Returns and Volatility (2) Date Closing price Expected Return 25 01 2008 26 64 24 01 2008 27 33 0 025900901 23 01 2008 26 36 0 035492133 22 01 2008 24 40 0 07435129 18 01 2008 24 45 0 002045074 17 01 2008 24 96 0 020858896 16 01 2008 26 24 0 051282051 15 01 2008 26 94 0 026676829 14 01 2008 29 06 0 078693393 11 01 2008 28 56 0 017205781 10 01 2008 28 11 0 015756303 09 01 2008 27 49 0 022056208 08 01 2008 27 14 0 012731903 07 01 2008 28 26 0 041267502 04 01 2008 28 24 0 000707714 03 01 2008 28 93 0 024433428 02 01 2008 28 92 0 000345662 31 12 2007 29 44 0 017980636 28 12 2007 29 29 0 005095109 27 12 2007 29 56 0 009218163 26 12 2007 30 45 0 030108254 24 12 2007 30 98 0 017405583 21 12 2007 30 24 0 023886378 20 12 2007 29 89 0 011574074 19 12 2007 30 21 0 010705922 18 12 2007 30 38 0 005627276 17 12 2007 30 77 0 012837393 14 12 2007 30 70 0 002274943 13 12 2007 31 01 0 01009772 12 12 2007 31 47 0 014833925 11 12 2007 33 23 0 055926279 10 12 2007 34 77 0 046343665 07 12 2007 34 31 0 013229796 06 12 2007 34 35 0 001165841 05 12 2007 33 69 0 019213974 04 12 2007 32 55 0 033837934 03 12 2007 33 06 0 015668203 30 11 2007 33 30 0 007259528 29 11 2007 32 29 0 03033033 28 11 2007 32 29 0 27 11 2007 30 32 0 0610096 26 11 2007 30 70 0 012532982 23 11 2007 31 70 0 03257329 21 11 2007 30 73 0 030599369 20 11 2007 31 40 0 021802799 19 11 2007 32 00 0 01910828 16 11 2007 34 00 0 0625 15 11 2007 34 58 0 017058824 14 11 2007 36 04 0 042220937 13 11 2007 35 90 0 003884573 12 11 2007 33 57 0 064902507 09 11 2007 33 10 0 014000596 08 11 2007 32 90 0 006042296 07 11 2007 33 41 0 01550152 06 11 2007 35 08 0 049985034 05 11 2007 35 90 0 023375143 02 11 2007 37 73 0 05097493 01 11 2007 38 51 0 020673204 31 10 2007 41 90 0 088029083 30 10 2007 42 11 0 005011933 29 10 2007 42 69 0 01377345 26 10 2007 42 63 0 001405481 25 10 2007 41 23 0 032840722 24 10 2007 41 82 0 014309968 23 10 2007 42 44 0 014825442 22 10 2007 42 61 0 004005655 19 10 2007 42 36 0 005867167 18 10 2007 43 83 0 03470255 17 10 2007 44 66 0 018936801 16 10 2007 44 79 0 002910882 15 10 2007 46 24 0 032373298 12 10 2007 47 87 0 035250865 11 10 2007 48 32 0 00940046 10 10 2007 47 14 0 02442053 09 10 2007 47 62 0 010182435 08 10 2007 47 80 0 003779924 05 10 2007 48 30 0 010460251 04 10 2007 47 63 0 013871636 03 10 2007 47 89 0 005458744 02 10 2007 47 86 0 000626436 01 10 2007 47 72 0 002925198 28 09 2007 46 67 0 022003353 27 09 2007 46 88 0 004499679 26 09 2007 46 55 0 007039249 25 09 2007 46 31 0 005155747 24 09 2007 46 59 0 00604621 21 09 2007 47 51 0 019746727 20 09 2007 47 20 0 006524942 19 09 2007 48 27 0 022669492 18 09 2007 48 37 0 00207168 17 09 2007 46 03 0 048377093 14 09 2007 46 64 0 013252227 13 09 2007 46 36 0 006003431 12 09 2007 45 76 0 012942192 11 09 2007 46 01 0 005463287 10 09 2007 45 30 0 015431428 07 09 2007 45 48 0 00397351 06 09 2007 45 66 0 003957784 05 09 2007 46 00 0 007446343 04 09 2007 47 21 0 026304348 31 08 2007 46 88 0 006990044 30 08 2007 46 23 0 013865188 29 08 2007 46 95 0 015574302 28 08 2007 46 14 0 017252396 27 08 2007 47 79 0 035760728 24 08 2007 48 50 0 014856665 23 08 2007 48 35 0 003092784 22 08 2007 48 43 0 001654602 21 08 2007 48 06 0 007639893 20 08 2007 48 39 0 006866417 17 08 2007 48 81 0 008679479 16 08 2007 47 55 0 025814382 15 08 2007 45 61 0 040799159 14 08 2007 45 66 0 001096251 13 08 2007 46 54 0 019272887 10 08 2007 47 00 0 009883971 09 08 2007 46 90 0 00212766 08 08 2007 49 49 0 055223881 07 08 2007 48 59 0 018185492 06 08 2007 48 35 0 004939288 03 08 2007 45 72 0 054395036 02 08 2007 47 24 0 033245844 01 08 2007 46 85 0 