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No offense but it will be much more complicated than what you think... I'm not even sure that you are familiar with risk-neutral pricing in the first place? I'll try to give you some clues. This security is called a basket option. On top of the multi-asset feature, there are non-trivial mechanisms embedded in the contract you mention: an auto-callable ...

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"Monte Carlo convergence" means that you've sampled enough individuals to represent (and understand) a general population. If the probability models behind your Monte Carlo simulation are accurate, then your results will match reality as you increase your sampling size. Monte Carlo convergence becomes difficult when you try to study a low-probability sub-...

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One way to do this would be to try to replicate the VIX calculation, which is calculated as the square root of a 30 day variance swap level. A variance swap can be replicated (in theory) using standard European calls and puts (you would need to convert American style stock option prices to European style prices using option models). The weighting scheme is ...

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For historical volatility I actually like this article: http://www.todaysgroep.nl/media/236846/measuring_historic_volatility.pdf it provides several of the better known methods for calculating historical vol, which of course could be done manually. Just being aware of the upsides and downsides of each method. As for implied vol, yes as onlyvix has said it'...

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Under GBM $$\frac {dS_t}{S_t} = \mu dt + \sigma dW_t$$ we get $$S_T = S_0 e^{(\mu - \frac{1}{2}\sigma^2)T + \sigma W_T}$$ suggesting that $$S_T \sim \text{ln}\mathcal {N} ( \tilde {\mu}, \tilde {\sigma})$$ where \begin{align} \tilde {\mu} &= \ln S_0 + (\mu - \frac{1}{2}\sigma^2)T \\ \tilde {\sigma} &= \sigma \sqrt {T} \end{align} Now if $X \... 2 I would definitely recommend Volopta as a reliable source of self-contained and commented financial engineering source codes (useful for prototyping/understanding but clearly not production code). I have for instance copy-pasted, the explicit PDE solver you are looking for (centred in space, backward in time) below (+ edited for clarity + improved ... 1 Practitioners tend to wear Black-Scholes glasses when dealing with European options: to them, quoting a certain option price today$V(S_0;T,K)$is equivalent to quoting the forward price of the underlying$F(0,T)$along with a relevant Black-Scholes volatility figure$\sigma(T,K)$(*) That being said, when you are asked to price a European option on a stock$...

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Measure change is still the most natural approach for such problems. We assume that, under the measure $P$, \begin{align*} dX_t &= \mu X_t dt + \sigma X_t dW_t^1,\\ dY_t &= \mu Y_t dt + \sigma Y_t \left(\rho dW_t^1 + \sqrt{1-\rho^2} dW_t^2 \right), \end{align*} based on the Cholesky decomposition, where $\{W_t^1, t \ge 0\}$ and $\{W_t^2, t \ge 0\}$ ...

