Question

Create a four-period binominal price tree and find the fair value of an European call and put options and an American put option on a nondividend-paying stock if the initial stock price is 82 PLN, the strike price of 80 PLN is expiring at the end of the fourth month, the compound risk-free interest rate is 12% per annum, and σ=0.1.

Answer 1

Parameters of Option Binomial Model –

There are three parameters of Option Binomial Pricing Model

• up factor (u)
• down factor (d)
• probability (P)

up factor and down factor used to calculate rise in price and fall in price of underlying assets in one period. Probability is measure probability of rise in price and (1-P) is probability of price fall.

Cox-Rox-Rubinstein has suggested following formula to calculate these factors,

$P = \frac{e^{rt}-d}{u-d}$

$u = e^{\sigma \sqrt{t}}$

$d = \frac{1}{u}$

Where,

r = risk free interest rate

t = one period time

$\sigma$ = risk

Please refer to below spreadsheets for calculations -

M23 Four Period Binomial Price tree Binomial Parameters Initial Stock Price (S1) Strike Price (K Maturit One Period(t) risk free interest rate volatalit 82 1.105170918 0.904837418 0.525187994 0.474812006 1.010050167 4 4 months P 1month (1-P 1% 0.1 10 Stock Price 0 2 4 Call Payoff at 4 Put Payoff at 4 12 13 122.33 42.33 0.00 110.69 16 17 18 19 20 21 100.16 100.16 20.16 0.00 90.62 90.62 82.00 82.00 82.00 2.00 0.00 74.20 74.20 67.14 67.14 0.00 12.86 60.75 54.97 0.00 25.03

Formula reference -

M23 Four Period Binomial Price tree Binomial Parameters Initial Stock Price (S1 Strike Price (K Maturit One Period risk free interest rate volatalit 82 EXP(C8 SORT(C6) months month ( EXP(C7"C6)-F4),(F3-F4) 1-F5 -EXP(C6 C7) 1-P 0.12/12 0.1 10 Stock Price Call Payoff at 4 Put Payoff at4 12 13 14 15 16 17 18 19 20 21 sC$4),0) MAX((F16-SC$4),0) MAX(F18-$c$4),0) -MAX((F20-SC$A),0) MAX(F22-$c$4),0) -MAX((SC$4-F14),o) MAXI(SC$A-F16),0) -MAX((SC$4-F18),0) -MAX((SC$4-F20),0) -MAX((SC$4-F22),0) MAX(F14- D16 F3 C17*F3 -B18 F3 D16F :C3 C19 F3 E19 F3 818 F4 D18 F4 =C19" F4 D20 F4 23

M23 23 25 Value of Europeon Call 0 2 4 27 42.33 29 31.48 21.74 20.16 31 14.32 11.42 32 9.14 6.43 2.00 3.60 1.04 0.54 0.00 0.00 0.00 37 38 Value of Europeon Put 0 2 4 42 0.00 43 0.00 0.00 0.00 45 1.34 0.00 4.00 2.84 0.00 47 7.03 6.05 11.82 12.86 49 18.46 25.03 51

formula reference-

832 23 24 25 26 27 28 29 30 31 Value of Europeon Call 4 42.3296252065842 20.1550261691339 (D30*SF$5+D32 $F$6)/SF$7 1+33 SF (F34 SF$5+F36 $F$6)/SF$7 35 36 37 38 39 Value of Europeon Put 42 0 (E43 SF$5+E45 SF 45 46 47 D46*SFS5+D48 SF6)SFS7 12.8640782476055 (F48 SF$5+F50 $F$6)/$F$7 25.0337562250776 49 50 51

Thus, Value of European call is 9.14 and Value of European Put is 4.0

formula reference-

Thus, Value of American Put is 4.23 at 0 period.

American Put can be exercised early.