Question: 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

The Excel file provides real data for call option contracts on SPY with exercise date on 5/2/2022. a. What 
SPY Call Option Data, exercise date 5/2/2022
Contract NameStrikeLast PriceVolumeOpen InterestImplied Volatility
SPY220502C00355000355103.82-10.682
SPY220502C0036000036083.3220.6462
SPY220502C0037000037090.07-10.5769
SPY220502C0039000039071.48-10.4592
SPY220502C0039500039547.64120.4253
SPY220502C0040000040038.2780.3923
SPY220502C0040400040438-10.3669
SPY220502C0040500040535.3120.3601
SPY220502C0040600040634.38-20.3549
SPY220502C0040800040831.39-50.3431
SPY220502C0041000041029.482920.3319
SPY220502C0041400041439.69110.3099
SPY220502C0041500041526.85150.3051
SPY220502C0041600041625.49-50.2999
SPY220502C0042000042020.7936270.2815
SPY220502C0042200042219.163990.2706
SPY220502C0042400042417.51-30.2627
SPY220502C0042500042514.521680.2606
SPY220502C0042600042619.8-170.2566
SPY220502C0042800042812.9212760.2473
SPY220502C0043000043010.473820.2414
SPY220502C004320004329.51621170.2324
SPY220502C004340004348.874291240.2249
SPY220502C004350004357.814444110.2211
SPY220502C004360004367.953002870.2142
SPY220502C004380004386.998892640.2067
SPY220502C004400004405.8515555370.1997
SPY220502C004420004424.075366060.1923
SPY220502C004440004443.464745430.1851
SPY220502C004450004453.342727550.1814
SPY220502C004460004462.932423120.1777
SPY220502C004470004472.32086070.1744
SPY220502C004480004481.712442360.1708
SPY220502C004490004491.461454280.1674
SPY220502C004500004501.62221814380.1646
SPY220502C004510004511.321414230.1613
SPY220502C004520004521.111964390.1587
SPY220502C004530004530.762995370.1563
SPY220502C004540004540.6121815300.1542
SPY220502C004550004550.624509690.152
SPY220502C004560004560.371292650.1506
SPY220502C004570004570.34613640.1489
SPY220502C004580004580.333875900.1475
SPY220502C004590004590.191174180.146
SPY220502C004600004600.1971318650.1453
SPY220502C004610004610.174306910.1436
SPY220502C004620004620.14774860.1436
SPY220502C004630004630.12222450.1426
SPY220502C004640004640.081234450.1426
SPY220502C004650004650.073256100.1421
SPY220502C004660004660.04661880.1436
SPY220502C004670004670.06124450.1436
SPY220502C004680004680.0625940.1436
SPY220502C004690004690.05316420.1426
SPY220502C004700004700.0342812430.1465
SPY220502C004710004710.02302930.1445
SPY220502C004720004720.0363810.1475
SPY220502C004730004730.0214480.1514
SPY220502C004740004740.0174840.1543
SPY220502C004750004750.0225240.1582
SPY220502C004760004760.0136020.1524
SPY220502C004780004780.0134400.1582
SPY220502C004800004800.018414980.1641
SPY220502C004820004820.0152660.1719
SPY220502C004840004840.011330.1797
SPY220502C004850004850.013650090.1816
SPY220502C004860004860.0215500.1836
SPY220502C004880004880.01334130.1914
SPY220502C004900004900.0111880.1992
SPY220502C004920004920.013390.2031
SPY220502C004940004940.015610.2109
SPY220502C004950004950.011602070.2149
SPY220502C004980004980.01202190.2227
SPY220502C005000005000.011005830.2305
SPY220502C005020005020.01160.2344
SPY220502C005040005040.0140520.2422
SPY220502C005060005060.01110.25
SPY220502C005080005080.01220.2539
SPY220502C005100005100.02-1130.2617
SPY220502C005200005200.01391100.2891
SPY220502C005300005300.011002000.3203
SPY220502C005400005400.01-70.3477
SPY Call Option Data, exercise date 5/2/2022
Contract NameStrikeLast PriceVolumeOpen InterestImplied Volatility
SPY220502C00355000355103.82-10.682
SPY220502C0036000036083.3220.6462
SPY220502C0037000037090.07-10.5769
