I just need to get question #1 done for my assignment with steps, details & graphs (if any). Many thanks
EPM947-CW-2021 Coursework 2021 Question 1 (Marks: 20) A Credit Default Swaps (CDS)1 is a contract where one party (credit protection buyer) pays the other one (credit protection seller) a fixed periodic coupon for the life of the contract on a specified reference asset. The party paying the premium is effectively buying insurance against specific credit events, such as default, bankruptcy or failure-to-pay or debt restructuring. If such a credit event occurs, the party receiving the premium makes a payment to the protection buyer, and the swap then terminates. Consider now that party A wishes to get covered from a potential loss of the face value (?") of an asset in case of a credit event. Hence, party A decides to purchase today (?$ = 0) some protection from party B that lasts until some specified maturity date ?. To pay for this protection, party A makes a regular stream of payments to party B. The size of these payments is a fixed percentage of the face value of the asset being insured and it is based on the yearly contractual spread ?*+, which represents the percentage used to determine the payments’ amount over one year. The payments are made every 3 months until maturity of the contract or until a credit event occurs, whichever occurs first. Assume that the credit event occurs as the first event of a Poisson counting process2 and hence default time is exponentially distributed with parameter ?. Denote the short rate with ?. The aim is to value the premium leg, i.e. to write a mathematical expression for this stream of payments taking into account both the appropriate discounting and the probabilities of default events. a) Illustrate the problem with a sketch representing the various payments occurring over the considered time period. Make sure you include the time at which the payments are made and the size of each undiscounted payments. b) Express the discounting factor at time ?/, where ? ∈ {0,1, … , ?}. c) Express the probability that a credit event occurs before time ?/ (?9: ;) and the survival probability at time ?/, i.e. the probability that no credit event has occurred before time ?/ (?9: <;). d)="" using="" the="" above,="" write="" down="" the="" full="" expression="" for="" the="" premium="" leg.="" e)="" using="" the="" values="" that="" correspond="" to="" your="" group,="" calculate="" the="" premium="" leg="" and="" price="" the="" cds.="" 1="" https://en.wikipedia.org/wiki/credit_default_swap="" 2="" https://en.wikipedia.org/wiki/poisson_point_process="" question="" 2="" (marks:="" 30)="" let="" 9="" be="" standard="" brownian="" motion.="" consider="" the="" following="" stochastic="" processes:="" i.="" 9="?$" +="" +="" 9="" ii.="" 9="?$???" ef?="" −="" *="" h="" hi="" +="" 9j="" a)="" use="" itô's="" lemma="" to="" derive="" the="" stochastic="" differential="" equation="" (sde)="" for="" (i)="" and="" (ii).="" discuss="" the="" main="" difference="" of="" the="" two="" models.="" b)="" obtain="" monthly="" (recent)="" data3="" for="" the="" price="" of="" the="" stock="" that="" corresponds="" to="" your="" group="" for="" three="" years.="" calculate="" the="" continuously="" compounded="" return="" of="" each="" month="" and="" then="" find="" the="" mean="" and="" the="" standard="" deviation="" of="" the="" return.="" c)="" determine="" the="" historical="" volatility="" and="" estimate="" the="" expected="" return="" by="" assuming="" it="" equal="" to="" the="" annualized="" average="" return.="" d)="" select="" an="" option4="" (near="" the="" money,="" middle="" june)="" with="" underlying="" asset="" the="" stock="" that="" corresponds="" to="" your="" group.="" given="" the="" spot="" price="" of="" the="" stock="" (?$)="" and="" the="" duration="" to="" expiry="" (???)="" of="" the="" option,="" simulate="" ten="" different="" paths="" for="" each="" of="" the="" models="" (i)="" and="" (ii)="" using="" 100="" discretisation="" points.