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5

By definition the fair value of an option is given by an expectation value of the payoff, $\mathbf{E}\left[\textrm{payoff}(\textit{paths})\right]$. The probability distribution of the paths is the risk neutral measure. This is just an integral expression of the form you wrote. This applies to all option prices. Many options are, of course, special in the ...


4

The method described in Hallerbach (2004) always worked well for me. We derive an estimator for Black-Scholes-Merton implied volatility that, when compared to the familiar Corrado & Miller [JBaF, 1996] estimator, has substantially higher approximation accuracy and extends over a wider region of moneyness.


3

I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector $z \sim N(\mathbb{0},I_n)$ (where $z,0 \in \mathbb{R}^{n\times1}$ and $I_n \in \mathbb{R}^{n\times n}$ is the identity matrix) and you want to transform it into a multivariate normal $x \sim N(\mu,\Sigma)$ you do it the ...


2

There is a qualitative shift in the shape of the density. When V is small it is monotone decaying. When V is large it looks more like a Gaussian. Another reason he uses two schemes is that he wants match two moments of the density. When V is small, the moment matching equations for the quadratic Gaussian are unsolvable. When V is large they are unsolvable ...


2

Peter Jaeckel wrote a paper just on how to solve this problem: By Implication (July 2006; Wilmott, pages 60-66, November 2006). Probably the most complicated trivial issue in financial mathematics: how to compute Black's implied volatility robustly, simply, efficiently, and fast downloadable from jaeckel.org In my experience the most important thing is to ...


2

To keep things simple let's assume you have a perfect random number generator (i.e. I will discuss only the statistics not the numerics of the problem). I will also focus on the practical matter and gloss over some mathematical details. From a practical perspective "convergence" means that you will never get an exact answer from Monte-Carlo but ...


2

the output of an MC simulation depends on the random numbers used and if the distribution used is not too weird, after 10,000 runs you will get an answer that is distributed $$ \mu + \frac{\sigma}{\sqrt{n}} Z, $$ with $Z$ a standard normal. Here $n=10,000.$ With $\mu$ the quantity you want and $\sigma$ the standard deviation. So you won't get precisely the ...


1

Rather than thinking about the steps, think about the piecewise regions where your value is constant. When using the explicit scheme, time zero option value at any stock price for your simple digital option is basically just a function of which antecedent nodes (accounting for backwards timestepping) were above or below the strike. Slight modifications of ...


1

Presumably you are trying to use a finite difference method to solve a differential equation. The non-uniformity of the grid has an impact on accuracy. Hence, it is useful to include a parameter in the grid-generation algorithm that controls the rate at which the spacing increases away from the boundary. There are many approaches for generating non-uniform ...


1

Try Finite Differences to calculate your Greeks, it will give all the greeks for that specific underlying moviment. In order to back out the dollar change in your pnl just multiply each greek by the amount held in that position.


1

It appears that you are plotting your analytical delta as a % of the delta of the underlying. This is why the delta converges to 100% As for the numerical delta, it could be that you are not adjusting for the DV01 of the underlying. This would explain why the numerical delta still increases as the option gets more in the money and why the distortion is ...


1

Below is the root search algorithm code I wrote in college. This is written in octave. It's simple to understand and re-write in C++. Develop numerical methods algos as a separate module and integrate with your pricing and other code I want to WARN you to re-check for bugs. It always converges for my objective functions First function is Dekker method ...


1

Bracketing methods such as Bisection and Regula Falsi are always known to converge but they are very slow. Newton Raphson and secant methods are fast (quadratic convergence) but has convergence problems. Google for Newton Raphson convergence pitfalls. Classical ones such as"Trapped in local minima", "Diverge instead of converge" etc Algorithms such as ...


1

Use your total wealth allocated to the trades as denominator. Total wealth allocated would include all collateral. In this way you (or your broker) make sure that the denominator is always positive. Presumably this would also reflect what you really want to track. The only problem that remains is what amount of your wealth needs to be allocated. But this is ...



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