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7

Your analysis is correct. From the risk neutral process $$dS_t = rS_tdt + \sigma S_tdW_t$$ we get $$\mathbb{E}(S_\tau|\mathbb{F}_t) = S_te^{r(\tau-t)}$$ and $$\mathbb{E}(\int_{t}^{T}S_\tau d\tau|\mathbb{F}_t) = \frac{S_t}{r}[e^{r(T-t)}-1]$$ Hence, as $S \rightarrow \infty$ C(S,A,t) \sim \frac{S_te^{-r(T-t)}}{r(T-T_0)}[e^{r(T-t)}-1] = \frac{S_t}{r(... 6 Instead of simulating the spot price, simulate its logarithm since the latter can be simulated exactly for any time step. $$\ln S_{t + \Delta t} = \ln S_t + \left( r - \frac{1}{2} \sigma^2 \right) \Delta t + \sigma Z,$$ where Z \sim \mathcal{N}(0, \Delta t). You then just simply take the exponential of the simulated logarithmic ... 5 Let \mathbb{Q} be the risk-neutral probability measure which uses the risk-free bank account (B_t) as numeraire. In general, \mathrm{d}B_t=r_tB_t\mathrm{d}t. In the Black-Scholes setting, r_t\equiv r, we have B_t=e^{rt}. The stock measure \mathbb{Q}_S uses the compounded stock price S_te^{qt} as numeraire and is defined via the Radon Nikodym ... 5 Note that \begin{align*} \int_0^T\ln S_u du &= \int_0^T\Big[\big(r-\frac{1}{2}\sigma^2\big)u + \sigma W_u \Big] du\\ &=\frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T^2 + \sigma\int_0^T\int_0^u dW_s \,du\\ &=\frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T^2 + \sigma\int_0^T\int_s^T du \,dW_s\\ &=\frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T^2 + \sigma\... 4 Yes it is known in closed form. See https://www.rocq.inria.fr/mathfi/Premia/free-version/doc/premia-doc/pdf_html/asian_doc/asian_doc.html section 5.1 which references an older Geman-Yor paper. 4 The Asian option is cash settled, so the bank will transfer you 0.5. Delivering the shares and doing some trades is not possible. You can't buy the spot for the average price over a period, you just pay the spot price. Since you're into Asian options, I assume asian option valuation is useful for you to assess whether you're not paying too much. 4 For the Geometric Average Asian Option in BS, there is an arithmetic formula for the price - in fact, it is possible to price it using a BS vanilla options calculator, if you adjust the parameters slightly - as discussed in this blog post: http://www.quantopia.net/asian-options-iii-geometric-asian/ As the pricing formula is the same, the Greeks can be ... 3 Whether arithmetic or geometric averaging, you always get \begin{align*} \mathrm{AsianCall} - \mathrm{AsianPut} = e^{-rT} (\mathbb{E}[\bar{S}]-K). \end{align*} So, let’s compute the expectation. You know that \bar{S}=\exp\left( \frac{1}{T} \int_0^T \ln(S_t)\mathrm{d}t \right) where \ln(S_t) =\ln(S_0)+\left(r-q-\frac{1}{2}\sigma^2\right)t+ \sigma W_t. ... 3 They are not traded, even Over-The-Counter (OTC). Asian options with arithmetic averaging are traded. The geometric Asian may be used to derive a closed-form approximation for the arithmetic variety, or as a control variate in a Monte-Carlo simulation to significantly reduce the variance of the estimate. Arithmetic Asian options are interesting, not only ... 3 Asian options are based on the average price of something during a period. The average price of a stock is not very interesting, so Asian options on stock are not traded. The average price of oil (or other commodity) during a season is important because it determines the cost of heating or transport when you burn fuel constantly during the season. ... 3 This is an example of an exotic option. These are not listed and traded on any exchange. Rather they are traded in what is called the over the counter market. The dealers will trade these by entering into a contract with clients. 3 If W is a standard Brownian motion then \frac{1}{T}\int_0^T W_t dt has standard deviation \sqrt{\frac{T}{3}}. For this reason if S_t=S_0e^{(\alpha -\frac{1}{2}\sigma^2)t + \sigma W_t} is the GBM spot price then the average spot price \frac{1}{T}\int_0^T S_t dt has approximate volatility \frac{\sigma}{\sqrt{3}}. 3 Welcome to Quant SE. Unfortunately there is no closed form formula for computing the american contract value \max_{\tau}E^P\left[e^{-r\tau}(A_{\tau} - K)\right], so you have to resort to an american monte carlo method or a 2 dimensional PDE finite differences scheme for the joint dynamics dS_t/S_t = (r - q) dt + \sigma dW_t \\ dA_t = d\left(\frac{1}{t} ...

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Asian options are more common in the FX market where corporate hedgers are concerned with the average exchange rate that affects regular streams of foreign denominated revenue. Bermudan exercise is most common for interest rate swaptions. They provide flexibility in choosing when to exercise for cancelling a swap without the added cost of an unnecessary ...

