Black Scholes Problem Assesment Report

BLACK-SCHOLES
Problem 1.a)
The price of a 2-year 75-strike cash-or-nothing-call option on the stock is
given by;
Ke-σ(T-t)N(d2)
80e-0.26(2)
=50.40
Problem 1.b)
The Price of a 2-year 75 strike asset-or-nothing call option on the work.
This is given by:
ST-K,when ST >1
And
K-ST
When ST<1
Problem 1.c)
The Price of a 2 –year 75-strike cash-on nothing call option on the stock;
This is given by;
Se-σ(T-t)
Problem 1.d)
The price of a 2-year 75-strike asset-or-nothing put option on the stock;

This is given by;
Se-σ(T-t)N(-d1)
Problem 1.e
The price of a 1-year 85-strike European call Option.
BScall(S,ɽ,K,r,σ)=SN(d1(S,ɽ,K,rσ)-KN(d2(S,ɽ,K,r,σ)
Problem 1.f
The Price of a 1-year 85-strike European put option on the stock;
BS (European Put Option)=Ke-r(T-t)N(-d)-Se-σ(T-t)N(-d1)
Problem 2.a)
Deriving The Greek Terms of a Put Option.
Delta
∆=ϭBScall/ϭS
∆=N(d1)
Plug negative σ into the call option price and subtract the put-option price.
BScall (S(t),r,K,r,-σ)=SN(-d1)-e-rtKN(-d2)
=S(1-N(d1)-e-rtK(1-d2))
=S-e-rtK-BScall(S(t),r,k,ɽ,σ)
=-BSput(S(t),r,K,ɽ,σ)
Problem 2. I)
Gamma
ɽ = φ (d 1)
Sσ √ t

Where φ(d1)=e-d2/2/√ 2 ⫪
d 1=
¿ ( S
k )+ ¿
¿
¿t|
θPt
θSt = θCt/θSt-1
→√ P = θ2 Pt
θ S2 t
Θ^2Ct/θS^2t=N’(d1) θd 1
θSt
1
2 ⫪ e-d^2/2* 1
σSt √T −t
This is because the gamma of the call option
┌c = θ2Ct
θ S2 t =N’(d1) θd 1
θSt
Problem 2.iii)
For a European put option, the Vega is given by ;
^p=θPt/θc=^c= QCt
θc =StN’(d1)√ r >0
According to the put-call-parity;
Pt=Ctθ + Xe-r(T-t) -St
Problem 2.iv)
Rho
From the put-call –Parity;
Pt=Ct +Xe-r(T-t) -St
For a European put option, the rho is given by ;