Dear Lech, at minute 16.33 line 35 of the code I can't understand why the value of the volatility, J, changes both with the path we consider and with time. It should not be constant over time for each simulation? Indedeed, when we consider the conditional expectation approach we use only the "last" value of J for each path, as if the value of J were random but constant for each realization. Thank you so much! Matteo
hello Matteo, you are completely right, if J is a random variable, and not a stochastic process then it should be "fixed" for every time "t". Well spotted. Thanks for the notification. If you like you can contribute to the github and suggest the fix. I will review it and include in the latest version of the code. Thanks again. Best wishes, Lech
Hi Professor, I have a question about slide 19. There can be three martingale processes on this slide under three measures. Does it mean we have an assumption of an incomplete market?
Great lecture, thank you for the fantastic videos. I will be happy if anyone can help me clarify the below two points: 1. At time 32 minutes you have listed the three possilble options for constructing different types of Numeraire. I am unclear what is the difference between Money Saving account (option 1) and ZCB Option 2 2. For the purpose of this concept: Numeraire, is there a distinction between made between non-dividend and non-coupon paying assets? a. Money savings account pays coupon but not dividend. So can we consider coupon paying assets as Numeraire? b. ZCB does not pay coupon no dividend. c. Stocks pay dividend, but not coupon. Do we choose only non-dividend paying stocks as Numeraire? Kindly help.
hello Prasad, thanks for the feedback. Good to hear you liked the content. Regarding your questions: 1) the difference is very subtle, in essence you can think that the zero coupon bond is simply an expectation of the money savings account (of the inverse). Thus for the money savings account we have M(T)= exp(int_0^T r(s) ds) while for P(0,T) = E(1/M(T)). So when you look at these two quantities you need to think what you know at time t_0 and time T. ZCB, P(0,T) simply gives you the expectation of the money saving account in the future time T. This is a market instrument that assigns a price & probability of that value. 2) In principle no, as both assets are tradable so both can be numeraires. I hope it clarifies, Best wishes, Lech
The lectures are awesome. Maybe these are the best courses on quant finance available online. And since these are given away for free, I have a feeling that I'm stealing something ) You may think about giving us a chance to compensate you. Maybe a Patreon account? Thank you, Lech!
Glad you like them Oleksandr! Appreciate. If you like the lectures and the materials I can only recommend to accquire the book that was used to create this course. Then you will have a complete package. Best wishes, Lech
Imho, can we still use tower property in case brownian motions in stock price and volatility processes correlate? Or we need to use brute MC in that case?
You may try to use the Cholesky decomposition, to ensure you are dealing with independent Brownian motion and then try the tower property. Does it make sense?
hello Thanh, it is because the second order derivative will be multiplied by (d M(t))^2, and since we have dM(t)=r(t)M(t)dt, this implies, by Ito's table, (dM(t)dM(t)) = 0 as dt*dt=0. I hope it explains. Best wishes, Lech