008255715 31 07 2007 46 57 0 005976521 30 07 2007 47 19 0 013313292 27 07 2007 46 97 0 004662005 26 07 2007 47 81 0 017883756 25 07 2007 49 21 0 029326501 24 07 2007 49 31 0 001989348 23 07 2007 50 86 0 031433786 20 07 2007 50 73 0 002556036 19 07 2007 51 13 0 007884881 18 07 2007 51 60 0 009192255 17 07 2007 52 46 0 016666667 16 07 2007 52 19 0 005146778 13 07 2007 52 52 0 00632305 12 07 2007 52 84 0 006092917 11 07 2007 51 41 0 027062831 10 07 2007 51 00 0 007975102 09 07 2007 51 60 0 011764706 06 07 2007 51 70 0 001937984 05 07 2007 51 50 0 003868472 03 07 2007 51 86 0 006990291 02 07 2007 51 64 0 004242191 29 06 2007 51 29 0 006777692 28 06 2007 51 81 0 010138429 27 06 2007 51 81 0 26 06 2007 51 15 0 012738854 25 06 2007 51 69 0 010557185 22 06 2007 52 41 0 013929193 21 06 2007 53 66 0 02385041 20 06 2007 53 44 0 004099888 19 06 2007 54 26 0 015344311 18 06 2007 53 96 0 005528935 15 06 2007 53 98 0 000370645 14 06 2007 53 67 0 005742868 13 06 2007 53 68 0 000186324 12 06 2007 52 60 0 020119225 11 06 2007 53 47 0 016539924 08 06 2007 53 33 0 002618291 07 06 2007 52 52 0 015188449 06 06 2007 53 33 0 015422696 05 06 2007 53 90 0 010688168 04 06 2007 54 15 0 004638219 01 06 2007 54 51 0 006648199 31 05 2007 54 49 0 000366905 30 05 2007 55 20 0 013029914 29 05 2007 54 91 0 005253623 25 05 2007 55 12 0 00382444 24 05 2007 54 93 0 003447025 23 05 2007 55 01 0 001456399 22 05 2007 55 08 0 001272496 21 05 2007 54 84 0 004357298 18 05 2007 55 00 0 002917578 17 05 2007 54 80 0 003636364 16 05 2007 54 91 0 002007299 15 05 2007 52 79 0 038608632 14 05 2007 52 86 0 001326009 11 05 2007 53 11 0 004729474 10 05 2007 53 20 0 001694596 09 05 2007 54 12 0 017293233 08 05 2007 53 72 0 007390983 07 05 2007 53 88 0 002978407 04 05 2007 54 09 0 00389755 03 05 2007 53 95 0 002588279 02 05 2007 54 30 0 006487488 01 05 2007 54 20 0 001841621 30 04 2007 53 62 0 010701107 27 04 2007 53 37 0 004662439 26 04 2007 53 56 0 003560052 25 04 2007 53 80 0 004480956 24 04 2007 52 81 0 018401487 23 04 2007 53 11 0 005680742 20 04 2007 53 42 0 005836942 19 04 2007 53 09 0 006177462 18 04 2007 52 99 0 001883594 17 04 2007 52 53 0 008680883 16 04 2007 52 93 0 007614696 13 04 2007 51 60 0 025127527 12 04 2007 51 65 0 000968992 11 04 2007 51 80 0 002904163 10 04 2007 52 40 0 011583012 09 04 2007 51 58 0 015648855 05 04 2007 51 57 0 000193874 04 04 2007 51 36 0 004072135 03 04 2007 51 41 0 00097352 02 04 2007 51 05 0 007002529 30 03 2007 51 34 0 005680705 29 03 2007 51 40 0 001168679 28 03 2007 50 96 0 008560311 27 03 2007 51 06 0 001962323 26 03 2007 51 54 0 009400705 23 03 2007 51 72 0 003492433 22 03 2007 51 84 0 002320186 21 03 2007 52 03 0 003665123 20 03 2007 50 64 0 026715357 19 03 2007 50 06 0 011453397 16 03 2007 49 53 0 010587295 15 03 2007 50 13 0 01211387 14 03 2007 49 08 0 020945542 13 03 2007 48 75 0 006723716 12 03 2007 50 36 0 033025641 09 03 2007 50 33 0 000595711 08 03 2007 50 50 0 003377707 07 03 2007 50 22 0 005544554 06 03 2007 50 58 0 007168459 05 03 2007 49 25 0 026294978 02 03 2007 49 97 0 014619289 01 03 2007 51 08 0 022213328 28 02 2007 50 37 0 013899765 27 02 2007 50 60 0 00456621 26 02 2007 52 68 0 041106719 23 02 2007 53 77 0 020690964 22 02 2007 53 59 0 003347592 21 02 2007 53 75 0 002985632 20 02 2007 54 19 0 008186047 16 02 2007 54 10 0 001660823 15 02 2007 54 21 0 002033272 14 02 2007 54 18 0 000553403 13 02 2007 53 71 0 008674788 12 02 2007 53 43 0 005213182 09 02 2007 53 40 0 000561482 08 02 2007 54 44 0 019475655 07 02 2007 54 95 0 009368112 06 02 2007 54 95 0 000000000 05 02 2007 54 75 0 003639672 02 02 2007 54 66 0 