romanmichaelpaolucci / q-fin

A Python library for mathematical finance

Q-Fin

A Python library for mathematical finance.

Installation

https://pypi.org/project/QFin/

pip install qfin

Version '0.1.20'

QFin is being reconstructed to leverage more principals of object-oriented programming. Several modules in this version are deprecated along with the solutions to PDEs/SDEs (mainly in the options module).

QFin now contains a module called 'stochastics' which will be largely responsible for model calibration and option pricing. A Cython/C++ equivalent to QFin is also being constructed so stay tuned!

Option Pricing (>= 0.1.20)

Stochastic differential equations that model underlying asset dynamics extend the 'StochasticModel' class and posses a list of model parameters and functions for pricing vanillas, calibrating to implied volatility surfaces, and Monte Carlo simulations (particularly useful after calibration for pricing path dependent options).

Below is a trivial example using ArithmeticBrownianMotion - first import the StochasticModel...

from qfin.stochastics import ArithmeticBrownianMotion

Next initialize the class object by parameterizing the model...

# abm parameterized by Bachelier vol = .3
abm = ArithmeticBrownianMotion([.3])

The abm may now be used to price a vanilla call/put option (prices default to "CALL") under the given parameter set...

# F0 = 101
# X = 100
# T = 1
abm.vanilla_pricing(101, 100, 1, "CALL")
# Call Price: 1.0000336233656906

Using call-put parity put prices may also be obtained...

# F0 = 99
# X = 100
# T = 1
abm.vanilla_pricing(99, 100, 1, "PUT")
# Put Price: 1.0000336233656952

Calibration and subsequent simulation of the process is also available - do note that some processes have a static volatility and can't be calibrated to an ivol surface.

The arithmetic Brownian motion may be simulated as follows...

# F0 = 100
# n (steps) = 10000
# dt = 1/252
# T = 1
abm.simulate(100, 10000, 1/252, 1)

Results of the simulation along with the simulation characteristics are stored under the tuple 'path_characteristics' : (paths, n, dt, T).

Using the stored path characteristics we may find the price of a call just as before by averaging each discounted path payoff (assuming a stock process) with zero-rates we can avoid discounting as follows and find the option value as follows...

# list of path payoffs
payoffs = []
# option strike price
X = 99

# iteration through terminal path values to identify payoff
for path in abm.path_characteristics[0]:
    # appending CALL payoff
    payoffs.append(max((path[-1] - X), 0))

# option value today
np.average(payoffs)

# Call Price:  1.0008974837343871

We can see here that the simulated price is converging to the price in close-form.

Option Pricing (deprecated <= 0.0.20)

Black-Scholes Pricing

Theoretical options pricing for non-dividend paying stocks is available via the BlackScholesCall and BlackScholesPut classes.

from qfin.options import BlackScholesCall
from qfin.options import BlackScholesPut
# 100 - initial underlying asset price
# .3 - asset underlying volatility
# 100 - option strike price
# 1 - time to maturity (annum)
# .01 - risk free rate of interest
euro_call = BlackScholesCall(100, .3, 100, 1, .01)
euro_put = BlackScholesPut(100, .3, 100, 1, .01)

print('Call price: ', euro_call.price)
print('Put price: ', euro_put.price)

Call price:  12.361726191532611
Put price:  11.366709566449416

Option Greeks

First-order and some second-order partial derivatives of the Black-Scholes pricing model are available.

Delta

First-order partial derivative with respect to the underlying asset price.

print('Call delta: ', euro_call.delta)
print('Put delta: ', euro_put.delta)

Call delta:  0.5596176923702425
Put delta:  -0.4403823076297575

Gamma

Second-order partial derivative with respect to the underlying asset price.

print('Call gamma: ', euro_call.gamma)
print('Put gamma: ', euro_put.gamma)

Call gamma:  0.018653923079008084
Put gamma:  0.018653923079008084

Vega

First-order partial derivative with respect to the underlying asset volatility.

print('Call vega: ', euro_call.vega)
print('Put vega: ', euro_put.vega)

Call vega:  39.447933090788894
Put vega:  39.447933090788894

Theta

First-order partial derivative with respect to the time to maturity.

print('Call theta: ', euro_call.theta)
print('Put theta: ', euro_put.theta)

Call theta:  -6.35319039407325
Put theta:  -5.363140560324083

Stochastic Processes

Simulating asset paths is available using common stochastic processes.

