Question

An investor wants to compare premium prices of a European call calculated with the Black–Scholes model with premium prices calculated with a binomial model. The call has strike price K = $19 and the underlying asset is currently selling for S = $20 . The yearly volatility of the underlying is estimated to be σ = 0.55 . The interest rate is r = 6% pa. The call expires in 90 days so T = 90/365 years.

(a) Calculate the call premium using the Black–Scholes model.

(b) Consider a ten-step binomial model.

(i) Assuming interest rates are constant over the life of the call, calculate the return R over one time step.
(ii) Calculate the up and down factors u and d in this ten-step model.
(iii) Calculate the risk neutral probability π in this ten-step model.
(iv) Construct a ten-step binomial pricing tree for the call and calculate its premium.

Answer 1

Black–Scholes model

strike price K = $19

underlying asset is currently selling for S = $20

yearly volatility (of the underlying is estimated to be) σ = 0.55

interest rate is r = 6% p.a

call expires in 90 days

T = 90/365 years

a) Calculate the call premium using the Black–Scholes model.

C (call premium)= SN(d1)-N(d2)Ke^(-rt)

C= call premium

S= Current stock price=$20

t= time until option exercised= 90 days

K= option strike price = $19

r= risk free interest rate =6%

N= cumulative std normal distribution=

e= exponential term

S= std dev=0.55

In= natural log

whereas, d1 calculated as= (In(s/k) +(r+s^2/2)t)/s- t^1/2 = 0.3785

                d2= d1- s- t^1/2 = 0.1054

C (call premium)= SN(d1)-N(d2)Ke^(-rt) = 2.803

(ii) Calculate the up and down factors u and d in this ten-step model.

S =current market price of stock

S*u= future prices for up moves t years later (20*0.55) =11

S*d =future prices for down moves t years later (20*0.45)= 8.8

Iv) here is the 2 step model, like wise goes the rest 8 steps

6.05 11 0.55 20 20 (same) 0.44 8.8 3.872