Question

Consider an at-the-money European call option with one year left to maturity written on a non-dividend paying stock. Let today's stock price be 40 kr and the strike price is 40 kr ,and the stock volatility be 30%. Furthermore let the risk free interest rate be 6%. Construct a one-year, two-step Binomial tree for the stock and calculate today’s price of the European call.

Answer 1

Answer.

Spot Price of the stock(S)= 40 kr

Since stock volatility is 30% so stock price either increase by 30% or decrease by 30%

So, U=1+0.30=1.30 D= 1-0.3=0.70

Price at the end of 6th months

SU= 40*1.30=52

SD= 40*0.70=28

Similarly price at the end of next 6 month is calculated

Risk free rate of return= 6%

Since it is two steps binomial tree so time period within 1 year would be 2 six months

Binomial Tree are as follows

Let "P" be the probability of stock going up then probability of stock going down would be (1-P)

According to formula

P=( ert-D)/(U-D)=( e0.06*1/2-0.70)/(1.3-0.70)= (1.03045-0.70)/0.60=0.5507 i.e. 55%

So probability of stock going down=1-0.55=0.45 i.e.. 45%

Now at the end of 1 year there are four cases

Case 1, When stock price = 67.60

Then Call option is exercise because strike price of call option is less than the price at the end of year 1

So, Net gain at the end of 1 year=Max(S-X,0)=MAX( 67.60-40,0)=27.60

As we know that if spot price at the time of expiry is less than the strike price, call option is not exercise

so, net gain from call option for remaining three cases is 0 each

Value of call option at the end of 1 year = 0.55*0.55*27.60+0.55*0.45*0+0.45*0.55*0+0.45*0.45*0

=8.349

Present value of call option = 8.349/e0.06*1=8.349/1.06183= 7.86 Kr (Ans)

Note:- Please note that it not how interest is applicable i.e.. continuous compounding or simple Interest

So we assume it would be continuous compounding however if it is simple Interest rate then you can replace e0.06*1/2= 1.03 and e0.06*1= 1.06