of) volatility of asset 2
`corr`	correlation
`intr`	interest rate
`divr1`	dividend rate of asset 1
`divr2`	dividend rate of asset 2
`cp`	call/put sign. `1` for call, `-1` for put.
`forward1`	forward price of asset 1. If given, overrides `spot1`
`forward2`	forward price of asset 2. If given, overrides `spot2`
`df`	discount factor. If given, `df` overrides `intr`

Value

option price

References

Kirk, E. (1995). Correlation in the energy markets. In Managing Energy Price Risk (First, pp. 71–78). Risk Publications.

Examples


FER::SpreadKirk((-2:2)*10, 100, 120, 1.3, 0.2, 0.3, -0.5)

FER::SpreadKirk((-2:2)*10, 100, 120, 1.3, 0.2, 0.3, -0.5)

Margrabe's formula for exhange option price

Description

The payout of the exchange option is max(S1_T - S2_T, 0) where S1_T and S2_T are the prices at expiry T of assets 1 and 2 respectively.

Usage

SwitchMargrabe(
  spot1,
  spot2,
  texp = 1,
  sigma1,
  sigma2,
  corr,
  intr = 0,
  divr1 = 0,
  divr2 = 0,
  cp = 1L,
  forward1 = spot1 * exp(-divr1 * texp)/df,
  forward2 = spot2 * exp(-divr2 * texp)/df,
  df = exp(-intr * texp)
)
SwitchMargrabe(
  spot1,
  spot2,
  texp = 1,
  sigma1,
  sigma2,
  corr,
  intr = 0,
  divr1 = 0,
  divr2 = 0,
  cp = 1L,
  forward1 = spot1 * exp(-divr1 * texp)/df,
  forward2 = spot2 * exp(-divr2 * texp)/df,
  df = exp(-intr * texp)
)

Arguments

`spot1`	(vector of) spot price of asset 1
`spot2`	(vector of) spot price of asset 2
`texp`	(vector of) time to expiry
`sigma1`	(vector of) volatility of asset 1
`sigma2`	(vector of) volatility of asset 2
`corr`	correlation
`intr`	interest rate
`divr1`	dividend rate of asset 1
`divr2`	dividend rate of asset 2
`cp`	call/put sign. `1` for call, `-1` for put.
`forward1`	forward price of asset 1. If given, overrides `spot1`
`forward2`	forward price of asset 2. If given, overrides `spot2`
`df`	discount factor. If given, `df` overrides `intr`

Value

option price

References

Margrabe, W. (1978). The value of an option to exchange one asset for another. The Journal of Finance, 33(1), 177–186.

Examples


FER::SwitchMargrabe(100, 120, 1.3, 0.2, 0.3, -0.5)

FER::SwitchMargrabe(100, 120, 1.3, 0.2, 0.3, -0.5)

Package 'FER'

Help Index

Calculate Bachelier model implied volatility

Description

Usage

Arguments

Value

References

See Also

Examples

Calculate Bachelier model option price

Description

Usage

Arguments

Value

References

See Also

Examples

Calculate Black-Scholes implied volatility

Description

Usage

Arguments

Value

References

See Also

Examples

Calculate Black-Scholes option price

Description

Usage

Arguments

Value

References

See Also

Examples

Calculate the mass at zero under the CEV model

Description

Usage

Arguments

Value

Examples

Calculate the constant elasticity of variance (CEV) model option price

Description

Usage

Arguments

Value

References

Examples

Calculate the option price under the NSVh model with lambda=1 (Choi et al. 2019)

Description

Usage

Arguments

Value

References

Examples

Calculate the equivalent BS volatility (Hagan et al. 2002) for the Stochatic-Alpha-Beta-Rho (SABR) model

Description

Usage

Arguments

Value

References

Examples

Spread option under the Bachelier model

Description

Usage

Arguments

Value

Examples

Spread option pricing method by Bjerksund & Stensland (2014)

Description

Usage

Arguments

Value

References

Examples

Kirk's approximation for spread option

Description

Usage

Arguments

Value

References