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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{\...


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 ...


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: ...


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.


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 ...


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 ...


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 ...


2

$d$ 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.


2

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 $\{W_t^2, t \ge 0\}$ ...


2

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

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' ...


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 $...


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 ...


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


1

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

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....


1

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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