Derivatives -- the View from the Trenches

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The Difference between P and Q Suppose we know that under P dS S dt dW and assume that 0.17 . Then we wish to estimate the drift as 1 St ( ) 1 t S(0) 2 2 ˆ (ln ) t We have std[ ˆ ] t This gives us the following table horizon std[ ˆ ] 1 17.0% 10 5.4% 20 3.8% 50 2.4% 100 1.7% 200 1.2% 400 0.9% To get within 1% error you need about 400 years of data. So with absence very very long time series of data it is extremely difficult to estimate the real distribution. This of course has the consequence that there essentially are as many P measures as there are agents in the economy! 20
The Difference between P and Q and Market Efficiency The difficulty of obtaining the "true" P measure aside, it is clear that the market generally prices in significant risk premia. There are tons of examples: - Implied volatility > Historical volatility. - Equity volatility skew. - Corporate bond spreads are much wider than historical default probabilities. - The long end of the yield curve is far to volatile to be consistent with the historical mean reversion of interest rates. - Long maturity volatilities are far to volatile to be consistent with historical mean reversion of volatility. This does not mean that the market is inefficient. On the contrary it means that the market is generally efficient but that there are significant risk premia. So the market is risk averse and financial theory works! What hedge funds do is to a 99% extend to collect risk premia. So hedge funds generally run in front of the steam engines! 21
Page 1 and 2: Derivatives -- the View from the Tr
Page 3 and 4: My Life as a Quant CV: - 1997: PhD
Page 5 and 6: The First Fundamental Theorem of De
Page 7 and 8: Theorem 1 and Option Markets Black
Page 9 and 10: Theorem 1 and Models Theorem 1 ext
Page 11 and 12: Theorem 1 and Technical Results A
Page 13 and 14: Theorem 2: The Gospel of the Jump
Page 15 and 16: But in essence the strategies invol
Page 17 and 18: Theorem 2 and the Equity Volatility
Page 19: So in front of you, you have an opt
Page 23 and 24: Model Philosophy Keep it simple --
Page 25 and 26: Foreign exchange - Black-Scholes wi
Page 27 and 28: An Interest Rate Model at Work in J

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