001643836 01 02 2007 54 73 0 001280644 31 01 2007 55 13 0 007308606 30 01 2007 54 27 0 015599492 29 01 2007 54 06 0 003869541 26 01 2007 54 67 0 011283759 25 01 2007 53 93 0 01353576 Ex 2 Citi dividend history Ex Dividend Date Dividend per Share ($) Closing Stock Price ($) 03 August 2006 0 49 48 47 02 Novembre 2006 0 49 49 74 01 February 2007 0 54 54 73 03 May 2007 0 54 53 95 02 August 2007 0 54 47 24 01 November 2007 0 54 38 51 Ex 3 Listed American Options 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 In task 2, you will use part of the Big Game Goldman Sachs' elephant hunt in Libya case The Goldman Sachs elephant trade in Libya refers to a series of complex financial transactions between Goldman Sachs and the Libyan Investment Authority (LIA) in 2008 The LIA, Libya's sovereign wealth fund, invested $1 2 billion in high risk derivatives recommended by Goldman Sachs These trades, involving bets on the shares of several major companies, including Citigroup, resulted in substantial losses for the LIA while Goldman Sachs profited significantly Goldman Sachs' role involved advising the LIA to invest in these complex financial products, which the LIA did not fully understand due to its limited financial expertise The LIA accused Goldman Sachs of exploiting their lack of sophistication, misrepresenting the risks, and taking advantage of their relationship to secure the deals The court case in 2016 ruled in favor of Goldman Sachs, stating that there was no undue influence or misrepresentation Citigroup's role in this context was as one of the six companies whose shares were included in the derivatives trades Although Citigroup was not directly implicated in the legal proceedings, it was part of the broader set of trades that led to the LIA's losses during the financial crisis This broader context underscores the complexity and risks of the financial products involved and the consequences of inadequate understanding of such investments Questions 1 Using simple and continuous compounding, estimate the volatility of Citigroup's stock return using stock price history provided in the spreadsheet template (10) A Estimate the standard deviation of daily returns using Excel's STDEV function B Annualize the standard deviation of daily returns by multiplying it by the square root of the approximate number of trading days in a year You should find an annualized return volatility for the past 12 months and for the past 6 months From now on, use the average of these estimates as volatility Volatility is a key input into option pricing models 2 Value the trade (i e , the equivalent call) at inception with a binomial tree (ignore dividends) Build a 3 step tree (1 step 1 year), creating a template in the Binomial Tree spreadsheet in the Excel file The up and down factors are u eTu e sigma sqrt Delta T u eT and d 1ud frac 1 u d u1 with sigma from Q1 as Citigroup's annual return volatility and T Delta TT a step in years (e g , T 1 Delta T 1T 1 if 1 step 1 year) (20) A Build a tree for Citigroup's stock, and another for the call What is the call's value B Value the trade at the time of the case (07 23 08) Citigroup's price was $21 12 3 Value the listed call and put options, again ignoring dividends (20) A Use the same tree to value the 1 and 2 year listed options To do so, cut the length of a step (e g , for 1 year options, 1 step 4 months 1 3 year) The options being American, the option tree should be adapted to allow early exercise (i e , before maturity) Show all the calculations in Excel B Might these options be exercised early

