Question

BCA stock price is currently $70. The risk free interest rate is 5% per annum with continuous compounding. Assume BCA's volatility is 25%. What is the difference in price of a of a 6-month American put option with a strike price of $75 and an identical European put option using 5--step binomial trees to calculate the price index today for both options? What is the new Black-Scholes price of this European option? LOOKING FOR ANSWER IN EXCEL FORMAT AND EXPLANATION PLEASE :)

Answer 1

Current Stock Price S₀ = $70
Risk Free Rate r = 5% = 0.05
BCA Stock Volatility σ = 25%
Total Time T = 6 months = 0.5 years
Strike Price K = $75
5 Step binomial tree, Δt = 0.5/5 = 0.1 years

Common Formulas for both European and American Options -

Size of up-movement u = e^σ(√Δt)
Size of down-movement d = e^-σ(√Δt) = 1/u
Probability of up movement p = (e^rΔt - d) / (u - d)
Probability of down movement = 1 - p
Stock Price in case of up-movement S_t+1 = S_t * u
Stock Price in case of down-movement S_t+1 = S_t * d

For European PUT Option -
Pay-off at the terminal node of price tree = max(k - ST, 0)
Pay-off at the intermediate and initial node of price tree =
(p * pay-off of up-movement node + (1-p)* pay-off at down movement node)/ e^rΔt

For American PUT Option -

Pay-off at the terminal node of price tree = max(k - ST, 0)
Pay-off at the intermediate and initial node of price tree at time t=
max(k - S_t, 0, (p * pay-off of up-movement node + (1-p)* pay-off at down movement node)/ e^rΔt

Difference with European PUT is that at every intermediate node we decide, if American PUT will be exercised immediately giving pay off max(k - S_t, 0) or kept for later giving pay-off as (p * pay-off of up-movement node + (1-p)* pay-off at down movement node)/ e^rΔt. Whichever has higher value, that thing is selected.

Information given in the question is as follows -

Values common to both calculations are given below -

Formula looks like below -

So, values will be as follows -

Price Tree formulas are shown below -

Price Tree values will look like below -

European Option Pay-off Tree formulas are given below -

European PUT pay-off values are given below -

American PUT Option Pay-off tree formulas are given below -

American PUT Option pay-off values are given below -

Hence,
value of European PUT at t = 0 is = $6.62
value of American PUT at t = 0 is = $6.99

Black-Scholes price of the European PUT Option -
p = K*e^-rT*N(-d₂) - S₀*N(-d₁)

Where,
d₁ = (ln(S0/k) + (r + σ²/2)*T)/σ(√Δt)
d₂ = (ln(S0/k) + (r - σ²/2)*T)/σ(√Δt) = d₁ - σ(√Δt)
N is cumulative Normal Distribution with mean 0 and standard deviation 1.

Formulas in excel are given below -


Values calculated are as below -

See that value given by Black Scholes is $6.77 which is very close to the one given by 5 step binomial tree($6.62)
As we increase number of steps in the binomial tree, value calculated by binomial tree method will start converging towards the one given by Black Scholes formula.