| Title: | Pricing Equity Derivatives with Extensions of Black-Scholes |
|---|---|
| Description: | Algorithms to price American and European equity options, convertible bonds and a variety of other financial derivatives. It uses an extension of the usual Black-Scholes model in which jump to default may occur at a probability specified by a power-law link between stock price and hazard rate as found in the paper by Takahashi, Kobayashi, and Nakagawa (2001) <doi:10.3905/jfi.2001.319302>. We use ideas and techniques from Andersen and Buffum (2002) <doi:10.2139/ssrn.355308> and Linetsky (2006) <doi:10.1111/j.1467-9965.2006.00271.x>. |
| Authors: | Brian K. Boonstra |
| Maintainer: | Brian K. Boonstra <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.1.1 |
| Built: | 2024-10-30 06:57:28 UTC |
| Source: | CRAN |
Compute "present" value as of time t for coupons that would otherwise have been paid up to time acceleration_t, in the case of accelerated coupon provisions for forced conversions (or sometimes even unforced ones).
accelerated_coupon_value( t, coupons_df, discount_factor_fcn, acceleration_t = Inf )accelerated_coupon_value( t, coupons_df, discount_factor_fcn, acceleration_t = Inf )
t |
The time toward which all coupons should be present valued |
coupons_df |
A data.frame of details for each coupon. It should have the
columns |
discount_factor_fcn |
A function specifying how the contract says future coupons should be discounted for this instrument in case the acceleration clause is triggered |
acceleration_t |
Time limit, up to which coupons will be accelerated |
Other Bond Coupons:
coupon_value_at_exercise(),
value_from_prior_coupons()
Other Bond Coupon Acceleration:
coupon_value_at_exercise()
Analyze dividends to find ones paid in the interval (t,t+dt]. Form
present value as of time t for them, and then use spline interpolation
to adjust instrument values accordingly.
adjust_for_dividends(grid_values, t, dt, r, h, S, S0, dividends)adjust_for_dividends(grid_values, t, dt, r, h, S, S0, dividends)
grid_values |
A |
t |
Time after this timestep has been taken |
dt |
Interval to end of timestep |
r |
risk-free interest rate |
h |
Default intensities |
S |
Underlying equity values for the grid |
S0 |
Time zero price of the base equity |
dividends |
A |
An object like grid_values with entries modified according to the dividends
Other Dividends:
shift_for_dividends(),
time_adj_dividends()
Use a control-variate scheme to simultaneously estimate the present values of a collection of one or more American-exercise options under a default model with survival probabilities not linked to equity prices.
american( callput, S0, K, time, const_short_rate = 0, const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, default_intensity_fcn = function(t, S, ...) { const_default_intensity + 0 * S }, ..., num_time_steps = 100, structure_constant = 2, std_devs_width = 5 )american( callput, S0, K, time, const_short_rate = 0, const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, default_intensity_fcn = function(t, S, ...) { const_default_intensity + 0 * S }, ..., num_time_steps = 100, structure_constant = 2, std_devs_width = 5 )
callput |
1 for calls, -1 for puts (may be a vector of the same) |
S0 |
initial underlying price |
K |
strike (may be a vector) |
time |
Time from |
const_short_rate |
A constant to use for the instantaneous interest rate in case |
const_default_intensity |
A constant to use for the instantaneous default intensity in case |
discount_factor_fcn |
A function for computing present values to
time |
survival_probability_fcn |
(Implied argument) A function for probability of survival, with
arguments |
default_intensity_fcn |
A function for computing default intensity
occurring at a given time, dependent on time and stock price, with
arguments |
... |
Further arguments passed on to |
num_time_steps |
Number of steps to use in the grid solver. Can usually be set quite low due to the control variate scheme. |
structure_constant |
The maximum ratio between time intervals |
std_devs_width |
The number of standard deviations, in |
The scheme uses find_present_value() to price the options and their European-exercise equivalents. It then compares the latter to black-scholes formula output and uses the results as an error correction on the prices of the American-exercise options.
A vector of estimated option present values
Other Equity Independent Default Intensity:
american_implied_volatility(),
black_scholes_on_term_structures(),
blackscholes(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other American Exercise Equity Options:
american_implied_volatility(),
control_variate_pairs()
american(PUT, S0=100, K=110, time=0.77, const_short_rate = 0.06, const_volatility=0.20, num_time_steps=200) american(callput=-1, S0=100, K=90, time=1, const_short_rate=0.025, variance_cumulation_fcn = function(T, t) { # Term structure of vola 0.45 ^ 2 * (T - t) + 0.15^2 * max(0, T-0.25) })american(PUT, S0=100, K=110, time=0.77, const_short_rate = 0.06, const_volatility=0.20, num_time_steps=200) american(callput=-1, S0=100, K=90, time=1, const_short_rate=0.025, variance_cumulation_fcn = function(T, t) { # Term structure of vola 0.45 ^ 2 * (T - t) + 0.15^2 * max(0, T-0.25) })
Use the grid solver to generate american option values under a default model with survival probabilities not linked to equity prices. and run them through a bisective root search method until a constant volatility matching the provided option price has been found.
american_implied_volatility( option_price, callput, S0, K, time, const_default_intensity = 0, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, default_intensity_fcn = function(t, S, ...) { const_default_intensity + 0 * S }, ..., num_time_steps = 30, structure_constant = 2, std_devs_width = 5, relative_tolerance = 1e-04, max.iter = 100, max_vola = 4 )american_implied_volatility( option_price, callput, S0, K, time, const_default_intensity = 0, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, default_intensity_fcn = function(t, S, ...) { const_default_intensity + 0 * S }, ..., num_time_steps = 30, structure_constant = 2, std_devs_width = 5, relative_tolerance = 1e-04, max.iter = 100, max_vola = 4 )
option_price |
Option price to match |
callput |
1 for calls, -1 for puts |
S0 |
An initial stock price, for setting grid scale |
K |
strike |
time |
Time from |
const_default_intensity |
A constant to use for the instantaneous default intensity in case |
survival_probability_fcn |
(Implied argument) A function for probability of survival, with
arguments |
default_intensity_fcn |
A function for computing default intensity
occurring at a given time, dependent on time and stock price, with
arguments |
... |
Additional arguments to be passed on to |
num_time_steps |
Minimum number of time steps in the grid |
structure_constant |
The maximum ratio between time intervals |
std_devs_width |
The number of standard deviations, in |
relative_tolerance |
Relative tolerance in instrument price defining the root-finder halting condition |
max.iter |
Maximum number of root-finder iterations allowed |
max_vola |
Maximum volatility to try |
Estimated volatility
implied_volatility_with_term_struct for implied volatility
of European options under the same conditions, american for the
underlying pricing algorithm
Other Implied Volatilities:
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
fit_variance_cumulation(),
implied_jump_process_volatility(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other Equity Independent Default Intensity:
american(),
black_scholes_on_term_structures(),
blackscholes(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other American Exercise Equity Options:
american(),
control_variate_pairs()
american_implied_volatility(25,CALL,S0=100,K=100,time=2.2, const_short_rate=0.03, num_time_steps=5) 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 american_implied_volatility(25,-1,100,100,2.2, discount_factor_fcn=df25, num_time_steps=5)american_implied_volatility(25,CALL,S0=100,K=100,time=2.2, const_short_rate=0.03, num_time_steps=5) 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 american_implied_volatility(25,-1,100,100,2.2, discount_factor_fcn=df25, num_time_steps=5)
A standard option contract allowing for early exercise at the choice of the option holder
optionality_fcn(v, ...)Return a version of v at time t corrected for any optionality conditions.
recovery_fcn(v, S, t, ...)Return recovery value, given non-default values v at time t. Subclasses may be more elaborate, this method simply returns 0.0.