The Answer is in the image, click to view ...

Question: Returns and Volatility (2) Date Closing price Expected Return 25/01/2008 26.64 24/01/2008 27.33 0.025900901 23/01/2008 26.36 -0.035492133 22/01/2008 24.40 -0.07435129 18/01/2008 24.45 0.002045074 17/01/2008 24.96

Returns and Volatility (2)

Date	Closing price	Expected Return
25/01/2008	26.64
24/01/2008	27.33	0.025900901
23/01/2008	26.36	-0.035492133
22/01/2008	24.40	-0.07435129
18/01/2008	24.45	0.002045074
17/01/2008	24.96	0.020858896
16/01/2008	26.24	0.051282051
15/01/2008	26.94	0.026676829
14/01/2008	29.06	0.078693393
11/01/2008	28.56	-0.017205781
10/01/2008	28.11	-0.015756303
09/01/2008	27.49	-0.022056208
08/01/2008	27.14	-0.012731903
07/01/2008	28.26	0.041267502
04/01/2008	28.24	-0.000707714
03/01/2008	28.93	0.024433428
02/01/2008	28.92	-0.000345662
31/12/2007	29.44	0.017980636
28/12/2007	29.29	-0.005095109
27/12/2007	29.56	0.009218163
26/12/2007	30.45	0.030108254
24/12/2007	30.98	0.017405583
21/12/2007	30.24	-0.023886378
20/12/2007	29.89	-0.011574074
19/12/2007	30.21	0.010705922
18/12/2007	30.38	0.005627276
17/12/2007	30.77	0.012837393
14/12/2007	30.70	-0.002274943
13/12/2007	31.01	0.01009772
12/12/2007	31.47	0.014833925
11/12/2007	33.23	0.055926279
10/12/2007	34.77	0.046343665
07/12/2007	34.31	-0.013229796
06/12/2007	34.35	0.001165841
05/12/2007	33.69	-0.019213974
04/12/2007	32.55	-0.033837934
03/12/2007	33.06	0.015668203
30/11/2007	33.30	0.007259528
29/11/2007	32.29	-0.03033033
28/11/2007	32.29	0
27/11/2007	30.32	-0.0610096
26/11/2007	30.70	0.012532982
23/11/2007	31.70	0.03257329
21/11/2007	30.73	-0.030599369
20/11/2007	31.40	0.021802799
19/11/2007	32.00	0.01910828
16/11/2007	34.00	0.0625
15/11/2007	34.58	0.017058824
14/11/2007	36.04	0.042220937
13/11/2007	35.90	-0.003884573
12/11/2007	33.57	-0.064902507
09/11/2007	33.10	-0.014000596
08/11/2007	32.90	-0.006042296
07/11/2007	33.41	0.01550152
06/11/2007	35.08	0.049985034
05/11/2007	35.90	0.023375143
02/11/2007	37.73	0.05097493
01/11/2007	38.51	0.020673204
31/10/2007	41.90	0.088029083
30/10/2007	42.11	0.005011933
29/10/2007	42.69	0.01377345
26/10/2007	42.63	-0.001405481
25/10/2007	41.23	-0.032840722
24/10/2007	41.82	0.014309968
23/10/2007	42.44	0.014825442
22/10/2007	42.61	0.004005655
19/10/2007	42.36	-0.005867167
18/10/2007	43.83	0.03470255
17/10/2007	44.66	0.018936801
16/10/2007	44.79	0.002910882
15/10/2007	46.24	0.032373298
12/10/2007	47.87	0.035250865
11/10/2007	48.32	0.00940046
10/10/2007	47.14	-0.02442053
09/10/2007	47.62	0.010182435
08/10/2007	47.80	0.003779924
05/10/2007	48.30	0.010460251
04/10/2007	47.63	-0.013871636
