API Reference
Module containing components for financial mathematics, including derivative pricing models.
Bases: DiscretePricingModel
A class modeling a binomial pricing model.
The base constructor is not meant to be called directly by users. Instead, the user should call the generate class method. See the Examples section below for usage.
Examples:
>>> from sigalg.finance import BinomialPricingModel
Given parameters of a binomial model, enumerate all length-3 price trajectories in dense enumeration mode.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... risk_free_rate=0.01,
... up_factor=1.1,
... up_prob=0.7,
... length=3,
... enum_mode="dense",
... mode="enum",
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.000000 121.000000 133.100000
1 100.0 110.000000 121.000000 110.000000
2 100.0 110.000000 100.000000 110.000000
3 100.0 110.000000 100.000000 90.909091
4 100.0 90.909091 100.000000 110.000000
5 100.0 90.909091 100.000000 90.909091
6 100.0 90.909091 82.644628 90.909091
7 100.0 90.909091 82.644628 75.131480
Enumerate all length-3 price trajectories in sparse enumeration mode.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... risk_free_rate=0.01,
... up_factor=1.1,
... up_prob=0.7,
... length=3,
... enum_mode="sparse",
... mode="enum",
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.000000 121.000000 133.100000
1 100.0 90.909091 100.000000 110.000000
2 100.0 90.909091 82.644628 90.909091
3 100.0 90.909091 82.644628 75.131480
Simulate ten length-3 trajectories in sim mode.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... risk_free_rate=0.01,
... up_factor=1.1,
... up_prob=0.7,
... length=3,
... mode="sim",
... n_trajectories=10,
... random_state=42,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 90.909091 100.000000 90.909091
1 100.0 110.000000 121.000000 110.000000
2 100.0 90.909091 82.644628 90.909091
3 100.0 110.000000 121.000000 110.000000
4 100.0 110.000000 100.000000 110.000000
5 100.0 110.000000 121.000000 133.100000
6 100.0 90.909091 100.000000 90.909091
7 100.0 110.000000 100.000000 90.909091
8 100.0 90.909091 100.000000 110.000000
9 100.0 110.000000 121.000000 133.100000
property
down_factor
Get the down-factor of the model.
Returns:
| Name | Type | Description |
|---|---|---|
down_factor |
Real
|
The down-factor of the model. |
property
down_prob
Get the probability of a downward price movement.
Returns:
| Name | Type | Description |
|---|---|---|
down_prob |
Real
|
The probability of a downward price movement. |
property
enum_mode
Get the enumeration mode of the model.
Returns:
| Name | Type | Description |
|---|---|---|
enum_mode |
Literal['dense', 'sparse']
|
The enumeration mode of the model. |
property
initial_price
Get the initial price of the asset.
Returns:
| Name | Type | Description |
|---|---|---|
initial_price |
Real
|
The initial price of the asset. |
property
risk_free_gross_return
Get the per-period risk-free gross return of the model.
Returns:
| Name | Type | Description |
|---|---|---|
risk_free_gross_return |
Real
|
The per-period risk-free gross return. |
property
risk_free_rate
Get the per-period risk-free rate.
Returns:
| Name | Type | Description |
|---|---|---|
risk_free_rate |
Real
|
The risk-free rate. |
property
up_factor
Get the up-factor of the model.
Returns:
| Name | Type | Description |
|---|---|---|
up_factor |
Real
|
The up-factor of the model. |
property
up_prob
Get the probability of an upward price movement.
Returns:
| Name | Type | Description |
|---|---|---|
up_prob |
Real
|
The probability of an upward price movement. |
B(claim, **kwargs)
Get the price process of the non-risky asset in the replicating portfolio of a claim.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The contingent claim whose price process of the non-risky claim is to be computed. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
B |
StochasticProcess
|
The price process of the non-risky asset of the claim. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the exercise time of the call option.
>>> print(S.B(claim=euro_call))
Stochastic process 'B':
t 0 1 2
omega
0 -61.564626 -77.108844 -95.238095
1 -61.564626 -77.108844 -95.238095
2 -61.564626 -77.108844 -38.142857
3 -61.564626 -77.108844 -38.142857
4 -61.564626 -27.244898 -38.142857
5 -61.564626 -27.244898 -38.142857
6 -61.564626 -27.244898 0.000000
7 -61.564626 -27.244898 0.000000
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the exercise time of the put option.
>>> print(S.B(claim=american_put))
Stochastic process 'B':
t 0 1 2
omega
0 49.73275 13.594104 0.000000
1 49.73275 13.594104 0.000000
2 49.73275 13.594104 57.095238
3 49.73275 13.594104 57.095238
4 49.73275 88.401361 57.095238
5 49.73275 88.401361 57.095238
6 49.73275 88.401361 95.238095
7 49.73275 88.401361 95.238095
continuation_process(claim, **kwargs)
Get the continuation/value process of the replicating portfolio of a claim.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The contingent claim whose continuation/price process is to be computed. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
continuation_process |
StochasticProcess
|
The continuation/value process of the replicating portfolio of the claim. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the risk-neutral continuation process of the call option.
>>> print(S.continuation_process(claim=euro_call))
Stochastic process 'V':
t 0 1 2 3
omega
0 15.306122 19.914966 25.761905 33.1
1 15.306122 19.914966 25.761905 8.9
2 15.306122 19.914966 6.357143 8.9
3 15.306122 19.914966 6.357143 0.0
4 15.306122 4.540816 6.357143 8.9
5 15.306122 4.540816 6.357143 0.0
6 15.306122 4.540816 0.000000 0.0
7 15.306122 4.540816 0.000000 0.0
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the risk-neutral continuation process of the put option.
>>> print(american_put.continuation_process())
Stochastic process 'V':
t 0 1 2 3
omega
0 2.822319 0.617914 0.000000 0.0
1 2.822319 0.617914 0.000000 0.0
2 2.822319 0.617914 2.595238 0.0
3 2.822319 0.617914 2.595238 10.9
4 2.822319 6.377551 2.595238 0.0
5 2.822319 6.377551 2.595238 10.9
6 2.822319 6.377551 14.238095 10.9
7 2.822319 6.377551 14.238095 27.1
delta(claim, **kwargs)
Get the delta process of the replicating portfolio of a claim.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The claim whose delta process is to be computed. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
delta |
StochasticProcess
|
The delta process of the claim. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the exercise time of the call option.
>>> print(S.delta(claim=euro_call))
Stochastic process 'Delta':
t 0 1 2
omega
0 0.768707 0.882035 1.000000
1 0.768707 0.882035 1.000000
2 0.768707 0.882035 0.449495
3 0.768707 0.882035 0.449495
4 0.768707 0.353175 0.449495
5 0.768707 0.353175 0.449495
6 0.768707 0.353175 -0.000000
7 0.768707 0.353175 -0.000000
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the exercise time of the put option.
>>> print(S.delta(claim=american_put))
Stochastic process 'Delta':
t 0 1 2
omega
0 -0.469104 -0.117965 -0.000000
1 -0.469104 -0.117965 -0.000000
2 -0.469104 -0.117965 -0.550505
3 -0.469104 -0.117965 -0.550505
4 -0.469104 -0.911376 -0.550505
5 -0.469104 -0.911376 -0.550505
6 -0.469104 -0.911376 -1.000000
7 -0.469104 -0.911376 -1.000000
emm(*args, **kwargs)
Get the equivalent martingale measure of the model.
Returns:
| Name | Type | Description |
|---|---|---|
emm |
ProbabilityMeasure
|
The equivalent martingale measure of the model. |
Examples:
>>> from sigalg.finance import BinomialPricingModel
Define the parameters of a binomial pricing model.
>>> S_0 = 100
>>> r = 0.05
>>> p = 0.55
>>> u = 1.1
>>> d = 0.9
>>> T = 2
Exhaustively generate all price trajectories and print them.
>>> S = BinomialPricingModel.generate(
... initial_price=S_0,
... risk_free_rate=r,
... up_prob=p,
... up_factor=u,
... down_factor=d,
... mode="enum",
... length=T,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2
omega
0 100.0 110.0 121.0
1 100.0 110.0 99.0
2 100.0 90.0 99.0
3 100.0 90.0 81.0
Print the equivalent martingale measure.
>>> Q = S.emm()
>>> print(Q)
Probability measure 'Q':
Q
omega
0 0.5625
1 0.1875
2 0.1875
3 0.0625
Check that the discounted model is a martingale with respect to the EMM.
>>> S.discount(r).is_martingale(prob_measure=Q)
True
exercise_time(claim, **kwargs)
Get the exercise time of a claim.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The contingent claim whose exercise times are to be compute. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
exercise_time |
StoppingTime
|
The exercise time of the claim. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the exercise time of the call option.
>>> print(S.exercise_time(claim=euro_call))
Stopping time 'tau':
tau
omega
0 3.0
1 3.0
2 3.0
3 inf
4 3.0
5 inf
6 inf
7 inf
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the exercise time of the put option.
