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1

You can take a derivative $\frac{\partial}{\partial m}\frac{\ln m}{m-1}$ at point $m=1$, so you will get $-\frac{1}{2}$. Yes, Black-Scholes volatility is log-normal volatility. In other terms it's comparison of Black-Scholes IV and Bachelier IV.


1

Try the code below. It consumes less memory and takes about 6 minutes to run. Please note that when simulating the stock price for purposes of pricing derivatives, we use the risk-neutral stock price process. The drift of the risk-neutral process becomes the risk free rate minus half the square of annualized volatility as indicated in the code. import numpy ...


2

The collar strategy combines one unit of stock with a (long) put option with strike $K_1$ and a (short) call option with strike $K_2$. The payoff of this strategy is exactly $K_1$ if $S_T\leq K_1$, $S_T$ if $K_1<S_T\leq K_2$ and $K_2$ if $S_T>K_2$. The easiest way to see that the statement is false is by comparing the payoff profiles of the collar and ...


1

This discussion has also confused me slightly, so I will add something that is possibly clarifying, although most likely will not be. It is also a reminder that I need to stop programming and brush up on options pricing theory. The Black Scholes hedge portfolio is given by: $$ \Pi_t = \frac{\partial V}{\partial S}(t,S_t)S_t + \left[1 - \frac{\partial V}{\...


1

I have not read the paper except for the Abstract and the Introduction but I completely agree with the OP: The statements by the authors are confusing. The convergence rate of crude Monte-Carlo is $\mathscr{O}(\frac{1}{\sqrt{n}})$ which is independent of the dimension of the problem. This is arguably THE greatest strength of Monte-Carlo: It avoids the curse ...


0

The paper takes each observed smile, bumps all of the strikes by a shift term to make them positive, and the fits a SABR smile to them. When I do the same thing with the dataset you've attached above (I remove the -150 point because it's vol of 0.0 breaks things) I get the following 'smiley' fit, which looks similar to the results presented above: This was ...


4

I'm no special expert on options and their Greeks. However, I have had a decade plus experience of almost-daily discussions with a bank derivatives desk, on pin risk and the behaviour of autocallable, cliquet etc. structures. You are correct that a gamma hedge would require an options as opposed to underlying hedge. However, the traders' obsession with gamma ...


4

SHORT STORY: forward Libor rates need not be assumed to be log-normally distributed. For example, they can be assumed to be normally distributed (and indeed, on Bloomberg, Swaption implied vols are quoted both, in terms of normal as well as log-normal models). The only condition required is that the forward Libor rate process needs to be a martingale under ...


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A lognormal distribution has three valuable properties (I) It ensures that the rate is only allowed to be positive; (II) the changes in the interest rate are proportional to the interest rate; and (III) the option price is analytically solvable. BTW, just to be precise, note that in Black's model, it is an assumption that the distribution of the interest ...


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