Assignment title: Information
1. Suppose there is an underlying asset with current price S which follows a lognormal
random walk:
dS=#Sdt+%SdX
where#isUIFdriftand% isUIFvolatility. 5IF Sisk-freeinterestraterisknown.
SupposethereBSFnotransactioncostsandnoarbitrageopportunities.
Write a program in C++ which computes the price of a European Put Option on
underlying asset with no dividends
(A)usingtheBlack-Scholesformula BOE
(B)usingBMonteCarlomethod
(C) 6sing either Black Scholes formula or B Monte Carlo method compute the price
of the European put option on
(i) the asset which pays dividend D
(ii) on foreign currency bond with interest rate rf
(iii) on commodity with storage cost q
(iv) on futures.
Comment on the accuracy of solutions and the computation time.
Please QSPWJEFcommentT BMPOHTJEF ZPVScode andexplaincarefullyall thecomponents
of the code. You should submit electronically all header and implementation
lessoUIBUtheprogramrunsanddoesnotrefertoexternallibraries.2. (a) Compare and contrast the results of using discrete and continuous time models:
(i) Suppose there is an underlying asset with current price S which follows a lognormal
random walk:
dS=#Sdt+%SdX
where#isdriftand% isvolatility. Risk-freeinterestrater isknown. Supposethereisno
transactioncostsandnoarbitrageopportunities.
Write a program in C++ which computes the price of a European Call Option on the
underlying asset with no dividends using the Black-Scholes formula
(ii)5IF$PY3PTT3VCJOTUFJO CRR
calloptionpricingformulaJTHJWFOCZ:
C0 = S0
TX ^k=k
"T k#p ^k(1 ! p ^)T !k ! (1 + Kr)T X T
k=k ^
"T k#p ~k(1 ! p ~)T !k
where
p ~ =
1 + r ! d
u ! d ; p ^ = 1 + pu ~ r
and k ^ is the smallest integer k such that
k log$u d% > log"S0 K dT # :
Use the CRR-option pricing formula to compute the price of a European call option.
Use the CRR-parametrization for u and d; that is, set u = e$p$t; d = 1 .
u
(b)computeë Greeksí for theEuropeancalloptionusingUIFcontinuoustimemodel
Please comment the code liberally and explain carefully all the components of the code.
You should submit electronically all header and implementation Öles so the program runs
and does not refer to external libraries.
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