>>> print(S.exercise_time(claim=american_put))
Stopping time 'tau':
tau
omega
0 inf
1 inf
2 inf
3 3.0
4 1.0
5 1.0
6 1.0
7 1.0
generate(
initial_price,
up_factor,
down_factor=None,
up_prob=0.5,
risk_free_rate=None,
risk_free_gross_return=None,
enum_mode="dense",
mode="sim",
n_trajectories=None,
domain_name=None,
index=None,
length=None,
name="S",
random_state=None,
)
Generate trajectories of the binomial pricing model by either exhaustive enumeration (in sparse or dense mode) or Monte Carlo simulation.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
initial_price
|
Real
|
The initial price of the asset. |
required |
up_factor
|
Real
|
The up-factor of the model. |
required |
down_factor
|
Real | None
|
The down-factor of the model. If |
None
|
up_prob
|
Real
|
The probability of an upward move in the price. |
0.5
|
risk_free_rate
|
Real | None
|
The per-period risk-free rate. If |
None
|
risk_free_gross_return
|
Real | None
|
The per-period risk-free gross return. If |
None
|
enum_mode
|
Literal['dense', 'sparse']
|
The mode of enumeration. If the generation mode is set to |
"dense"
|
mode
|
Literal['enum', 'sim']
|
Whether to generate trajectories by exhuastive enumeration or Monte Carlo simulation. |
'sim'
|
n_trajectories
|
int | None
|
The number of trajectories to simulate. If the generation mode is set to |
None
|
index
|
IndexLike | None
|
The index of the stochastic process. One of |
None
|
length
|
int | None
|
The length of the trajectories of the stochastic process. One of |
None
|
name
|
Hashable
|
The name of the stochastic process. |
"S"
|
random_state
|
int | Generator | None
|
An optional random state for reproducibility. |
None
|
Returns:
| Name | Type | Description |
|---|---|---|
info |
dict[str, object]
|
A dictionary containing the names of the BinomialPricingModel-specific parameters and their values. |
Examples:
>>> from sigalg.finance import BinomialPricingModel
Given parameters of a binomial model, enumerate all length-3 price trajectories in dense enumeration mode.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... risk_free_rate=0.01,
... up_factor=1.1,
... up_prob=0.7,
... length=3,
... enum_mode="dense",
... mode="enum",
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.000000 121.000000 133.100000
1 100.0 110.000000 121.000000 110.000000
2 100.0 110.000000 100.000000 110.000000
3 100.0 110.000000 100.000000 90.909091
4 100.0 90.909091 100.000000 110.000000
5 100.0 90.909091 100.000000 90.909091
6 100.0 90.909091 82.644628 90.909091
7 100.0 90.909091 82.644628 75.131480
Enumerate all length-3 price trajectories in sparse enumeration mode.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... risk_free_rate=0.01,
... up_factor=1.1,
... up_prob=0.7,
... length=3,
... enum_mode="sparse",
... mode="enum",
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.000000 121.000000 133.100000
1 100.0 90.909091 100.000000 110.000000
2 100.0 90.909091 82.644628 90.909091
3 100.0 90.909091 82.644628 75.131480
Simulate ten length-3 trajectories in sim mode.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... risk_free_rate=0.01,
... up_factor=1.1,
... up_prob=0.7,
... length=3,
... mode="sim",
... n_trajectories=10,
... random_state=42,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 90.909091 100.000000 90.909091
1 100.0 110.000000 121.000000 110.000000
2 100.0 90.909091 82.644628 90.909091
3 100.0 110.000000 121.000000 110.000000
4 100.0 110.000000 100.000000 110.000000
5 100.0 110.000000 121.000000 133.100000
6 100.0 90.909091 100.000000 90.909091
7 100.0 110.000000 100.000000 90.909091
8 100.0 90.909091 100.000000 110.000000
9 100.0 110.000000 121.000000 133.100000
price(claim, **kwargs)
Get the risk-neutral price of a claim.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The contingent claim to price. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
price |
Real
|
The risk-neutral price of the claim. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the risk-neutral price of the call option.
>>> S.price(claim=euro_call)
15.306122448979611
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the risk-neutral price of the put option.
>>> S.price(claim=american_put)
2.822319403952058
price_process(claim, **kwargs)
Get the risk-neutral price process of a claim.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The contingent claim whose price process is to be computed. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
price_process |
StochasticProcess
|
The risk-neutral price process of the claim. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the risk-neutral price process of the call option.
>>> print(S.price_process(claim=euro_call))
Stochastic process 'X':
t 0 1 2 3
omega
0 15.306122 19.914966 25.761905 33.1
1 15.306122 19.914966 25.761905 8.9
2 15.306122 19.914966 6.357143 8.9
3 15.306122 19.914966 6.357143 0.0
4 15.306122 4.540816 6.357143 8.9
5 15.306122 4.540816 6.357143 0.0
6 15.306122 4.540816 0.000000 0.0
7 15.306122 4.540816 0.000000 0.0
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the risk-neutral price process of the put option.
>>> print(S.price_process(claim=american_put))
Stochastic process 'X':
t 0 1 2 3
omega
0 2.822319 0.617914 0.000000 0.0
1 2.822319 0.617914 0.000000 0.0
2 2.822319 0.617914 2.595238 0.0
3 2.822319 0.617914 2.595238 10.9
4 2.822319 10.000000 2.595238 0.0
5 2.822319 10.000000 2.595238 10.9
6 2.822319 10.000000 19.000000 10.9
7 2.822319 10.000000 19.000000 27.1
replicating_portfolio(claim, **kwargs)
Compute the replicating portfolio for a given contingent claim.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The contingent claim for which to compute the replicating portfolio. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
replicating_portfolio |
tuple[StochasticProcess, StochasticProcess, StochasticProcess]
|
Three processes |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Generate the replicating portfolio for the European call.
>>> Delta, B, V = S.replicating_portfolio(claim=euro_call)
Print the Delta and B price processes of the hedging portfolio, along with its value process V.
>>> print(Delta)
Stochastic process 'Delta':
t 0 1 2
omega
0 0.768707 0.882035 1.000000
1 0.768707 0.882035 1.000000
2 0.768707 0.882035 0.449495
3 0.768707 0.882035 0.449495
4 0.768707 0.353175 0.449495
5 0.768707 0.353175 0.449495
6 0.768707 0.353175 -0.000000
7 0.768707 0.353175 -0.000000
>>> print(B)
Stochastic process 'B':
t 0 1 2
omega
0 -61.564626 -77.108844 -95.238095
1 -61.564626 -77.108844 -95.238095
2 -61.564626 -77.108844 -38.142857
3 -61.564626 -77.108844 -38.142857
4 -61.564626 -27.244898 -38.142857
5 -61.564626 -27.244898 -38.142857
6 -61.564626 -27.244898 0.000000
7 -61.564626 -27.244898 0.000000
>>> print(V)
Stochastic process 'V':
t 0 1 2 3
omega
0 15.306122 19.914966 25.761905 33.1
1 15.306122 19.914966 25.761905 8.9
2 15.306122 19.914966 6.357143 8.9
3 15.306122 19.914966 6.357143 0.0
4 15.306122 4.540816 6.357143 8.9
5 15.306122 4.540816 6.357143 0.0
6 15.306122 4.540816 0.000000 0.0
7 15.306122 4.540816 0.000000 0.0
Now define an Asian put option with early-exercise and the same strike.
>>> asian_put = Claim.asian_option(strike=100, model=S, type="put", early_exercise=True, name="asian_put")
>>> print(asian_put)
Asian option 'asian_put':
t 0 1 2 3
omega
0 0.0 0.0 0.000000 0.000
1 0.0 0.0 0.000000 0.000
2 0.0 0.0 0.000000 0.000
3 0.0 0.0 0.000000 0.475
4 0.0 5.0 3.666667 0.525
5 0.0 5.0 3.666667 5.475
6 0.0 5.0 9.666667 9.975
7 0.0 5.0 9.666667 14.025
Compute and print the replicating portfolio.
>>> Delta, B, V = S.replicating_portfolio(claim=asian_put)
>>> print(Delta)
Stochastic process 'Delta':
t 0 1 2
omega
0 -0.254181 -0.005141 -0.00000
1 -0.254181 -0.005141 -0.00000
2 -0.254181 -0.005141 -0.02399
3 -0.254181 -0.005141 -0.02399
4 -0.254181 -0.377646 -0.25000
5 -0.254181 -0.377646 -0.25000
6 -0.254181 -0.377646 -0.25000
7 -0.254181 -0.377646 -0.25000
>>> print(B)
Stochastic process 'B':
t 0 1 2
omega
0 26.654114 0.592404 0.000000
1 26.654114 0.592404 0.000000
2 26.654114 0.592404 2.488095
3 26.654114 0.592404 2.488095
4 26.654114 39.098639 26.428571
5 26.654114 39.098639 26.428571
6 26.654114 39.098639 30.714286
7 26.654114 39.098639 30.714286
>>> print(V)
Stochastic process 'V':
t 0 1 2 3
omega
0 1.23603 0.026927 0.000000 0.000
1 1.23603 0.026927 0.000000 0.000
2 1.23603 0.026927 0.113095 0.000
3 1.23603 0.026927 0.113095 0.475
4 1.23603 5.110544 1.678571 0.525
5 1.23603 5.110544 1.678571 5.475
6 1.23603 5.110544 10.464286 9.975
7 1.23603 5.110544 10.464286 14.025
risk_neutral_probs(*args, **kwargs)
Get the risk-neutral probabilities of the model.
Returns:
| Name | Type | Description |
|---|---|---|
risk_neutral_probs |
tuple[Real, Real]
|
The risk neutral probabilities as a tuple |
Examples:
>>> from sigalg.finance import BinomialPricingModel
Define the parameters of a binomial pricing model.
>>> S_0 = 100
>>> r = 0.05
>>> p = 0.55
>>> u = 1.1
>>> d = 0.9
>>> T = 2
Exhaustively generate all price trajectories and print them.
>>> S = BinomialPricingModel.generate(
... initial_price=S_0,
... risk_free_rate=r,
... up_prob=p,
... up_factor=u,
... down_factor=d,
... mode="enum",
... length=T,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2
omega
0 100.0 110.0 121.0
1 100.0 110.0 99.0
2 100.0 90.0 99.0
3 100.0 90.0 81.0
Get the risk-neutral probabilities of the model.