Price an option according to the famous Black-Scholes formula, with the
optional addition of a jump-to-default intensity and discrete dividends. Volatility
and rates may be provided as constants or as 2+ parameter functions with
first argument T corresponding to maturity and second argument t corresponding to
model date.
black_scholes_on_term_structures( callput, S0, K, time, const_volatility = 0.5, const_short_rate = 0, const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, variance_cumulation_fcn = function(T, t) { const_volatility^2 * (T - t) }, dividends = NULL, borrow_cost = 0, dividend_rate = 0 )black_scholes_on_term_structures( callput, S0, K, time, const_volatility = 0.5, const_short_rate = 0, const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, variance_cumulation_fcn = function(T, t) { const_volatility^2 * (T - t) }, dividends = NULL, borrow_cost = 0, dividend_rate = 0 )
callput |
1 for calls, -1 for puts |
S0 |
initial underlying price |
K |
strike |
time |
Time from |
const_volatility |
A constant to use for volatility in case |
const_short_rate |
A constant to use for the instantaneous interest rate in case |
const_default_intensity |
A constant to use for the instantaneous default intensity in case |
discount_factor_fcn |
A function for computing present values to
time |
survival_probability_fcn |
A function for probability of survival, with
arguments |
variance_cumulation_fcn |
A function for computing total stock variance
occurring during this timestep, with arguments |
dividends |
A |
borrow_cost |
A continuous rate for stock borrow costs |
dividend_rate |
A continuous rate for dividends and other cashflows such as foreign interest rates |
Any term structures will be converted to equivalent constant arguments by calling
them with the arguments (time, 0).
Other European Options:
blackscholes(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other Equity Independent Default Intensity:
american_implied_volatility(),
american(),
blackscholes(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
black_scholes_on_term_structures(callput=-1, S0=100, K=90, time=1, discount_factor_fcn = function(T, t, ...) { exp(-0.03 * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-0.07 * (T - t)) }, variance_cumulation_fcn = function(T, t) { 0.45 ^ 2 * (T - t) })black_scholes_on_term_structures(callput=-1, S0=100, K=90, time=1, discount_factor_fcn = function(T, t, ...) { exp(-0.03 * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-0.07 * (T - t)) }, variance_cumulation_fcn = function(T, t) { 0.45 ^ 2 * (T - t) })
Price options according to the famous Black-Scholes formula, with the optional addition of a jump-to-default intensity and discrete dividends.
blackscholes( callput, S0, K, r, time, vola, default_intensity = 0, divrate = 0, borrow_cost = 0, dividends = NULL )blackscholes( callput, S0, K, r, time, vola, default_intensity = 0, divrate = 0, borrow_cost = 0, dividends = NULL )
callput |
1 for calls, -1 for puts |
S0 |
initial underlying price |
K |
strike |
r |
risk-free interest rate |
time |
Time from |
vola |
Default-free volatility of the underlying |
default_intensity |
hazard rate of underlying default |
divrate |
A continuous rate for dividends and other cashflows such as foreign interest rates |
borrow_cost |
A continuous rate for stock borrow costs |
dividends |
A |
Note that if the default_intensity is set larger than zero then
put-call parity still holds. Greeks are reduced according to cumulated default
probability.
All inputs must either be scalars or have the same nonscalar shape.
A list with elements
PriceThe present value(s)
DeltaSensitivity to underlying price
VegaSensitivity to volatility
Other European Options:
black_scholes_on_term_structures(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other Equity Independent Default Intensity:
american_implied_volatility(),
american(),
black_scholes_on_term_structures(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
blackscholes(callput=-1, S0=100, K=90, r=0.03, time=1, # -1 is a PUT vola=0.5, default_intensity=0.07)blackscholes(callput=-1, S0=100, K=90, r=0.03, time=1, # -1 is a PUT vola=0.5, default_intensity=0.07)
Constant CALL for defining option contracts
CALLCALL
An object of class numeric of length 1.
When a bond is emphcallable, the issuer may choose to pay the call price to the bond holder and end the life of the contract.
When a bond is emphputable, the bond holder may choose to force the issuer pay the put price to the bond holder thus ending the life of the contract.
callsA data.frame of details for each call. It should have the columns call_price and effective_time.
putsA data.frame of details for each put. It should have the columns put_price and effective_time.
critical_times()Important times in the life of this instrument for simulation and grid solvers
Infer a reasonable structure for our implicit grid solver based on the voltime, structure constant, and requested grid width in standard deviations.
construct_implicit_grid_structure( tenors, M, S0, K, c, sigma, structure_constant, std_devs_width, min_z_width = 0 )construct_implicit_grid_structure( tenors, M, S0, K, c, sigma, structure_constant, std_devs_width, min_z_width = 0 )
tenors |
Tenors of instruments to be treated on this grid |
M |
Minimum number of timesteps on this grid |
S0 |
An initial stock price, for setting grid scale |
K |
An instrument reference stock price, for setting grid scale |
c |
A continuous stock drift rate |
sigma |
Volatility of diffusion process (without jumps to default) |
structure_constant |
The maximum ratio between time intervals |
std_devs_width |
The number of standard deviations, in |
min_z_width |
Minimum grid width, in log space |
Generally speaking pricing will be good to about 10bp of relative accuracy when the ratio of timesteps to voltime (in annualized units) is over 200.
Cases with pathologically low volatility may go awry (in the sense of yielding ultimately inaccurate PDE solutions), as the structure_constant will force a step in z space much bigger than the width in standard deviations.
A list with elements
TThe maximum time for this grid
dtLargest permissible timestep size
dzDistance between space grid points
z0Center of space grid
z_widthWidth in space
half_NA misnomer, actually
NThe number of space points
zLocations of space points
Other Implicit Grid Solver:
find_present_value(),
form_present_value_grid(),
infer_conforming_time_grid(),
integrate_pde(),
iterate_grid_from_timestep(),
take_implicit_timestep(),
timestep_instruments()
Matrix entries for implicit numerical differentiation using Neumann boundary conditions
construct_tridiagonals(sigma, structure_constant, drift)construct_tridiagonals(sigma, structure_constant, drift)
sigma |
Volatility of diffusion process (without jumps to default) |
structure_constant |
The ratio between time interval |
drift |
Vector of drift rate of underlying equity grid points, including induced drift from default intensity |
A list with elements super, diag and sub
containing the superdiagonal, diagonal and subdiagonal of the implicit
timestep differencing matrix
Form a list twice as long as the longest of the arguments
callput, K, time whose
first half consists of AmericanOption objects and
second half consists of EuropeanOption objects
having the same exercise specification
control_variate_pairs(callput, K, time)control_variate_pairs(callput, K, time)
callput |
1 for calls, -1 for puts |
K |
strike |
time |
Time from |
Other American Exercise Equity Options:
american_implied_volatility(),
american()
Convertible bond with exercise into stock
conversion_ratioThe number of shares, per bond, that result from exercise
dividend_ceilingThe level of dividend protection (if any) specified in terms and conditions
exercise_decision(v, S, t, discount_factor_fctn = discount_factor_fcn, ...)Find indexes where hold value v will be inferior to conversion value at each stock price level in S, adjusted to include all past coupons
optionality_fcn(v, S, t, discount_factor_fctn = discount_factor_fcn, ...)Return the greater of hold value v or exercise value at each stock price level in S. If the given date is beyond maturity, return value at maturity.
terminal_values(v, ...)Return a terminal value. defaults to simply calling optionality_fcn.
update_cashflows(
small_t,
big_t,
discount_factor_fctn = discount_factor_fcn,
include_notional = TRUE,
...
