Question

17.Consider a binary (digital) option on a stock currently trading at $100. The option pays a $1 if the stock price goes above $100 in three months. 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 to achieve a delta-neutral position?

Hint: Assume that the stock price evolves with respect to a geometric Brownian motion(GBM) under risk-neutral valuation, i.e. the drift is equal to the risk-free rate.
(a)4
(b)8
(c)32
(d)48

Answer 1

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

d1 = 0.05

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

Option d is correct