>>> S.risk_neutral_probs()
(0.7499999999999999, 0.2500000000000001)
Bases: StochasticProcess
Base class for contingent claims.
The base constructor is not meant to be called directly by users. Instead, the user should call the from_stochastic_process or `from_random_variables_ class methods. See the Examples section below for usage.
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Create a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_prob=0.7,
... up_factor=1.1,
... down_factor=0.9,
... risk_free_rate=0.05,
... enum_mode="dense",
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Create a stochastic process representing an American-style Asian call option.
>>> process = (S.cumavg() - 100).max(0)
>>> print(process)
Stochastic process 'max((S_cumavg - 100), 0)':
t 0 1 2 3
omega
0 0.0 5.0 10.333333 16.025
1 0.0 5.0 10.333333 9.975
2 0.0 5.0 3.000000 4.475
3 0.0 5.0 3.000000 0.000
4 0.0 0.0 0.000000 0.000
5 0.0 0.0 0.000000 0.000
6 0.0 0.0 0.000000 0.000
7 0.0 0.0 0.000000 0.000
Construct the contingent claim.
>>> Phi = Claim.from_stochastic_process(process=process, model=S, name="Phi")
>>> print(Phi)
Claim 'Phi':
t 0 1 2 3
omega
0 0.0 5.0 10.333333 16.025
1 0.0 5.0 10.333333 9.975
2 0.0 5.0 3.000000 4.475
3 0.0 5.0 3.000000 0.000
4 0.0 0.0 0.000000 0.000
5 0.0 0.0 0.000000 0.000
6 0.0 0.0 0.000000 0.000
7 0.0 0.0 0.000000 0.000
Create the payoff random variable of a European call with strike K=100.
>>> Phi_3 = (S[3] - 100).max(0)
>>> print(Phi_3)
Random variable 'max((S_3 - 100), 0)':
max((S_3 - 100), 0)
omega
0 33.1
1 8.9
2 8.9
3 0.0
4 8.9
5 0.0
6 0.0
7 0.0
Create an instance of Claim from the random variable.
>>> Phi = Claim.from_random_variables(rvs=[Phi_3], times=[3], model=S)
>>> print(Phi)
Claim 'Phi':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Create the payoff random variable of a European call with strike K=100 at time t=2.
>>> Phi_2 = (S[2] - 100).max(0)
>>> print(Phi_2)
Random variable 'max((S_2 - 100), 0)':
max((S_2 - 100), 0)
omega
0 21.0
1 21.0
2 0.0
3 0.0
4 0.0
5 0.0
6 0.0
7 0.0
Combine the random variables Phi_3 and Phi_2 into a single instance of Claim that allows exercise at times t=2 and t=3.
>>> Phi = Claim.from_random_variables(rvs=[Phi_2, Phi_3], times=[2, 3], model=S)
>>> print(Phi)
Claim 'Phi':
t 0 1 2 3
omega
0 0.0 0.0 21.0 33.1
1 0.0 0.0 21.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
property
is_early_exercise
Return whether the claim allows early exercise or not.
Returns:
| Name | Type | Description |
|---|---|---|
is_early_exercise |
bool
|
Whether the claim allows early exercise or not. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Create a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_prob=0.7,
... up_factor=1.1,
... down_factor=0.9,
... risk_free_rate=0.05,
... enum_mode="dense",
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Create the payoff random variable of a European call with strike K=100.
>>> Phi_3 = (S[3] - 100).max(0)
>>> print(Phi_3)
Random variable 'max((S_3 - 100), 0)':
max((S_3 - 100), 0)
omega
0 33.1
1 8.9
2 8.9
3 0.0
4 8.9
5 0.0
6 0.0
7 0.0
Create an instance of Claim from the random variable and check whether it allows early exercise.
>>> Phi = Claim.from_random_variables(rvs=[Phi_3], times=[3], model=S)
>>> print(Phi)
Claim 'Phi':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
>>> Phi.is_early_exercise
False
Create the payoff random variable of a European call with strike K=100 at time t=2.
>>> Phi_2 = (S[2] - 100).max(0)
>>> print(Phi_2)
Random variable 'max((S_2 - 100), 0)':
max((S_2 - 100), 0)
omega
0 21.0
1 21.0
2 0.0
3 0.0
4 0.0
5 0.0
6 0.0
7 0.0
Combine the random variables Phi_3 and Phi_2 into a single instance of Claim that allows exercise at times t=2 and t=3.
>>> Phi = Claim.from_random_variables(rvs=[Phi_2, Phi_3], times=[2, 3], model=S)
>>> print(Phi)
Claim 'Phi':
t 0 1 2 3
omega
0 0.0 0.0 21.0 33.1
1 0.0 0.0 21.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
>>> Phi.is_early_exercise
True
dataclass
Data class for pricing data for contingent claims.
B(**kwargs)
Get the price process of the non-risky asset of the claim.
Returns:
| Name | Type | Description |
|---|---|---|
B |
StochasticProcess
|
The price process of the non-risky asset of the claim. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the exercise time of the call option.
>>> print(euro_call.B())
Stochastic process 'B':
t 0 1 2
omega
0 -61.564626 -77.108844 -95.238095
1 -61.564626 -77.108844 -95.238095
2 -61.564626 -77.108844 -38.142857
3 -61.564626 -77.108844 -38.142857
4 -61.564626 -27.244898 -38.142857
5 -61.564626 -27.244898 -38.142857
6 -61.564626 -27.244898 0.000000
7 -61.564626 -27.244898 0.000000
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the exercise time of the put option.
>>> print(american_put.B())
Stochastic process 'B':
t 0 1 2
omega
0 49.73275 13.594104 0.000000
1 49.73275 13.594104 0.000000
2 49.73275 13.594104 57.095238
3 49.73275 13.594104 57.095238
4 49.73275 88.401361 57.095238
5 49.73275 88.401361 57.095238
6 49.73275 88.401361 95.238095
7 49.73275 88.401361 95.238095
classmethod
american_option(strike, model, type='call', name='Phi')
Create an Asian option.
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_prob=0.7,
... up_factor=1.1,
... down_factor=0.9,
... risk_free_rate=0.05,
... enum_mode="dense",
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Create an Asian call option with strike K=100 and no early exercise.
>>> K = 100
>>> Phi_call_no_early = Claim.asian_option(
... strike=100, model=S, type="call", early_exercise=False, name="Phi_call_no_early"
... )
>>> print(Phi_call_no_early)
Asian option 'Phi_call_no_early':
t 0 1 2 3
omega
0 0.0 0.0 0.0 16.025
1 0.0 0.0 0.0 9.975
2 0.0 0.0 0.0 4.475
3 0.0 0.0 0.0 0.000
4 0.0 0.0 0.0 0.000
5 0.0 0.0 0.0 0.000
6 0.0 0.0 0.0 0.000
7 0.0 0.0 0.0 0.000
Create an Asian call with the same strike and with early exercise.
>>> Phi_call_early = Claim.asian_option(
... strike=100, model=S, type="call", early_exercise=True, name="Phi_call_early"
... )
>>> print(Phi_call_early)
Asian option 'Phi_call_early':
t 0 1 2 3
omega
0 0.0 5.0 10.333333 16.025
1 0.0 5.0 10.333333 9.975
2 0.0 5.0 3.000000 4.475
3 0.0 5.0 3.000000 0.000
4 0.0 0.0 0.000000 0.000
5 0.0 0.0 0.000000 0.000
6 0.0 0.0 0.000000 0.000
7 0.0 0.0 0.000000 0.000
Create an Asian put with the same strike and no early exercise.
>>> Phi_put_no_early = Claim.asian_option(
... strike=100, model=S, type="put", early_exercise=False, name="Phi_put_no_early"
... )
>>> print(Phi_put_no_early)
Asian option 'Phi_put_no_early':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.000
1 0.0 0.0 0.0 0.000
2 0.0 0.0 0.0 0.000
3 0.0 0.0 0.0 0.475
4 0.0 0.0 0.0 0.525
5 0.0 0.0 0.0 5.475
6 0.0 0.0 0.0 9.975
7 0.0 0.0 0.0 14.025
Create an Asian put with the same strike and early exercise.
>>> Phi_put_early = Claim.asian_option(
... strike=100, model=S, type="put", early_exercise=True, name="Phi_put_early"
... )
>>> print(Phi_put_early)
Asian option 'Phi_put_early':
t 0 1 2 3
omega
0 0.0 0.0 0.000000 0.000
1 0.0 0.0 0.000000 0.000
2 0.0 0.0 0.000000 0.000
3 0.0 0.0 0.000000 0.475
4 0.0 5.0 3.666667 0.525
5 0.0 5.0 3.666667 5.475
6 0.0 5.0 9.666667 9.975
7 0.0 5.0 9.666667 14.025
classmethod
asian_option(
strike,
model,
type="call",
early_exercise=False,
name="Phi",
)
Pass.
continuation_process(**kwargs)
Get the continuation process of the claim.
Returns:
| Name | Type | Description |
|---|---|---|
continuation_process |
StochasticProcess
|
The continuation process of the claim relative to the underlying price model. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the risk-neutral continuation process of the call option.
>>> print(euro_call.continuation_process())
Stochastic process 'V':
t 0 1 2 3
omega
0 15.306122 19.914966 25.761905 33.1
1 15.306122 19.914966 25.761905 8.9
2 15.306122 19.914966 6.357143 8.9
3 15.306122 19.914966 6.357143 0.0
4 15.306122 4.540816 6.357143 8.9
5 15.306122 4.540816 6.357143 0.0
6 15.306122 4.540816 0.000000 0.0
7 15.306122 4.540816 0.000000 0.0
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the risk-neutral continuation process of the put option.