)Update last_computed_cash and return cashflow information for the given time period, valued at big_t
Compute "present" value as of time t for coupons that
would otherwise have been paid up to time acceleration_t, in the
case of accelerated coupon provisions for forced conversions (or
sometimes even unforced ones).
coupon_value_at_exercise( t, coupons_df, discount_factor_fcn, model_t = 0, accelerate_future_coupons = FALSE, acceleration_discount_factor_fcn = discount_factor_fcn, acceleration_t = Inf )coupon_value_at_exercise( t, coupons_df, discount_factor_fcn, model_t = 0, accelerate_future_coupons = FALSE, acceleration_discount_factor_fcn = discount_factor_fcn, acceleration_t = Inf )
t |
The time toward which all coupons should be present valued |
coupons_df |
A data.frame of details for each coupon. It should have the
columns |
discount_factor_fcn |
A function specifying how future cashflows should generally be discounted for this instrument |
model_t |
Model timestamp passed to |
accelerate_future_coupons |
If |
acceleration_discount_factor_fcn |
A function specifying how future coupons should be discounted for this instrument under coupon acceleration conditions |
acceleration_t |
The maximum time up to which future coupons will be counted for acceleration, passed on to |
A scalar equal to the present value
Other Bond Coupons:
accelerated_coupon_value(),
value_from_prior_coupons()
Other Bond Coupon Acceleration:
accelerated_coupon_value()
A coupon bond is treated here as the entire collection of cashflows. In particular, coupons are included in the package even after they have been paid, accruing at the risk-free rate.
couponsA data.frame of details for each coupon. It should have the
columns payment_time and payment_size.
accumulate_coupon_values_before(t, discount_factor_fctn = discount_factor_fcn)Compute the sum of coupon present values as of t according to discount_factor_fctn
critical_times()Important times in the life of this instrument for simulation and grid solvers
optionality_fcn(v, S, t, ...)Return the notional value in the shape of S at any time on or after maturity, otherwise just return v
total_coupon_values_between(
small_t,
big_t,
discount_factor_fctn = discount_factor_fcn
)Compute the sum (as of big_t) of present values of coupons paid between small_t and big_t
update_cashflows(
small_t,
big_t,
discount_factor_fctn = discount_factor_fcn,
include_notional = TRUE,
...
)Update last_computed_cash and return cashflow information for the given time period, valued at big_t
The BondValuation package provides day count convention treatments superior to quantmod or any other R package known (as of May 2019). This function takes output from BondValuation::AnnivDates(...) and parses it into notionals, maturity time, and coupon times and sizes.
detail_from_AnnivDates( anvdates, as_of = Sys.time(), normalization_factor = 365.25 )detail_from_AnnivDates( anvdates, as_of = Sys.time(), normalization_factor = 365.25 )
anvdates |
Output of BondValuation::AnnivDates(), which must have included a 'Coup' argument so that the resulting list contains an entry for 'PaySched' |
as_of |
Date or time from whose perspective times should be computed |
normalization_factor |
Factor by which raw R time differences should be multiplied. If volatilites are going to be annualized, then this should typically be 365 or so. |
Note: volatilities used in 'ragtop' must have compatible time units to these times.
A list with some of the arguments appropriate for defining a Bond as follows: maturity - maturity notional - notional amount coupons - 'data.frame' with 'payment_time', 'payment_size'
An option contract with call or put terms
strikeA decision price for the contract
callputEither 1 for a call or -1 for a put
Find Black-Scholes volatility based on known interest rates and hazard rates, using an at-the-money put option at the given tenor to set the standard price.
equivalent_bs_vola_to_jump( jump_process_vola, time, const_short_rate = 0, const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, dividends = NULL, borrow_cost = 0, dividend_rate = 0, relative_tolerance = 1e-06, max.iter = 100 )equivalent_bs_vola_to_jump( jump_process_vola, time, const_short_rate = 0, const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, dividends = NULL, borrow_cost = 0, dividend_rate = 0, relative_tolerance = 1e-06, max.iter = 100 )
jump_process_vola |
Volatility of default-free process |
time |
Time to expiration of associated option contracts |
const_short_rate |
A constant to use for the instantaneous interest rate in case |
const_default_intensity |
A constant to use for the instantaneous default intensity in case |
discount_factor_fcn |
A function for computing present values to
time |
survival_probability_fcn |
(Implied argument) A function for probability of survival, with
arguments |
dividends |
A |
borrow_cost |
A continuous rate for stock borrow costs |
dividend_rate |
A continuous accumulation rate for the stock, affecting the drift |
relative_tolerance |
Relative tolerance in instrument price defining the root-finder halting condition |
max.iter |
Maximum number of root-finder iterations allowed |
A scalar defaultable volatility of an option
Other Implied Volatilities:
american_implied_volatility(),
equivalent_jump_vola_to_bs(),
fit_variance_cumulation(),
implied_jump_process_volatility(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other Equity Independent Default Intensity:
american_implied_volatility(),
american(),
black_scholes_on_term_structures(),
blackscholes(),
equivalent_jump_vola_to_bs(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Find default-free volatility (i.e. volatility of a Wiener process with a companion jump process to default) based on known interest rates and hazard rates, using and at-the-money put option at the given tenor to set the standard price.
equivalent_jump_vola_to_bs( bs_vola, time, const_short_rate = 0, const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, dividends = NULL, borrow_cost = 0, dividend_rate = 0, relative_tolerance = 1e-06, max.iter = 100 )equivalent_jump_vola_to_bs( bs_vola, time, const_short_rate = 0, const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, dividends = NULL, borrow_cost = 0, dividend_rate = 0, relative_tolerance = 1e-06, max.iter = 100 )
bs_vola |
BlackScholes volatility of an option with no default assumption |
time |
Time to expiration of associated option contracts |
const_short_rate |
A constant to use for the instantaneous interest rate in case |
const_default_intensity |
A constant to use for the instantaneous default intensity in case |
discount_factor_fcn |
A function for computing present values to
time |
survival_probability_fcn |
(Implied argument) A function for probability of survival, with
arguments |
dividends |
A |
borrow_cost |
Stock borrow cost, affecting the drift rate |
dividend_rate |
A continuous accumulation rate for the stock, affecting the drift |
relative_tolerance |
Relative tolerance in instrument price defining the root-finder halting condition |
max.iter |
Maximum number of root-finder iterations allowed |
A scalar volatility
Other Implied Volatilities:
american_implied_volatility(),
equivalent_bs_vola_to_jump(),
fit_variance_cumulation(),
implied_jump_process_volatility(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other Equity Independent Default Intensity:
american_implied_volatility(),
american(),
black_scholes_on_term_structures(),
blackscholes(),
equivalent_bs_vola_to_jump(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
At maturity, the call option holder will "exercise", i.e. choose stock, with value S, if the
stock price is above the strike K, paying K to the option issuer,
realizing value S-K. The put option holder will exercise, receiving K while surrendering
stock worth S, if the stock price is below K.
Therefore the value at maturity is equal to max(0,callput*(S-K))
optionality_fcn(v, ...)Return a version of v at time t corrected for any optionality conditions.
recovery_fcn(v, S, t, ...)Return recovery value, given non-default values v at time t. Subclasses may be more elaborate, this method simply returns 0.0.
Use a finite difference scheme to form estimates of present values for a variety
of stock prices. Once the grid has been created, interpolate to obtain the
value of each instrument at the present stock price S0
find_present_value( S0, num_time_steps, instruments, const_volatility = 0.5, const_short_rate = 0, const_default_intensity = 0, override_Tmax = NA, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, default_intensity_fcn = function(t, S, ...) { const_default_intensity + 0 * S }, variance_cumulation_fcn = function(T, t) { const_volatility^2 * (T - t) }, dividends = NULL, borrow_cost = 0, dividend_rate = 0, structure_constant = 2, std_devs_width = 3 )find_present_value( S0, num_time_steps, instruments, const_volatility = 0.5, const_short_rate = 0, const_default_intensity = 0, override_Tmax = NA, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, default_intensity_fcn = function(t, S, ...) { const_default_intensity + 0 * S }, variance_cumulation_fcn = function(T, t) { const_volatility^2 * (T - t) }, dividends = NULL, borrow_cost = 0, dividend_rate = 0, structure_constant = 2, std_devs_width = 3 )
S0 |
An initial stock price, for setting grid scale |
num_time_steps |
Minimum number of time steps in the grid |
instruments |
A list of instruments to be priced. Each
one must have a |
const_volatility |
A constant to use for volatility in case |
const_short_rate |
A constant to use for the instantaneous interest rate in case |
const_default_intensity |
A constant to use for the instantaneous default intensity in case |
override_Tmax |
A different maximum time on the grid to enforce |
discount_factor_fcn |
A function for computing present values to
time |
default_intensity_fcn |
A function for computing default intensity
occurring during this timestep, dependent on time and stock price, with
arguments |
variance_cumulation_fcn |
A function for computing total stock variance
occurring during this timestep, with arguments |
dividends |
A |
borrow_cost |
Stock borrow cost, affecting the drift rate |
dividend_rate |
Continuous dividend rate, affecting the drift rate |
structure_constant |
The maximum ratio between time intervals |
std_devs_width |
The number of standard deviations, in |
A list of present values, with the same names as instruments
Other Equity Dependent Default Intensity:
fit_to_option_market_df(),
fit_variance_cumulation(),
form_present_value_grid(),
implied_jump_process_volatility()
Other Implicit Grid Solver:
construct_implicit_grid_structure(),
form_present_value_grid(),
infer_conforming_time_grid(),
integrate_pde(),
iterate_grid_from_timestep(),
take_implicit_timestep(),
timestep_instruments()
Given derivative instruments (subclasses of
GridPricedInstrument, though typically either AmericanOption
or EuropeanOption objects), along with their prices and spreads, calibrate
variance cumulation (the
at-the-money volatility of the continuous process) and equity linked default
intensity of the form $h(s + (1-s)(S0/S_t)^p)$.