03/10/2007	47.89	0.005458744
02/10/2007	47.86	-0.000626436
01/10/2007	47.72	-0.002925198
28/09/2007	46.67	-0.022003353
27/09/2007	46.88	0.004499679
26/09/2007	46.55	-0.007039249
25/09/2007	46.31	-0.005155747
24/09/2007	46.59	0.00604621
21/09/2007	47.51	0.019746727
20/09/2007	47.20	-0.006524942
19/09/2007	48.27	0.022669492
18/09/2007	48.37	0.00207168
17/09/2007	46.03	-0.048377093
14/09/2007	46.64	0.013252227
13/09/2007	46.36	-0.006003431
12/09/2007	45.76	-0.012942192
11/09/2007	46.01	0.005463287
10/09/2007	45.30	-0.015431428
07/09/2007	45.48	0.00397351
06/09/2007	45.66	0.003957784
05/09/2007	46.00	0.007446343
04/09/2007	47.21	0.026304348
31/08/2007	46.88	-0.006990044
30/08/2007	46.23	-0.013865188
29/08/2007	46.95	0.015574302
28/08/2007	46.14	-0.017252396
27/08/2007	47.79	0.035760728
24/08/2007	48.50	0.014856665
23/08/2007	48.35	-0.003092784
22/08/2007	48.43	0.001654602
21/08/2007	48.06	-0.007639893
20/08/2007	48.39	0.006866417
17/08/2007	48.81	0.008679479
16/08/2007	47.55	-0.025814382
15/08/2007	45.61	-0.040799159
14/08/2007	45.66	0.001096251
13/08/2007	46.54	0.019272887
10/08/2007	47.00	0.009883971
09/08/2007	46.90	-0.00212766
08/08/2007	49.49	0.055223881
07/08/2007	48.59	-0.018185492
06/08/2007	48.35	-0.004939288
03/08/2007	45.72	-0.054395036
02/08/2007	47.24	0.033245844
01/08/2007	46.85	-0.008255715
31/07/2007	46.57	-0.005976521
30/07/2007	47.19	0.013313292
27/07/2007	46.97	-0.004662005
26/07/2007	47.81	0.017883756
25/07/2007	49.21	0.029326501
24/07/2007	49.31	0.001989348
23/07/2007	50.86	0.031433786
20/07/2007	50.73	-0.002556036
19/07/2007	51.13	0.007884881
18/07/2007	51.60	0.009192255
17/07/2007	52.46	0.016666667
16/07/2007	52.19	-0.005146778
13/07/2007	52.52	0.00632305
12/07/2007	52.84	0.006092917
11/07/2007	51.41	-0.027062831
10/07/2007	51.00	-0.007975102
09/07/2007	51.60	0.011764706
06/07/2007	51.70	0.001937984
05/07/2007	51.50	-0.003868472
03/07/2007	51.86	0.006990291
02/07/2007	51.64	-0.004242191
29/06/2007	51.29	-0.006777692
28/06/2007	51.81	0.010138429
27/06/2007	51.81	0
26/06/2007	51.15	-0.012738854
25/06/2007	51.69	0.010557185
22/06/2007	52.41	0.013929193
21/06/2007	53.66	0.02385041
20/06/2007	53.44	-0.004099888
19/06/2007	54.26	0.015344311
18/06/2007	53.96	-0.005528935
15/06/2007	53.98	0.000370645
14/06/2007	53.67	-0.005742868
13/06/2007	53.68	0.000186324
12/06/2007	52.60	-0.020119225
11/06/2007	53.47	0.016539924
08/06/2007	53.33	-0.002618291
07/06/2007	52.52	-0.015188449
06/06/2007	53.33	0.015422696
05/06/2007	53.90	0.010688168
04/06/2007	54.15	0.004638219
01/06/2007	54.51	0.006648199
31/05/2007	54.49	-0.000366905
