Black - Scholes -- Option Pricing Models

You have purchased a call option at Rs 7 and the strike price is Rs 70, if the market price at the maturity is 90 what is the profit you make? Value of a Call= Max(S-k,0) Value of a call=Max(90-70,0) =20 The profit we will make is 20 minus Rs 7 = 13 You have sold a call option for a premium of Rs 10 strike price is Rs 50 on the maturity date the stock is trading at 40 what is your profit? The holder of the call will not excise, he will allow the option to expire we can keep our Rs 10/You have purchased a put option on an stock at Rs 5 the strike price is Rs 70 and the current stock price is Rs 40 what is your profit? Value of the Put= Max(K-S,0) Value of the Put= Max(70-40,0) 30 Our profit is Rs 30 minus Rs 5 = Rs 25

Cc=

8.405

Current Price of the Underlying Asset

So

50

Strike Price ( Excise Price of the Option)

K

52

d1=

Annualized Standard Deviation (Volatility) Time to maturity

σ T

59% 0.5

d1=

Continously compounded risk free rate of return

r

9%

d2=

exponent

e

2.72

N(D1) N(D2)

Black Scholes Model Pp= Put Call Parity

8.117 So + 50 58.12

Stock Price+PutPrice=PV of the S

29.41 minus

Cc= d1=

21.01

-0.04 plus

d2= Pp=

0.13 0.42

Pp

0.223 -d1=

-0.223

-0.195 -d2=

0.195

0.5882 N(-D1) 0.4226 N(-D2)

0.4118 0.5774

28.706 minus

20.59

= 8.117

Ke^-(in) 49.71

+

S0*N(D1)-Ke^-(in)N(D2) LN(S/K) plus (r+ σ^2/2)T σ * (T)^0.5 d1- σ * (T)^0.5 Ke^-(in)*N(-D2)-SoN(-D1)

Imbibed in the above partial differential equations are the properties of Marakov Stochasic Movement of Variables Martingale. all from Quantative finance. Cc

plus 58.12

Stock Price+PutPrice=PV of the Strike Price+CostofCall

8.405 -d1=-.2229 -d2=.195349 0.41 N(-D1) 0.58 N(-D2)

From the following data using the B/S option pricing model value the Call option S 50 K 52 i 9% Variance 35% Time 0.5 d1= -0.04 plus 0.22 0.42 d2= -0.2 N(D1)= 0.5882 N(D2)= 0.4226 Cc= S0*N(D1)-Ke^-(in)N(D2) 8.41

29.41 minus

ng the B/S option

0.13

21.01

206.91 Cc= S

450

K

375

i

9%

Variance

6%

Cc= d1= d2= Pp=

278.99 minus

S0*N(D1)-Ke^-(in)N(D2) LN(S/K)plus (r+ σ^2/2)T σ * (T)^0.5 d1- σ * (T)^0.5 Ke^-(in)*N(-D2)-SoN(-D1)

n 7 Compute the Cc and Pp and Prove the Put Call Parity. -43.37 D1= 0.31 0.18 plus N(D1)= 0.62 0.66 -D1 -0.31 D2= -0.36 -D2 0.36 N(D2)= 0.36 So + Pp = Ke^-(in) + Cc 450 (43.37) = 199.72 plus 406.63

72.08

0.02 0.38 0.64

206.91

406.63

S K i e Variance t

75 Compute the Cc and Pp using B/S option Pricing Model 85 Prove the Put Call Parity Relationship. 7% 2.72 Cc= SND1-Ke^-in*ND2 0.45 6.83 0.25 -0.15 d1= ln(S/K)+(r+v/2)t -5.141314295400600% stddev * t^.5 0.34

N(D1) N(D2)

0.44 0.31

-d1 -d2 N(-D1) N(-D2)

0.15 0.49 0.56 0.69

-0.49 d2= Pp=

d1- stddev*t^.5 Ke^-in*(-ND2)-S(-ND1) 15.35

So+Pp=Ke^-in+Cc 90.35 83.53 plus

6.83 90.35