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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 timeseries (Excels function =STDEV does the job). For the whole timeperiod (19632011) 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 timevarying, 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 timeperiod. This is probably the simplest possible way to look for potential timevariation in volatility. 1 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 20day volatility estimates Time Volatility 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 40day volatility estimates Time Volatility 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 60day volatility estimates Time Volatility 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 120day volatility estimates Time Volatility 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 252day volatility estimates Time Volatility 1970 1980 1990 2000 2010 0.1 0.2 0.3 0.4 0.5 4year volatility estimates Time Volatility Figure 1: Estimates of volatility using different rolling windows, from 20days (topleft) to 4years (bottomright). 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, KenanFlagler 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 19992003 and 19871991 peridos are significantly more volatile than 19761980 or 19931998.19992003 and 19871991 peridos are significantly more volatile than 19761980 or 19931998....
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 Fall '10
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 Derivatives, Pricing, Options, Volatility

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