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I give you a brief outline about some key properties of Lévy processes. Lévy processes have three components: linear drift Brownian diffusion jumps. ($\to$ Lévy–Itô decomposition) You can model big jumps which occur as rare events (finite activity models) or consider a process which has infinitely many jumps during any finite time interval. Such ...


8

There's no easy answer to your question, as noob2 pointed out. You can look online for info from Universa. That fund does exactly what you are asking: https://www.universa.net/riskmitigation.html Of course, post a crash, such as the one we just experienced, the cost of hedges is larger than it is prior to such events. Understand that you aren't going ...


5

In my opinion there is one modern author on the subject of practical options trading who stands head and shoulders above the rest, and that is Euan Sinclair. His most recent book is Volatility Trading, and he has another book out soon. Sinclair covers actual strategies, along with questions of liquidity and position sizing that I never see in other options ...


5

I have also been looking into this stuff for a while. Apparently there are not that many tail risk funds, the prominent ones being Taleb and Spitznagel's Universa and Bhansali's LongTail Alpha. Obviously, all these guys have tons of papers and books on this topic. Spitznagel provides some nice case study on prototypical tail hedging in his book called The ...


3

Into the first equation we can substitute $C$ and $P$ as given by the other two equations, we get: $(S-K)^+ -(K-S)^+ +TV_C - TV_P = S-PV(K)$ $S-K+TV_C-TV_P=S-PV(K)$ $TV_C-TV_P=K-PV(K)$ If interest rates are zero then $PV(K)=K$ and then we indeed have $TV_C=TV_P$ Note: as suggested in the comments above a slightly different definition of Intrinsic Value ...


2

This is an issue that arises in the calculation of currency forward rates: You could simply take the Forward Rate from the FX Forward market, as "the market is always right" ;) You could calculate the Forward Rate from the Spot Rate and the Interest Rates in the 2 countries. This relies on the CIP Formula (Covered Interest Parity) which until 2008 was ...


2

It is poorly worded. Replace "options markets" with "stock markets" and it becomes clear that they're just noting typical spot vol correlation. When stock markets trade up volatility tends to fall and when stock markets trade down volatility tends to rise. This has been the case, on average, historically. For more on why, see here: Why does implied ...


2

One of the most straighforward way to hedge tail risk and buy insurance is just buying the Vix. A few hedge funds made a lot of money on the VIX lately. What's the downside? Well in normal times (which are most of them) you actually need to pay for that insurance. That's called the variance risk premium. So imagine, that you think this are normal times. ...


2

Cant talk specifically to stock pricing models but in foreign exchange the list in order of use goes: Geometric Brownian motion with time dependent vol and drift Local Volatility, either SABR or some other parametric or cubic-spline+Dupire Heston's stochastic volatility model Stochastic-Local hybrid volatility models, usually some from of parametric local ...


2

By put-call parity, put and call must have the same vega : \begin{align} & c - p = PV\left(F_T - K\right) \\ \Rightarrow & \partial_\sigma c - \partial_\sigma p = \partial_\sigma PV\left(F_T - K\right) = 0 \\ \Rightarrow & \partial_\sigma c \equiv \partial_\sigma p \end{align}


2

It’s a nice story and it makes for nice headlines and good reading (it’s a great book and movie) but it’s not necessarily that simple. You may or may not have the same view as your counterparty regarding the probability of some remote event and one of you may be “mispricing it” but the fact that it (or a few of them) happens and makes your fund $30m doesn’t ...


1

You can consider the CPI index like a stock, and treat the floor like a put option. You can measure the historical volatility of this index by looking at the monthly data. In the case of the US, we are talking about the non seasonally adjusted CPI index CPURNSA. You should find historical volatility around 2pct per annum, and like many markets I think the ...


1

In an incomplete market model, some risk factors are not traded. Hence, while you have a clear idea of how the market rewards traded risk factors (assuming parameter values and the data generating process are known), you do not know exactly how untraded risk factors are priced. One way to close the loop and fall on a unique price is to posit a ...


1

Take a look at the following note which explains that indeed the dynamic position in the underlying is from delta hedging the options. I actually still have to complete the note with showing that the aggregate delta of the options portfolio is $1/S_t$, but I have not had time yet. F. Rolloos, Delta Hedging and Variance Swap Replication


1

There is a difference between extrapolation and interpolation, and strike-wise and time-wise components. I will consider all 4 cases. If you are interpolating for a different strike between strikes of existing options, for single expiration, then any parametric or non-parametric methods mentioned by @raptor22 will work fine, and you can trivially check for ...


1

You would usually use the following steps: Express your quotes in the Black-Scholes (BS) parametrization: by inverting BS formula, you now have implied volatilities for each strike and tenor that you observe. Then, you can interpolate/extrapolate the whole implied volatility surface using a methodology that guarantees absence of arbitrage in the resulting ...


1

Assuming the options are European (they should be since the underlying is an index) and assuming the prices you have are synchronous so that the whole exercise makes sense in the first place, then provided the forward you are using is that which the market implies you should find the same implied vols for calls and puts. So you need to start by finding the ...


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