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Pricing

stochPathSim.pricing

Pricing: payoffs, Monte Carlo, least-squares Monte Carlo and binomial trees.

MCResult dataclass

Monte Carlo price estimate.

Attributes:

Name Type Description
price float

Estimated present value.
stderr float

Standard error of the estimate.
n_paths int

Number of simulated paths.
samples (ndarray, optional)

The independent discounted samples the estimate averages (pair averages under antithetic sampling, control-variate adjusted if one was used).

conf_int 

conf_int(level=0.95)

Two-sided normal confidence interval for the price.

AsianArithmetic

Bases: _Asian

Fixed-strike Asian option on the arithmetic average of the fixings.

Parameters:

Name Type Description Default
K float

Strike.

 required
option_type ('call', 'put')
"call"
fixing_times array_like

Averaging times (must be grid times). By default every grid time from start_index on (start_index=1 excludes t = 0).

 None

AsianGeometric

Bases: _Asian

Fixed-strike Asian option on the geometric average of the fixings (see :class:AsianArithmetic).

Barrier

Bases: Payoff

Single-barrier knock-in or knock-out option, monitored at every grid time.

Parameters:

Name Type Description Default
K float

Strike.

 required
barrier float

Barrier level.

 required
kind ('up-and-out', 'up-and-in', 'down-and-out', 'down-and-in')
"up-and-out"
option_type ('call', 'put')
"call"
rebate float

Paid at maturity if the option is knocked out, or never knocked in.

 0.0
sigma float

Volatility for the Broadie-Glasserman-Kou continuity correction. When given, the barrier is moved towards the spot by the factor exp(0.5826 sigma sqrt(dt)), so that discrete monitoring on the grid approximates a continuously monitored barrier at barrier.

 None

effective_barrier 

effective_barrier(times)

Barrier used on the grid, after the optional continuity correction.

knocked 

knocked(paths, times)

Boolean per path: whether the barrier was touched on the grid.

Basket

Bases: Payoff

Option on a weighted sum of terminal values of several state variables.

Parameters:

Name Type Description Default
weights array_like

One weight per state.

 required
K float

Strike.

 required
option_type ('call', 'put')
"call"
states sequence of str

State names, in the order of weights. Defaults to every state of the result, in order. Requires a :class:SimulationResult.

 None

Digital

Bases: Payoff

Cash-or-nothing digital paying cash if S_T > K (call) or S_T < K (put).

European

Bases: Payoff

European call or put on the terminal value: max(S_T - K, 0) or max(K - S_T, 0).

EuropeanCall

Bases: European

European call max(S_T - K, 0).

EuropeanPut

Bases: European

European put max(K - S_T, 0).

LookbackFixed

Bases: Payoff

Fixed-strike lookback: max(max S - K, 0) (call) or max(K - min S, 0) (put), over the grid.

LookbackFloating

Bases: Payoff

Floating-strike lookback: S_T - min S (call) or max S - S_T (put), over the grid.

Payoff

Base class for path-dependent payoffs.

Parameters:

Name Type Description Default
state str

Name of the state variable to read; defaults to the primary state.

 None

__call__ 

__call__(result, times=None)

Undiscounted payoff per path.

Parameters:

Name Type Description Default
result SimulationResult or ndarray

Simulated paths. A raw array must have shape (n_paths, n_times).

 required
times array_like

Grid times for a raw array (needed by time-aware payoffs).

 None

evaluate 

evaluate(paths, times)

Payoff per path from a (n_paths, n_times) array.

Spread

Bases: Basket

Spread option on S_a(T) - S_b(T): max(S_a - S_b - K, 0) for a call.

longstaff_schwartz 

longstaff_schwartz(process, payoff_fn, T, *, r, n_steps=50, n_paths=100000, basis='laguerre', degree=3, seed=None, antithetic=False, state=None, grid=None)

Price an American (Bermudan) option with the Longstaff-Schwartz algorithm.

Exercise is allowed at every grid time after t = 0; the continuation value is regressed on in-the-money paths only.

Parameters:

Name Type Description Default
process StochasticProcess

Risk-neutral process for the underlying.

 required
payoff_fn callable

Intrinsic value as a function of the underlying, e.g. lambda s: np.maximum(K - s, 0).

 required
T float

Maturity in years.

 required
r float

Continuously compounded discount rate.

 required
n_steps int

Number of exercise dates (equal steps).

 50
n_paths int

Number of paths.

 100000
basis ('laguerre', 'poly')

Regression basis, evaluated on the spot scaled by its initial value.

 "laguerre"
degree int

Degree of the regression basis.

 3
seed

As in :meth:StochasticProcess.simulate.

 None
antithetic

As in :meth:StochasticProcess.simulate.

 None
grid

As in :meth:StochasticProcess.simulate.

 None
state str

State variable to exercise on (default: primary state).

