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by Alan
August 1st, 2019, 2:48 pm
Forum: General Forum
Topic: Circular barrier
Replies: 14
Views: 588

Re: Circular barrier

To me, a circle requires two dimensions. What are the two dimensions? 
by Alan
July 30th, 2019, 11:31 pm
Forum: Student Forum
Topic: Derivation for two independent brownian motions
Replies: 4
Views: 464

Re: Derivation for two independent brownian motions

Thanks.

No, first, I said "n is large but not infinite", so the puzzle is to provide an asymptotic formula that depends upon n. Keep thinking!

Second, if n were infinite, the expected max is [$]+\infty[$].

Hint: start from bearish's comment, generalize it, and start calculating. 
by Alan
July 30th, 2019, 11:24 pm
Forum: Numerical Methods Forum
Topic: New Approximation to the Normal Distribution Quantile Function
Replies: 14
Views: 36418

Re: New Approximation to the Normal Distribution Quantile Function

The late Graeme West, Wilmott mag, Fig. 2 quoting an alogorithm of Hart (accurate to machine double precision everywhere). 
by Alan
July 30th, 2019, 4:34 am
Forum: Numerical Methods Forum
Topic: Explicit Formula for computing IV
Replies: 15
Views: 758

Re: Explicit Formula for computing IV

Please justify the need for extreme speed. What is the application?
by Alan
July 26th, 2019, 5:56 pm
Forum: Student Forum
Topic: Black-Scholes from Random Walk Derivation - Issue
Replies: 8
Views: 462

Re: Black-Scholes from Random Walk Derivation - Issue

Or, the cookbook recipe to working with stochastic differential equations is the following multiplication table: [$]dt \times dt = dt \times dW_t = dW_t \times dt = 0 [$] [$]dW_t \times dW_t = dt[$] (All higher powers are zero). After all, to drive a car, you don't have to be a mechanic.   Of course...
by Alan
July 26th, 2019, 5:48 pm
Forum: Numerical Methods Forum
Topic: High Accuracy Greeks computation under trinomial model
Replies: 12
Views: 588

Re: High Accuracy Greeks computation under trinomial model

If you set up the option value pde problem using x = log S with solution [$]f(x,t)[$], you can differentiate it to get the pde for [$]g(x,t) = f_x[$]. Then perhaps simply solve that second pde for [$]g[$] on a trinomial lattice, and convert to delta. You probably need two passes, the first to get t...
by Alan
July 26th, 2019, 4:01 am
Forum: Numerical Methods Forum
Topic: High Accuracy Greeks computation under trinomial model
Replies: 12
Views: 588

Re: High Accuracy Greeks computation under trinomial model

If you set up the option value pde problem using x = log S with solution [$]f(x,t)[$], you can differentiate it to get the pde for [$]g(x,t) = f_x[$]. Then perhaps simply solve that second pde for [$]g[$] on a trinomial lattice, and convert to delta. You probably need two passes, the first to get th...
by Alan
July 24th, 2019, 12:57 am
Forum: Technical Forum
Topic: Why is Bellman Equation solved by backwards?
Replies: 14
Views: 846

Re: Why is Bellman Equation solved by backwards?

It's usually solved backwards because of the "principle of optimality", which you can google if you don't know it. The standard finance example is determining an optimal exercise strategy for an American-style put. It's day 1 and the put expires on day N. Whatever the "strategy" on day 1, it should ...
by Alan
July 22nd, 2019, 6:41 pm
Forum: Technical Forum
Topic: Why is Bellman Equation solved by backwards?
Replies: 14
Views: 846

Re: Why is Bellman Equation solved by backwards?

I cannot make a specific example, because I read that in a book. 

page1 https://ibb.co/1fhhK2m
page2 https://ibb.co/3WW69NW
page3 https://ibb.co/Zfxt1Ty

I looked briefly. I think I was right: with stationary transition densities, it's a trivial relabeling: [$]n \rightarrow n' = N - n[$].
by Alan
July 22nd, 2019, 5:28 am
Forum: Numerical Methods Forum
Topic: Smoothing splines (clamped spline)
Replies: 22
Views: 969

Re: Smoothing splines (clamped spline)

You could take a look at http://www.netlib.org/dierckx/. concur.f implements the clamped smoothing spline. Disclaimer: I haven’t used this library myself, but Dierckx’s book is quite good. Thank you -- it looks like Dierckx's book may indeed have what I am searching for -- ordered a decently priced...
by Alan
July 21st, 2019, 7:11 pm
Forum: Technical Forum
Topic: Why is Bellman Equation solved by backwards?
Replies: 14
Views: 846

Re: Why is Bellman Equation solved by backwards?

I will take a guess that it is a "trivial" rewrite. That is, if you a have a stationary PDE problem with a terminal condition at T, you can always trivially rewrite it with [$]\tau = T - t[$] and the terminal condition becomes an initial condition. Ditto for the Bellman problem.
by Alan
July 19th, 2019, 4:46 pm
Forum: Numerical Methods Forum
Topic: Smoothing splines (clamped spline)
Replies: 22
Views: 969

Re: Smoothing splines (clamped spline)

For Alan specific problem with the smoothing parameter, the problem was already solved in Carl de Boor (1978), A Practical Guide to Splines (Chapter XIV). While it is solved with "natural" BC (allowing to have a tridiagonal matrix and adding 2 equations to the linear system), solving for your speci...
by Alan
July 18th, 2019, 5:30 pm
Forum: Numerical Methods Forum
Topic: Smoothing splines (clamped spline)
Replies: 22
Views: 969

Re: Smoothing splines (clamped spline)

Thanks, guys but it looks like you are all citing either (i) cubic spline interpolation (the interpolant goes through the data points) or (ii) standard smoothing splines (Green & Silverman, for example, which have a so-called "natural" boundary condition [$]g''(a) = g''(b) = 0[$]). I am asking about...
by Alan
July 18th, 2019, 12:47 am
Forum: Numerical Methods Forum
Topic: Smoothing splines (clamped spline)
Replies: 22
Views: 969

Re: Smoothing splines (clamped spline)

Hi Daniel, Thank you for the code. I haven't opened it yet. Let me know if I will find the solution to the following problem, which will answer your question. I want to find the piecewise cubic polynomial [$]g(t)[$] that solves the following problem: (*) minimize [$] \left( \sum_{i=1}^n \{ y_i - g(t...
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