Stochastic Calculus for Quants | Risk-Neutral Pricing for Derivatives | Option Pricing Explained

Подписаться 73 тыс.

Просмотров 29 тыс.

50% 1

In this tutorial we will learn the basics of risk-neutral options pricing and attempt to further our understanding of Geometric Brownian Motion (GBM) dynamics. First, we will learn why risk-neutral valuation is an important tool for quantitative financial analysts, and its advantages over other valuation methods. This leads us to discussing the one period binomial asset pricing model and the derivation of risk-neutral expectations under this simple model.
We will understand how this method links back to the fundamental theorem of asset pricing and discuss change of measures while introducing the radon-nikodym derivative. We then present this change of measure from real world to risk-neutral world through Girsanov’s Theorem. We then work through a practical example using Black-scholes model stock and bank account dynamics to show the usefulness of risk-neutral pricing.
We briefly discuss the generalisation of risk-neutral pricing to the risk neutral expectation pricing formula and establish its usefulness in valuing complex derivatives by using the Monte Carlo method of discounted risk-neutral path simulations.
00:00 Intro
01:30 Why risk-neutral pricing?
03:43 1-period Binomial Model
06:18 Fundamental Theorem of Asset Pricing
07:00 Radon-Nikodym derivative
10:20 Geometric Brownian Motion Dynamics
12:00 Change of Measures - Girsanov’s Theorem
14:00 Example of Girsanov’s Theorem on GBM
19:45 Risk-Neutral Expectation Pricing Formula
★ ★ QuantPy GitHub ★ ★
Collection of resources used on QuantPy RU-vid channel. github.com/thequantpy
★ ★ Discord Community ★ ★
Join a small niche community of like-minded quants on discord. / discord
★ ★ Support our Patreon Community ★ ★
Get access to Jupyter Notebooks that can run in the browser without downloading python.
/ quantpy
★ ★ ThetaData API ★ ★
ThetaData's API provides both realtime and historical options data for end-of-day, and intraday trades and quotes. Use coupon 'QPY1' to receive 20% off on your first month.
www.thetadata.net/
★ ★ Online Quant Tutorials ★ ★
WEBSITE: quantpy.com.au
★ ★ Contact Us ★ ★
EMAIL: pythonforquants@gmail.com
Disclaimer: All ideas, opinions, recommendations and/or forecasts, expressed or implied in this content, are for informational and educational purposes only and should not be construed as financial product advice or an inducement or instruction to invest, trade, and/or speculate in the markets. Any action or refraining from action; investments, trades, and/or speculations made in light of the ideas, opinions, and/or forecasts, expressed or implied in this content, are committed at your own risk an consequence, financial or otherwise. As an affiliate of ThetaData, QuantPy Pty Ltd is compensated for any purchases made through the link provided in this description.

Опубликовано:

22 июл 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 33

@midoalpha9667 2 года назад

We Appreciate you a lot ❤❤

@matthowell6230 2 года назад

Great stuff as always! Just finished reading a paper on the viability of historically long-term regime-switching models and would love to see a video / hear your opinion on regime-switching models where n > 3 since it appears you have the industry knowledge to talk about it.

@QuantPy 2 года назад

Thanks Matt, Yes I plan to head towards this direction in the future and transition to more Machine Learning and Modelling videos, discussing Markov chains and transition probabilities for regime swicthing is very relevant there. In the meantime really focusing over the next few months to complete a solid block of videos in Financial Mathematics Theory & Application

@anisamalik542 Год назад

Hi, why do we have to calculate to the derivative of s(t)/b(t) ?

@_marcopk2434 8 месяцев назад

Sorry, studying this i am very confused about the approach using risk neutral probability and feynmanc kac formula. Are they the same thing, or linked in somewhay?

@MADgamer9212 2 года назад

Swear you go from 0 to 100 so quick here haha.

@BB-ok1jt 2 года назад

Please apply to real world markets. Examples would draw out the concepts. TY

@War4Skills Год назад

He literally says he does that in the next video.

@Maximus18.6 4 месяца назад

This is oriented to options and to be honest anyone can predict the future.

