ragtop-package: Pricing schemes for derivatives using equity-linked default...

In ragtop: Pricing Equity Derivatives with Extensions of Black-Scholes

Pricing schemes for derivatives using equity-linked default intensity

Description

Using numerical integration, we price convertible bonds, straight bonds, equity options and various other derivatives consistently using a jump-diffusion model in which default intensity can vary with equity price in a user-specified deterministic manner.

Details

We apply the stochastic model

dS/S=(r+h-q) dt + \sigma dZ - dJ

where r and q play their usual roles, h is a deterministic function of stock price and time, and J is a Poisson jump process adapted to the default intensity or hazard rate h. This model is a jump-diffusion extension of Black-Scholes, with the jump process J representing default, compensated by extra drift in the equity at rate h.

Volatilities, default intensities and risk-free rates may all be represented with arbitrary term structures. Default intensity term structures may also take the underlying equity price into account.

Pricing in the standard Black-Scholes model is a special case with default intensity set to zero. Therefore this package also serves to price securities in the standard Black-Scholes model, while still allowing risk-free rates and volatilites have nontrivial term structures.

Important Features

Black-Scholes

The standard model is automatically supported as a special case, but also has optimized routines

Term Structures

The package allows for any kind of instrument to be priced with time-varying rates, volatility and default intensity

Dividends

Allows for discrete dividends in an arbitrary combination of fixed and proportional amounts. The difference between fixed and proprtional can be up to 10 percent in implied volatility terms.

Calibration

Model calibration routines are included

Bankruptcy Realism

A parsimonious deterministic model of default intensity gives rich behavior and conforms reasonably well to observed market data

Algorithm Parameters

Default parameters for the algorithm work well for a very wide variety of pricing and implied volatility scenarios

Author(s)

Maintainer: Brian K. Boonstra [e-mail]

Authors:

• Brian K. Boonstra [e-mail]

Examples

## Vanilla European exercise
blackscholes(callput=-1, S0=100, K=90, r=0.03, time=1, vola=0.5)
blackscholes(PUT, S0=100, K=90, r=0.03, time=1, vola=0.5,
             default_intensity=0.07, borrow_cost=0.005)
## With a term structure of volatility
## Not run: 
black_scholes_on_term_structures(callput=-1, S0=100, K=90, time=1,
                                 const_short_rate=0.025,
                                 variance_cumulation_fcn = function(T, t) {
                                   0.45 ^ 2 * (T - t) + 0.15^2 * max(0, T-0.25)
                                 })

## End(Not run)

## Vanilla American exercise
## Not run: 
american(PUT, S0=100, K=110, time=0.77, const_short_rate = 0.06,
         const_volatility=0.20, num_time_steps=200)

## End(Not run)
## With a term structure of volatility
## Not run: 
american(callput=-1, S0=100, K=90, time=1, const_short_rate=0.025,
         variance_cumulation_fcn = function(T, t) {
             0.45 ^ 2 * (T - t) + 0.15^2 * max(0, T-0.25)
         })

## End(Not run)
## With discrete dividends, combined fixed and proportional
divs = data.frame(time=seq(from=0.11, to=2, by=0.25),
                  fixed=seq(1.5, 1, length.out=8),
                  proportional = seq(1, 1.5, length.out=8))
## Not run: 
american(callput=-1, S0=100, K=90, time=1, const_short_rate=0.025,
         const_volatility=0.20, dividends=divs)

## End(Not run)

## American Exercise Implied Volatility
american_implied_volatility(25,CALL,S0=100,K=100,time=2.2, const_short_rate=0.03)
df250 =  function(t) ( exp(-0.02*t)*exp(-0.03*max(0,t-1.0))) # Simple term structure
df25 = function(T,t){df250(T)/df250(t)} # Relative discount factors
## Not run: 
american_implied_volatility(25,-1,100,100,2.2,discount_factor_fcn=df25)

## End(Not run)

## Convertible Bond
## Not Run
pct4 = function(T,t=0) { exp(-0.04*(T-t)) }
cb = ConvertibleBond(conversion_ratio=3.5, maturity=1.5, notional=100,
                     discount_factor_fcn=pct4, name='Convertible')
S0 = 10; p = 6.0; h = 0.10
h_fcn = function(t, S, ...){0.9 * h + 0.1 * h * (S0/S)^p }  # Intensity linked to equity price
## Not run: 
find_present_value(S0=S0, instruments=list(Convertible=cb), num_time_steps=250,
                   default_intensity_fcn=h_fcn,
                   const_volatility = 0.4, discount_factor_fcn=pct4,
                   std_devs_width=5)

## End(Not run)

## Fitting Term Structure of Volatility
## Not Run
opts = list(m1=AmericanOption(callput=-1, strike=9.9, maturity=1/12, name="m1"),
            m2=AmericanOption(callput=-1, strike=9.8, maturity=1/6, name="m2"))
## Not run: 
vfit = fit_variance_cumulation(S0, opts, c(0.6, 0.8), default_intensity_fcn=h_fcn)
print(vfit$volatilities)

## End(Not run)

ragtop documentation built on June 20, 2026, 5:06 p.m.