SPY220502C0039000039071.48-10.4592
SPY220502C0039500039547.64120.4253
SPY220502C0040000040038.2780.3923
SPY220502C0040400040438-10.3669
SPY220502C0040500040535.3120.3601
SPY220502C0040600040634.38-20.3549
SPY220502C0040800040831.39-50.3431
SPY220502C0041000041029.482920.3319
SPY220502C0041400041439.69110.3099
SPY220502C0041500041526.85150.3051
SPY220502C0041600041625.49-50.2999
SPY220502C0042000042020.7936270.2815
SPY220502C0042200042219.163990.2706
SPY220502C0042400042417.51-30.2627
SPY220502C0042500042514.521680.2606
SPY220502C0042600042619.8-170.2566
SPY220502C0042800042812.9212760.2473
SPY220502C0043000043010.473820.2414
SPY220502C004320004329.51621170.2324
SPY220502C004340004348.874291240.2249
SPY220502C004350004357.814444110.2211
SPY220502C004360004367.953002870.2142
SPY220502C004380004386.998892640.2067
SPY220502C004400004405.8515555370.1997
SPY220502C004420004424.075366060.1923
SPY220502C004440004443.464745430.1851
SPY220502C004450004453.342727550.1814
SPY220502C004460004462.932423120.1777
SPY220502C004470004472.32086070.1744
SPY220502C004480004481.712442360.1708
SPY220502C004490004491.461454280.1674
SPY220502C004500004501.62221814380.1646
SPY220502C004510004511.321414230.1613
SPY220502C004520004521.111964390.1587
SPY220502C004530004530.762995370.1563
SPY220502C004540004540.6121815300.1542
SPY220502C004550004550.624509690.152
SPY220502C004560004560.371292650.1506
SPY220502C004570004570.34613640.1489
SPY220502C004580004580.333875900.1475
SPY220502C004590004590.191174180.146
SPY220502C004600004600.1971318650.1453
SPY220502C004610004610.174306910.1436
SPY220502C004620004620.14774860.1436
SPY220502C004630004630.12222450.1426
SPY220502C004640004640.081234450.1426
SPY220502C004650004650.073256100.1421
SPY220502C004660004660.04661880.1436
SPY220502C004670004670.06124450.1436
SPY220502C004680004680.0625940.1436
SPY220502C004690004690.05316420.1426
SPY220502C004700004700.0342812430.1465
SPY220502C004710004710.02302930.1445
SPY220502C004720004720.0363810.1475
SPY220502C004730004730.0214480.1514
SPY220502C004740004740.0174840.1543
SPY220502C004750004750.0225240.1582
SPY220502C004760004760.0136020.1524
SPY220502C004780004780.0134400.1582
SPY220502C004800004800.018414980.1641
SPY220502C004820004820.0152660.1719
SPY220502C004840004840.011330.1797
SPY220502C004850004850.013650090.1816
SPY220502C004860004860.0215500.1836
SPY220502C004880004880.01334130.1914
SPY220502C004900004900.0111880.1992
SPY220502C004920004920.013390.2031
SPY220502C004940004940.015610.2109
SPY220502C004950004950.011602070.2149
SPY220502C004980004980.01202190.2227
SPY220502C005000005000.011005830.2305
SPY220502C005020005020.01160.2344
SPY220502C005040005040.0140520.2422
SPY220502C005060005060.01110.25
SPY220502C005080005080.01220.2539
SPY220502C005100005100.02-1130.2617
SPY220502C005200005200.01391100.2891
SPY220502C005300005300.011002000.3203
SPY220502C005400005400.01-70.3477

The Excel file provides real data for call option contracts on SPY with exercise date on 5/2/2022. a. What pattern do implied volatilities follow across exercise prices? Make a graph and describe the pattern. (4 points) b. Calculate annualized sigma based on past daily returns of SPY. (3 points) c. 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. (2 points) d. 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. (4 points) e. 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? (5 points) f. 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? (5 points)

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