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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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If you assume the payoff is paid at time T, you just have to compute P(tau < T). In this case, you have everything you need to do it. If the payoff is paid at time tau, you need to compute the density of the stopping time.


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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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The exercise is really not about replicating a call with asset or nothing. It is simply about the PDE of the delta of a call. The usual derivation of the BS equation starts by considering a portfolio short the option $$ \Pi_t = \delta^0_t B_t + \delta_t S_t - V(t,S_t) $$ Assuming the portfolio is selfinancing (and interest rate = 0), we get $$ d\Pi_t ...


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I think the title here is misleading. Let's go back to the BS world with $r=0$ to $a(S_t)=S_t \sigma.$ In that case, all you are saying is that you can replicate a call option by holding $N(d_1)$ units of stock at time $t.$ What does this have to do with the second equation? I am guessing that this is the price process of an asset of nothing option with ...


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Forward implied volatility smile is implied from forward start options. For example call options have payoff $$ g_{T+\theta} = \left( \frac{S_{T+\theta}}{S_T} -K\right)_+ $$ If you are in a stochastic volatility model this can be rewritten $$ g_{T+\theta} = \left( e^{ \int_T^{T+\theta} r - \frac{1}{2}\sigma_t^2 dt + \int_T^{T+\theta}\sigma_tdW^S_t } ...


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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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I found and answer to my own question. So, I post it here for people who maybe have the same problem. The answer, however, is quite intuitive. The last observation used for the estimation of the physical density is also the time point where the investors know the most about the physical density because at this point the most possible historical observations ...


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Without getting into all the Math one thing should be clear that: Call option is equivalent to: long asset or nothing AND short cash or nothing options. You cannot replicate a call option without asset or nothing since replicating portfolio for long call requires holding N(d1) quantity of the underlying. Asset or nothing gives you this exposure directly. ...


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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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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 ...


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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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I think the yield curve is not what you need here. The idea is to have a model for the dynamics of the bond process $dB(t,T)$ (which you can compute by having dynamics for short-term interest rate $dr_t$. A common assumption is to use Black 76 model with $F = B(0,T)$ if I remember well. You will also need to know the volatility $\sigma$ of your bond prices. ...


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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 ...



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