>>> print(american_put.continuation_process())
Stochastic process 'V':
t 0 1 2 3
omega
0 2.822319 0.617914 0.000000 0.0
1 2.822319 0.617914 0.000000 0.0
2 2.822319 0.617914 2.595238 0.0
3 2.822319 0.617914 2.595238 10.9
4 2.822319 6.377551 2.595238 0.0
5 2.822319 6.377551 2.595238 10.9
6 2.822319 6.377551 14.238095 10.9
7 2.822319 6.377551 14.238095 27.1
delta(**kwargs)
Get the delta process of the claim.
Returns:
| Name | Type | Description |
|---|---|---|
delta |
StochasticProcess
|
The delta process of the claim relative to the underlying price model. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the exercise time of the call option.
>>> print(euro_call.delta())
Stochastic process 'Delta':
t 0 1 2
omega
0 0.768707 0.882035 1.000000
1 0.768707 0.882035 1.000000
2 0.768707 0.882035 0.449495
3 0.768707 0.882035 0.449495
4 0.768707 0.353175 0.449495
5 0.768707 0.353175 0.449495
6 0.768707 0.353175 -0.000000
7 0.768707 0.353175 -0.000000
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the exercise time of the put option.
>>> print(american_put.delta())
Stochastic process 'Delta':
t 0 1 2
omega
0 -0.469104 -0.117965 -0.000000
1 -0.469104 -0.117965 -0.000000
2 -0.469104 -0.117965 -0.550505
3 -0.469104 -0.117965 -0.550505
4 -0.469104 -0.911376 -0.550505
5 -0.469104 -0.911376 -0.550505
6 -0.469104 -0.911376 -1.000000
7 -0.469104 -0.911376 -1.000000
classmethod
european_option(strike, model, type='call', name='Phi')
Create a European option.
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_prob=0.7,
... up_factor=1.1,
... down_factor=0.9,
... risk_free_rate=0.05,
... enum_mode="dense",
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Create a European call option with strike K=100.
>>> K = 100
>>> Phi_call = Claim.european_option(strike=K, model=S, type="call", name="Phi_call")
>>> print(Phi_call)
European option 'Phi_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Create a European put option with the same strike.
>>> Phi_put = Claim.european_option(strike=K, model=S, type="put", name="Phi_put")
>>> print(Phi_put)
European option 'Phi_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 0.0 0.0
3 0.0 0.0 0.0 10.9
4 0.0 0.0 0.0 0.0
5 0.0 0.0 0.0 10.9
6 0.0 0.0 0.0 10.9
7 0.0 0.0 0.0 27.1
exercise_time(**kwargs)
Get the exercise time of the claim.
Returns:
| Name | Type | Description |
|---|---|---|
exercise_time |
StoppingTime
|
The exercise time of the claim relative to the underlying price model. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the exercise time of the call option.
>>> print(euro_call.exercise_time())
Stopping time 'tau':
tau
omega
0 3.0
1 3.0
2 3.0
3 inf
4 3.0
5 inf
6 inf
7 inf
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the exercise time of the put option.
>>> print(american_put.exercise_time())
Stopping time 'tau':
tau
omega
0 inf
1 inf
2 inf
3 3.0
4 1.0
5 1.0
6 1.0
7 1.0
classmethod
from_random_variables(rvs, times, model, name='Phi')
Construct a contingent claim from a list of random variables and a list of times.
The random variables are inserted into the claim at the specified times. Any unspecified times are filled with the 0 random variable. See the Examples section below.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
rvs
|
list[RandomVariable]
|
The list of random variables to include as the components of the contingent claim. |
required |
times
|
list[Real]
|
The list of times corresponding to the random variables. |
required |
model
|
DiscretePricingModel
|
The pricing model. |
required |
name
|
Hashable
|
The name for the claim. |
"Phi"
|
Returns:
| Name | Type | Description |
|---|---|---|
claim |
Claim
|
The contingent claim created from the random variables. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Create a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_prob=0.7,
... up_factor=1.1,
... down_factor=0.9,
... risk_free_rate=0.05,
... enum_mode="dense",
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Create the payoff random variable of a European call with strike K=100.
>>> Phi_3 = (S[3] - 100).max(0)
>>> print(Phi_3)
Random variable 'max((S_3 - 100), 0)':
max((S_3 - 100), 0)
omega
0 33.1
1 8.9
2 8.9
3 0.0
4 8.9
5 0.0
6 0.0
7 0.0
Create an instance of Claim from the random variable.
>>> Phi = Claim.from_random_variables(rvs=[Phi_3], times=[3], model=S)
>>> print(Phi)
Claim 'Phi':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Create the payoff random variable of a European call with strike K=100 at time t=2.
>>> Phi_2 = (S[2] - 100).max(0)
>>> print(Phi_2)
Random variable 'max((S_2 - 100), 0)':
max((S_2 - 100), 0)
omega
0 21.0
1 21.0
2 0.0
3 0.0
4 0.0
5 0.0
6 0.0
7 0.0
Combine the random variables Phi_3 and Phi_2 into a single instance of Claim that allows exercise at times t=2 and t=3.
>>> Phi = Claim.from_random_variables(rvs=[Phi_2, Phi_3], times=[2, 3], model=S)
>>> print(Phi)
Claim 'Phi':
t 0 1 2 3
omega
0 0.0 0.0 21.0 33.1
1 0.0 0.0 21.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
classmethod
from_stochastic_process(process, model, name=None)
Construct a contingent claim from a stochastic process.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
process
|
StochasticProcess
|
The stochastic process used to generate the contingent claim. |
required |
model
|
DiscretePricingModel
|
The pricing model. |
required |
name
|
Hashable | None
|
The name for the claim. If |
None
|
Returns:
| Name | Type | Description |
|---|---|---|
claim |
Claim
|
The contingent claim created from the stochastic process |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Create a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_prob=0.7,
... up_factor=1.1,
... down_factor=0.9,
... risk_free_rate=0.05,
... enum_mode="dense",
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Create a stochastic process representing an American-style Asian call option.
>>> process = (S.cumavg() - 100).max(0)
>>> print(process)
Stochastic process 'max((S_cumavg - 100), 0)':
t 0 1 2 3
omega
0 0.0 5.0 10.333333 16.025
1 0.0 5.0 10.333333 9.975
2 0.0 5.0 3.000000 4.475
3 0.0 5.0 3.000000 0.000
4 0.0 0.0 0.000000 0.000
5 0.0 0.0 0.000000 0.000
6 0.0 0.0 0.000000 0.000
7 0.0 0.0 0.000000 0.000
Construct the contingent claim.
>>> Phi = Claim.from_stochastic_process(process=process, model=S, name="Phi")
>>> print(Phi)
Claim 'Phi':
t 0 1 2 3
omega
0 0.0 5.0 10.333333 16.025
1 0.0 5.0 10.333333 9.975
2 0.0 5.0 3.000000 4.475
3 0.0 5.0 3.000000 0.000
4 0.0 0.0 0.000000 0.000
5 0.0 0.0 0.000000 0.000
6 0.0 0.0 0.000000 0.000
7 0.0 0.0 0.000000 0.000
price(**kwargs)
Get the price of the claim.
Returns:
| Name | Type | Description |
|---|---|---|
price |
Real
|
The price of the claim relative to the underlying price model. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the risk-neutral price of the call option.
>>> euro_call.price()
15.306122448979611
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the risk-neutral price of the put option.
>>> american_put.price()
2.822319403952058
price_process(**kwargs)
Get the price process of the claim.
Returns:
| Name | Type | Description |
|---|---|---|
price_process |
StochasticProcess
|
The price process of the claim relative to the underlying price model. |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Get the risk-neutral price process of the call option.
>>> print(euro_call.price_process())
Stochastic process 'X':
t 0 1 2 3
omega
0 15.306122 19.914966 25.761905 33.1
1 15.306122 19.914966 25.761905 8.9
2 15.306122 19.914966 6.357143 8.9
3 15.306122 19.914966 6.357143 0.0
4 15.306122 4.540816 6.357143 8.9
5 15.306122 4.540816 6.357143 0.0
6 15.306122 4.540816 0.000000 0.0
7 15.306122 4.540816 0.000000 0.0
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 10.9
4 0.0 10.0 1.0 0.0
5 0.0 10.0 1.0 10.9
6 0.0 10.0 19.0 10.9
7 0.0 10.0 19.0 27.1
Get the risk-neutral price process of the put option.
>>> print(american_put.price_process())
Stochastic process 'X':
t 0 1 2 3
omega
0 2.822319 0.617914 0.000000 0.0
1 2.822319 0.617914 0.000000 0.0
2 2.822319 0.617914 2.595238 0.0
3 2.822319 0.617914 2.595238 10.9
4 2.822319 10.000000 2.595238 0.0
5 2.822319 10.000000 2.595238 10.9
6 2.822319 10.000000 19.000000 10.9
7 2.822319 10.000000 19.000000 27.1
replicating_portfolio(**kwargs)
Compute the replicating portfolio of the claim.
Returns:
| Name | Type | Description |
|---|---|---|
replicating_portfolio |
tuple[StochasticProcess, StochasticProcess, StochasticProcess]
|
Three processes |
Examples:
>>> from sigalg.finance import BinomialPricingModel, Claim
Define a binomial pricing model.
>>> S = BinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... down_factor=0.9,
... up_prob=0.7,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Binomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 108.9
2 100.0 110.0 99.0 108.9
3 100.0 110.0 99.0 89.1
4 100.0 90.0 99.0 108.9
5 100.0 90.0 99.0 89.1
6 100.0 90.0 81.0 89.1
7 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 8.9
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 8.9
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 0.0
7 0.0 0.0 0.0 0.0
Generate the replicating portfolio for the European call.