fit_to_option_market( variance_instruments, variance_instrument_prices, variance_instrument_spreads, fit_instruments, fit_instrument_prices, fit_instrument_spreads, fit_instrument_weights, S0, num_time_steps = 30, const_short_rate = 0, discount_factor_fcn = function(T, t) { exp(-const_short_rate * (T - t)) }, ..., base_default_intensity = 0.05, relative_spread_tolerance = 0.15, num_variance_time_steps = 30 )fit_to_option_market( variance_instruments, variance_instrument_prices, variance_instrument_spreads, fit_instruments, fit_instrument_prices, fit_instrument_spreads, fit_instrument_weights, S0, num_time_steps = 30, const_short_rate = 0, discount_factor_fcn = function(T, t) { exp(-const_short_rate * (T - t)) }, ..., base_default_intensity = 0.05, relative_spread_tolerance = 0.15, num_variance_time_steps = 30 )
variance_instruments |
A list of instruments in strictly increasing order of maturity, from which the volatility term structure will be inferred. Once the calibration is finished, the chosen parameters will reproduce the prices of these instruments with fairly high precision. |
variance_instrument_prices |
Central price targets for the variance instruments |
variance_instrument_spreads |
Bid-offer spreads used to normalize errors in variance instrument prices during term structure fitting |
fit_instruments |
A list of instruments in any order, from which the mispricing penalties used for judging fit quality will be computed |
fit_instrument_prices |
Central price targets for the variance instruments |
fit_instrument_spreads |
Bid-offer spreads used to normalize errors in fit instrument prices during default intensity |
fit_instrument_weights |
Weights applied to relative errors in fit instrument prices before summing to form the penalty |
S0 |
Current underlying price |
num_time_steps |
Time step count passed on to |
const_short_rate |
A constant to use for the instantaneous interest rate in case |
discount_factor_fcn |
A function for computing present values to
time |
... |
Further arguments passed to |
base_default_intensity |
Overall default intensity (in natural units) |
relative_spread_tolerance |
Tolerance to apply in
calling |
num_variance_time_steps |
Number of time steps to use in
calling |
In its present form, this function uses a brain-dead grid search.
penalty_with_intensity_link for the penalty function used
as an optimization target
This is a convenience function for calibrating variance cumulation (the
at-the-money volatility of the continuous process) and equity linked default
intensity of the form $h(s + (1-s)(S0/S_t)^p)$, using a data.frame of
option market data.
fit_to_option_market_df( S0 = ragtop::TSLAMarket$S0, discount_factor_fcn = spot_to_df_fcn(ragtop::TSLAMarket$risk_free_rates), options_df = ragtop::TSLAMarket$options, min_maturity = 1/12, min_moneyness = 0.8, max_moneyness = 1.2, base_default_intensity = 0.05 )fit_to_option_market_df( S0 = ragtop::TSLAMarket$S0, discount_factor_fcn = spot_to_df_fcn(ragtop::TSLAMarket$risk_free_rates), options_df = ragtop::TSLAMarket$options, min_maturity = 1/12, min_moneyness = 0.8, max_moneyness = 1.2, base_default_intensity = 0.05 )
S0 |
Current underlying price |
discount_factor_fcn |
A function for computing present values to
time |
options_df |
A data frame of American option details. It should
have columns |
min_maturity |
Minimum option maturity to allow in calibration |
min_moneyness |
Maximum option strike as a proportion of S0 to allow in calibration |
max_moneyness |
Maximum option strike as a proportion of S0 to allow in calibration |
base_default_intensity |
Overall default intensity (in natural units) |
fit_to_option_market the underlying fit algorithm
Other Equity Dependent Default Intensity:
find_present_value(),
fit_variance_cumulation(),
form_present_value_grid(),
implied_jump_process_volatility()
Given a set of equity options with increasing tenors, along with target prices for those options, and a set of equity-lined default SDE parameters, fit a vector of piecewise constant volatilities and an associated cumulative variance function to them.
fit_variance_cumulation( S0, eq_options, mid_prices, spreads = NULL, initial_vols_guess = 0.55 + 0 * mid_prices, use_impvol = TRUE, relative_spread_tolerance = 0.01, force_same_grid = FALSE, num_time_steps = 40, const_short_rate = 0, const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, default_intensity_fcn = function(t, S, ...) { const_default_intensity + 0 * S }, dividends = NULL, borrow_cost = 0, dividend_rate = 0, ... )fit_variance_cumulation( S0, eq_options, mid_prices, spreads = NULL, initial_vols_guess = 0.55 + 0 * mid_prices, use_impvol = TRUE, relative_spread_tolerance = 0.01, force_same_grid = FALSE, num_time_steps = 40, const_short_rate = 0, const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T, t, ...) { exp(-const_default_intensity * (T - t)) }, default_intensity_fcn = function(t, S, ...) { const_default_intensity + 0 * S }, dividends = NULL, borrow_cost = 0, dividend_rate = 0, ... )
S0 |
Current stock price |
eq_options |
A list of options to find prices for. Each must have fields |
mid_prices |
Prices to match |
spreads |
Spreads within which any match is tolerable |
initial_vols_guess |
Initial set of volatilities to try in the root finder |
use_impvol |
Judge fit quality on implied vol distance rather than price distance |
relative_spread_tolerance |
Tolerance multiplier on bid-ask spreads taken from vol normalization |
force_same_grid |
Price all options on the same grid, rather than having smaller timestep sizes for earlier maturities |
num_time_steps |
Minimum number of time steps in the grid |
const_short_rate |
A constant to use for the instantaneous interest rate in case |
const_default_intensity |
A constant to use for the instantaneous default intensity in case |
discount_factor_fcn |
A function for computing present values to
time |
survival_probability_fcn |
A function for probability of survival, with
arguments |
default_intensity_fcn |
A function for computing default intensity
occurring during this timestep, dependent on time and stock price, with
arguments |
dividends |
A |
borrow_cost |
Stock borrow cost, affecting the drift rate |
dividend_rate |
Continuous dividend rate, affecting the drift rate |
... |
Futher arguments to |
By default, the fitting happens in implied Black-Scholes volatility
space for better normalization. That is to say, the fitting does pricing
using the full SDE and PDE solver via find_present_value, but
judges fit quality on the basis of running resulting prices through a
nonlinear transformation that just
happens to come from the straight Black-Scholes model.
