What is the most elegant Black Scholes proof that you have seen from martingale theory?Thankslas

What do you mean by Black-Scholes? The differential equation? The concept of risk neutrality? The formulae for calls and puts?P

- paullee123
**Posts:**26**Joined:**

The most elegant derivation is the Harrison and Pliska approach that involved two change of meausres using Girsanov, followed by derivation into PDE using Feymann-Kac. It made very few assumptions aside from equity being a geometric Brownian motion. See: www.russell.co.uk/images/pdf/The-Black- ... -Model.pdf (p11-13.)For details, enroll in CQF course and it is explained in details it in lecture 3.2.

What did Black and Scholes actually prove? The term "proof" should be reserved for things that are true, always and everywhere. What Black and Scholes describe with their formula is not true, it is a hypothetical mechanism that only works in la la land.

An elegant derivation using the martingale approach, please.@ paullee123 That reference looks great, thanks.@ Gmike2000 I am happy that we all live in la la Land since 30 years now... at least until the current credit crunch.

@ Paul: I would like to get from sde to the closed form solution for a call option.

QuoteOriginally posted by: lasAn elegant derivation using the martingale approach, please.@ Gmike2000 I am happy that we all live in la la Land since 30 years now... at least until the current credit crunch.hey, there is nothing wrong with training your gray cells in the mathematical tools that are used to derive financial models. this is very good, and it will set you apart from lesser educated MBA airheads.but keep in mind, these are untested and unverified models with dubious assumptions. The math may be correct, but the model itself can still be wrong.that's all.

There several financial and mathematical drawbacks made by B&S and next developers. Financial Drawbacks: They did not defined what they meant using notion 'price'. From the proof one could only imply what they meant. Indeed when they used their portfolio they wish that rate of return of the expected value of the portfolio would be equal to risk free rate. This is like a hypothetical definition which one must tested based on formal logic laws. Until test this hypothetical definition is not good not bad yet. The First question why expected value. This is common for stochastic finance modeling misleading that in turn out forced to make others unnatural assumptions.The test that checks correctness of the BS result: assume that an underlying stock a has expected return +5% and other -6% other parameters volatility and strike and maturity are the same. In this case the model suggests the same price for the good and bad stocks? Thus paying a one price you can receive the bad and the good stocks for the same price. Good one suggests your profit and the bad one suggest your losses in future. This fact call immediately for asking what is the price definition used? Because if we used the price definition and intermediate mathematics is correct then the only price definition could be wrong. One could argue that this example is not realistic. But we are talking about the theory in which all values for returns volatility and other parameters are admissible.Other interpretation of the price that I used for construction derivatives price and that does not contains such explicit drawbacks based on the definition that 2 investment are equal at the moment if they promised equal immediate rate of return and they are equal over an interval if they promised the same rate of return at any point of this interval. Using such definition it is possible to develop derivatives pricing theory. In This setting an option price is defined by the market by balancing risk reward possibilities.Unfortunately the people who control the situation do not have any relation to finance and the only appropriate solution was chosen about 8 years ago was removing me from financial business. I have a satisfaction feeling that in spite of all difficulties I could make a list:http://papers.ssrn.com/sol3/cf_dev/AbsB ... d=365639If you are interested to hear about math mistakes I could send you another message.

GZIP: On