case09sol

case09sol - Introduction to derivatives, Fall 2011 BUSI...

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Unformatted text preview: Introduction to derivatives, Fall 2011 BUSI 588, Case 9 solutions Solutions to Galitzin - pricing American options 1. (*) The first issue was to deal with what volatility number to use. The data was in daily format, and in prive levels, so a reasonable first cut was to create a return series using r t +1 = log ( S t +1 /S t ) and then compute the standard estimate of the variance of this time-series (Excels function =STDEV does the job). For the whole time-period (1963-2011) my estimate assuming 252 trading days per year is = 0 . 162, which I will round up to 15% in what follows (recall the relationship between the annual volatility and the daily one: a = 252 d ). This could have been a good starting point for our binomial model. Nonetheless, as discussed in class, volatility is significantly time-varying, and quite persistent. Figure 1 plots several estimates of volatility. Essentially, I compute estimates based on x days windows and roll- over these windows over the whole time-period. This is probably the simplest possible way to look for potential time-variation in volatility. 1 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 20-day volatility estimates Time Volatility 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 40-day volatility estimates Time Volatility 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 60-day volatility estimates Time Volatility 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 120-day volatility estimates Time Volatility 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 252-day volatility estimates Time Volatility 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 4-year volatility estimates Time Volatility Figure 1: Estimates of volatility using different rolling windows, from 20-days (top-left) to 4-years (bottom-right). 1 The industry standard is to use a class of models that follow under the GARCH label (generalized autoregressive conditional heteroskedasticity). We shall not pursue these methods in this course. c Diego Garc a, Kenan-Flagler Business School Page 1 of 4 Introduction to derivatives, Fall 2011 BUSI 588, Case 9 solutions As it is apparent from the graph, there is significant variation in the volatility of the S&P500 over the last 30 years (the data provided in the case corresponds indeed to the S&P500). The 1999-2003 and 1987-1991 peridos are significantly more volatile than 1976-1980 or 1993-1998.1999-2003 and 1987-1991 peridos are significantly more volatile than 1976-1980 or 1993-1998....
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case09sol - Introduction to derivatives, Fall 2011 BUSI...

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