Geometric Brownian Motion

Standard model for implementing geometric Brownian motion.

from qfin.simulations import GeometricBrownianMotion
# 100 - initial underlying asset price
# 0 - underlying asset drift (mu)
# .3 - underlying asset volatility
# 1/52 - time steps (dt)
# 1 - time to maturity (annum)
gbm = GeometricBrownianMotion(100, 0, .3, 1/52, 1)

print(gbm.simulated_path)

[107.0025048205179, 104.82320056538235, 102.53591127422398, 100.20213816642244, 102.04283245358256, 97.75115579923988, 95.19613943526382, 96.9876745495834, 97.46055174410736, 103.93032659279226, 107.36331603194304, 108.95104494118915, 112.42823319947456, 109.06981862825943, 109.10124426285238, 114.71465058375804, 120.00234814086286, 116.91730159923688, 118.67452601825876, 117.89233466917202, 118.93541257993591, 124.36106523035058, 121.26088015675688, 120.53641952983601, 113.73881043255554, 114.91724168548876, 112.94192281337791, 113.55773877160591, 107.49491796151044, 108.0715118831013, 113.01893111071472, 110.39204535739405, 108.63917240906524, 105.8520395233433, 116.2907247951675, 114.07340779267213, 111.06821275009212, 109.65530380775077, 105.78971667172465, 97.75385009989282, 97.84501925249452, 101.90695475825825, 106.0493833583297, 105.48266575656817, 106.62375752876223, 112.39829297429974, 111.22855058562658, 109.89796974828265, 112.78068777325248, 117.80550869036715, 118.4680557054793, 114.33258212280838]

Stochastic Variance Process

Stochastic volatility model based on Heston's paper (1993).

from qfin.simulations import StochasticVarianceModel
# 100 - initial underlying asset price
# 0 - underlying asset drift (mu)
# .01 - risk free rate of interest
# .05 - continuous dividend
# 2 - rate in which variance reverts to the implied long run variance
# .25 - implied long run variance as time tends to infinity
# -.7 - correlation of motion generated
# .3 - Variance's volatility
# 1/52 - time steps (dt)
# 1 - time to maturity (annum)
svm = StochasticVarianceModel(100, 0, .01, .05, 2, .25, -.7, .3, .09, 1/52, 1)

print(svm.simulated_path)

[98.21311553503577, 100.4491317019877, 89.78475515902066, 89.0169762497475, 90.70468848525869, 86.00821802256675, 80.74984494892573, 89.05033807013137, 88.51410029337134, 78.69736798230346, 81.90948751054125, 83.02502248913251, 83.46375102829755, 85.39018282900138, 78.97401642238059, 78.93505221741903, 81.33268688455111, 85.12156706038515, 79.6351983987908, 84.2375291273571, 82.80206517176038, 89.63659376223292, 89.22438477640516, 89.13899271995662, 94.60123239511816, 91.200165507022, 96.0578905115345, 87.45399399599378, 97.908745925816, 97.93068975065052, 103.32091104292813, 110.58066464778392, 105.21520242908348, 99.4655106985056, 106.74882010453683, 112.0058519886151, 110.20930861932342, 105.11835510815085, 113.59852610881678, 107.13315204738092, 108.36549026977205, 113.49809943785571, 122.67910031073885, 137.70966794451425, 146.13877267735612, 132.9973784430374, 129.75750117504984, 128.7467891695649, 127.13115959080305, 130.47967713110302, 129.84273088908265, 129.6411527208744]

Simulation Pricing

Exotic Options

Simulation pricing for exotic options is available under the assumptions associated with the respective stochastic processes. Geometric Brownian motion is the base underlying stochastic process used in each Monte Carlo simulation. However, should additional parameters be provided, the appropriate stochastic process will be used to generate each sample path.

Vanilla Options

from qfin.simulations import MonteCarloCall
from qfin.simulations import MonteCarloPut
# 100 - strike price
# 1000 - number of simulated price paths
# .01 - risk free rate of interest
# 100 - initial underlying asset price
# 0 - underlying asset drift (mu)
# .3 - underlying asset volatility
# 1/52 - time steps (dt)
# 1 - time to maturity (annum)
call_option = MonteCarloCall(100, 1000, .01, 100, 0, .3, 1/52, 1)
# These additional parameters will generate a Monte Carlo price based on a stochastic volatility process
# 2 - rate in which variance reverts to the implied long run variance
# .25 - implied long run variance as time tends to infinity
# -.5 - correlation of motion generated
# .02 - continuous dividend
# .3 - Variance's volatility
put_option = MonteCarloPut(100, 1000, .01, 100, 0, .3, 1/52, 1, 2, .25, -.5, .02, .3)

print(call_option.price)
print(put_option.price)

12.73812121792851
23.195814963576286

Binary Options

from qfin.simulations import MonteCarloBinaryCall
from qfin.simulations import MonteCarloBinaryPut
# 100 - strike price
# 50 - binary option payout
# 1000 - number of simulated price paths
# .01 - risk free rate of interest
# 100 - initial underlying asset price
# 0 - underlying asset drift (mu)
# .3 - underlying asset volatility 
# 1/52 - time steps (dt)
# 1 - time to maturity (annum)
binary_call = MonteCarloBinaryCall(100, 50, 1000, .01, 100, 0, .3, 1/52, 1)
binary_put = MonteCarloBinaryPut(100, 50, 1000, .01, 100, 0, .3, 1/52, 1)

print(binary_call.price)
print(binary_put.price)

