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.
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
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!
this is the exact point that's throwing me as well, I feel like there's some logical step missing. I think it's "you can set mu to zero and say St is a martingale, but it's not arbitrage-free since a zero-risk Bt exists. To be arbitrage free, St/Bt has to be a martingale under Q, not just St. We first find the SDE for St/Bt under P, which is not a martingale. But if we multiply by some R-N derivative that we don't even need to find such that the resulting drift is zero, we found the Q measure. Let's just define the zero-drift brownian motion under Q as d~Wt~ that happened after the R-N derivative. If we do that, then, under Q, d~Wt~ = dWt(underP) + (mu-r)/sigma*t. go back to the original equation for St, and we'll transform St using the P process in to St using the Q process." I think that logically step probably should have been said, and I'm probably muddling it up as well!
Well you can find a probability measure under which your stock price is a martingale, but since what interests us is the discounted value of the option under risk neutral pricing it wouldn’t help you much to have the stock price by itself to be a brownian motion
@@will-ti2qs Thank you so much for your advice! I think I can follow your logic, and agree. The target is the option value. But why here we use St/Bt as the process to study? That's not the option value. Still quite abstract to me. Really appreciate your help anyway! Let me try read more materials.
@@scotthoward8308 Thank you very much for your explain Sir! Two things I still don't get here: 1) "To be arbitrage free, St/Bt has to be a martingale under Q, not just St." why St/Bt has to be martingale here? how does that guarantee arbitrage free? 2) "go back to the original equation for St, and we'll transform St using the P process in to St using the Q process." How? and as @will-ti2qs pointed 2 weeks ago, we need option value to be martingale, not stock price St. Anyway really appreciate your reply and help!
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?
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
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.
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?
@@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
@@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.
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
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
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 .
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
![Lp. Сердце Вселенной #60 РОЖДЕНИЕ ЛОЛОЛОШКИ [Финал] • Майнкрафт](http://i.ytimg.com/vi/YoR0pAV9FVQ/mqdefault.jpg)
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.
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
Swear you go from 0 to 100 so quick here haha.
We Appreciate you a lot ❤❤
Please apply to real world markets. Examples would draw out the concepts. TY
He literally says he does that in the next video.
This is literally like my Quantum Computing degree.
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!
this is the exact point that's throwing me as well, I feel like there's some logical step missing. I think it's "you can set mu to zero and say St is a martingale, but it's not arbitrage-free since a zero-risk Bt exists. To be arbitrage free, St/Bt has to be a martingale under Q, not just St. We first find the SDE for St/Bt under P, which is not a martingale. But if we multiply by some R-N derivative that we don't even need to find such that the resulting drift is zero, we found the Q measure. Let's just define the zero-drift brownian motion under Q as d~Wt~ that happened after the R-N derivative. If we do that, then, under Q, d~Wt~ = dWt(underP) + (mu-r)/sigma*t. go back to the original equation for St, and we'll transform St using the P process in to St using the Q process." I think that logically step probably should have been said, and I'm probably muddling it up as well!
Well you can find a probability measure under which your stock price is a martingale, but since what interests us is the discounted value of the option under risk neutral pricing it wouldn’t help you much to have the stock price by itself to be a brownian motion
@@will-ti2qs Thank you so much for your advice! I think I can follow your logic, and agree. The target is the option value. But why here we use St/Bt as the process to study? That's not the option value. Still quite abstract to me. Really appreciate your help anyway! Let me try read more materials.
@@scotthoward8308 Thank you very much for your explain Sir! Two things I still don't get here: 1) "To be arbitrage free, St/Bt has to be a martingale under Q, not just St." why St/Bt has to be martingale here? how does that guarantee arbitrage free?
2) "go back to the original equation for St, and we'll transform St using the P process in to St using the Q process." How? and as @will-ti2qs pointed 2 weeks ago, we need option value to be martingale, not stock price St.
Anyway really appreciate your reply and help!
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?
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
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.
Yes, this will be in following videos. I will work through some examples of risk-neutral pricing using monte carlo simulations.
This is oriented to options and to be honest anyone can predict the future.
You legit make no sense in the context of this video.
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?
Hi, why do we have to calculate to the derivative of s(t)/b(t) ?
What did you study at university?
Chemical Engineering, then Masters of Financial Mathematics
@@QuantPy damn that’s impressive, I’ll never be that smart
@@QuantPy where did you study for your masters, and would you recommend that uni?
@@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
@@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.
How is it gonna work if you use made up probabilities and not the real probabilities?
Cuz we have assumed No-arbitrage or market equilibrium.
:)
Why is d(1/Bt) = - r *1/(Bt) *dt
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
@@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?
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
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 .
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
This one is definitely not for babies (like me).
💦 💦