I'm thinking about taking a course on Linear and Convex Programming, but I don't know how useful it is in the real world finance. Which areas in finance is mathematical programming used?
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2$\begingroup$ It is useful. Basic understanding of optimisation is required for many areas of quant finance. $\endgroup$– user2763361Sep 27, 2014 at 12:56
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$\begingroup$ Welcome to Quant.SE! Thank you for this question. If you find the answers helpful it would be great if you upvoted them and accepted one of them :-) $\endgroup$– vonjdOct 7, 2014 at 6:22
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3 Answers
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To price financial instruments such as options, bonds and stocks must be priced so as to be "arbitrage free". The concept of arbitrage can be made precise by one of the fundamental ideas of quantitative finance, the so called Arbitrage Theorem.
Put differently the Arbitrage Theorem provides a very elegant and general method for pricing derivative instruments. The result from which the Arbitrage Theorem is normally derived is the Duality Theorem from linear programming. So in a way it could be argued that linear programming forms the theoretical basis of derivatives pricing.
A very good introduction can be found here (from An Undergraduate Introduction fo Financial Mathematics by J. Robert Buchanan):
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Optimization is definitely important in Quantitative Finance, especially for portfolio optimization where we maximize utility of the return of a portfolio as linear weighted vector of asset returns subject to a desired risk level:
$$ \max_{w\in[0,1]^n} U(\mu_p(w),\sigma_p(w))\quad s.t. \sum_{i=1}^n w_i=1$$
where $w$ being the portfolio weights, and $U$ utility function.
CAPM assumes investors with concave utility function $U=\mu_p-\frac{1}{2}\sigma_p^2$, from which then follows that all investors mix the market portfolio with the riskfree asset according to their desired minimum risk/maximum return level.
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1$\begingroup$ +1 for covering the P part, see my answer for the Q part :-) $\endgroup$– vonjdOct 6, 2014 at 8:14
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Does optimizing a solution for a given set of parameters sound like something quants would need to know how to do?
Yep! Indeed many things quants do revolve around optimization, and linear programming (and integer programming, multi-integer programming, quadratic programming, etc.) is all about finding the optimal solution to something given some set of constraints.
For instance, think of the problem of finding multi-dimensional vectors for creating a mean-reverting portfolio (i.e., cointegrating vectors). If mean reversion is something new, read this. Essentially, a mean reverting portfolio is a basket of assets with some associated cointegration relationship among them that allow for easy-to-trade signals when the portfolio is above / down some standard deviation of the norm.
If you think about this entails; you're looking a vector of correlated securities given some optimal constraints—mainly size. This is a problem of quadratic programming. There are other things to consider of course w.r. to constraints: min / max position sizes, whether you can short assets, etc...