A list with two elements, volatilities and cumulation_function. The cumulation_function will
be a 2-parameter function giving cumulated variances, as created by codevariance_cumulation_from_vols
Other Implied Volatilities:
american_implied_volatility(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
implied_jump_process_volatility(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other Equity Dependent Default Intensity:
find_present_value(),
fit_to_option_market_df(),
form_present_value_grid(),
implied_jump_process_volatility()
Use a finite difference scheme to form estimates of present values for a variety
of stock prices on a grid of initial underlying prices, determined by constructing
a logarithmic equivalent conforming to the grid parameters structure_constant
and structure_constant
form_present_value_grid( S0, num_time_steps, instruments, const_volatility = 0.5, const_short_rate = 0, const_default_intensity = 0, override_Tmax = NA, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, default_intensity_fcn = function(t, S, ...) { const_default_intensity + 0 * S }, variance_cumulation_fcn = function(T, t) { const_volatility^2 * (T - t) }, dividends = NULL, borrow_cost = 0, dividend_rate = 0, structure_constant = 2, std_devs_width = 3, grid_center = NA )form_present_value_grid( S0, num_time_steps, instruments, const_volatility = 0.5, const_short_rate = 0, const_default_intensity = 0, override_Tmax = NA, discount_factor_fcn = function(T, t, ...) { exp(-const_short_rate * (T - t)) }, default_intensity_fcn = function(t, S, ...) { const_default_intensity + 0 * S }, variance_cumulation_fcn = function(T, t) { const_volatility^2 * (T - t) }, dividends = NULL, borrow_cost = 0, dividend_rate = 0, structure_constant = 2, std_devs_width = 3, grid_center = NA )
S0 |
An initial stock price, for setting grid scale |
num_time_steps |
Minimum number of time steps in the grid |
instruments |
A list of instruments to be priced. Each
one must have a |
const_volatility |
A constant to use for volatility in case |
const_short_rate |
A constant to use for the instantaneous interest rate in case |
const_default_intensity |
A constant to use for the instantaneous default intensity in case |
override_Tmax |
A different maximum time on the grid to enforce |
discount_factor_fcn |
A function for computing present values to
time |
default_intensity_fcn |
A function for computing default intensity
occurring during this timestep, dependent on time and stock price, with
arguments |
variance_cumulation_fcn |
A function for computing total stock variance
occurring during this timestep, with arguments |
dividends |
A |
borrow_cost |
Stock borrow cost, affecting the drift rate |
dividend_rate |
Continuous dividend rate, affecting the drift rate |
structure_constant |
The maximum ratio between time intervals |
std_devs_width |
The number of standard deviations, in |
grid_center |
A reasonable central value for the grid, defaults to S0 or an instrument strike |
If any instrument in the instruments has a strike, then the grid will be
normalized to the last such instrument's strike.
Other Equity Dependent Default Intensity:
find_present_value(),
fit_to_option_market_df(),
fit_variance_cumulation(),
implied_jump_process_volatility()
Other Implicit Grid Solver:
construct_implicit_grid_structure(),
find_present_value(),
infer_conforming_time_grid(),
integrate_pde(),
iterate_grid_from_timestep(),
take_implicit_timestep(),
timestep_instruments()
Our basic instrument defines a tenor/maturity, a method to provide values in case of default, and a method to correct instrument prices in light of exercise decisions.
maturityThe tenor, expiration date or terminal date by which the value of this security will be certain.
last_computed_gridThe most recently computed set of values from a grid pricing scheme. Used internally for pricing chains of derivatives.
nameA mnemonic name for the instrument, not used by ragtop
optionality_fcn(v, ...)Return a version of v at time t corrected for any optionality conditions.
recovery_fcn(v, S, t, ...)Return recovery value, given non-default values v at time t. Subclasses may be more elaborate, this method simply returns 0.0.
terminal_values(v, ...)Return a terminal value. defaults to simply calling optionality_fcn.
Use the grid solver to generate instrument prices via find_present_value and run them
through a bisective root search method until a constant volatility matching the provided
instrument price has been found.
implied_jump_process_volatility( instrument_price, instrument, ..., starting_volatility_estimate = 0.85, relative_tolerance = 0.005, max.iter = 100, max_vola = 4 )implied_jump_process_volatility( instrument_price, instrument, ..., starting_volatility_estimate = 0.85, relative_tolerance = 0.005, max.iter = 100, max_vola = 4 )
instrument_price |
Target price for root finder |
instrument |
Instrument to search for the target price on, passed as
the sole instrument to |
... |
Additional arguments to be passed on to |
starting_volatility_estimate |
Bisection method original guess |
relative_tolerance |
Relative tolerance in instrument price defining the root-finder halting condition |
max.iter |
Maximum number of root-finder iterations allowed |
max_vola |
Maximum volatility to try |
Unlike american_implied_volatility, this routine allows for any legal
term structures and equity-linked default intensities. For that reason, it eschews
the control variate tricks that make american_implied_volatility so much faster.
Note that equity-linked default intensities can result in instrument prices that are not monotonic in volatility. This bisective root finder will find a solution but not necessarily any particular one.
A list of present values, with the same names as instruments
find_present_value for the underlying
pricing algorithm, implied_volatility_with_term_struct for European options
without equity dependence of default intensity, american_implied_volatility for the same on American options
Other Implied Volatilities:
american_implied_volatility(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
fit_variance_cumulation(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other Equity Dependent Default Intensity:
find_present_value(),
fit_to_option_market_df(),
fit_variance_cumulation(),
form_present_value_grid()
implied_jump_process_volatility( 25, AmericanOption(maturity=1.1, strike=100, callput=-1), S0=100, num_time_steps=50, relative_tolerance=1.e-3)implied_jump_process_volatility( 25, AmericanOption(maturity=1.1, strike=100, callput=-1), S0=100, num_time_steps=50, relative_tolerance=1.e-3)
Find default-free volatilities based on known interest rates and hazard rates, using a given option price.
implied_volatilities( option_price, callput, S0, K, r, time, const_default_intensity = 0, divrate = 0, borrow_cost = 0, dividends = NULL, relative_tolerance = 1e-06, max.iter = 100, max_vola = 4 )implied_volatilities( option_price, callput, S0, K, r, time, const_default_intensity = 0, divrate = 0, borrow_cost = 0, dividends = NULL, relative_tolerance = 1e-06, max.iter = 100, max_vola = 4 )
option_price |
Present option values (may be a vector) |
callput |
1 for calls, -1 for puts (may be a vector) |
S0 |
initial underlying price (may be a vector) |
K |
strike (may be a vector) |
r |
risk-free interest rate (may be a vector) |
time |
Time from |
const_default_intensity |
hazard rate of underlying default (may be a vector) |
divrate |
A continuous rate for dividends and other cashflows such as foreign interest rates (may be a vector) |
borrow_cost |
A continuous rate for stock borrow costs (may be a vector) |
dividends |
A |
relative_tolerance |
Relative tolerance in option price to achieve before halting the search |
max.iter |
Number of iterations to try before abandoning the search |
max_vola |
Maximum volatility to try in the search |
Scalar volatilities
Other Implied Volatilities:
american_implied_volatility(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
fit_variance_cumulation(),
implied_jump_process_volatility(),
implied_volatilities_with_rates_struct(),
implied_volatility_with_term_struct(),
implied_volatility()
Other European Options:
black_scholes_on_term_structures(),
blackscholes(),
implied_volatilities_with_rates_struct(),
implied_volatility_with_term_struct(),
implied_volatility()
Other Equity Independent Default Intensity:
american_implied_volatility(),
american(),
black_scholes_on_term_structures(),
blackscholes(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
implied_volatilities_with_rates_struct(),
implied_volatility_with_term_struct(),
implied_volatility()
Use the provided discount factor function to infer constant short rates applicable to each expiration time, then use the Black-Scholes formula to generate European option values and run them through Newton's method until a constant volatility matching each provided option price has been found.