="" present="" your="" findings="" in="" two="" separate="" figures,="" where="" you="" may="" also="" incorporate="" a="" short="" part="" the="" stock="" price’s="" recent="" history="" (not="" more="" that="" 2⁄="" ).="" e)="" use="" the="" black-scholes-merton="" model="" to="" calculate="" the="" implied="" volatility="" for="" the="" stock="" that="" corresponds="" to="" your="" group="" using="" the="" details="" of="" the="" option="" you="" selected="" in="" (d).="" discuss="" the="" difference="" between="" historical="" and="" implied="" volatility.="" note:="" for="" the="" risk-free="" rate="" you="" can="" use="" the="" united="" states="" 52="" week="" bill="" yield5.="" 3="" go="" to="" https://finance.yahoo.com/,="" search="" for="" the="" symbol,="" go="" on="" the="" “historical="" data”="" tab,="" select="" “monthly”="" frequency="" and="" download="" the="" data.="" 4="" go="" to="" https://finance.yahoo.com/,="" search="" for="" the="" symbol,="" go="" on="" the="" “options”="" tab,="" filter="" for="" the="" proper="" dates.="" 5="" https://tradingeconomics.com/bonds="" general="" remarks="" for="" the="" coursework:="" •="" the="" report="" must="" be="" created="" using="" ms="" word="" and="" you="" are="" advised="" to="" use="" ms="" excel.="" •="" you="" should="" submit="" your="" report="" in="" pdf="" format="" and="" a="" supplementary="" xls(x)="" file,="" containing="" all="" the="" required="" calculations.="" •="" mathematical="" expression="" should="" be="" typed="" using="" “equation="" writer”="" (insert="" -=""> Equation). • In the cover page of the report include personal details, coursework details, university logo, date. • Include a summary page of the key data where you have based your calculations along with their source and the retrieval dates. Include also screenshots of the websites in an Appendix. • Take the required screenshots before you start working with data (Q2 & Q3) • Start answering each question on a new page. • The structure and organization of the report and the supplementary file are among the quantitative indicators. Question 3 (Marks: 50) For the stock that corresponds to your Group, select an option expiry date among the available ones in the market so that ? > 0.5 years. Given the spot price of the stock (?$), select three strike prices ?*, ?H and ?S near the money such that ?* < h="">< ?s6. a) pricing call and put options with strike price ?h and maturity ? (in detail calculations). the time horizon should be divided into 5 intervals of duration ??w each. consider that the stock price may increase by ?w and decrease by ?w on each time interval. the volatility of the stock price is the historical volatility found in question 2, part (c). - calculate analytically the ?∗ probability - draw the binomial tree for the stock price evolution - provide and explain the formula of calculating the probabilities of reaching each node - price the european call and put options (show the required trees; explain) - price the american call and put options (show the required trees; explain) - comment on the difference of the premiums b) pricing call and put options with strike prices ?*, ?h and ?s and maturity ?. the time horizon should be divided into 15 intervals now of duration ??[ each. consider that the stock price may increase by ?[ and decrease by ?[ on each time interval. the volatility of the stock price is again the historical volatility found in question 2, part (c). - calculate analytically the ?∗ probability - price the european call and put options of strike price ?*7. - price the european call and put options of strike price ?h. - price the european call and put options of strike price ?s. - compare the european call and put premiums for strike price ?h of (a) and (b) - explain the difference between the calculated premiums and those in the market. c) given the premiums ?/ and ?/ (? = 1,2,3) that correspond to the options with strike prices ?*, ?h and ?s that have been already calculated, built the following trading strategies and discuss their suitability: - option (?h) and underlying asset. i) covered call and ii) protective put - multiple options (?