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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 ...


6

The point is the following: Delta, $\Delta$, is defined as $\frac{\partial C}{\partial S}$, where $C$ is the value of the call option, and $S$ is the price of the underlying asset. So, given that the value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is $$C = N(d_{1})S - N(d_{2})Ke^{-rT},$$ $$\Delta ...


4

there are a number of ways to do this. You do have to make some modelling assumptions, however. eg continuity, BS model holds, or log stock price process is independent of level. The most common way is to take the pay-off and geometrically reflect in the barrier. (i.e. pass to log coordinates and reflect). i.e. write the function as $f(x)$ where $x= \log ...


4

Due to the lack of a carry arbitrage, VIX futures are actually the direct hedge for VIX Index options


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 ...


3

The first portfolio is what you obtain when you delta hedge an option position (here short, but could be long without loss of generality) using the underlying asset. The second portfolio usually figures a replicating portfolio. The option position is 'dynamically replicated' using a self-financing strategy involving shares of the underlying asset and ...


2

He's saying that if you know the volatility, and you hedge continuously, you can lock in the exact Black-Scholes price. Any deviation from that delta hedging scheme must result in noise. ie the replicated price must have a distribution with some width around the theoretical value. This noise does not create systematic profit or loss, because it's just ...


2

$\alpha_t$ must be chosen prior to stock price movements so the expression $S_t d\alpha $ does not make sense: we can't take a position in a stock based off information that we don't know yet. The missing step is that the replicating portfolio is required to be self financing: that is, for all $t$ the following equations hold: $$X_t=\Delta S_t+\Gamma M_t$$ ...


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 ...


2

if you hedge it means that your USD return equals (neglecting hedging cost) your EUR return. You just change the name. If you want to know what the return measured in EUR is, then you either calculate the price of S&P in EUR and then take returns or equivalently you calculate the product of the local return and the return of the USD in EUR in the ...


1

Commonly used procedures: A) hedge when a 1 sd move has happened B). Hedge when your delta position exceeds some risk limit. C) hedge once a day D) hedge based on your desired delta position All are used. I personally prefer B.


1

There has been a lot of work in recent years on the pricing and hedging of volatility derivatives, leading to some non-obvious, even startling results. It is summarized in Mark Joshi's book More Mathematical Finance among other places. It all started with the work of Anthony Neuberger on the Log Contract in 1994, which seemed to be a theoretical result ...


1

The EUR is normally quoted as EURUSD, i.e. the value of one euro measured in dollars, currently about 1.1281. If the S&P index is $sp_t$ and the EURUSD rate is $eu_t$ then the S&P converted into Euros is $sp(t)/eu(t)$. The arithmetic 1 day return on this is $-1+\frac{sp_t}{sp_{t-1}}\frac{eu_{t-1}}{eu_t}$. The logarithmic return is ...


1

You need to hedge future cash flows (not future value) using a fixed for fixed currency swap (equivalent to a series of forwards). This translates into a "cash flow hedge". Hedging present value would be hedging the "fair value" of the bond with a fixed-for-float currency swap. Using a fixed for fixed swap will convert your cash flows into desired currency ...


1

In a Black-Scholes world a portfolio of options (some calls, some puts) of different maturities and strikes on the same underlying still has one delta and one gamma, which can be calculated by summing over the deltas and gammas. So you still have the same setup as with a single option situation.


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

The most rigorous approach I have seen so far eliminating the risk premium is this one: Emanuel Derman: The Perception of Time, Risk and Return During Periods of Speculation (2002) Equation 2.23 on page 11 derives $\mu$ ~ $r$ but it only holds in the limit when you hypothesize countless uncorrelated stocks in a diversifiable market. Still an interesting ...



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