implied_volatilities_with_rates_struct( option_price, callput, S0, K, discount_factor_fcn, time, const_default_intensity = 0, divrate = 0, borrow_cost = 0, dividends = NULL, relative_tolerance = 1e-06, max.iter = 100, max_vola = 4 )implied_volatilities_with_rates_struct( option_price, callput, S0, K, discount_factor_fcn, time, const_default_intensity = 0, divrate = 0, borrow_cost = 0, dividends = NULL, relative_tolerance = 1e-06, max.iter = 100, max_vola = 4 )
option_price |
Present option values (may be a vector) |
callput |
1 for calls, -1 for puts (may be a vector) |
S0 |
initial underlying prices (may be a vector) |
K |
strikes (may be a vector) |
discount_factor_fcn |
A function for computing present values to
time |
time |
Time from |
const_default_intensity |
hazard rates of underlying default (may be a vector) |
divrate |
A continuous rate for dividends and other cashflows such as foreign interest rates (may be a vector) |
borrow_cost |
A continuous rate for stock borrow costs (may be a vector) |
dividends |
A |
relative_tolerance |
Relative tolerance in option price to achieve before halting the search |
max.iter |
Number of iterations to try before abandoning the search |
max_vola |
Maximum volatility to try in the search |
Differs from implied_volatility_with_term_struct by first computing constant interest rates
for each option, and then calling implied_volatilities
Scalar volatilities
implied_volatility for simpler cases with constant
parameters, implied_volatilities for the underlying
algorithm with constant rates, implied_volatility_with_term_struct when
volatilities or survival probabilities also have a nontrivial term structure
Other Implied Volatilities:
american_implied_volatility(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
fit_variance_cumulation(),
implied_jump_process_volatility(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other European Options:
black_scholes_on_term_structures(),
blackscholes(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
Other Equity Independent Default Intensity:
american_implied_volatility(),
american(),
black_scholes_on_term_structures(),
blackscholes(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
implied_volatilities(),
implied_volatility_with_term_struct(),
implied_volatility()
d_fcn = function(T,t) {exp(-0.03*(T-t))} implied_volatilities_with_rates_struct(c(23,24,25), c(-1,1,1), 100, 100, discount_factor_fcn=d_fcn, time=c(4,4,5))d_fcn = function(T,t) {exp(-0.03*(T-t))} implied_volatilities_with_rates_struct(c(23,24,25), c(-1,1,1), 100, 100, discount_factor_fcn=d_fcn, time=c(4,4,5))
Find default-free volatility (not necessarily just Black-Scholes) based on known interest rates and hazard rates, using a given option price.
implied_volatility( option_price, callput, S0, K, r, time, const_default_intensity = 0, divrate = 0, borrow_cost = 0, dividends = NULL, relative_tolerance = 1e-06, max.iter = 100, max_vola = 4 )implied_volatility( option_price, callput, S0, K, r, time, const_default_intensity = 0, divrate = 0, borrow_cost = 0, dividends = NULL, relative_tolerance = 1e-06, max.iter = 100, max_vola = 4 )
option_price |
Present option value |
callput |
1 for calls, -1 for puts |
S0 |
initial underlying price |
K |
strike |
r |
risk-free interest rate |
time |
Time from |
const_default_intensity |
hazard rate of underlying default |
divrate |
A continuous rate for dividends and other cashflows such as foreign interest rates |
borrow_cost |
A continuous rate for stock borrow costs |
dividends |
A |
relative_tolerance |
Relative tolerance in option price to achieve before halting the search |
max.iter |
Number of iterations to try before abandoning the search |
max_vola |
Maximum volatility to try in the search |
To get a straight Black-Scholes implied volatility, simply call this function with
const_default_intensity set to zero (the default).
A scalar volatility
Other Implied Volatilities:
american_implied_volatility(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
fit_variance_cumulation(),
implied_jump_process_volatility(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct()
Other Equity Independent Default Intensity:
american_implied_volatility(),
american(),
black_scholes_on_term_structures(),
blackscholes(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct()
Other European Options:
black_scholes_on_term_structures(),
blackscholes(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility_with_term_struct()
implied_volatility(2.5, 1, 100, 105, 0.01, 0.75) implied_volatility(option_price = 17, callput = CALL, S0 = 250, K=245, r = 0.005, time = 2, const_default_intensity = 0.03)implied_volatility(2.5, 1, 100, 105, 0.01, 0.75) implied_volatility(option_price = 17, callput = CALL, S0 = 250, K=245, r = 0.005, time = 2, const_default_intensity = 0.03)
Use the Black-Scholes formula to generate European option values and run them through Newton's method until a constant volatility matching the provided option price has been found.
implied_volatility_with_term_struct( option_price, callput, S0, K, time, ..., starting_volatility_estimate = 0.5, relative_tolerance = 1e-06, max.iter = 100, max_vola = 4 )implied_volatility_with_term_struct( option_price, callput, S0, K, time, ..., starting_volatility_estimate = 0.5, relative_tolerance = 1e-06, max.iter = 100, max_vola = 4 )
option_price |
Option price to match |
callput |
1 for calls, -1 for puts |
S0 |
initial underlying price |
K |
strike |
time |
Time to expiration |
... |
Further arguments to be passed on to |
starting_volatility_estimate |
The Newton method's original guess |
relative_tolerance |
Relative tolerance in instrument price defining the root-finder halting condition |
max.iter |
Maximum number of root-finder iterations allowed |
max_vola |
Maximum volatility to try |
Differs from implied_volatility by calling black_scholes_on_term_structures for
pricing, thereby allowing term structures of rates, and a nontrivial survival_probability_fcn
Estimated volatility
implied_volatility for simpler cases with constant
parameters, black_scholes_on_term_structures for the underlying
pricing algorithm, implied_volatilities_with_rates_struct when
neither volatilities nor survival probabilities have a nontrivial term structure
Other Implied Volatilities:
american_implied_volatility(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
fit_variance_cumulation(),
implied_jump_process_volatility(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility()
Other Equity Independent Default Intensity:
american_implied_volatility(),
american(),
black_scholes_on_term_structures(),
blackscholes(),
equivalent_bs_vola_to_jump(),
equivalent_jump_vola_to_bs(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility()
Other European Options:
black_scholes_on_term_structures(),
blackscholes(),
implied_volatilities_with_rates_struct(),
implied_volatilities(),
implied_volatility()
## Dividends 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)) surv_prob_fcn = function(T, t, ...) { exp(-0.07 * (T - t)) } disc_factor_fcn = function(T, t, ...) { exp(-0.03 * (T - t)) } implied_volatility_with_term_struct( option_price = 12, S0 = 150, callput=PUT, K = 147.50, time=1.5, discount_factor_fcn=disc_factor_fcn, survival_probability_fcn=surv_prob_fcn, dividends=divs)## Dividends 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)) surv_prob_fcn = function(T, t, ...) { exp(-0.07 * (T - t)) } disc_factor_fcn = function(T, t, ...) { exp(-0.03 * (T - t)) } implied_volatility_with_term_struct( option_price = 12, S0 = 150, callput=PUT, K = 147.50, time=1.5, discount_factor_fcn=disc_factor_fcn, survival_probability_fcn=surv_prob_fcn, dividends=divs)
At its base, this function chooses a time grid with 1+min_num_time_steps
elements from 0 to Tmax. Any coupon, call, or put times occurring in
one of the supplied instruments are also inserted.
infer_conforming_time_grid(min_num_time_steps, Tmax, instruments = NULL)infer_conforming_time_grid(min_num_time_steps, Tmax, instruments = NULL)
min_num_time_steps |
The minimum number of timesteps the output vector should have |
Tmax |
The maximum time on the grid |
instruments |
A set of instruments whose maturity and terms
and conditions can introduce extra timesteps. Each will be queried for the output of
a |
A vector of times at which the grid should have nodes
Other Implicit Grid Solver:
construct_implicit_grid_structure(),
find_present_value(),
form_present_value_grid(),
integrate_pde(),
iterate_grid_from_timestep(),
take_implicit_timestep(),
timestep_instruments()
Use an implicit integration scheme to numerically integrate
the pricing differential equation for each of the given instruments,
backwardating from time Tmax to time 0.
integrate_pde( z, min_num_time_steps, S0, Tmax, instruments, stock_level_fcn, discount_factor_fcn, default_intensity_fcn, variance_cumulation_fcn, dividends = NULL )integrate_pde( z, min_num_time_steps, S0, Tmax, instruments, stock_level_fcn, discount_factor_fcn, default_intensity_fcn, variance_cumulation_fcn, dividends = NULL )
z |
Space grid value morphable to stock prices using |
min_num_time_steps |
The minimum number of timesteps used. Calls, puts and coupons may result in extra timesteps taken. |
S0 |
Time zero price of the base equity |
Tmax |
The maximum time on the grid, from which all backwardation steps will take place. |
instruments |
A list of instruments to be priced. Each
one must have a |
stock_level_fcn |
A function for changing space grid value to stock
prices, with arguments |
discount_factor_fcn |
A function for computing present values to
time |
default_intensity_fcn |
A function for computing default intensity
occurring during this timestep, dependent on time and stock price, with
arguments |
variance_cumulation_fcn |
A function for computing total stock variance
occurring during this timestep, with arguments |
dividends |
A |
A grid of present values of derivative prices, adapted to z at
each timestep. Time zero value will appear in the first index.