*, ?s) of the same type. i) bull spread with calls and ii) bear spread with puts - combinations. i) strip (?h) and butterfly spread (?*, ?h, ?s) note: for the risk-free rate you can use the united states 52 week bill yield8. 6 go to https://finance.yahoo.com/, search for the symbol, go on the “options” tab, filter for the proper dates. 7 include also a table with values of the terminal nodes (stock price, probability, option value). 8 https://tradingeconomics.com/bonds inputdataallocation2021 va (£) t (months) w1y λ r symbol name group a1 42000 36 0.0177 0.01824 0.0244 msft microsoft corp group a2 31000 27 0.0169 0.01744 0.0184 aapl apple inc group a3 43000 36 0.0248 0.02531 0.0233 bac bank of america corp group a4 33000 36 0.0202 0.02073 0.0285 ba boeing company group a5 37000 39 0.0182 0.01874 0.0236 tsla tesla inc group a6 37000 36 0.0201 0.02063 0.0271 nke nike inc group a7 27000 33 0.0214 0.02193 0.0265 fb facebook inc group a8 30000 39 0.0224 0.02292 0.0280 dis walt disney company group a9 28000 36 0.0202 0.02073 0.0207 uber uber technologies inc group b1 36000 30 0.0168 0.01734 0.0194 aal american airlines gp group b2 45000 33 0.0235 0.02401 0.0203 nflx netflix inc group b3 44000 33 0.0184 0.01894 0.0293 ge general electric company group b4 45000 27 0.0227 0.02323 0.0346 amzn amazon.com inc group b5 30000 30 0.0235 0.02401 0.0246 pypl paypal holdings group b6 32000 30 0.0165 0.01705 0.0322 googl alphabet cl a group b7 38000 33 0.0154 0.01595 0.0257 nvda nvidia corp group b8 27000 33 0.0242 0.02471 0.0188 dal delta air lines inc group b9 27000 30 0.0246 0.02511 0.0313 f ford motor company group c1 43000 39 0.0230 0.02352 0.0325 jnj johnson & johnson group c2 30000 33 0.0217 0.02222 0.0254 twtr twitter inc group c3 38000 30 0.0229 0.02342 0.0256 sbux starbucks corp group c4 40000 36 0.0167 0.01724 0.0246 pep pepsico inc group c5 39000 27 0.0152 0.01575 0.0325 zm zoom video communications cl a group c6 31000 30 0.0155 0.01604 0.0183 ko coca-cola company group c7 30000 33 0.0164 0.01695 0.0301 intc intel corp group c8 35000 27 0.0232 0.02371 0.0211 t at&t inc group c9 29000 30 0.0159 0.01645 0.0257 baba alibaba group holding question 1 question 2 & 3 input data allocation 2021 group s6.="" a)="" pricing="" call="" and="" put="" options="" with="" strike="" price="" h="" and="" maturity="" (in="" detail="" calculations).="" the="" time="" horizon="" should="" be="" divided="" into="" 5="" intervals="" of="" duration="" w="" each.="" consider="" that="" the="" stock="" price="" may="" increase="" by="" w="" and="" decrease="" by="" w="" on="" each="" time="" interval.="" the="" volatility="" of="" the="" stock="" price="" is="" the="" historical="" volatility="" found="" in="" question="" 2,="" part="" (c).="" -="" calculate="" analytically="" the="" ∗="" probability="" -="" draw="" the="" binomial="" tree="" for="" the="" stock="" price="" evolution="" -="" provide="" and="" explain="" the="" formula="" of="" calculating="" the="" probabilities="" of="" reaching="" each="" node="" -="" price="" the="" european="" call="" and="" put="" options="" (show="" the="" required="" trees;="" explain)="" -="" price="" the="" american="" call="" and="" put="" options="" (show="" the="" required="" trees;="" explain)="" -="" comment="" on="" the="" difference="" of="" the="" premiums="" b)="" pricing="" call="" and="" put="" options="" with="" strike="" prices="" *,="" h="" and="" s="" and="" maturity="" .="" the="" time="" horizon="" should="" be="" divided="" into="" 15="" intervals="" now="" of="" duration="" [="" each.="" consider="" that="" the="" stock="" price="" may="" increase="" by="" [="" and="" decrease="" by="" [="" on="" each="" time="" interval.="" the="" volatility="" of="" the="" stock="" price="" is="" again="" the="" historical="" volatility="" found="" in="" question="" 2,="" part="" (c).="" -="" calculate="" analytically="" the="" ∗="" probability="" -="" price="" the="" european="" call="" and="" put="" options="" of="" strike="" price="" *7.="" -="" price="" the="" european="" call="" and="" put="" options="" of="" strike="" price="" h.