Question

please fast

A bank has just sold 500,000 call options on shares of a stock. The strike price is 40; the stock price is 40; the continuously risk-free rate is 5%; the volatility is 30%; and the time to maturity is 0.25. What position should the company take in the stock for delta neutrality?

Answer 1

Note:- I have assumed the lot size of 1 call option as 100 shares

d₁ = {ln(S₀/X) + t * [r - q + (Volatility²/2)]} / [Volatility * (t^(1/2))]
where
ln = Log
S₀ = Current Spot Price
X = Strike Price
r = risk-free rate
q = Divdend Yield

d₁ = 15.81%

N(d₁) = Normal Distribution of d₁ (Done using Excel)
N(d₁) = 0.56

Delta of Call Option = e^-q*t * N(d₁)
= e^-0*0.25 * 0.56
Delta of Call Option = 0.56

Since, the call options were shorted, therefore the delta of call option will be -0.56 whereas delta of shares is 1
Total Call Options shorted = 500,000

Total shares to be bought = Delta of one call option * no. of call options shorted * Option lot size
= -0.56 * -500,000 * 100
= 28,000,000

In order to have a delta neutral strategy, the bank needs to buy 28,000,000 shares against 500,000 call options shorted