 None

discount_factors 

discount_factors(result, discount, T)

Discount factor(s) to maturity T.

discount may be a continuously compounded rate, a :class:ZeroCurve, or None; with None the pathwise "discount" state of the result is used when the process provides one, otherwise payoffs are left undiscounted.

price_mc 

price_mc(process, payoff, T, *, discount=None, n_steps=252, n_paths=100000, seed=None, antithetic=False, control_variate=None, grid=None, keep_samples=False)

Price a payoff by Monte Carlo simulation of a process.

Parameters:

Name Type Description Default
process StochasticProcess

Process to simulate (e.g. a risk-neutral :class:~stochPathSim.processes.GBM).

 required
payoff callable

Maps a :class:~stochPathSim.core.SimulationResult to undiscounted payoffs.

 required
T float

Maturity in years.

 required
discount (float, ZeroCurve or None)

Discounting: a continuously compounded rate, a curve, or None to use the process's pathwise "discount" state if present (undiscounted otherwise).

 None
n_steps int

Time steps and paths.

 252
n_paths int

Time steps and paths.

 252
seed int

Seed for reproducible results.

 None
antithetic bool

Antithetic variates; the standard error is computed on pair averages.

 False
control_variate tuple(payoff, exact_value)

A second payoff with known present value. Its discounted samples are used as a linear control with the variance-minimising coefficient.

 None
grid TimeGrid

Custom simulation grid (overrides T/n_steps for the simulation).

 None
keep_samples bool

Store the samples in the result.

 False

crr_price 

crr_price(S, K, T, r, sigma, option_type='call', *, american=False, q=0.0, n_steps=500)

Cox-Ross-Rubinstein binomial price of a vanilla option.

Parameters:

Name Type Description Default
S float

Spot, strike, maturity (years), continuously compounded rate and volatility.

 required
K float

Spot, strike, maturity (years), continuously compounded rate and volatility.

 required
T float

Spot, strike, maturity (years), continuously compounded rate and volatility.

 required
r float

Spot, strike, maturity (years), continuously compounded rate and volatility.

 required
sigma float

Spot, strike, maturity (years), continuously compounded rate and volatility.

 required
option_type ('call', 'put')
"call"
american bool

Allow early exercise at every node.

 False
q float

Continuous dividend yield.

 0.0
n_steps int

Number of tree steps.

 500

american

American and Bermudan options by least-squares Monte Carlo.

longstaff_schwartz 

longstaff_schwartz(process, payoff_fn, T, *, r, n_steps=50, n_paths=100000, basis='laguerre', degree=3, seed=None, antithetic=False, state=None, grid=None)

Price an American (Bermudan) option with the Longstaff-Schwartz algorithm.

Exercise is allowed at every grid time after t = 0; the continuation value is regressed on in-the-money paths only.

Parameters:

Name Type Description Default
process StochasticProcess

Risk-neutral process for the underlying.

 required
payoff_fn callable

Intrinsic value as a function of the underlying, e.g. lambda s: np.maximum(K - s, 0).

 required
T float

Maturity in years.

 required
r float

Continuously compounded discount rate.

 required
n_steps int

Number of exercise dates (equal steps).

 50
n_paths int

Number of paths.

 100000
basis ('laguerre', 'poly')

Regression basis, evaluated on the spot scaled by its initial value.

 "laguerre"
degree int

Degree of the regression basis.

 3
seed

As in :meth:StochasticProcess.simulate.

 None
antithetic

As in :meth:StochasticProcess.simulate.

 None
grid

As in :meth:StochasticProcess.simulate.

 None
state str

State variable to exercise on (default: primary state).

 None

montecarlo

Monte Carlo pricing of path-dependent payoffs.

MCResult dataclass

Monte Carlo price estimate.

Attributes:

Name Type Description
price float

Estimated present value.
stderr float

Standard error of the estimate.
n_paths int

Number of simulated paths.
samples (ndarray, optional)

The independent discounted samples the estimate averages (pair averages under antithetic sampling, control-variate adjusted if one was used).
conf_int 
conf_int(level=0.95)

Two-sided normal confidence interval for the price.

discount_factors 

discount_factors(result, discount, T)

Discount factor(s) to maturity T.

discount may be a continuously compounded rate, a :class:ZeroCurve, or None; with None the pathwise "discount" state of the result is used when the process provides one, otherwise payoffs are left undiscounted.

price_mc 

price_mc(process, payoff, T, *, discount=None, n_steps=252, n_paths=100000, seed=None, antithetic=False, control_variate=None, grid=None, keep_samples=False)

Price a payoff by Monte Carlo simulation of a process.

Parameters:

Name Type Description Default
process StochasticProcess

Process to simulate (e.g. a risk-neutral :class:~stochPathSim.processes.GBM).

 required
payoff callable

Maps a :class:~stochPathSim.core.SimulationResult to undiscounted payoffs.

 required
T float

Maturity in years.

 required
discount (float, ZeroCurve or None)

Discounting: a continuously compounded rate, a curve, or None to use the process's pathwise "discount" state if present (undiscounted otherwise).