Other Implicit Grid Solver:
construct_implicit_grid_structure(),
find_present_value(),
form_present_value_grid(),
infer_conforming_time_grid(),
iterate_grid_from_timestep(),
take_implicit_timestep(),
timestep_instruments()
Return TRUE if the argument is empty, NULL or NA
is.blank(x, false.triggers = FALSE)is.blank(x, false.triggers = FALSE)
x |
Argument to test |
false.triggers |
Whether to allow nonempty vectors of all FALSE to trigger this condition |
Timestep an implicit integration scheme to numerically integrate
the pricing differential equation for each of the given instruments,
backwardating from time Tmax to time 0.
iterate_grid_from_timestep( starting_time_step, time_pts, z, S0, instruments, stock_level_fcn, discount_factor_fcn, default_intensity_fcn, variance_cumulation_fcn, dividends = NULL, grid = NULL, original_grid_values = as.matrix(grid[1 + starting_time_step, , ]) )iterate_grid_from_timestep( starting_time_step, time_pts, z, S0, instruments, stock_level_fcn, discount_factor_fcn, default_intensity_fcn, variance_cumulation_fcn, dividends = NULL, grid = NULL, original_grid_values = as.matrix(grid[1 + starting_time_step, , ]) )
starting_time_step |
The index into time_pts of the first timestep to be emplyed. This must be no larger than the length of time_pts, minus one |
time_pts |
Time nodes to be treated on the grid |
z |
Space grid value morphable to stock prices using |
S0 |
Time zero price of the base equity |
instruments |
A list of instruments to be priced. Each
one must have a |
stock_level_fcn |
A function for changing space grid value to stock
prices, with arguments |
discount_factor_fcn |
A function for computing present values to
time |
default_intensity_fcn |
A function for computing default intensity
occurring during this timestep, dependent on time and stock price, with
arguments |
variance_cumulation_fcn |
A function for computing total stock variance
occurring during this timestep, with arguments |
dividends |
A |
grid |
An optional grid into which results at each timestep will
be written. Its size should be at least
|
original_grid_values |
Grid values to timestep from |
Either a populated grid of present values of derivative prices, or a matrix
of values at the first time point, adapted to z at
each timestep. Time zero value will appear in the first index of any grid.
Other Implicit Grid Solver:
construct_implicit_grid_structure(),
find_present_value(),
form_present_value_grid(),
infer_conforming_time_grid(),
integrate_pde(),
take_implicit_timestep(),
timestep_instruments()
Given a set SDE parameters, form a volatility term structure that fairly precisely matches
the supplied prices of the variance_instruments. Then use that term structure and
the default intensity to price all the fit_instruments, and compare them to the
fit_instrument_prices.
penalty_with_intensity_link( p, s, h, variance_instruments, variance_instrument_prices, variance_instrument_spreads, fit_instruments, fit_instrument_prices, fit_instrument_spreads, fit_instrument_weights, S0, num_time_steps = 30, const_short_rate = 0, discount_factor_fcn = function(T, t) { exp(-const_short_rate * (T - t)) }, ..., relative_spread_tolerance = 0.15, num_variance_time_steps = 30 )penalty_with_intensity_link( p, s, h, variance_instruments, variance_instrument_prices, variance_instrument_spreads, fit_instruments, fit_instrument_prices, fit_instrument_spreads, fit_instrument_weights, S0, num_time_steps = 30, const_short_rate = 0, discount_factor_fcn = function(T, t) { exp(-const_short_rate * (T - t)) }, ..., relative_spread_tolerance = 0.15, num_variance_time_steps = 30 )
p |
Power of default intensity |
s |
Proportion of constant default intensity |
h |
Base default intensity |
variance_instruments |
A list of instruments in strictly increasing order of maturity, from which the volatility term structure will be inferred. Once the calibration is finished, the chosen parameters will reproduce the prices of these instruments with fairly high precision. |
variance_instrument_prices |
Central price targets for the variance instruments |
variance_instrument_spreads |
Bid-offer spreads used to normalize errors in variance instrument prices during term structure fitting |
fit_instruments |
A list of instruments in any order, from which the mispricing penalties used for judging fit quality will be computed |
fit_instrument_prices |
Central price targets for the variance instruments |
fit_instrument_spreads |
Bid-offer spreads used to normalize errors in fit instrument prices during default intensity |
fit_instrument_weights |
Weights applied to relative errors in fit instrument prices before summing to form the penalty |
S0 |
Current underlying price |
num_time_steps |
Time step count passed on to |
const_short_rate |
A constant to use for the instantaneous interest rate in case |
discount_factor_fcn |
A function for computing present values to
time |
... |
Further arguments passed to |
relative_spread_tolerance |
Tolerance to apply in
calling |
num_variance_time_steps |
Number of time steps to use in
calling |
Forms implied Black-Scholes volatilities from all supplied mid prices, and their implied bid and offer prices, as well as from the prices computed by the grid solver. Each instrument is then assigned an error term component in proportion to its weight and the pricing error (in implied vol terms) divided by the spread (also in implied vol terms).
price_with_intensity_link for the pricing function
Given derivative instruments (subclasses of
GridPricedInstrument, though typically either AmericanOption
or EuropeanOption objects), along with their prices and spreads, calibrate
variance cumulation (the
at-the-money volatility of the continuous process) and then price the instruments via equity linked default
intensity of the form $h(s + (1-s)(S0/S_t)^p)$.
price_with_intensity_link( p, s, h, variance_instruments, variance_instrument_prices, variance_instrument_spreads, fit_instruments, S0, num_time_steps = 30, ..., relative_spread_tolerance = 0.15, num_variance_time_steps = 30 )price_with_intensity_link( p, s, h, variance_instruments, variance_instrument_prices, variance_instrument_spreads, fit_instruments, S0, num_time_steps = 30, ..., relative_spread_tolerance = 0.15, num_variance_time_steps = 30 )
p |
Power of default intensity |
s |
Proportion of constant default intensity |
h |
Base default intensity |
variance_instruments |
A list of instruments in strictly increasing order of maturity, from which the volatility term structure will be inferred. Once the calibration is finished, the chosen parameters will reproduce the prices of these instruments with fairly high precision. |
variance_instrument_prices |
Central price targets for the variance instruments |
variance_instrument_spreads |
Bid-offer spreads used to normalize errors in variance instrument prices during term structure fitting |
fit_instruments |
A list of instruments in any order, from which the mispricing penalties used for judging fit quality will be computed |
S0 |
Current underlying price |
num_time_steps |
Time step count passed on to |
... |
Further arguments passed to both
|
relative_spread_tolerance |
Tolerance to apply in
calling |
num_variance_time_steps |
Number of time steps to use in
calling |
Constant PUT for defining option contracts
PUTPUT
An object of class numeric of length 1.
