03_european_call_option_pricing: adding Black Scholes formulas

[alonso-pena]

Hello, I am writing with a small suggestion regarding the Qiskit Tutorial: 03 European Call Option Pricing.

I have noticed that inside this notebook the exact price of the European Call option is computed from:

exact_value = np.dot(uncertainty_model.probabilities, y)

and is given as "Exact value: 0.1623", which compares with the estimate from the IQAE "Estimated value: 0.1708".

However, I would like to suggest to refer to this price as an numerical approximation because it depends on the number of qubits used to discretize the uncertainty model.

In fact, via the Black Scholes model (1973), we have the analytical exact price of a European Call Option (please see below).
With the same parameters, the Black Scholes price is 0.169695, which is in fact much closer to the IQAE estimate of 0.1708.

Kindly find the formulas below, as well as a short python code implementing the Black Scholes Call and Put formulas.

I hope this proves useful.

With best wishes,

Alonso

............... BS Python code ...............

"""
Black-Scholes Option Model: Call & Put
[email protected]
bs_call(S,K,T,r,sigma)

Example:
bs_call(2,1.896,40/365,0.05,0.4)
0.16969509974913577
"""
from math import sqrt, log, pi, exp
from scipy.stats import norm

def bs_call(S,K,T,r,sigma):
d1 = (log(S/K)+(r+sigmasigma/2.)T)/(sigmasqrt(T))
d2 = d1-sigmasqrt(T)
return Snorm.cdf(d1)-Kexp(-r*T)*norm.cdf(d2)

def bs_put(S,K,T,r,sigma):
d1 = (log(S/K)+(r+sigmasigma/2.)T)/(sigmasqrt(T))
d2 = d1-sigmasqrt(T)
return Kexp(-rT)norm.cdf(-d2)-Snorm.cdf(-d1)

PLEASE SEE BELOW FOR FORMULAS:

BS formulas.docx

[Cryoris]

Hi @alonso-pena,

the "exact value" in the tutorial references to the exact expectation value that QAE approximates. It does not correspond to the true/analytic value of the european call option. Even though we're lucky and IQAE is closer to the analytic value of the option, it's error is actually the difference to the exact expectation value.

[alonso-pena]

Dear Julien, Thanks for your response. The distinction should be clarified. I understand that for professionals in quantum computing the numerical expectation value will be the benchmark. However, as the tutorial is aimed for solving a problem in finance, the benchmark there is certainly the Black Scholes price. Regards, Alonso Pena Get Outlook for Android<https://aka.ms/AAb9ysg>

…

________________________________ From: Julien Gacon ***@***.***> Sent: Thursday, October 21, 2021 12:47:33 PM To: Qiskit/qiskit-finance ***@***.***> Cc: Alonso Pena ***@***.***>; Mention ***@***.***> Subject: Re: [Qiskit/qiskit-finance] 03_european_call_option_pricing: adding Black Scholes formulas (#65) You don't often get email from ***@***.*** Learn why this is important<http://aka.ms/LearnAboutSenderIdentification> Hi @alonso-pena<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fgithub.com%2Falonso-pena&data=04%7C01%7Calonso.pena%40sdabocconi.it%7C072b652d9a3c46d8438a08d994803160%7C6bf3b57a9fb447c29ada51156518f52f%7C1%7C0%7C637704100588885660%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=ySzYP7cjiyGuPWUAAZ6llwzmB4HpDBASPBCLWcdcOpg%3D&reserved=0>, the "exact value" in the tutorial references to the exact expectation value that QAE approximates. It does not correspond to the true/analytic value of the european call option. Even though we're lucky and IQAE is closer to the analytic value of the option, it's error is actually the difference to the exact expectation value. — You are receiving this because you were mentioned. Reply to this email directly, view it on GitHub<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fgithub.com%2FQiskit%2Fqiskit-finance%2Fissues%2F65%23issuecomment-948487369&data=04%7C01%7Calonso.pena%40sdabocconi.it%7C072b652d9a3c46d8438a08d994803160%7C6bf3b57a9fb447c29ada51156518f52f%7C1%7C0%7C637704100588895655%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=vgDsrZg7rF33g0AY5ac%2FgcUiD2noTFqpFGehZImlBf4%3D&reserved=0>, or unsubscribe<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fgithub.com%2Fnotifications%2Funsubscribe-auth%2FAUVVMY6QXOVH3JSQDKENAUTUH7VULANCNFSM47SK5JVA&data=04%7C01%7Calonso.pena%40sdabocconi.it%7C072b652d9a3c46d8438a08d994803160%7C6bf3b57a9fb447c29ada51156518f52f%7C1%7C0%7C637704100588895655%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=XucLaqIdjlFYbCsuKItXtp93W1aNMBgmtE6hdQY%2FU5E%3D&reserved=0>. Triage notifications on the go with GitHub Mobile for iOS<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fapps.apple.com%2Fapp%2Fapple-store%2Fid1477376905%3Fct%3Dnotification-email%26mt%3D8%26pt%3D524675&data=04%7C01%7Calonso.pena%40sdabocconi.it%7C072b652d9a3c46d8438a08d994803160%7C6bf3b57a9fb447c29ada51156518f52f%7C1%7C0%7C637704100588905646%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=q4Qd6BBJxt%2B3afF1QXkh0K837P1prN5PYThO%2B%2FRvQ1c%3D&reserved=0> or Android<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fplay.google.com%2Fstore%2Fapps%2Fdetails%3Fid%3Dcom.github.android%26referrer%3Dutm_campaign%253Dnotification-email%2526utm_medium%253Demail%2526utm_source%253Dgithub&data=04%7C01%7Calonso.pena%40sdabocconi.it%7C072b652d9a3c46d8438a08d994803160%7C6bf3b57a9fb447c29ada51156518f52f%7C1%7C0%7C637704100588915644%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=k4h2m8%2BBUHZhFo1Q9BojFgxck4iLqItx%2FIOWdEg1wm8%3D&reserved=0>.