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06-RettoLin(f)

# 06-RettoLin(f) - From the Return Form to the Linear...

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Primbs, MS&E 345 1 From the Return Form to the Linear Functional Form of Arbitrage

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Primbs, MS&E 345 2 Pricing Theory: Optimization Return form (pdes) Linear function form (risk neutral)
Primbs, MS&E 345 3 Pricing Theory: Return form (pdes) Linear function form (risk neutral) Optimization

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Primbs, MS&E 345 4 The Linear Functional Form of Arbitrage: What is the return form lacking? r μ rdt B dB = B B(1+r) π σ μ d dt S dS + = S ) 1 ( σ μ + + S ) 1 ( μ + S There is a clear arbitrage opportunity! The return form doesn’t capture it! Consider the following example:
Primbs, MS&E 345 5 In our example, does there exist a λ such that: λσ μ = - r Yes! of course! Something is missing from the return form. The return form is a necessary condition for no arbitrage, but it is not sufficient. Let’s return to the basics... The return form doesn’t capture it!

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Primbs, MS&E 345 6 What is an arbitrage opportunity? t v u dt t v + d dt t v + price payoff If we plot all the cash flows associated with the purchase of a portfolio v , there must be some positive and some negative cash flows. If all cash flows were positive (actually non-negative), then this would represent an arbitrage opportunity.
Primbs, MS&E 345 7 Portfolio: x

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06-RettoLin(f) - From the Return Form to the Linear...

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