 None
n_steps int

Time steps and paths.

 252
n_paths int

Time steps and paths.

 252
seed int

Seed for reproducible results.

 None
antithetic bool

Antithetic variates; the standard error is computed on pair averages.

 False
control_variate tuple(payoff, exact_value)

A second payoff with known present value. Its discounted samples are used as a linear control with the variance-minimising coefficient.

 None
grid TimeGrid

Custom simulation grid (overrides T/n_steps for the simulation).

 None
keep_samples bool

Store the samples in the result.

 False

payoffs

Payoffs evaluated on simulated paths.

Every payoff maps simulated paths to one undiscounted payoff per path at maturity. Call a payoff with a :class:~stochPathSim.core.SimulationResult, or with a raw (n_paths, n_times) array plus the matching times.

Payoff

Base class for path-dependent payoffs.

Parameters:

Name Type Description Default
state str

Name of the state variable to read; defaults to the primary state.

 None
__call__ 
__call__(result, times=None)

Undiscounted payoff per path.

Parameters:

Name Type Description Default
result SimulationResult or ndarray

Simulated paths. A raw array must have shape (n_paths, n_times).

 required
times array_like

Grid times for a raw array (needed by time-aware payoffs).

 None
evaluate 
evaluate(paths, times)

Payoff per path from a (n_paths, n_times) array.

European

Bases: Payoff

European call or put on the terminal value: max(S_T - K, 0) or max(K - S_T, 0).

EuropeanCall

Bases: European

European call max(S_T - K, 0).

EuropeanPut

Bases: European

European put max(K - S_T, 0).

Digital

Bases: Payoff

Cash-or-nothing digital paying cash if S_T > K (call) or S_T < K (put).

AsianArithmetic

Bases: _Asian

Fixed-strike Asian option on the arithmetic average of the fixings.

Parameters:

Name Type Description Default
K float

Strike.

 required
option_type ('call', 'put')
"call"
fixing_times array_like

Averaging times (must be grid times). By default every grid time from start_index on (start_index=1 excludes t = 0).

 None

AsianGeometric

Bases: _Asian

Fixed-strike Asian option on the geometric average of the fixings (see :class:AsianArithmetic).

Barrier

Bases: Payoff

Single-barrier knock-in or knock-out option, monitored at every grid time.

Parameters:

Name Type Description Default
K float

Strike.

 required
barrier float

Barrier level.

 required
kind ('up-and-out', 'up-and-in', 'down-and-out', 'down-and-in')
"up-and-out"
option_type ('call', 'put')
"call"
rebate float

Paid at maturity if the option is knocked out, or never knocked in.

 0.0
sigma float

Volatility for the Broadie-Glasserman-Kou continuity correction. When given, the barrier is moved towards the spot by the factor exp(0.5826 sigma sqrt(dt)), so that discrete monitoring on the grid approximates a continuously monitored barrier at barrier.

 None
effective_barrier 
effective_barrier(times)

Barrier used on the grid, after the optional continuity correction.

knocked 
knocked(paths, times)

Boolean per path: whether the barrier was touched on the grid.

LookbackFixed

Bases: Payoff

Fixed-strike lookback: max(max S - K, 0) (call) or max(K - min S, 0) (put), over the grid.

LookbackFloating

Bases: Payoff

Floating-strike lookback: S_T - min S (call) or max S - S_T (put), over the grid.

Basket

Bases: Payoff

Option on a weighted sum of terminal values of several state variables.

Parameters:

Name Type Description Default
weights array_like

One weight per state.

 required
K float

Strike.

 required
option_type ('call', 'put')
"call"
states sequence of str

State names, in the order of weights. Defaults to every state of the result, in order. Requires a :class:SimulationResult.

 None

Spread

Bases: Basket

Spread option on S_a(T) - S_b(T): max(S_a - S_b - K, 0) for a call.

trees

Binomial trees for European and American options.

crr_price 

crr_price(S, K, T, r, sigma, option_type='call', *, american=False, q=0.0, n_steps=500)

Cox-Ross-Rubinstein binomial price of a vanilla option.

Parameters:

Name Type Description Default
S float

Spot, strike, maturity (years), continuously compounded rate and volatility.

 required
K float

Spot, strike, maturity (years), continuously compounded rate and volatility.

 required
T float

Spot, strike, maturity (years), continuously compounded rate and volatility.

 required
r float

Spot, strike, maturity (years), continuously compounded rate and volatility.

 required
sigma float

Spot, strike, maturity (years), continuously compounded rate and volatility.

 required
option_type ('call', 'put')
"call"
american bool

Allow early exercise at every node.

 False
q float

Continuous dividend yield.

 0.0
n_steps int

Number of tree steps.

 500