30/05/2007	55.20	0.013029914
29/05/2007	54.91	-0.005253623
25/05/2007	55.12	0.00382444
24/05/2007	54.93	-0.003447025
23/05/2007	55.01	0.001456399
22/05/2007	55.08	0.001272496
21/05/2007	54.84	-0.004357298
18/05/2007	55.00	0.002917578
17/05/2007	54.80	-0.003636364
16/05/2007	54.91	0.002007299
15/05/2007	52.79	-0.038608632
14/05/2007	52.86	0.001326009
11/05/2007	53.11	0.004729474
10/05/2007	53.20	0.001694596
09/05/2007	54.12	0.017293233
08/05/2007	53.72	-0.007390983
07/05/2007	53.88	0.002978407
04/05/2007	54.09	0.00389755
03/05/2007	53.95	-0.002588279
02/05/2007	54.30	0.006487488
01/05/2007	54.20	-0.001841621
30/04/2007	53.62	-0.010701107
27/04/2007	53.37	-0.004662439
26/04/2007	53.56	0.003560052
25/04/2007	53.80	0.004480956
24/04/2007	52.81	-0.018401487
23/04/2007	53.11	0.005680742
20/04/2007	53.42	0.005836942
19/04/2007	53.09	-0.006177462
18/04/2007	52.99	-0.001883594
17/04/2007	52.53	-0.008680883
16/04/2007	52.93	0.007614696
13/04/2007	51.60	-0.025127527
12/04/2007	51.65	0.000968992
11/04/2007	51.80	0.002904163
10/04/2007	52.40	0.011583012
09/04/2007	51.58	-0.015648855
05/04/2007	51.57	-0.000193874
04/04/2007	51.36	-0.004072135
03/04/2007	51.41	0.00097352
02/04/2007	51.05	-0.007002529
30/03/2007	51.34	0.005680705
29/03/2007	51.40	0.001168679
28/03/2007	50.96	-0.008560311
27/03/2007	51.06	0.001962323
26/03/2007	51.54	0.009400705
23/03/2007	51.72	0.003492433
22/03/2007	51.84	0.002320186
21/03/2007	52.03	0.003665123
20/03/2007	50.64	-0.026715357
19/03/2007	50.06	-0.011453397
16/03/2007	49.53	-0.010587295
15/03/2007	50.13	0.01211387
14/03/2007	49.08	-0.020945542
13/03/2007	48.75	-0.006723716
12/03/2007	50.36	0.033025641
09/03/2007	50.33	-0.000595711
08/03/2007	50.50	0.003377707
07/03/2007	50.22	-0.005544554
06/03/2007	50.58	0.007168459
05/03/2007	49.25	-0.026294978
02/03/2007	49.97	0.014619289
01/03/2007	51.08	0.022213328
28/02/2007	50.37	-0.013899765
27/02/2007	50.60	0.00456621
26/02/2007	52.68	0.041106719
23/02/2007	53.77	0.020690964
22/02/2007	53.59	-0.003347592
21/02/2007	53.75	0.002985632
20/02/2007	54.19	0.008186047
16/02/2007	54.10	-0.001660823
15/02/2007	54.21	0.002033272
14/02/2007	54.18	-0.000553403
13/02/2007	53.71	-0.008674788
12/02/2007	53.43	-0.005213182
09/02/2007	53.40	-0.000561482
08/02/2007	54.44	0.019475655
07/02/2007	54.95	0.009368112
06/02/2007	54.95	0.000000000
05/02/2007	54.75	-0.003639672
02/02/2007	54.66	-0.001643836
01/02/2007	54.73	0.001280644
31/01/2007	55.13	0.007308606
30/01/2007	54.27	-0.015599492
29/01/2007	54.06	-0.003869541
26/01/2007	54.67	0.011283759
25/01/2007	53.93	-0.01353576

