Question

5. Consider an at-the-money call option that is two weeks to maturity on a stock with a local standard deviation of 40%/year. Assume the stock is selling for $50 and the risk-free rate is 5%/year straight interest. Write your answer below – Show all calculations.

5-1. What are the variables S, B, σ, and T to be used in the option formula?

5-2. What is the call price from the approximate formula?

5-3. What is the corresponding European put price?

Answer 1

n this pricing model is not given but we are that we are supposed to find the approximate price which means that we can use the binomial model.

part a

S = current stock price = 50

B = strike price =50 (why? because for an at the money option S =B)

(Sigma ) = volatility(V) =Standered deviation SD= 40%annual

T = option's time to maturity in years = 2/52 ( as there are 52 weeks in year and option is of 2 weeks)

r = risk free rate =5% annual

part b

we will use the risk-neutral approach to value the option under this approach we first need to calculate upside factor (u) and downside factor(d)

u= e^(SD*(T)^(1/2)) = e^(.40*(2/52)^(1/2)) = 1.081605

d= 1/u = 1/1.081605=.9245519

now we need expected stock prices after 2 weeks

upside price(Su)= S*u = 50*1.081605 =54.08

downside price(Sd)= S*d = 50*.9245519 = 46.23

now we need to find the call payoff on expirey which is given by

call payoff = MAX(0,(St -B)

i.e. for upside payoff = Cu = Max(0,(Su-50)) = Max(0,(54.08-50)) = 4.08

for downside payoff = Cd = Max(0,(Sd -50)) = max(0,(46.23-50)) =0

now we need risk nutral probability p and q

p= probability that stock price will go up = ((1+r)^T - d)/(u - d) =((1.05)^(2/52) - .9245519)/(1.081605-.9245519) = .492358

q= probabilty that stock price will go down = 1-p = 1-.492358 =.507642

now finally the current price of call option is given by

C0 = present value of expected value of future payoffs = (1+r)^(-T) *(p*Cu +Cd*q)

C0 = 1.05^(-2/52)*(.492358*4.08 + .507642*0) =2.005 this is the european call price as per binomial model.

part c

put call parity for europeanoption is given by

C0 +B*(1+r)^T = P0 +S0

where P0 = putprice=??

2.005 +50*(1.005)^(2/52) = P0 + 50

solving for P0 we get P0 = 2.099 why is this higher then call price because call is at the money which market is expecting the stock prices to go down which automatically make the put in the money and in the money option has comperatively higher demand that is why its price is higher.