This is a caching wrapper for Quandl_df_fcn_UST_raw
Quandl_df_fcn_UST(..., envir = parent.frame())Quandl_df_fcn_UST(..., envir = parent.frame())
... |
Arguments passed to |
envir |
Environment passed to |
A function taking two time arguments, which returns the discount factor from the second to the first
Get a US Treasury curve discount factor function
Quandl_df_fcn_UST_raw(on_date)Quandl_df_fcn_UST_raw(on_date)
on_date |
Date for which to query Quandl for the curve |
A function taking two time arguments, which returns the discount factor from the second to the first
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.
We apply the stochastic model
where and play their usual roles, is a
deterministic function of stock price and time, and
is a Poisson jump process adapted to the default intensity
or hazard rate . 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 .
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.
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
## 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)## 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)
Shift a set of grid values for dividends paid, using spline interpolation
shift_for_dividends(grid_values_before_shift, stock_prices, div_sum)shift_for_dividends(grid_values_before_shift, stock_prices, div_sum)
grid_values_before_shift |
Values on grid before accounting for expected dividends |
stock_prices |
Stock prices for which to shift the grid |
div_sum |
Sum of dividend values at each grid point |
An object like grid_values_before_shift with entries shifted
according to the dividend sums
Other Dividends:
adjust_for_dividends(),
time_adj_dividends()
Use a piecewise constant approximation to the given spot curve to generate a function capable of returning corresponding discount factors
spot_to_df_fcn(yield_curve)spot_to_df_fcn(yield_curve)
yield_curve |
A data.frame with numeric columns |
A function taking two time arguments, which returns the discount factor from the second to the first
disct_fcn = ragtop::spot_to_df_fcn( data.frame(time=c(1, 5, 10, 15), rate=c(0.01, 0.02, 0.03, 0.05))) print(disct_fcn(1, 0.5))disct_fcn = ragtop::spot_to_df_fcn( data.frame(time=c(1, 5, 10, 15), rate=c(0.01, 0.02, 0.03, 0.05))) print(disct_fcn(1, 0.5))
Take one timestep of an implicit solver for a given instrument
take_implicit_timestep( t, S, full_discount_factor, local_discount_factor, discount_factor_fcn, prev_grid_values, survival_probabilities, tridiag_matrix_entries, instrument = NULL, dividends = NULL, instr_name = "this instrument" )take_implicit_timestep( t, S, full_discount_factor, local_discount_factor, discount_factor_fcn, prev_grid_values, survival_probabilities, tridiag_matrix_entries, instrument = NULL, dividends = NULL, instr_name = "this instrument" )
t |
Time after this timestep has been taken |
S |
Underlying equity values for the grid |
full_discount_factor |
A discount factor for the transform from grid values to actual derivative prices |
local_discount_factor |
A discount factor to apply to recovery values |
discount_factor_fcn |
A function for computing present values to
time |
prev_grid_values |
A vector of space grid values from the previously calculated timestep |
survival_probabilities |
Vector of probabilities of survival for each space grid node |
tridiag_matrix_entries |
Diagonal, superdiagonal and subdiagonal of tridiagonal matrix from the numerical integrator |
instrument |
If not NULL/NA, must have a |
dividends |
A |
instr_name |
Name of instrument to use in log messages |
Grid values for the instrument after taking the implicit timestep
Other Implicit Grid Solver:
construct_implicit_grid_structure(),
find_present_value(),
form_present_value_grid(),
infer_conforming_time_grid(),
integrate_pde(),
iterate_grid_from_timestep(),
timestep_instruments()
For each of the N elements of S/h find the sum of the
given M dividends, discounted to t_final by r and h
time_adj_dividends(relevant_divs, t_final, r, h, S, S0)time_adj_dividends(relevant_divs, t_final, r, h, S, S0)
relevant_divs |
A |
t_final |
Time beyond which to ignore dividends |
r |
risk-free interest rate |
h |
Default intensities |
S |
Stock prices |
S0 |
initial underlying price |
Sum of dividends, at each grid node
Other Dividends:
adjust_for_dividends(),
shift_for_dividends()
Constant to define when times are considered so close to each other that they should be treated as simultaneous
TIME_RESOLUTION_FACTORTIME_RESOLUTION_FACTOR
An object of class numeric of length 1.
Constant to define when times are considered so close to each other that they should be treated as simultaneous, in terms of significant digits
TIME_RESOLUTION_SIGNIF_DIGITSTIME_RESOLUTION_SIGNIF_DIGITS
An object of class numeric of length 1.
Backwardate grid values for all the given instruments from a set of grid
values matched to time t+dt to form a new set of grid value as
of time t.
timestep_instruments( z, prev_grid_values, t, dt, S0, instruments, stock_level_fcn, discount_factor_fcn, default_intensity_fcn, variance_cumulation_fcn, dividends = NULL )timestep_instruments( z, prev_grid_values, t, dt, S0, instruments, stock_level_fcn, discount_factor_fcn, default_intensity_fcn, variance_cumulation_fcn, dividends = NULL )
z |
Space grid value morphable to stock prices using |
prev_grid_values |
A matrix with one column for each
instrument and one row for each of the |
t |
Time after this timestep has been taken |
dt |
Interval to the end of this timestep |
S0 |
Time zero price of the base equity |
instruments |
Instruments corresponding to layers of the value grid in |
stock_level_fcn |
A function for changing space grid value to stock
prices, with arguments |
discount_factor_fcn |
A function for computing present values to
time |
default_intensity_fcn |
A function for computing default intensity
occurring during this timestep, dependent on time and stock price, with
arguments |
variance_cumulation_fcn |
A function for computing total stock variance
occurring during this timestep, with arguments |
dividends |
A |
Grid values after applying an implicit timestep
Other Implicit Grid Solver:
construct_implicit_grid_structure(),
find_present_value(),
form_present_value_grid(),
infer_conforming_time_grid(),
integrate_pde(),
iterate_grid_from_timestep(),
take_implicit_timestep()
A dataset containing option contract details and a snapshot of market prices for Tesla Motors (TSLA) equity options, interest rates and an equity price.
TSLAMarketTSLAMarket
An object of class list of length 3.
The TSLAMarket list contains three elements:
S0: The stock price as of snapshot time
risk_free_rates: The spot risk-free rate curve as of snapshot time
options: A data frame with details of the options market
Present value as of time t for coupons paid since the model_t
value_from_prior_coupons(t, coupons_df, discount_factor_fcn, model_t = 0)value_from_prior_coupons(t, coupons_df, discount_factor_fcn, model_t = 0)
t |
The time toward which all coupons should be present valued |
coupons_df |
A data.frame of details for each coupon. It should have the
columns |
discount_factor_fcn |
A function specifying how the contract says future coupons should be discounted for this instrument in case the acceleration clause is triggered |
model_t |
The payment time beyond which coupons will be included in this computation |
Other Bond Coupons:
accelerated_coupon_value(),
coupon_value_at_exercise()
Given a volatility term structure, create a corresponding variance cumulation function. The function assumes piecewise constant forward volatility, with the final such forward volatility extending to infinity.
variance_cumulation_from_vols(vols_df)variance_cumulation_from_vols(vols_df)
vols_df |
A data.frame with numeric columns |
A function taking two time arguments, which returns the cumulated variance from the second to the first
vc = variance_cumulation_from_vols( data.frame(time=c(0.1,2,3), volatility=c(0.2,0.5,1.2))) vc(1.5, 0)vc = variance_cumulation_from_vols( data.frame(time=c(0.1,2,3), volatility=c(0.2,0.5,1.2))) vc(1.5, 0)
notional amount at the maturity
A simple contract paying the notional amount at the maturity
notionalThe amount that will be paid at maturity, conditional on survival
recovery_rateThe proportion of notional that would be expected to be paid to bond holders after bankruptcy court proceedings
discount_factor_fcnA function specifying how cashflows should generally be discounted for this instrument
optionality_fcn(v, ...)Return a version of v at time t corrected for any optionality conditions.
recovery_fcn(v, S, t, ...)Return recovery value, given non-default values v at time t. Subclasses may be more elaborate, this method simply returns 0.0.