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Expected volatility in the underlying price over the life of the option is a major component of the BSM option pricing model. When you calculate the volatility based on the current market price, you're figuring out what the market thinks the volatility would be, that's why it's called implied volatility. So to answer your question, you can either assume a ...


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@Quantuple had a good answer in the comments. Basically, I was not using a fine enough grid in my replication. Click here to see the original comment that answers the question.


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I'll answer both of your questions in one go: Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric Brownian motion as stock price model is misspecified and we need more sophisticated models (sto vol, jumps etc), in particular if we want to price more ...


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The correlation of USDGBP and GBPUSD is -1! If your sample and measurement thereof, realised or implied, but suggests <>-1, jour jargonistic problem is “Siegel’s Paradox” :-) In log terms, ie transforming returns such that they are additive, there is no difference. They sum to zero. The assumption with implieds is also lognormal returns, so these too (...


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Using usual FX modelling techniques, let us assume $\text{USDGBP}_t$ follows Geometric Brownian Motion under the domestic risk-neutral measure, when the domestic currency is USD: $$d\text{USDGBP}_t=(r_{USD}-r_{GBP})\text{USDGBP}_tdt+\color{blue}{\sigma}\text{USDGBP}_tdW_t$$ $r_{USD}$ and $r_{GBP}$ are the USD and GBP risk-free rates respectively. By Itô's ...


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Jumps are an attempt to solve a math mistake in Modern Portfolio Theory. In the 19502-70s, economists were working on solving the variance-mean tradeoff. Furthermore, they needed to do so with punchcard computing. That radically restricted the set of computable, potential solutions. Both the normal distribution and the log-normal distribution are ...


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Jumps do not imply fat tails. See the simulation in R. Note that the excess kurtosis of [normal variable + jump] is negative. > set.seed(1) > Normal_Variable <- rnorm(1e8) > kurtosis(Normal_Variable) [1] -0.000628316 > Jump <- 2 * ((runif(1e8) < 0.5) * 2 - 1) > kurtosis(Normal_Variable + Jump) [1] -1.280009


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The argument given in the OP doesn’t convince me. Yes, the dollar gamma of ATM options is the largest. But so is the Vega. Therefore the amount of implied volatility increase necessary to compensate is not clear. For me, the intuition is simply that jumps-> the distribution of log returns has fat tails relative to a normal distribution. And then it ...


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Actually, I do not think it's true. Jumps, when added to the Black-Scholes (BS) dynamics, do modify the volatility surface. However, the volatility skew may get inverted: the implied BS volatility may be higher when the strike is closer to the current value $S(0)$ of the underlying asset $S$. Consider an idealized example: $$ \log(S(t+dt) / S(t)) ={\rm[...


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FX differs from other asset classes in that certain amount of market manupulation by cebtral banks is the norm. For almost any currency, if its exchange rate versus other currencies moves outside a certain band, the central banks will try to intervene, usually by just buying the currency in the market. The bank's goal is not to make money by speculation, ...


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For a very short answer, given that the event is scheduled, the implied vol for a fixed future expiry date decreases, and the historical volality increases at event time. This could seem a bit counterintuitive but the implied vol factors in all scheduled forthoming events up to expiry. As the event has hit the market and its impact is priced into the ...


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I do not think this is allowed in this forum, but anything that has to do with using option implied volatility and skewness to estimate market betas or expected returns. Here's a few references: Measuring Equity Risk with Option-implied Correlations The Skew Risk Premium in the Equity Index Market What is the Expected Return on a Stock? The Term-Structure ...


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A comprehensive review of how people deal with negative interest rates in SABR / LMM and similar models could make a good thesis.


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