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22

Many of them are on my website at emanuelderman.com. Others I probably have anyway. Feel free to email me


8

I had read some of them; actually, it does not exist an on-line library that collected them (or, better, it existed here, but it seems the website does not work anymore). I reported here below some of them that you did not find: More Than You Ever Wanted To Know* About Volatility Swaps Model Risk The Volatility Smile And Its implied Tree Enhanced ...


6

the problem is that the pay-off has discontinuous first derivative. Try a contract with pay-off that is twice differentiable and it will probably work. The problem is that all the value comes from the tiny number of paths within $\Delta S$ of the strike, and these paths have huge value. This is a well-known problem. As the bump size goes to zero, the ...


3

As the manager of a mutual fund (not a hedge fund) you can only short treasury futures. So you take the one that is clostest in duration, look for an optimal hedge ratio and that's it. In my experience you have to leave liquidity risk open.


3

The differential equation has a trend due to the interest rate. When you discount you take this trend away: $$ \frac{d}{dt} (e^{-rt}Z_t) = -re^{-rt}Z_t + e^{-rt} \frac{d}{dt}Z_t = e^{-rt}\frac{1}{2}S_t^2\Gamma_t(\hat{\sigma}^2-\beta_t^2) $$ $Z$ doesn't appear on the rhs anymore and you can integrate $$ e^{-rT}Z_T - e^{-r0}Z_0 = \int_0^T ...


2

Assuming zero interest, the put option has the price \begin{align*} KN(-d_2)-S_0N(-d_1), \end{align*} and delta $-N(-d_1)$. When $N(-d_1)$ units of stocks are shorted and invested in bonds, the total value in bonds is $KN(-d_2)$, which is indeed greater than the option price. However, as you have shorted $N(-d_1)$ units of stocks, your portfolio value is ...


1

Instead of just considering a parallel shift of the whole volatility surface, you can decompose the surface into maturities/strikes domains, so called buckets and consider Vega buckets which are sensitivities wrt to bumps of each of these domains. The vol smile is often inter/extra-polated using a model calibrated to market prices, e.g. the SABR model or ...


1

we should first define some notation before discussing pricing. Let $t_0$ be initial time and $ t_1, . . . , t_M$ be pre-specified exercise dates with $t_0 < t_1 < · · · < t_M = T$ , the final maturity, and $Δt = t_m−t_{m−1}$. Without a loss of generality it is assumed exercise dates are equidistant. To price a Bermudan option, its value is split ...


1

It's a combination of too few sample paths and/or too small an increment. Your estimation error on the price is magnified by the $dS^2$. Try using a larger sample or a larger increment. Alternatively, you can use a multiplier instead of a fixed increment; in my experience, it usually yields better results.


1

You have already agreed to pay $QK$ EUR at $T$ to receive $Q$ units of A. If you sell $Q$ lots of $F^A(t,T)$ then you will receive $Q F^A(t,T)$ EUR and deliver $Q$ units of A. The combined flow is now just in EUR: at $T$ you receive a net of $Q(F^A(t,T)-K)$ EUR. You can hedge that by selling $Q(F^A(t,T)-K)$ of $F^{FX}(t,T).$ Then with both hedges, the net ...


1

That seems to be a nice paper but I haven't worked through it completely yet. As I understand it, the goal is to replicate the holding (by an investor) of an European option using an American option, stock and bonds in a self-financing manner. As the value of the underlying changes this requires rebalancing of the option and the bond, i.e. hedging. Since ...


1

This sounds like quadratic hedging. If you have the return of the assets $r_X$ and $r_Y$ with negative correlation $\rho$ between the two (we could think of bonds and stocks) and more variance in one of them then the problem of weighting the two by $w$ is (assume zero expected returns for ease of presentation) $$ \text{risk} = E[(w r_X + (1-w) r_Y)^2] ...



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