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Relatively quick Solution If $U$ and $V$ be normally distributed with means $\mu_u\,,\,\mu_v$, variances $\sigma^2_u\,,\,\sigma^2_v$ and correlation $\rho$ then we can show ( by definition of expectation and apply joint density function ) $$\mathbb{E}\left[\left(e^U-e^V\right)^+\right]={\large{e^{\mu_u+\frac{1}{2}\sigma_u^2}}}\Phi\left(d_1\right)-{\large{e^... 1 Assuming deterministic interest rates, the price of an American call option struck at K and expiring at T is given by$$ V_0 = \text{sup}_{\tau \in \mathcal{T}[0,T]} \mathbb{E}_0^\mathbb{Q}\left[ e^{-r\tau} \max(S_{\tau}-K, 0) \right] $$where \mathcal{T}[0,T] denotes a family of stopping times with values in [0,T] and where, under the risk-neutral ... 0 I would suggest to use :$$f(t,S^\theta_t)=\max(K-S^\theta_t,0)\exp(-\theta W_t\color{red}{\mathbf{-}}\frac{1}{2}\theta^2t)$$where dS^\theta_t=(r+\sigma\theta)S^\theta_t dt + \sigma S^\theta_t dW_t 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 ... 0 With respect I think that this issue was associated Martingale properties AND dominated convergence theorem.(May be Wrong) Let L\in(0,K) a fixed price, we can consider the following choices for the exercise of a put option with strike price K: If S_t\le K, then we exercise contract at time t, and were delighted. O.W. we should wait until the ... 2 To elaborate on the explanation provided by @Alex, the reasoning is because when we look at the PDE we notice that the S terms appear in pairs with the \dfrac{\partial}{\partial S}, i.e. S\dfrac{\partial}{\partial S} and S^2\dfrac{\partial^2}{\partial S^2}. What this says it that if we were to try a polynomial function of S then after applying ... 1 I would tend to do the following: If, under your working modelling assumptions, there exist closed form formulas, then compare your results to them. "The Complete Guide to Option Pricing Formulas" in @Student T is indeed a nice reference for that. Beware of true formulas vs. approximations though. Now if it's not the case: Compare different pricers' ... 4 1: Follow the calculations in The Complete Guide to Option Pricing Formulas. The book has many formulas, sample values and outputs. Highly recommended for validating your results. Apparently, this is one of most popular books used by real-world quants (simple and fast). 2: You can still use QuantLib to price with year fractions. I have an example: ... 1 two things I would try...and this is really off the top of my head... is 1). to use put-call parity to check that your work makes financial sense. Call = Spot + Put - (strike price)/(1+risk_free_rate)^Time 2). see if you can recreate anything close to present/past market (Yahoo finance?) data prices, i.e. testing your model against reality. good luck 2 It's only true if the claim can be replicated by dynamically hedging with the tradeable assets. So any proof should certainly refer to that property. My proof would be: There is a dynamic portfolio that replicates the claim, i.e. which is self-financing, pre-visible, and has terminal value equal to the value of the call option The value of any portfolio, ... 2 Generally we consider this issue for every T-claim contingent \Pi(t,X). However, there are two main approach in this context. As you mentioned, for first approach we should demand that the extended market \Pi(.,X)\,,\,S_0\,,S_1,...,S_N is free of arbitrage possibilities. Indeed we demand that there should exist a martingale measure Q for the ... 5 Thanks to @Phun and @oliversm I solved the problem. So I'm posting here the solution in case someone will need it. Under Black-Scholes assets dynamics are determined by a Geometric Brownian Motion, and we can define the price of a security at time t+\Delta t as:$$S_{t+\Delta t}=S_{t}\exp\left(\left(r-\frac{1}{2}\sigma^{2}\right)\Delta t+\sigma\sqrt{\...

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you can write the pay-off as $$(S_T-K)_+ I_{\min S_t > L} + RI_{\min S_t < L}$$ for down and out call. The first term is the standard call. The second is the rebate. Its value is $$Re^{-rT} P( \min S_t < L).$$ There is a standard formula for this probability. See eg my book Concepts.

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$d$ is a vector that collapses the $n$-dimensional vector into a real number. In the BS case $d=1$. There is nothing to be estimated. Also not that in practice affine pricing is done through FFT (and variants) rather than the direct transform you quote.

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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.... 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. 5 Starting from the Black-Scholes model that \dfrac{dS}{S} = \mu \:dt + \sigma\:dW_t $$where W_t is a standard Brownian motion, and \sigma and \mu are constant where \sigma > 0. Here W_t is a Brownian motion under the physical measure \mathbb{P}. We can then use Girsanov's theorem to change the measure to risk neutral measure \mathbb{Q} ... 5 stochastic vol and Levy process models are popular. Jump diffusion less so. FT techniques are definitely used. These days most of the focus is on valuation adjustments for vanilla products rather than how to price structured products. It tends to use both MC and lattice methods. If you want to be topical, I'd advise something related to valuation ... 0 I guess for you to obtain the value for the real world probability measure P, the expected rate of return should be given...else the value of P should be given. 0 I just made things clearer hoping it would help. Let define \mathbb{Q}_\theta as$$\frac{d\mathbb{Q}_\theta}{d\mathbb{P}}|_{\mathcal{F}_t}=\exp(\theta W_t -\frac{1}{2}\theta^2 t)=Z^\theta_t$$By girsanov, if W is a brownian motion under \mathbb{P}, then W^\theta_t=W_t-\theta t is a brownian motion under \mathbb{Q}^\theta$$\begin{split} \mathbb{...

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