@War4Skills 3 месяца назад

You legit make no sense in the context of this video.

@twistedsector2708 2 года назад

Thanks for the very nice explanation! Just one question: at 16:07, do we really need the term dS * d(1/B) ? It seems that Itô's lemma will only apply to dS and not to the d(1/B) piece. This is because there is no Wiener process present in B. Another way to see this is that the term dS * d(1/B) is of order (dt)^{3/2}. Furthermore, this is precisely the reason why the cross terms marked in yellow in the next slide do not contribute eventually. Am I correct?

@QuantPy 2 года назад

Yes you’re right you could as you’ve said avoid writing out the terms to begin with. However as we’re going through this for the first time on the channel best to show all the steps, so people include it in case they deal with stochastic money market dynamics

@kevinshao9148 26 дней назад

One question please 17:47, I can do the same algebra on dSt = St(mu*dt + sigma*dWt), then I can also say St also martingale. why is that not correct? Thank you!

@rupeshpoudel3468 2 года назад

Do you mind providing us with some historical and current options data to work along? Say, in your case here: A panel data on non dividend paying European call, with many strikes and maturities.

@QuantPy 2 года назад

Yes, this will be in following videos. I will work through some examples of risk-neutral pricing using monte carlo simulations.

@retinapeg1846 2 года назад

This is literally like my Quantum Computing degree.

@Math4Pears-u2p 2 года назад

@gutefrage9425 2 года назад

How is it gonna work if you use made up probabilities and not the real probabilities?

@xp_money7847 Год назад

Cuz we have assumed No-arbitrage or market equilibrium.

@retinapeg1846 2 года назад

Why is d(1/Bt) = - r *1/(Bt) *dt

@QuantPy 2 года назад

I skipped over this, sorry. for d(1/B) you need to apply Ito's rule to function f(x) = 1/x. so df/dt = 0, df/dx = (-1)/x^2, d^2f/dx^2 = (2)/x^3. That leaves you with d(1/B) = d(f(B)) = (-1/B^2)dB + (1/2)*(2/B^3)dB^2. Sub. dB into equation and you're left with d(1/B) = (--1/B ) r dt. Hopefully that makes sense

@retinapeg1846 2 года назад

@@QuantPy No idea why this is confusing me so much. Two questions: Is Bt a function of t? And where is the r term coming from?

@QuantPy 2 года назад

No don’t worry it’s a tricky step and I shouldn’t have skipped over it in the video. Bank account Bt is a function of time. r is the risk free rate. And it’s appearing from the original bank account dynamics SDE dB = rBdt

@twistedsector2708 2 года назад

I think d(1/B) = - r *1/(B) *dt should directly follow from dB = rBdt without the need of Itô's lemma. Note that there is no volatility in B and B does not obey a SDE. In other words dB is a Newtonian differential. The steps are: d(1/B) = - dB / B^2 = - (r B dt) / B^2 = - (r / B ) dt .

@JackSmith-cd6eo 2 года назад

What did you study at university?

@QuantPy 2 года назад

Chemical Engineering, then Masters of Financial Mathematics

@JackSmith-cd6eo 2 года назад

@@QuantPy damn that’s impressive, I’ll never be that smart

@piepieicecream 2 года назад

@@QuantPy where did you study for your masters, and would you recommend that uni?

@QuantPy 2 года назад

@@piepieicecream Studied at The University of Queensland - it was alright, good structure of base courses. I would recommend getting a job in the industry before studying Quantitative Finance Masters

@BigDog-dw5ns 2 года назад

@@JackSmith-cd6eo it's all about desire and hard work. As a "smart guy" it always makes me sad watching people as smart as me underachieve because they got the message they were "average" or even "dumb" as young kids.

@thomase4592 6 месяцев назад

It is totally wrong to hedge risk neutral you have to BUY a fraction of the underlying to follow the long option contract. Indeed if you are a banker you sell a call option to somebody, price of the Underlying rising and rising, you have to pay him a lot so of course you needed to buy the same amount to be in profit as well. Not to short the underlying