34. Consider a binary (digital) option on a stock currently trading at $100. The option pays a $1 if the stock price goes below $100 three months from now. The annualized standard deviation of the stock is 20%, and the risk-free rate is 0%. Suppose you sold a 100 of these binary options. How many shares of the underlying stock you need to long/short to achieve a delta-neutral position

48

Explanation:

N (d2) : probability of call option being exercised

So current stock price = 100

K strike price = 100

r risk free rate = 0% = 0.05

s: standard deviation = 20%

t: time to maturity = 3month = 0.25 year

di - In (So/K) + (r + 0.5 * 5%) * * S*t0.5

d1 = 0.05

d2 = dl - 5*10.5

d2 = - 0.05

N (d2) = normsdist (d2) = 0.48

Pay-off per option = 1

No. of options sold = 100

Expected pay-off = - 0.48*1*100 = - 48

Therefore go long on 48 shares so that if stock price becomes 101, pay-off from stocks = 48 * (101-100) = 48