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3

Very simply, Ross' framework assumes a great deal to extract the true pricing kernel. Time homogeneity, additively separable state dependent utility, (discrete time Markovian structure - though these have been relaxed.) In particular, there are two schools of criticism, one is that time homogeneity makes little sense in the real market. In fact, the Recovery ...


3

Think of moving volatility in the other direction. As volatility approaches zero, any call strike strictly smaller than the ATM strike, $K<K_{ATM}$, will have zero probability of ending in the money, and the corresponding option value will be zero. An infinitesimally small change in stock price will not move $K$ past $K_{ATM}$, so the option value ...


2

Victor123, let's start from $\Delta$. This is the expected change in the price of an option if the underlying asset moves by a currency unit, say 1 USD. For the case of a call option, the Delta varies between 0 and 1. Everything else been equal, the Delta of OTM calls will approach to 0 as the price moves out of the target barrier. Conversely for the case ...


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Here is to continue the above answer of Emcor to make it more explicit. Note that the fact given in the question should instead be \begin{align*} P(\inf \big\{t \in [0, T], B_t +ct = a \big\} \geq T) = 1- \Phi\Big(\frac{a-cT}{\sqrt{T}}\Big) + e^{2ac}\Phi\Big(\frac{-a-cT}{\sqrt{T}}\Big). \end{align*} Then, for $0<t_0\leq T$, \begin{align*} P(\tau \leq ...


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$S_t$ is already under $Q$ (riskfree drift), so you not need to change the measure here. Note that $c:=\left(\frac{r}{\sigma}-\frac{1}{2}\sigma\right)$ and $E\left(1_A\right)=P(A)$. So one computes the European option price as the discounted payoff expectation: $$C=e^{-rT}E\left(1_{\tau\leq T}\right)=e^{-rT}P(\tau\leq T).$$ The option price equals the ...


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Dupire model is just one way of generating a local volatility surface from an implied volatility surface. There are many other ways to generate a local volatility surface. One critical aspect of Dupire model is that the input implied volatility (IV) surface should be arbitrage free. If not, you will negative instantaneous variance when generating the local ...


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No, if you are referring to the famous Dupire Model (there are others), then they are the same. It is usually referred to as the Local Volatility Model and the Dupire Equation. I would disentagle those with the concept of Local Volatility, which is model independent and a fairly deep result.


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In the Merton jump diffusion model, the stock price process consists of a continuous part and a discrete part (this one represents the jumps). While deriving the PDE for the riskless portfolio and imposing the riskless evolution, the discrete part can't be instantaneously hedged. In fact, you can assume that the effects of jumps can be nullified on average, ...


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I am guessing that the argument is as follows. They certainly have the same value at time t since they are both worth $S_t$ then. If they have the same value at $t$ they should have the same value at time $0.$ So if we are pricing by expectation our measure has to give the same discounted expectation price to both portfolios. So we must have $$ e^{-rT} ...


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it depends on how it's converted. There are three different possibilities. the pay-off is $(K-S_T)_+$ with $K$ and $S_T$ in USD but the pay-off is converted to EUR as a predetermined rate. This called a quanto and is widely discussed in books. (eg my book Concepts...) the pay-off is $(K-S_T)_+$ with $K$ in EUR and $S_T$ in USD. Then you have to model the ...



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