>>> Delta, B, V = euro_call.replicating_portfolio()
Print the Delta and B price processes of the hedging portfolio, along with its value process V.
>>> print(Delta)
Stochastic process 'Delta':
t 0 1 2
omega
0 0.768707 0.882035 1.000000
1 0.768707 0.882035 1.000000
2 0.768707 0.882035 0.449495
3 0.768707 0.882035 0.449495
4 0.768707 0.353175 0.449495
5 0.768707 0.353175 0.449495
6 0.768707 0.353175 -0.000000
7 0.768707 0.353175 -0.000000
>>> print(B)
Stochastic process 'B':
t 0 1 2
omega
0 -61.564626 -77.108844 -95.238095
1 -61.564626 -77.108844 -95.238095
2 -61.564626 -77.108844 -38.142857
3 -61.564626 -77.108844 -38.142857
4 -61.564626 -27.244898 -38.142857
5 -61.564626 -27.244898 -38.142857
6 -61.564626 -27.244898 0.000000
7 -61.564626 -27.244898 0.000000
>>> print(V)
Stochastic process 'V':
t 0 1 2 3
omega
0 15.306122 19.914966 25.761905 33.1
1 15.306122 19.914966 25.761905 8.9
2 15.306122 19.914966 6.357143 8.9
3 15.306122 19.914966 6.357143 0.0
4 15.306122 4.540816 6.357143 8.9
5 15.306122 4.540816 6.357143 0.0
6 15.306122 4.540816 0.000000 0.0
7 15.306122 4.540816 0.000000 0.0
Now define an Asian put option with early-exercise and the same strike.
>>> asian_put = Claim.asian_option(strike=100, model=S, type="put", early_exercise=True, name="asian_put")
>>> print(asian_put)
Asian option 'asian_put':
t 0 1 2 3
omega
0 0.0 0.0 0.000000 0.000
1 0.0 0.0 0.000000 0.000
2 0.0 0.0 0.000000 0.000
3 0.0 0.0 0.000000 0.475
4 0.0 5.0 3.666667 0.525
5 0.0 5.0 3.666667 5.475
6 0.0 5.0 9.666667 9.975
7 0.0 5.0 9.666667 14.025
Compute and print the replicating portfolio.
>>> Delta, B, V = asian_put.replicating_portfolio()
>>> print(Delta)
Stochastic process 'Delta':
t 0 1 2
omega
0 -0.254181 -0.005141 -0.00000
1 -0.254181 -0.005141 -0.00000
2 -0.254181 -0.005141 -0.02399
3 -0.254181 -0.005141 -0.02399
4 -0.254181 -0.377646 -0.25000
5 -0.254181 -0.377646 -0.25000
6 -0.254181 -0.377646 -0.25000
7 -0.254181 -0.377646 -0.25000
>>> print(B)
Stochastic process 'B':
t 0 1 2
omega
0 26.654114 0.592404 0.000000
1 26.654114 0.592404 0.000000
2 26.654114 0.592404 2.488095
3 26.654114 0.592404 2.488095
4 26.654114 39.098639 26.428571
5 26.654114 39.098639 26.428571
6 26.654114 39.098639 30.714286
7 26.654114 39.098639 30.714286
>>> print(V)
Stochastic process 'V':
t 0 1 2 3
omega
0 1.23603 0.026927 0.000000 0.000
1 1.23603 0.026927 0.000000 0.000
2 1.23603 0.026927 0.113095 0.000
3 1.23603 0.026927 0.113095 0.475
4 1.23603 5.110544 1.678571 0.525
5 1.23603 5.110544 1.678571 5.475
6 1.23603 5.110544 10.464286 9.975
7 1.23603 5.110544 10.464286 14.025
Bases: DiscretePricingModel
A class modeling a trinomial pricing model.
The base constructor is not meant to be called directly by users. Instead, the user should call the generate class method. See the Examples section below for usage.
Examples:
>>> from sigalg.finance import TrinomialPricingModel
Given parameters of a trinomial model, generate all length-2 price trajectories in enum mode.
>>> S_0 = 4
>>> u = 1.2
>>> m = 1.1
>>> d = 0.9
>>> p_u = 0.6
>>> p_d = 0.1
>>> r = 0.01
>>> S = TrinomialPricingModel.generate(
... mode="enum",
... initial_price=S_0,
... up_factor=u,
... middle_factor=m,
... down_factor=d,
... up_prob=p_u,
... down_prob=p_d,
... risk_free_rate=r,
... length=2,
... )
>>> print(S)
Trinomial price process 'S':
t 0 1 2
omega
0 4.0 4.8 5.76
1 4.0 4.8 5.28
2 4.0 4.8 4.32
3 4.0 4.4 5.28
4 4.0 4.4 4.84
5 4.0 4.4 3.96
6 4.0 3.6 4.32
7 4.0 3.6 3.96
8 4.0 3.6 3.24
Simulate ten length-2 trajectories in sim mode.
>>> S.n_trajectories = 10
>>> S.random_state = 42
>>> S.mode = "sim"
>>> S.regenerate()
>>> print(S)
Trinomial price process 'S':
t 0 1 2
omega
0 4.0 4.4 5.28
1 4.0 4.4 4.84
2 4.0 4.8 4.32
3 4.0 4.4 4.84
4 4.0 4.8 5.76
5 4.0 4.8 4.32
6 4.0 4.4 4.84
7 4.0 4.8 5.76
8 4.0 4.8 5.76
9 4.0 4.4 4.84
property
down_factor
Get the down-factor of the model.
Returns:
| Name | Type | Description |
|---|---|---|
down_factor |
Real
|
The down-factor of the model. |
property
down_prob
Get the probability of a downward price movement.
Returns:
| Name | Type | Description |
|---|---|---|
down_prob |
Real
|
The probability of a downward price movement. |
property
initial_price
Get the initial price of the asset.
Returns:
| Name | Type | Description |
|---|---|---|
initial_price |
Real
|
The initial price of the asset. |
property
middle_factor
Get the middle-factor of the model.
Returns:
| Name | Type | Description |
|---|---|---|
middle_factor |
Real
|
The middle-factor of the model. |
property
middle_prob
Get the probability of a middle price movement.
Returns:
| Name | Type | Description |
|---|---|---|
down_prob |
Real
|
The probability of a middle price movement. |
property
risk_free_gross_return
Get the per-period risk-free gross return of the model.
Returns:
| Name | Type | Description |
|---|---|---|
risk_free_gross_return |
Real
|
The per-period risk-free gross return. |
property
risk_free_rate
Get the per-period risk-free rate.
Returns:
| Name | Type | Description |
|---|---|---|
risk_free_rate |
Real
|
The risk-free rate. |
property
up_factor
Get the up-factor of the model.
Returns:
| Name | Type | Description |
|---|---|---|
up_factor |
Real
|
The up-factor of the model. |
property
up_prob
Get the probability of an upward price movement.
Returns:
| Name | Type | Description |
|---|---|---|
up_prob |
Real
|
The probability of an upward price movement. |
B(*args, **kwargs)
Pass.
continuation_process(claim, theta, **kwargs)
Get the continuation/value process of the replicating portfolio of a claim given an equivalent martingale measure.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The contingent claim whose continuation/price process is to be computed. |
required |
theta
|
Real
|
A parameter in the open interval (0, 1) yielding a collection of risk-neutral probabilities of the model. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
continuation_process |
StochasticProcess
|
The continuation/value process of the replicating portfolio of the claim. |
Examples:
>>> from sigalg.finance import TrinomialPricingModel, Claim
Define a trinomial pricing model.
>>> S = TrinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... middle_factor=1.0,
... down_factor=0.9,
... up_prob=0.7,
... down_prob=0.2,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Trinomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 121.0
2 100.0 110.0 121.0 108.9
3 100.0 110.0 110.0 121.0
4 100.0 110.0 110.0 110.0
5 100.0 110.0 110.0 99.0
6 100.0 110.0 99.0 108.9
7 100.0 110.0 99.0 99.0
8 100.0 110.0 99.0 89.1
9 100.0 100.0 110.0 121.0
10 100.0 100.0 110.0 110.0
11 100.0 100.0 110.0 99.0
12 100.0 100.0 100.0 110.0
13 100.0 100.0 100.0 100.0
14 100.0 100.0 100.0 90.0
15 100.0 100.0 90.0 99.0
16 100.0 100.0 90.0 90.0
17 100.0 100.0 90.0 81.0
18 100.0 90.0 99.0 108.9
19 100.0 90.0 99.0 99.0
20 100.0 90.0 99.0 89.1
21 100.0 90.0 90.0 99.0
22 100.0 90.0 90.0 90.0
23 100.0 90.0 90.0 81.0
24 100.0 90.0 81.0 89.1
25 100.0 90.0 81.0 81.0
26 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 21.0
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 21.0
4 0.0 0.0 0.0 10.0
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 8.9
7 0.0 0.0 0.0 0.0
8 0.0 0.0 0.0 0.0
9 0.0 0.0 0.0 21.0
10 0.0 0.0 0.0 10.0
11 0.0 0.0 0.0 0.0
12 0.0 0.0 0.0 10.0
13 0.0 0.0 0.0 0.0
14 0.0 0.0 0.0 0.0
15 0.0 0.0 0.0 0.0
16 0.0 0.0 0.0 0.0
17 0.0 0.0 0.0 0.0
18 0.0 0.0 0.0 8.9
19 0.0 0.0 0.0 0.0
20 0.0 0.0 0.0 0.0
21 0.0 0.0 0.0 0.0
22 0.0 0.0 0.0 0.0
23 0.0 0.0 0.0 0.0
24 0.0 0.0 0.0 0.0
25 0.0 0.0 0.0 0.0
26 0.0 0.0 0.0 0.0
Get the risk-neutral continuation process of the call option for a specified parameter corresponding to risk-neutral probabilities.
