Financial Derivatives: Black-Scholes Model Problems and Solutions

This document presents solutions to a series of problems related to the Black-Scholes option pricing model. The solutions cover various types of options, including cash-or-nothing call options, asset-or-nothing call and put options, and European call and put options. The solutions demonstrate the application of the Black-Scholes formula to calculate option prices under different scenarios. Furthermore, the assignment delves into the derivation of the Greek terms (delta, gamma, vega, rho, and theta) for a put option, explaining how each Greek measures the sensitivity of the option price to changes in underlying parameters such as the stock price, volatility, and time to maturity. The solutions provide a detailed breakdown of the formulas and calculations involved in determining the Greeks, offering insights into the risk management aspects of options trading.

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 ;
θPt
θr = θCt
θr –(T-t)Xe-r(T-t)
=[(T-t)Xe-r(T-t)]N(d2) –(T-t)Xe-r
=(T-t)Xe-r(T-t)[N(d2)<0
Since;
θC t /θr=(T −t) Xe−r (T −t )N (d 2)
❑
Problem 2 .v)
θCt/θr=StN’(d1) θd 1
θr + rXp-rTN(d2)-XP-r(T-t)N’(d2) θd 2
θdr
Consider that;
θd 2
θr = θd 1
θr
= σ
2┌ r
θCt
θr =StN’(d1) θd 1
θr + rXe-rTN(d2)
-Xe-r(T-t)N’(d2) θd 1
θr + Xe-r(T-t)N’(d2) σ
2 √r

But θC
θr summarizes to
θCt
θr =+rXe-rTN(d2) +nXe-r(T-t)N’(d2)
σ
2 √ r >0
Φ= θCt
θt =- θCt
θr <0
Theta tends to be less valuable as the time to maturity decreases to maturity.