22.42462873441866
27.869902820039087

Barrier Options

from qfin.simulations import MonteCarloBarrierCall
from qfin.simulations import MonteCarloBarrierPut
# 100 - strike price
# 50 - binary option payout
# 1000 - number of simulated price paths
# .01 - risk free rate of interest
# 100 - initial underlying asset price
# 0 - underlying asset drift (mu)
# .3 - underlying asset volatility
# 1/52 - time steps (dt)
# 1 - time to maturity (annum)
# True/False - Barrier is Up or Down
# True/False - Barrier is In or Out
barrier_call = MonteCarloBarrierCall(100, 1000, 150, .01, 100, 0, .3, 1/52, 1, up=True, out=True)
barrier_put = MonteCarloBarrierCall(100, 1000, 95, .01, 100, 0, .3, 1/52, 1, up=False, out=False)

print(binary_call.price)
print(binary_put.price)

4.895841997908933
5.565856754630819

Asian Options

from qfin.simulations import MonteCarloAsianCall
from qfin.simulations import MonteCarloAsianPut
# 100 - strike price
# 1000 - number of simulated price paths
# .01 - risk free rate of interest
# 100 - initial underlying asset price
# 0 - underlying asset drift (mu)
# .3 - underlying asset volatility
# 1/52 - time steps (dt)
# 1 - time to maturity (annum)
asian_call = MonteCarloAsianCall(100, 1000, .01, 100, 0, .3, 1/52, 1)
asian_put = MonteCarloAsianPut(100, 1000, .01, 100, 0, .3, 1/52, 1)

print(asian_call.price)
print(asian_put.price)

6.688201154529573
7.123274528125894

Extendible Options

from qfin.simulations import MonteCarloExtendibleCall
from qfin.simulations import MontecarloExtendiblePut
# 100 - strike price
# 1000 - number of simulated price paths
# .01 - risk free rate of interest
# 100 - initial underlying asset price
# 0 - underlying asset drift (mu)
# .3 - underlying asset volatility
# 1/52 - time steps (dt)
# 1 - time to maturity (annum)
# .5 - extension if out of the money at expiration
extendible_call = MonteCarloExtendibleCall(100, 1000, .01, 100, 0, .3, 1/52, 1, .5)
extendible_put = MonteCarloExtendiblePut(100, 1000, .01, 100, 0, .3, 1/52, 1, .5)

print(extendible_call.price)
print(extendible_put.price)

13.60274931789973
13.20330578685724

q-fin's Issues

Problem with the Geometric Brownian Motion Code from YT video

I get the following error:

---

TypeError Traceback (most recent call last)
Cell In [19], line 1
----> 1 sample_paths = geometric_Brownian_motion (S_0, mu, sigma, T, dt, n)

Cell In [17], line 13, in geometric_Brownian_motion(S_0, mu, sigma, T, dt, n)
10 prices.append( prices[-1]np.exp((mu - .5(sigma2))dt + sigmanp.random.normal(0, np.sqrt(dt))))
11 time += dt
---> 13 if T (time) > 0:
14 prices.append(prices[-1]np.exp((mu - .5(sigma2))(T-time) + sigmanp.random.normal(0, np.sqrt(T-time))))
16 paths.append(prices)

TypeError: 'int' object is not callable

under the following code:
sample_paths = geometric_Brownian_motion (S_0, mu, sigma, T, dt, n)

Backtesting code error - no 'modules.isolation_model' found

I'm enjoying your Medium article and have implemented the code but its giving errors.

Are these suppsed to be part of one huge class or is there a function not included? I've expanded upon your code in my repo here for reference since I didnt see it in yours. Thanks for writing these articles they are good reference material.

box/PycharmProjects/Financial_Backtrader/backtesting_b.py", line 3, in <module>
    from models.isolation_model import IsolationModel
ModuleNotFoundError: No module named 'models'

datatalking/Financial_Backtrader#1

Negative Put Prices

The library produces negative prices for put options having small delta. Calls don't have this behaviour (price tends to zero, as expected).

i.e compare:

BlackScholesCall(4600, 0.19, 4400, float(5)/365, 0.01).price # 201.4
BlackScholesPut(4600, 0.19, 4400, float(5)/365, 0.01).price # -190.6

or see (where price and delta columns are derived from q-fin BlackScholesPut)

I'm wondering what I should make of this -- whether there's an error with the library, or a quirk of BS model (in which case I can take a floor of zero of put price value).