The Excel file provides real data for call option contracts on SPY with exercise date on 5/2/2022.

1. What pattern do implied volatilities follow across exercise prices? Make a graph and describe the pattern.

2. Calculate annualized sigma based on past daily returns of SPY.

3. The current SPY price is $437.97, exercise price is $438, 3-month treasury bill rate is

0.78%, dividend yield is 1.3%, there are 12 days before the exercise date. Calculate the

call price based on B-S-Merton model.

4. Make a data table showing the BSMerton prices for call options with different exercise

prices. Calculate the difference between market price of option and BSMerton prices. Make a graph that shows actual option prices and BSMerton prices for different options. Describe the pattern you observe.

5. Black-Scholes-Merton option pricing models assume that the implied volatility (IV) of an option for the same underlying stock and expiration date should be identical, regardless of the strike price. Define function BSMCallVolatility(Stock, Exercise, Time, Interest, Dividend, Target) to calculate the volatility implied by Black-Scholes-Merton model. Set target price=BSMerton price calculated in c. Based on your calculation, are volatilities identical for options with different strike prices under B-S-M model?

6. Volatility smiles started occurring in options pricing after the 1987 stock market crash. Explain why implied volatility is higher for deep out-of-the-money and in-the-money options. Why the pattern is not symmetric? In other words, why there is a smirk instead of smile?

SPY Call Option Data, exercise date 5/2/2022
Contract Name	Strike	Last Price	Volume	Open Interest	Implied Volatility
SPY220502C00355000	355	103.82	-	1	0.682
SPY220502C00360000	360	83.3	2	2	0.6462
SPY220502C00370000	370	90.07	-	1	0.5769
SPY220502C00390000	390	71.48	-	1	0.4592
SPY220502C00395000	395	47.64	1	2	0.4253
SPY220502C00400000	400	38.2	7	8	0.3923
SPY220502C00404000	404	38	-	1	0.3669
SPY220502C00405000	405	35.3	1	2	0.3601
SPY220502C00406000	406	34.38	-	2	0.3549
SPY220502C00408000	408	31.39	-	5	0.3431
SPY220502C00410000	410	29.48	29	2	0.3319
SPY220502C00414000	414	39.69	1	1	0.3099
SPY220502C00415000	415	26.85	1	5	0.3051
SPY220502C00416000	416	25.49	-	5	0.2999
SPY220502C00420000	420	20.79	36	27	0.2815
SPY220502C00422000	422	19.16	3	99	0.2706
SPY220502C00424000	424	17.51	-	3	0.2627
SPY220502C00425000	425	14.52	1	68	0.2606
SPY220502C00426000	426	19.8	-	17	0.2566
SPY220502C00428000	428	12.92	12	76	0.2473
SPY220502C00430000	430	10.4	73	82	0.2414
SPY220502C00432000	432	9.51	62	117	0.2324
SPY220502C00434000	434	8.87	429	124	0.2249
SPY220502C00435000	435	7.81	444	411	0.2211
SPY220502C00436000	436	7.95	300	287	0.2142
SPY220502C00438000	438	6.99	889	264	0.2067
SPY220502C00440000	440	5.85	1555	537	0.1997
SPY220502C00442000	442	4.07	536	606	0.1923
SPY220502C00444000	444	3.46	474	543	0.1851
SPY220502C00445000	445	3.34	272	755	0.1814
SPY220502C00446000	446	2.93	242	312	0.1777
SPY220502C00447000	447	2.3	208	607	0.1744
SPY220502C00448000	448	1.71	244	236	0.1708
SPY220502C00449000	449	1.46	145	428	0.1674
SPY220502C00450000	450	1.62	2218	1438	0.1646
SPY220502C00451000	451	1.32	141	423	0.1613
SPY220502C00452000	452	1.11	196	439	0.1587
SPY220502C00453000	453	0.76	299	537	0.1563
SPY220502C00454000	454	0.61	218	1530	0.1542
SPY220502C00455000	455	0.62	450	969	0.152
SPY220502C00456000	456	0.37	129	265	0.1506
SPY220502C00457000	457	0.34	61	364	0.1489
SPY220502C00458000	458	0.33	387	590	0.1475
SPY220502C00459000	459	0.19	117	418	0.146
SPY220502C00460000	460	0.19	713	1865	0.1453
SPY220502C00461000	461	0.17	430	691	0.1436
SPY220502C00462000	462	0.14	77	486	0.1436
SPY220502C00463000	463	0.12	22	245	0.1426
SPY220502C00464000	464	0.08	123	445	0.1426
SPY220502C00465000	465	0.07	325	610	0.1421
SPY220502C00466000	466	0.04	66	188	0.1436
SPY220502C00467000	467	0.06	12	445	0.1436
SPY220502C00468000	468	0.06	2	594	0.1436
SPY220502C00469000	469	0.05	31	642	0.1426
SPY220502C00470000	470	0.03	428	1243	0.1465
SPY220502C00471000	471	0.02	30	293	0.1445
SPY220502C00472000	472	0.03	6	381	0.1475
SPY220502C00473000	473	0.02	1	448	0.1514
SPY220502C00474000	474	0.01	7	484	0.1543
SPY220502C00475000	475	0.02	2	524	0.1582
SPY220502C00476000	476	0.01	3	602	0.1524
SPY220502C00478000	478	0.01	3	440	0.1582
SPY220502C00480000	480	0.01	84	1498	0.1641
SPY220502C00482000	482	0.01	5	266	0.1719
SPY220502C00484000	484	0.01	1	33	0.1797
SPY220502C00485000	485	0.01	36	5009	0.1816
SPY220502C00486000	486	0.02	15	50	0.1836
SPY220502C00488000	488	0.01	33	413	0.1914
SPY220502C00490000	490	0.01	1	188	0.1992
SPY220502C00492000	492	0.01	3	39	0.2031
SPY220502C00494000	494	0.01	5	61	0.2109
SPY220502C00495000	495	0.01	160	207	0.2149
SPY220502C00498000	498	0.01	20	219	0.2227
SPY220502C00500000	500	0.01	100	583	0.2305
SPY220502C00502000	502	0.01	1	6	0.2344
SPY220502C00504000	504	0.01	40	52	0.2422
SPY220502C00506000	506	0.01	1	1	0.25
SPY220502C00508000	508	0.01	2	2	0.2539
SPY220502C00510000	510	0.02	-	113	0.2617
SPY220502C00520000	520	0.01	39	110	0.2891
SPY220502C00530000	530	0.01	100	200	0.3203
SPY220502C00540000	540	0.01	-	7	0.3477