>>> print(S.continuation_process(claim=euro_call, theta=0.2))
Stochastic process 'V':
t 0 1 2 3
omega
0 13.97158 19.358050 25.761905 33.1
1 13.97158 19.358050 25.761905 21.0
2 13.97158 19.358050 25.761905 8.9
3 13.97158 19.358050 14.809524 21.0
4 13.97158 19.358050 14.809524 10.0
5 13.97158 19.358050 14.809524 0.0
6 13.97158 19.358050 4.661905 8.9
7 13.97158 19.358050 4.661905 0.0
8 13.97158 19.358050 4.661905 0.0
9 13.97158 9.752834 14.809524 21.0
10 13.97158 9.752834 14.809524 10.0
11 13.97158 9.752834 14.809524 0.0
12 13.97158 9.752834 5.238095 10.0
13 13.97158 9.752834 5.238095 0.0
14 13.97158 9.752834 5.238095 0.0
15 13.97158 9.752834 0.000000 0.0
16 13.97158 9.752834 0.000000 0.0
17 13.97158 9.752834 0.000000 0.0
18 13.97158 2.441950 4.661905 8.9
19 13.97158 2.441950 4.661905 0.0
20 13.97158 2.441950 4.661905 0.0
21 13.97158 2.441950 0.000000 0.0
22 13.97158 2.441950 0.000000 0.0
23 13.97158 2.441950 0.000000 0.0
24 13.97158 2.441950 0.000000 0.0
25 13.97158 2.441950 0.000000 0.0
26 13.97158 2.441950 0.000000 0.0
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 0.0 0.0
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 0.0
5 0.0 0.0 0.0 1.0
6 0.0 0.0 1.0 0.0
7 0.0 0.0 1.0 1.0
8 0.0 0.0 1.0 10.9
9 0.0 0.0 0.0 0.0
10 0.0 0.0 0.0 0.0
11 0.0 0.0 0.0 1.0
12 0.0 0.0 0.0 0.0
13 0.0 0.0 0.0 0.0
14 0.0 0.0 0.0 10.0
15 0.0 0.0 10.0 1.0
16 0.0 0.0 10.0 10.0
17 0.0 0.0 10.0 19.0
18 0.0 10.0 1.0 0.0
19 0.0 10.0 1.0 1.0
20 0.0 10.0 1.0 10.9
21 0.0 10.0 10.0 1.0
22 0.0 10.0 10.0 10.0
23 0.0 10.0 10.0 19.0
24 0.0 10.0 19.0 10.9
25 0.0 10.0 19.0 19.0
26 0.0 10.0 19.0 27.1
Get the risk-neutral continuation process of the put option.
>>> print(S.continuation_process(claim=american_put, theta=0.2))
Stochastic process 'V':
t 0 1 2 3
omega
0 0.770651 0.065760 0.000000 0.0
1 0.770651 0.065760 0.000000 0.0
2 0.770651 0.065760 0.000000 0.0
3 0.770651 0.065760 0.047619 0.0
4 0.770651 0.065760 0.047619 0.0
5 0.770651 0.065760 0.047619 1.0
6 0.770651 0.065760 0.900000 0.0
7 0.770651 0.065760 0.900000 1.0
8 0.770651 0.065760 0.900000 10.9
9 0.770651 0.682540 0.047619 0.0
10 0.770651 0.682540 0.047619 0.0
11 0.770651 0.682540 0.047619 1.0
12 0.770651 0.682540 0.476190 0.0
13 0.770651 0.682540 0.476190 0.0
14 0.770651 0.682540 0.476190 10.0
15 0.770651 0.682540 5.238095 1.0
16 0.770651 0.682540 5.238095 10.0
17 0.770651 0.682540 5.238095 19.0
18 0.770651 5.238095 0.900000 0.0
19 0.770651 5.238095 0.900000 1.0
20 0.770651 5.238095 0.900000 10.9
21 0.770651 5.238095 5.238095 1.0
22 0.770651 5.238095 5.238095 10.0
23 0.770651 5.238095 5.238095 19.0
24 0.770651 5.238095 14.238095 10.9
25 0.770651 5.238095 14.238095 19.0
26 0.770651 5.238095 14.238095 27.1
delta(*args, **kwargs)
Pass.
emm(theta=None, **kwargs)
Get the equivalent martingale measures of the model.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
theta
|
Real | None
|
An optional real number in the open interval |
None
|
Returns:
| Name | Type | Description |
|---|---|---|
emm |
ProbabilityMeasure | ParametrizedProbabilityMeasure
|
The equivalent martingale measures of the model. |
Examples:
>>> from sigalg.finance import TrinomialPricingModel
Define the parameters of a trinomial pricing model.
>>> S_0 = 4
>>> r = 0.01
>>> p_u = 0.6
>>> p_d = 0.1
>>> u = 1.2
>>> m = 1.1
>>> d = 0.9
>>> T = 2
Exhaustively generate all price trajectories and print them.
>>> S = TrinomialPricingModel.generate(
... initial_price=S_0,
... risk_free_rate=r,
... up_prob=p_u,
... down_prob=p_d,
... up_factor=u,
... middle_factor=m,
... down_factor=d,
... mode="enum",
... length=T,
... )
>>> print(S)
Trinomial price process 'S':
t 0 1 2
omega
0 4.0 4.8 5.76
1 4.0 4.8 5.28
2 4.0 4.8 4.32
3 4.0 4.4 5.28
4 4.0 4.4 4.84
5 4.0 4.4 3.96
6 4.0 3.6 4.32
7 4.0 3.6 3.96
8 4.0 3.6 3.24
Get the equivalent martingale measures of the model as an instance of ParametrizedProbabilityMeasure.
>>> Q = S.emm()
>>> print(Q)
ParametrizedProbabilityMeasure(parameters=(theta), domain_vars=(omega), sig_alg=R, name=Q)
Get the EMM corresponding to a specific parameter value.
>>> print(Q(theta=0.2))
Probability measure 'Q(theta=0.2)':
Q(theta=0.2)
omega
0 0.005378
1 0.032267
2 0.035689
3 0.032267
4 0.193600
5 0.214133
6 0.035689
7 0.214133
8 0.236844
Alternativaly, pass the parameter value directly into the emm method to get the same EMM.
>>> print(S.emm(theta=0.2))
Probability measure 'Q':
Q
omega
0 0.005378
1 0.032267
2 0.035689
3 0.032267
4 0.193600
5 0.214133
6 0.035689
7 0.214133
8 0.236844
Check that the discounted model is a martingale with repsect to an EMM.
>>> S.discount(r).is_martingale(prob_measure=Q(theta=0.2))
True
exercise_time(claim, theta, **kwargs)
Get the exercise time of a claim given an equivalent martingale measure.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The contingent claim whose exercise times are to be compute. |
required |
theta
|
Real
|
A parameter in the open interval (0, 1) yielding a collection of risk-neutral probabilities of the model. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
exercise_time |
StoppingTime
|
The exercise time of the claim. |
Examples:
>>> from sigalg.finance import TrinomialPricingModel, Claim
Define a trinomial pricing model.
>>> S = TrinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... middle_factor=1.0,
... down_factor=0.9,
... up_prob=0.7,
... down_prob=0.2,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Trinomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 121.0
2 100.0 110.0 121.0 108.9
3 100.0 110.0 110.0 121.0
4 100.0 110.0 110.0 110.0
5 100.0 110.0 110.0 99.0
6 100.0 110.0 99.0 108.9
7 100.0 110.0 99.0 99.0
8 100.0 110.0 99.0 89.1
9 100.0 100.0 110.0 121.0
10 100.0 100.0 110.0 110.0
11 100.0 100.0 110.0 99.0
12 100.0 100.0 100.0 110.0
13 100.0 100.0 100.0 100.0
14 100.0 100.0 100.0 90.0
15 100.0 100.0 90.0 99.0
16 100.0 100.0 90.0 90.0
17 100.0 100.0 90.0 81.0
18 100.0 90.0 99.0 108.9
19 100.0 90.0 99.0 99.0
20 100.0 90.0 99.0 89.1
21 100.0 90.0 90.0 99.0
22 100.0 90.0 90.0 90.0
23 100.0 90.0 90.0 81.0
24 100.0 90.0 81.0 89.1
25 100.0 90.0 81.0 81.0
26 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 21.0
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 21.0
4 0.0 0.0 0.0 10.0
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 8.9
7 0.0 0.0 0.0 0.0
8 0.0 0.0 0.0 0.0
9 0.0 0.0 0.0 21.0
10 0.0 0.0 0.0 10.0
11 0.0 0.0 0.0 0.0
12 0.0 0.0 0.0 10.0
13 0.0 0.0 0.0 0.0
14 0.0 0.0 0.0 0.0
15 0.0 0.0 0.0 0.0
16 0.0 0.0 0.0 0.0
17 0.0 0.0 0.0 0.0
18 0.0 0.0 0.0 8.9
19 0.0 0.0 0.0 0.0
20 0.0 0.0 0.0 0.0
21 0.0 0.0 0.0 0.0
22 0.0 0.0 0.0 0.0
23 0.0 0.0 0.0 0.0
24 0.0 0.0 0.0 0.0
25 0.0 0.0 0.0 0.0
26 0.0 0.0 0.0 0.0
Get the exercise time of the call option for a specified parameter corresponding to risk-neutral probabilities.
