date dividend price 20070125 53 93 20070126 54 67 20070129 54 06 20070130 54 27 20070131 55 13 20070201 0 54 54 73 20070202 54 66 20070205 54 75 20070206 54 95 20070207 54 95 20070208 54 44 20070209 53 4 20070212 53 43 20070213 53 71 20070214 54 18 20070215 54 21 20070216 54 1 20070220 54 19 20070221 53 751 Estimate the volatility of Citigroup Calculate the daily returns of Citigroup Use the excel function STDEV to estimate the volatility of the daily returns within the last year Multiply the daily volatility by the square root of 252 (number of trading days per year) to get the annualized volatility Further, estimate the annualized volatility within the last 6 months (July 26, 2007 to Jan 25, 2008) Construct the volatility surface using the options in Exhibit 3 Assume the dividend yield for options with 1 year to maturity is 3 6 and for options with 2 years to maturity 2 6 Use the Excel Option Pricing Tool from our class Use the excel Solver Add in to find a volatility (cell C6) such that the option prices in Exhibit 3 match the American option prices in cells R17 and R18 in a Binomial tree with n 250 steps How do the option implied volatilities compare to the volatility estimated from the past 1 year or 6 months of daily data Exhibit 3 Listed American Options on Citigroup Stock on January 25, 2008 Expiration Date Option Symbol Type Strike Price ($) Bid ($) Ask ($) Open Interest 17 Jan 2009 VRN AE Call 25 4 70 4 70 39,297 VRN AF Call 30 2 60 2 61 87,923 VRN ME Put 25 3 40 3 45 69,211 VRN MF Put 30 6 25 6 25 105,537 16 Jan 2010 WRV AE Call 25 6 10 6 40 19,821 WRV AF Call 30 4 15 4 45 44,015 WRV ME Put 25 4 75 4 90 23,785 WRV MF Put 30 7 55 7 90 49,501 No options with a longer maturity were listed The bid price or bid (ask price or ask) is the price received (paid) by option sellers (buyers) The open interest for a given option is the number of options outstanding, i e , the number of long positions (which equals that of short positions)

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date dividend price 20070125 53.93 20070126 54.67 20070129 54.06 20070130 54.27 20070131 55.13 20070201 0.54 54.73 20070202 54.66 20070205 54.75 20070206 54.95 20070207 54.95 20070208

date	dividend	price
20070125		53.93
20070126		54.67
20070129		54.06
20070130		54.27
20070131		55.13
20070201	0.54	54.73
20070202		54.66
20070205		54.75
20070206		54.95
20070207		54.95
20070208		54.44
20070209		53.4
20070212		53.43
20070213		53.71
20070214		54.18
20070215		54.21
20070216		54.1
20070220		54.19
20070221		53.751

Estimate the volatility of Citigroup.
1. Calculate the daily returns of Citigroup.
2. Use the excel function STDEV to estimate the volatility of the daily returns within the last year. Multiply the daily volatility by the square root of 252 (number of trading days per year) to get the annualized volatility. Further, estimate the annualized volatility within the last 6 months (July 26, 2007 to Jan 25, 2008).
3. Construct the volatility surface using the options in Exhibit 3. Assume the dividend yield for options with 1 year to maturity is 3.6% and for options with 2 years to maturity 2.6%. Use the Excel Option Pricing Tool from our class. Use the excel Solver Add-in to find a volatility (cell C6) such that the option prices in Exhibit 3 match the American option prices in cells R17 and R18 in a Binomial tree with n = 250 steps. How do the option implied volatilities compare to the volatility estimated from the past 1 year or 6 months of daily data?

Exhibit 3

Listed American Options on Citigroup Stock on January 25, 2008

Expiration Date	Option Symbol	Type	Strike Price ($)	Bid ($)	Ask ($)	Open Interest
17 Jan 2009	VRN.AE	Call	25	4.70	4.70	39,297
	VRN.AF	Call	30	2.60	2.61	87,923
	VRN.ME	Put	25	3.40	3.45	69,211
	VRN.MF	Put	30	6.25	6.25	105,537
16 Jan 2010	WRV.AE	Call	25	6.10	6.40	19,821
	WRV.AF	Call	30	4.15	4.45	44,015
	WRV.ME	Put	25	4.75	4.90	23,785
	WRV.MF	Put	30	7.55	7.90	49,501

No options with a longer maturity were listed. The bid price or bid (ask price or ask) is the price received (paid) by option sellers (buyers). The open interest for a given option is the number of options outstanding, i.e., the number of long positions (which equals that of short positions).