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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 BlackÔÇôScholes 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


4

The first portfolio $\Pi^{(1)}_t$ is a self-financing hedging portfolio. It is typically what you get when you delta hedge an option position (here short hence the minus sign, but it could be long without loss of generality) with shares of the underlying asset. If the only source of risk comes from the randomness of the underlying asset price $S_t$, then one ...


3

So, you simulate the pnl one month in advance in a scenario where the Index has moved down by 20%. This is for options which are 30% + out of the money. In your example this would be August expiration and 1400 strike not the 1600 strike. So if you are long X index shares, as you said then you would lose 400x in one month's time. You buy Y puts to ...


3

just take a call and a put struck at $K$ and add them together. For the hedge just add the hedges together as well.


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 e^{-rt}\frac{1}{2}...


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

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 ...


2

Just a heads up, I'm not going to go through all the mathematical caveats of using this approach. Let $\Sigma$ be your covariance matrix, and $X$ a random vector of daily returns. So $$\text{Var}(X) = \Sigma.$$ You have a bug in your code. In your code you call it pxCov, but you probably meant to use cov() insted of cor(). Check out the documentation to ...


1

Another approach as follow. The $T$-Straddle option $X$, i.e. $$X=\left\{ \begin{align} & K-S(T)\quad ,\quad 0<S(T)\le K \\ & S(T)-K\quad ,\quad S(T)>K \\ \end{align} \right. $$ has then following contract function $$\Phi (x)=\left\{ \begin{align} & K-x\quad ,\quad 0<x\le K \\ & x-K\quad ,\quad x>K \\ \end{align} \right....


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 $\ln(sp_t)-\ln(sp_{...


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

Well stock prices change all the time when markets are open. American options give you the opportunity to exercise it at any time up until maturity, whereas a European option only allows you to exercise it at a specific date and time. A simple example is to compare an American option that matures in 1 day and European option where it matures at the last ...


1

One example could be someone using option strategies and its underlying dividends. In these cases, the trader could use early excersise to capture the dividend value. Google it for more information.


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

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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