Question

Barrier option

For 3 month options on a stock with strike price $50, you are given: i. The stock's price is $50. ii. The stock pays dividends continuously proportional to its price. The dividend yield is 2%. iii. The continuously compounded risk{free rate of interest is 6%. iv. The price of a European put option is $1.6. v. The price of a down and in call option with barrier $45 and strike price $50 is $0.85. Determine the price of a down and out call option with barrier $45 and strike price $50.

Answer 1

The value of a regular call equals the value of a down-and-in call plus the value of a down-and-out call, so the value of a down-and-out call is given by

c_do = c - c_di ...................................(1)

First, we’ll calculate the value of regular call option with the help of given values, then we’ll place it in equation (1) above. And the value of regular call option can be found by put-call parity relationship.

For a dividend paying stock, the put-call parity equation is

c+K*e^-(r-q)T= p+S₀ .................................(2)

c = regular call price, K = strike price (50), r = risk free rate (.06), q = dividend yield (.02), T = time in years (3/12), p = put price (1.6), S₀ = initial stock price (50)

Substituting in equation (2), we get

c + 50*e^{-(.06-.02)*3/12} = 1.6 + 50

c + 49.5025 = 51.6

c = 51.6 - 49.5025 = 2.0975 ie 2.10

Substituting c and c_di in equation (1) above,

c_do = 2.10 - 0.85 = $1.25