Ex.2 - Citi dividend history

Ex-Dividend Date	Dividend per Share ($)	Closing Stock Price ($)
03 August 2006	0.49	48.47
02 Novembre 2006	0.49	49.74
01 February 2007	0.54	54.73
03 May 2007	0.54	53.95
02 August 2007	0.54	47.24
01 November 2007	0.54	38.51

Ex.3-Listed American Options

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

In task 2, you will use part of the Big Game: Goldman Sachs' elephant hunt in Libya case.

The Goldman Sachs "elephant" trade in Libya refers to a series of complex financial transactions between Goldman Sachs and the Libyan Investment Authority (LIA) in 2008. The LIA, Libya's sovereign wealth fund, invested $1.2 billion in high-risk derivatives recommended by Goldman Sachs. These trades, involving bets on the shares of several major companies, including Citigroup, resulted in substantial losses for the LIA while Goldman Sachs profited significantly.

Goldman Sachs' role involved advising the LIA to invest in these complex financial products, which the LIA did not fully understand due to its limited financial expertise. The LIA accused Goldman Sachs of exploiting their lack of sophistication, misrepresenting the risks, and taking advantage of their relationship to secure the deals. The court case in 2016 ruled in favor of Goldman Sachs, stating that there was no undue influence or misrepresentation.

Citigroup's role in this context was as one of the six companies whose shares were included in the derivatives trades. Although Citigroup was not directly implicated in the legal proceedings, it was part of the broader set of trades that led to the LIA's losses during the financial crisis. This broader context underscores the complexity and risks of the financial products involved and the consequences of inadequate understanding of such investments.

Questions:

1. Using simple and continuous compounding, estimate the volatility of Citigroup's stock return using stock price history provided in the spreadsheet template. (10)

A. Estimate the standard deviation of daily returns using Excel's STDEV function.

B. Annualize the standard deviation of daily returns by multiplying it by the square root of the approximate number of trading days in a year. You should find an annualized return volatility for the past 12 months and for the past 6 months. From now on, use the average of these estimates as volatility. Volatility is a key input into option pricing models.

2. Value the trade (i.e., the equivalent call) at inception with a binomial tree (ignore dividends). Build a 3-step tree (1 step = 1 year), creating a template in the Binomial Tree spreadsheet in the Excel file. The up and down factors are u=eTu = e^{\sigma \sqrt{\Delta T}}u=eT and d=1ud = \frac{1}{u}d=u1 with \sigma from Q1 as Citigroup's annual return volatility and T\Delta TT a step in years (e.g., T=1\Delta T = 1T=1 if 1 step = 1 year). (20)

A. Build a tree for Citigroup's stock, and another for the call. What is the call's value?

B. Value the trade at the time of the case (07/23/08). Citigroup's price was $21.12.

3. Value the listed call and put options, again ignoring dividends. (20)

A. Use the same tree to value the 1- and 2-year listed options. To do so, cut the length of a step (e.g., for 1-year options, 1 step = 4 months = 1/3 year). The options being American, the option tree should be adapted to allow early exercise (i.e., before maturity). Show all the calculations in Excel.

B. Might these options be exercised early?

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