>>> print(S.exercise_time(claim=euro_call, theta=0.2))
Stopping time 'tau':
tau
omega
0 3.0
1 3.0
2 3.0
3 3.0
4 3.0
5 inf
6 3.0
7 inf
8 inf
9 3.0
10 3.0
11 inf
12 3.0
13 inf
14 inf
15 inf
16 inf
17 inf
18 3.0
19 inf
20 inf
21 inf
22 inf
23 inf
24 inf
25 inf
26 inf
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 0.0 0.0
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 0.0
5 0.0 0.0 0.0 1.0
6 0.0 0.0 1.0 0.0
7 0.0 0.0 1.0 1.0
8 0.0 0.0 1.0 10.9
9 0.0 0.0 0.0 0.0
10 0.0 0.0 0.0 0.0
11 0.0 0.0 0.0 1.0
12 0.0 0.0 0.0 0.0
13 0.0 0.0 0.0 0.0
14 0.0 0.0 0.0 10.0
15 0.0 0.0 10.0 1.0
16 0.0 0.0 10.0 10.0
17 0.0 0.0 10.0 19.0
18 0.0 10.0 1.0 0.0
19 0.0 10.0 1.0 1.0
20 0.0 10.0 1.0 10.9
21 0.0 10.0 10.0 1.0
22 0.0 10.0 10.0 10.0
23 0.0 10.0 10.0 19.0
24 0.0 10.0 19.0 10.9
25 0.0 10.0 19.0 19.0
26 0.0 10.0 19.0 27.1
Get the exercise time of the put option.
>>> print(S.exercise_time(claim=american_put, theta=0.2))
Stopping time 'tau':
tau
omega
0 inf
1 inf
2 inf
3 inf
4 inf
5 3.0
6 2.0
7 2.0
8 2.0
9 inf
10 inf
11 3.0
12 inf
13 inf
14 3.0
15 2.0
16 2.0
17 2.0
18 1.0
19 1.0
20 1.0
21 1.0
22 1.0
23 1.0
24 1.0
25 1.0
26 1.0
generate(
initial_price,
up_factor,
middle_factor,
down_factor,
up_prob=1 / 3,
down_prob=1 / 3,
risk_free_rate=None,
risk_free_gross_return=None,
mode="sim",
n_trajectories=None,
domain_name=None,
index=None,
length=None,
name="S",
random_state=None,
)
Generate trajectories of the trinomial pricing model by either exhaustive enumeration or Monte Carlo simulation.
See the Notes section below for the mathematical details.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
initial_price
|
Real
|
The initial price of the asset. |
required |
up_factor
|
Real
|
The up-factor of the model. |
required |
middle_factor
|
Real
|
The middle-factor of the model. |
required |
down_factor
|
Real
|
The down-factor of the model. |
required |
up_prob
|
Real
|
The probability of an upward move in the price. |
1 / 3
|
down_prob
|
Real
|
The probability of an upward move in the price. |
1 / 3
|
risk_free_rate
|
Real | None
|
The per-period risk-free rate. If |
None
|
risk_free_gross_return
|
Real | None
|
The per-period risk-free gross return. If |
None
|
mode
|
Literal['enum', 'sim']
|
Whether to generate trajectories by exhuastive enumeration or Monte Carlo simulation. |
'sim'
|
n_trajectories
|
int | None
|
The number of trajectories to simulate. If the generation mode is set to |
None
|
index
|
IndexLike | None
|
The index of the stochastic process. One of |
None
|
length
|
int | None
|
The length of the trajectories of the stochastic process. One of |
None
|
name
|
Hashable
|
The name of the stochastic process. |
"S"
|
random_state
|
int | Generator | None
|
An optional random state for reproducibility. |
None
|
Returns:
| Name | Type | Description |
|---|---|---|
info |
dict[str, object]
|
A dictionary containing the names of the BinomialPricingModel-specific parameters and their values. |
Examples:
>>> from sigalg.finance import TrinomialPricingModel
Given parameters of a trinomial model, generate all length-2 price trajectories in enum mode.
>>> S_0 = 4
>>> u = 1.2
>>> m = 1.1
>>> d = 0.9
>>> p_u = 0.6
>>> p_d = 0.1
>>> r = 0.01
>>> S = TrinomialPricingModel.generate(
... mode="enum",
... initial_price=S_0,
... up_factor=u,
... middle_factor=m,
... down_factor=d,
... up_prob=p_u,
... down_prob=p_d,
... risk_free_rate=r,
... length=2,
... )
>>> print(S)
Trinomial price process 'S':
t 0 1 2
omega
0 4.0 4.8 5.76
1 4.0 4.8 5.28
2 4.0 4.8 4.32
3 4.0 4.4 5.28
4 4.0 4.4 4.84
5 4.0 4.4 3.96
6 4.0 3.6 4.32
7 4.0 3.6 3.96
8 4.0 3.6 3.24
Simulate ten length-2 trajectories in sim mode.
>>> S.n_trajectories = 10
>>> S.random_state = 42
>>> S.mode = "sim"
>>> S.regenerate()
>>> print(S)
Trinomial price process 'S':
t 0 1 2
omega
0 4.0 4.4 5.28
1 4.0 4.4 4.84
2 4.0 4.8 4.32
3 4.0 4.4 4.84
4 4.0 4.8 5.76
5 4.0 4.8 4.32
6 4.0 4.4 4.84
7 4.0 4.8 5.76
8 4.0 4.8 5.76
9 4.0 4.4 4.84
price(claim, theta, **kwargs)
Get the risk-neutral price of a claim given an equivalent martingale measure.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The contingent claim to price. |
required |
theta
|
Real
|
A parameter in the open interval (0, 1) yielding a collection of risk-neutral probabilities of the model. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
price |
Real
|
The risk-neutral price of the claim. |
Examples:
>>> from sigalg.finance import TrinomialPricingModel, Claim
Define a trinomial pricing model.
>>> S = TrinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... middle_factor=1.0,
... down_factor=0.9,
... up_prob=0.7,
... down_prob=0.2,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Trinomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 121.0
2 100.0 110.0 121.0 108.9
3 100.0 110.0 110.0 121.0
4 100.0 110.0 110.0 110.0
5 100.0 110.0 110.0 99.0
6 100.0 110.0 99.0 108.9
7 100.0 110.0 99.0 99.0
8 100.0 110.0 99.0 89.1
9 100.0 100.0 110.0 121.0
10 100.0 100.0 110.0 110.0
11 100.0 100.0 110.0 99.0
12 100.0 100.0 100.0 110.0
13 100.0 100.0 100.0 100.0
14 100.0 100.0 100.0 90.0
15 100.0 100.0 90.0 99.0
16 100.0 100.0 90.0 90.0
17 100.0 100.0 90.0 81.0
18 100.0 90.0 99.0 108.9
19 100.0 90.0 99.0 99.0
20 100.0 90.0 99.0 89.1
21 100.0 90.0 90.0 99.0
22 100.0 90.0 90.0 90.0
23 100.0 90.0 90.0 81.0
24 100.0 90.0 81.0 89.1
25 100.0 90.0 81.0 81.0
26 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 21.0
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 21.0
4 0.0 0.0 0.0 10.0
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 8.9
7 0.0 0.0 0.0 0.0
8 0.0 0.0 0.0 0.0
9 0.0 0.0 0.0 21.0
10 0.0 0.0 0.0 10.0
11 0.0 0.0 0.0 0.0
12 0.0 0.0 0.0 10.0
13 0.0 0.0 0.0 0.0
14 0.0 0.0 0.0 0.0
15 0.0 0.0 0.0 0.0
16 0.0 0.0 0.0 0.0
17 0.0 0.0 0.0 0.0
18 0.0 0.0 0.0 8.9
19 0.0 0.0 0.0 0.0
20 0.0 0.0 0.0 0.0
21 0.0 0.0 0.0 0.0
22 0.0 0.0 0.0 0.0
23 0.0 0.0 0.0 0.0
24 0.0 0.0 0.0 0.0
25 0.0 0.0 0.0 0.0
26 0.0 0.0 0.0 0.0
Get the risk-neutral price of the call option for a specified parameter corresponding to risk-neutral probabilities.
>>> S.price(claim=euro_call, theta=0.2)
13.971579743008329
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 0.0 0.0
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 0.0
5 0.0 0.0 0.0 1.0
6 0.0 0.0 1.0 0.0
7 0.0 0.0 1.0 1.0
8 0.0 0.0 1.0 10.9
9 0.0 0.0 0.0 0.0
10 0.0 0.0 0.0 0.0
11 0.0 0.0 0.0 1.0
12 0.0 0.0 0.0 0.0
13 0.0 0.0 0.0 0.0
14 0.0 0.0 0.0 10.0
15 0.0 0.0 10.0 1.0
16 0.0 0.0 10.0 10.0
17 0.0 0.0 10.0 19.0
18 0.0 10.0 1.0 0.0
19 0.0 10.0 1.0 1.0
20 0.0 10.0 1.0 10.9
21 0.0 10.0 10.0 1.0
22 0.0 10.0 10.0 10.0
23 0.0 10.0 10.0 19.0
24 0.0 10.0 19.0 10.9
25 0.0 10.0 19.0 19.0
26 0.0 10.0 19.0 27.1
Get the risk-neutral price of the put option.
