Question

A non-paying dividend stock price is currently 40 US$. Over each of the next two three-month periods it is expected to go either up by 10% or down by 10%. The riskless interest rate is 12% per annum with continuous compounding. What is the value of a six-month European put option with a strike price of 42 US$?

Given the information above find the relevant call and put price of that European non-paying dividend stock option using the Black-Scholes formula

Answer 1

Follow the steps below to obtain the value of six-month European put option Step 1 Calculate the probability of option moving upwards (p)and probability of option moving downwards (1-p) ea -d p= TAt u-d Here, The risk-free rate is represented by r Time to expiration is denoted by Ar Probability of upward movement of stock is denoted by u Probability of downward movement of stock is denoted by d -d е р: u-d 12% -(1-10%) (1+10%)-(1-10%) e 0.03 1.10-0.90 1.030454534-0.90 = 0.20 0.6523 1-p 1-0.6523 0.3477 Step 2

Construct binomial tree to obtain the terminal payoffs at each node. $48.40 Ge. $44*(1+10% ) r0 (As the exercise pice $42 is less than $48.40) $44 ie. $44 (1+10% В $40 A. $39.60 (ie. $44 (1-10 %) f2.40 (i.e. $42-$39.60 $36.00 ie. $40*(1-10% ) S32.40 Ge. $36.00 (1-10%) fad =$9.60 Ge. $42-532.40) Step 3 Calculate the value of six month European put option p2p-) +(1-p) fdd f e -217 (0.6523) x0 +2 x 0.6523 x(0.3477) x 2.40 + ( 0.3477) x9.60 = e 0.941764534x[0+1.0886626+1.1605948 X - 2.118271 or 2.118 Calculate the call price and put price Formulas used:

A C 1 Time to maturity 2 Volatility 3 Strike price 4 Risk free rate, 5 Stock price (ie. 6 months) 180 0.1 42 0.12 40 6 - 180/ 365 =SQRT(B7) -B4*B7 -EXP(-B9) =B2*B8 |-LN(B5 / (B3 *B10) 7 Time duration T 8 Sqrt(T) 9 r T 10 exp(-r*T) 11 Volatility*sqrt(T) 12 In(S/PV(K) 13 -B12 / B11 + 0.5*B11 -B14- B11 -NORMSDIST(B14) -NORMSDIST(B15) 14 dl 15 d2 16 N(dl) 17 N(d2) 18 19 C (Black Scholes Call price)-B5* B16-B3 B10 B17 B16 - B20* B5/ B19 - B19-B5+ B10 B3 20 Delta 21 Elasticity 22 P (Black Scholes put price) Results obtained:

A B C 1 Time to maturity 2 Volatility 3 Strike price 4 Risk free rate, r 5 Stock price 180 (ie. 6 months) 10% 42 12% 40 6 7 Time duration T 0.49315 8 Sqrt(T) 9 r*T 0.70225 0.05918 10 exp(-r*T) 11 Volatility*sqrt(T) 12 In(S/PV(K) 0.94254 0.07022 0.01039 13 14 dl 0.18304 15 d2 0.11281 16 N(dl) 17 N(d2) 0.57262 0.54491 18 19 C (Black Scholes Call price) 20 Delta 1.33345 0.57262 21 Elasticity 22 P (Black Scholes put price) 17.17689 0.92009