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I am trying to develop a short course on financial theory, covering the fundamentals of forward and options pricing, and 'efficient market' theory. I want to reduce the amount of mathematics to a minimum. This is not because the audience does not include mathematicians (it does) but rather because in my view mathematics generally detracts from the simplicity and beauty of a subject. Mathematics also focuses on the process of derivation from assumptions, rather than the assumptions themselves.

My question is whether this would be possible for financial theory. In particular (a) are there any basic principles of financial theory that cannot be grasped without complex mathematics and (2) are there any important results (i.e. derived results) which cannot be explained except by complex mathematics?

For a sense of where I am coming from, this page http://mathworld.wolfram.com/PythagoreanTheorem.html on beautiful versus complicated proofs of Pythagoras.

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You realize that it doesn't have to be that way. Mathematics are a tool that allows you to formalize and be precise about what you mean. I personally understand any concept much better when someone writes some math. Before than that, everything looks fuzzy or arbitrary. Mathematics are a great tool to have and understand, and I feel that is responsibility of the professor to translate that intuition to mathematics. In other words, if you can explain something without the maths, you can explain the same with the maths (unless you don't understand them) and it will be even better. –  FKaria Feb 13 at 0:27
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2 Answers

For the binary tree model the full replication property of all possible options can be shown using basic algebra and the no-arbitrage argument. It's beautiful how simple it is actually.

You can find the complete derivation in Shreve's Stochastic Calculus for Finance I: The Binomial Asset Pricing Model.

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Thank you Anna. I also found "Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model" by Lars Tyge Nielsen very helpful. (Revue Finance (Journal of the French Finance Association) 14 (1993), 95-106. ltnielsen.com/wp-content/uploads/Understanding.pdf ). This has a neat explanation of what d1 and d2 are, which I always found a bit of a mystery. –  quis est ille Feb 12 at 15:13
    
Reminded me there was a time when the professor insisted a guy to describe the algorithm in form of a "Formula" - my friend wrote pseudo code –  Joanne Feb 13 at 16:11
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A very good book covering such fundamentals with no or only a minimal amount of maths — highly recommended!

The topics that are covered here are:

  • Siegel's Paradox
  • Likelihood of Loss
  • Time Diversification
  • Why the Expected Return Is Not To Be Expected
  • Half Stocks All the Time or All Stocks Half the Time?
  • The Irrelevance of Expected Return on Option Valuation

Another well received title, perhaps even more fitting here, is the following:

From the preface:

My primary objective in a book like this is to create something about derivatives that is easy to read. Derivatives can be a painful subject to learn, and many legal pads are used up, sometimes frustratingly, in working through some of the principles covered in technical derivatives books. This book is different. While I do not advise that you curl up with it by a warm fire, a loyal dog, and a loved one, I do think you can relax in an easy chair and read it without pen and paper at your side. To that extent, this book is unique. Rarely will you find a derivatives book without equations.

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+1 currently reading. –  Anna Feb 12 at 18:27
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@Anna: I haven't seen you before here, so a very warm welcome to you! What is your background? I see that you are located in Frankfurt. I am a professor in Aschaffenburg, near Frankfurt. If you would like to expand your quant network you can drop me a line here: h-ab.de/nc/eng/… –  vonjd Feb 12 at 19:08
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@vondj Thanks, I will. –  Anna Feb 12 at 19:18
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I feel compelled to read any book with a chapter called "Why the Expected Return Is Not To Be Expected". –  quis est ille Feb 13 at 10:45
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