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Below is an example of how you could plot a "call" option value with RQuantLib: library(RQuantLib) library(ggplot2) call_price <- sapply(seq(365,0,-1), function(x) AmericanOption("call", 100, 100, 0.2, 0.03, x/365, 0.4)$value) qplot(day, call_price, data=data.frame(day=0:365, call_price=call_price), geom="line") The code output: Another useful package ... 3 Chan, Jiun Hong and Joshi, Mark S. and Zhu, Dan, First and Second Order Greeks in the Heston Model (December 26, 2010). Available at SSRN: https://ssrn.com/abstract=1718102 or http://dx.doi.org/10.2139/ssrn.1718102 should just about cover it. 3 QuantLib does have an FD pricing engine for asian options ql.FdBlackScholesAsianEngine(stochProcess, tGrid=100, xGrid=100, aGrid=50), but I've just discovered it only prices Discrete, Arithmetic payoffs! Moving from Continuous to Discrete (documented here) doesn't change the price of the option much, if you pass in something like asianFixingDates = [ql.... 2 The$Pdynamics of the underlying asset are: \begin{align*} dS=S(\mu dt+\sigma dB_t) \end{align*} That has the following solution under the\mathcal{Q}dynamics: \begin{align*} S_t=S_0 e^{(r-\frac{\sigma^2}{2})t+\sigma W_t} \end{align*} WhereW_t$is the equivalent martingale with respect to the original geometric brownian motion. Define$Y_t=\int_0^t ...

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A few tips. First note that $e^{-rt}S_t$ is a martingale. So make it appear and then integrate by part to rewrite $\int S_u du$ as a stochastic integral. Finally use the Ito isometry property.

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You want to work directly with $\overline{X}$, and not some other r.v. with the same distribution, since equivalence in distribution doesn't imply that correlation remains the same. For ease of notation, I'll assume that $\mu = 0$ and $\sigma = 1$. I claim that $$\text{cov}\left(\overline{X},X \right) = \frac{1}{t} \int_0^t s \ ds.$$ Note that this is ...

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This is the standard textbook definition for the payoff of an arithmetic average Asian call option. In reality the average will be based on a daily average of stock prices over some period at the end of the life of the option. The first "idea" behind Asian options is that the payoff is harder to manipulate by dealers - the story is that in certain small ...

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Kemna and Vorst (1990) [ download ] is a classic in Monte Carlo method for Asian option. Geometric mean, which can be analytically computed, is used as a control variate to reduce MC noise.

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Looks good to me, although idk why you have (T-t) in the discounting... isn't big T the total time to maturity? What is little t in the equation? Shouldn't it just be exp[-rT] because you discount from the time of payoff which is the expiration of the option.

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The paper is reliable and the formula is correct. However as you mention yourself there is an error. $$\frac{\log \left(\frac{e^{0.01}-1}{0.01}\right)}{0.01} = 0.500417 \neq 0.498$$

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It would be an Asian strike call option, with the Asianing being computed over some period $[0;\tau]$. Not a big deal that the average is not computed from 0 to T. It even seems more natural to be done that way in practice. Before $\tau$, pricing would be similar to your option Asianing until T. For $t>=\tau$, you already know the value of the Asian ...

1

Showing that $(A_t^*)$ is a martingale does not really help you in understanding the distribution of $A_T^*$. Instead, the key is your previous question. Under $\mathbb{Q}\sim\mathbb{P}$, you have \begin{align*} S_t=S_0\exp\left(\left(r-\frac{1}{2}\sigma^2\right)t+\sigma W_t\right). \end{align*} Under $\mathbb{Q}_S\sim\mathbb{Q}$, you have \begin{align*} ...

1

This won't transform a version of the heat equation that can be solved analytically. The extra term results in the time-integral of Geometric Brownian Motion, which has no known analytical transform. The sum (or equally, the arithmetic average) of lognormally distributed variables is believe to result in a Bessel process that converges to an inverse gamma ...

1

I don't think there is any good approximation to the american option $\max_{\tau}E^P\left[e^{-r \tau}(S_{\tau} - M_{\tau})^+\right]$ where $M_t = \frac{1}{t}\int_0^t S_u du$ is the running average, but you can compute it quite efficiently by noting that  \max_{\tau}E^P\left[e^{-r \tau}(S_{\tau} - M_{\tau})^+\right] =S_0\max_{\tau}E^P\left[\frac{e^{-r \tau}...

1

Not possible for us to debug your code to find out the cause. Possible reasons: You have a bug You don't have a bug but your understanding of the model is incorrect Both your code and FinCAD are correct, the difference is due to assumptions I don't think FinCAD would make a mistake in such simple model, they have a team of PhD quants for model validation. ...

1

numpy.max wouldn't work in this case. Try Math.max. If you don't believe me, try this: print(numpy.max(-100, 0)) gives -100 You're supposed to draw from random Gaussian, not uniform. https://docs.scipy.org/doc/numpy/reference/generated/numpy.random.rand.html Create an array of the given shape and populate it with random samples from a uniform ...

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