="" -="" price="" the="" european="" call="" and="" put="" options="" of="" strike="" price="" s.="" -="" compare="" the="" european="" call="" and="" put="" premiums="" for="" strike="" price="" h="" of="" (a)="" and="" (b)="" -="" explain="" the="" difference="" between="" the="" calculated="" premiums="" and="" those="" in="" the="" market.="" c)="" given="" the="" premiums="" and="" (?="1,2,3)" that="" correspond="" to="" the="" options="" with="" strike="" prices="" *,="" h="" and="" s="" that="" have="" been="" already="" calculated,="" built="" the="" following="" trading="" strategies="" and="" discuss="" their="" suitability:="" -="" option="" (?h)="" and="" underlying="" asset.="" i)="" covered="" call="" and="" ii)="" protective="" put="" -="" multiple="" options="" (?*,="" s)="" of="" the="" same="" type.="" i)="" bull="" spread="" with="" calls="" and="" ii)="" bear="" spread="" with="" puts="" -="" combinations.="" i)="" strip="" (?h)="" and="" butterfly="" spread="" (?*,="" h,="" s)="" note:="" for="" the="" risk-free="" rate="" you="" can="" use="" the="" united="" states="" 52="" week="" bill="" yield8.="" 6="" go="" to="" https://finance.yahoo.com/,="" search="" for="" the="" symbol,="" go="" on="" the="" “options”="" tab,="" filter="" for="" the="" proper="" dates.="" 7="" include="" also="" a="" table="" with="" values="" of="" the="" terminal="" nodes="" (stock="" price,="" probability,="" option="" value).="" 8="" https://tradingeconomics.com/bonds="" inputdataallocation2021="" va="" (£)="" t="" (months)="" w1y="" λ="" r="" symbol="" name="" group="" a1="" 42000="" 36="" 0.0177="" 0.01824="" 0.0244="" msft="" microsoft="" corp="" group="" a2="" 31000="" 27="" 0.0169="" 0.01744="" 0.0184="" aapl="" apple="" inc="" group="" a3="" 43000="" 36="" 0.0248="" 0.02531="" 0.0233="" bac="" bank="" of="" america="" corp="" group="" a4="" 33000="" 36="" 0.0202="" 0.02073="" 0.0285="" ba="" boeing="" company="" group="" a5="" 37000="" 39="" 0.0182="" 0.01874="" 0.0236="" tsla="" tesla="" inc="" group="" a6="" 37000="" 36="" 0.0201="" 0.02063="" 0.0271="" nke="" nike="" inc="" group="" a7="" 27000="" 33="" 0.0214="" 0.02193="" 0.0265="" fb="" facebook="" inc="" group="" a8="" 30000="" 39="" 0.0224="" 0.02292="" 0.0280="" dis="" walt="" disney="" company="" group="" a9="" 28000="" 36="" 0.0202="" 0.02073="" 0.0207="" uber="" uber="" technologies="" inc="" group="" b1="" 36000="" 30="" 0.0168="" 0.01734="" 0.0194="" aal="" american="" airlines="" gp="" group="" b2="" 45000="" 33="" 0.0235="" 0.02401="" 0.0203="" nflx="" netflix="" inc="" group="" b3="" 44000="" 33="" 0.0184="" 0.01894="" 0.0293="" ge="" general="" electric="" company="" group="" b4="" 45000="" 27="" 0.0227="" 0.02323="" 0.0346="" amzn="" amazon.com="" inc="" group="" b5="" 30000="" 30="" 0.0235="" 0.02401="" 0.0246="" pypl="" paypal="" holdings="" group="" b6="" 32000="" 30="" 0.0165="" 0.01705="" 0.0322="" googl="" alphabet="" cl="" a="" group="" b7="" 38000="" 33="" 0.0154="" 0.01595="" 0.0257="" nvda="" nvidia="" corp="" group="" b8="" 27000="" 33="" 0.0242="" 0.02471="" 0.0188="" dal="" delta="" air="" lines="" inc="" group="" b9="" 27000="" 30="" 0.0246="" 0.02511="" 0.0313="" f="" ford="" motor="" company="" group="" c1="" 43000="" 39="" 0.0230="" 0.02352="" 0.0325="" jnj="" johnson="" &="" johnson="" group="" c2="" 30000="" 33="" 0.0217="" 0.02222="" 0.0254="" twtr="" twitter="" inc="" group="" c3="" 38000="" 30="" 0.0229="" 0.02342="" 0.0256="" sbux="" starbucks="" corp="" group="" c4="" 40000="" 36="" 0.0167="" 0.01724="" 0.0246="" pep="" pepsico="" inc="" group="" c5="" 39000="" 27="" 0.0152="" 0.01575="" 0.0325="" zm="" zoom="" video="" communications="" cl="" a="" group="" c6="" 31000="" 30="" 0.0155="" 0.01604="" 0.0183="" ko="" coca-cola="" company="" group="" c7="" 30000="" 33="" 0.0164="" 0.01695="" 0.0301="" intc="" intel="" corp="" group="" c8="" 35000="" 27="" 0.0232="" 0.02371="" 0.0211="" t="" at&t="" inc="" group="" c9="" 29000="" 30="" 0.0159="" 0.01645="" 0.0257="" baba="" alibaba="" group="" holding="" question="" 1="" question="" 2="" &="" 3="" input="" data="" allocation="" 2021="">