>>> S.price(claim=american_put, theta=0.2)
0.7706511175898927
price_process(claim, theta, **kwargs)
Get the risk-neutral price process of a claim given an equivalent martingale measure.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
claim
|
Claim
|
The contingent claim whose price process is to be computed. |
required |
theta
|
Real
|
A parameter in the open interval (0, 1) yielding a collection of risk-neutral probabilities of the model. |
required |
Returns:
| Name | Type | Description |
|---|---|---|
price_process |
StochasticProcess
|
The risk-neutral price process of the claim. |
Examples:
>>> from sigalg.finance import TrinomialPricingModel, Claim
Define a trinomial pricing model.
>>> S = TrinomialPricingModel.generate(
... initial_price=100,
... up_factor=1.1,
... middle_factor=1.0,
... down_factor=0.9,
... up_prob=0.7,
... down_prob=0.2,
... risk_free_rate=0.05,
... mode="enum",
... length=3,
... )
>>> print(S)
Trinomial price process 'S':
t 0 1 2 3
omega
0 100.0 110.0 121.0 133.1
1 100.0 110.0 121.0 121.0
2 100.0 110.0 121.0 108.9
3 100.0 110.0 110.0 121.0
4 100.0 110.0 110.0 110.0
5 100.0 110.0 110.0 99.0
6 100.0 110.0 99.0 108.9
7 100.0 110.0 99.0 99.0
8 100.0 110.0 99.0 89.1
9 100.0 100.0 110.0 121.0
10 100.0 100.0 110.0 110.0
11 100.0 100.0 110.0 99.0
12 100.0 100.0 100.0 110.0
13 100.0 100.0 100.0 100.0
14 100.0 100.0 100.0 90.0
15 100.0 100.0 90.0 99.0
16 100.0 100.0 90.0 90.0
17 100.0 100.0 90.0 81.0
18 100.0 90.0 99.0 108.9
19 100.0 90.0 99.0 99.0
20 100.0 90.0 99.0 89.1
21 100.0 90.0 90.0 99.0
22 100.0 90.0 90.0 90.0
23 100.0 90.0 90.0 81.0
24 100.0 90.0 81.0 89.1
25 100.0 90.0 81.0 81.0
26 100.0 90.0 81.0 72.9
Define a European call option with strike K=100.
>>> K = 100
>>> euro_call = Claim.european_option(model=S, strike=K, type="call", name="euro_call")
>>> print(euro_call)
European option 'euro_call':
t 0 1 2 3
omega
0 0.0 0.0 0.0 33.1
1 0.0 0.0 0.0 21.0
2 0.0 0.0 0.0 8.9
3 0.0 0.0 0.0 21.0
4 0.0 0.0 0.0 10.0
5 0.0 0.0 0.0 0.0
6 0.0 0.0 0.0 8.9
7 0.0 0.0 0.0 0.0
8 0.0 0.0 0.0 0.0
9 0.0 0.0 0.0 21.0
10 0.0 0.0 0.0 10.0
11 0.0 0.0 0.0 0.0
12 0.0 0.0 0.0 10.0
13 0.0 0.0 0.0 0.0
14 0.0 0.0 0.0 0.0
15 0.0 0.0 0.0 0.0
16 0.0 0.0 0.0 0.0
17 0.0 0.0 0.0 0.0
18 0.0 0.0 0.0 8.9
19 0.0 0.0 0.0 0.0
20 0.0 0.0 0.0 0.0
21 0.0 0.0 0.0 0.0
22 0.0 0.0 0.0 0.0
23 0.0 0.0 0.0 0.0
24 0.0 0.0 0.0 0.0
25 0.0 0.0 0.0 0.0
26 0.0 0.0 0.0 0.0
Get the risk-neutral price process of the call option for a specified parameter corresponding to risk-neutral probabilities.
>>> print(S.price_process(claim=euro_call, theta=0.2))
Stochastic process 'X':
t 0 1 2 3
omega
0 13.97158 19.358050 25.761905 33.1
1 13.97158 19.358050 25.761905 21.0
2 13.97158 19.358050 25.761905 8.9
3 13.97158 19.358050 14.809524 21.0
4 13.97158 19.358050 14.809524 10.0
5 13.97158 19.358050 14.809524 0.0
6 13.97158 19.358050 4.661905 8.9
7 13.97158 19.358050 4.661905 0.0
8 13.97158 19.358050 4.661905 0.0
9 13.97158 9.752834 14.809524 21.0
10 13.97158 9.752834 14.809524 10.0
11 13.97158 9.752834 14.809524 0.0
12 13.97158 9.752834 5.238095 10.0
13 13.97158 9.752834 5.238095 0.0
14 13.97158 9.752834 5.238095 0.0
15 13.97158 9.752834 0.000000 0.0
16 13.97158 9.752834 0.000000 0.0
17 13.97158 9.752834 0.000000 0.0
18 13.97158 2.441950 4.661905 8.9
19 13.97158 2.441950 4.661905 0.0
20 13.97158 2.441950 4.661905 0.0
21 13.97158 2.441950 0.000000 0.0
22 13.97158 2.441950 0.000000 0.0
23 13.97158 2.441950 0.000000 0.0
24 13.97158 2.441950 0.000000 0.0
25 13.97158 2.441950 0.000000 0.0
26 13.97158 2.441950 0.000000 0.0
Define an American put option with the same strike.
>>> american_put = Claim.american_option(model=S, strike=K, type="put", name="american_put")
>>> print(american_put)
American option 'american_put':
t 0 1 2 3
omega
0 0.0 0.0 0.0 0.0
1 0.0 0.0 0.0 0.0
2 0.0 0.0 0.0 0.0
3 0.0 0.0 0.0 0.0
4 0.0 0.0 0.0 0.0
5 0.0 0.0 0.0 1.0
6 0.0 0.0 1.0 0.0
7 0.0 0.0 1.0 1.0
8 0.0 0.0 1.0 10.9
9 0.0 0.0 0.0 0.0
10 0.0 0.0 0.0 0.0
11 0.0 0.0 0.0 1.0
12 0.0 0.0 0.0 0.0
13 0.0 0.0 0.0 0.0
14 0.0 0.0 0.0 10.0
15 0.0 0.0 10.0 1.0
16 0.0 0.0 10.0 10.0
17 0.0 0.0 10.0 19.0
18 0.0 10.0 1.0 0.0
19 0.0 10.0 1.0 1.0
20 0.0 10.0 1.0 10.9
21 0.0 10.0 10.0 1.0
22 0.0 10.0 10.0 10.0
23 0.0 10.0 10.0 19.0
24 0.0 10.0 19.0 10.9
25 0.0 10.0 19.0 19.0
26 0.0 10.0 19.0 27.1
Get the risk-neutral price process of the put option.
>>> print(S.price_process(claim=american_put, theta=0.2))
Stochastic process 'X':
t 0 1 2 3
omega
0 0.770651 0.06576 0.000000 0.0
1 0.770651 0.06576 0.000000 0.0
2 0.770651 0.06576 0.000000 0.0
3 0.770651 0.06576 0.047619 0.0
4 0.770651 0.06576 0.047619 0.0
5 0.770651 0.06576 0.047619 1.0
6 0.770651 0.06576 1.000000 0.0
7 0.770651 0.06576 1.000000 1.0
8 0.770651 0.06576 1.000000 10.9
9 0.770651 0.68254 0.047619 0.0
10 0.770651 0.68254 0.047619 0.0
11 0.770651 0.68254 0.047619 1.0
12 0.770651 0.68254 0.476190 0.0
13 0.770651 0.68254 0.476190 0.0
14 0.770651 0.68254 0.476190 10.0
15 0.770651 0.68254 10.000000 1.0
16 0.770651 0.68254 10.000000 10.0
17 0.770651 0.68254 10.000000 19.0
18 0.770651 10.00000 1.000000 0.0
19 0.770651 10.00000 1.000000 1.0
20 0.770651 10.00000 1.000000 10.9
21 0.770651 10.00000 10.000000 1.0
22 0.770651 10.00000 10.000000 10.0
23 0.770651 10.00000 10.000000 19.0
24 0.770651 10.00000 19.000000 10.9
25 0.770651 10.00000 19.000000 19.0
26 0.770651 10.00000 19.000000 27.1
replicating_portfolio(*args, **kwargs)
Pass.
risk_neutral_probs(theta, **kwargs)
Get the risk-neutral probabilities of the model given a parameter.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
theta
|
Real
|
A real number in the open interval |
required |
Returns:
| Name | Type | Description |
|---|---|---|
risk_neutral_probs |
tuple[Real, Real, Real]
|
The risk neutral probabilities as a tuple |
Examples:
>>> from sigalg.finance import TrinomialPricingModel
Define the parameters of a trinomial pricing model.
>>> S_0 = 4
>>> r = 0.01
>>> p_u = 0.6
>>> p_d = 0.1
>>> u = 1.2
>>> m = 1.1
>>> d = 0.9
>>> T = 2
Exhaustively generate all price trajectories and print them.
>>> S = TrinomialPricingModel.generate(
... initial_price=S_0,
... risk_free_rate=r,
... up_prob=p_u,
... down_prob=p_d,
... up_factor=u,
... middle_factor=m,
... down_factor=d,
... mode="enum",
... length=T,
... )
>>> print(S)
Trinomial price process 'S':
t 0 1 2
omega
0 4.0 4.8 5.76
1 4.0 4.8 5.28
2 4.0 4.8 4.32
3 4.0 4.4 5.28
4 4.0 4.4 4.84
5 4.0 4.4 3.96
6 4.0 3.6 4.32
7 4.0 3.6 3.96
8 4.0 3.6 3.24
The risk-neutral probabilities are a one-parameter family of probabilities. Get these probabilities for a specific parameter.
>>> S.risk_neutral_probs(theta=0.42)
(0.15400000000000003, 0.31899999999999973, 0.5270000000000001)