Question

An at-the-money call option with expiry 100 trading days later (for calculations consider that there are 250 trading days per year) is written on a stock XYZ. The volatility of the stock is 26% and the stock pays no dividends during the life of the option. The risk free rate is 5%. Can you find the delta of the option? Suppose that you own 5,000 shares of the stock XYZ, how can you use the available call options to make sure that your portfolio is not affected by changes in the stock in the short run?

Answer 1

Answer ) The question is consider as case of European call option , so, the calculation will be based out on black and scholes model .

ATM call option means , Strike price= Current price., so neither profit nor loss.

Black and scholes model of option valuation ,

S = Current value of the underlying asset

K = Strike price of the option

t = Life to expiration of the option

r = Riskless interest rate corresponding to the life of the option

?2 = Variance in the ln (value) of the underlying asset

Input Data
Stock Price now (S)		1
Exercise Price of Option (K)		1
Enter Start Date
Enter End date
Number of periods to Exercise in years (t)		0.40 --(100/250)
Compounded Risk-Free Interest Rate (rf)		5.00%
Standard Deviation (annualized s)		26.00%
Output Data
Present Value of Exercise Price (PV(K))		0.9802
s*t^.5		0.1644
d1		0.2038
d2		0.0394
Delta N(d1) Normal Cumulative Density Function		0.5808
N(d2)*PV(K)		0.5055
Value of Call		0.0753

Here Delta = 0.5808 ,

It means $1 change in stock price will lead to change of $ 0.5808 in call price .As number of shares in call contract is not given in question , I assumed 1 contract is made of 1 share only .

In the case , Position delta = Delta * number of shares = 0.5808 * 5000 =2904 shares in call contracts as position.