Question

a) Write the payoff functions for the European and American up-and-in barrier put option with barrier B.

b) Use the 4-step binomial tree to price the up-and-in barrier put options (European style and American style, separately). Assume that the spot price is $105. The strike is $101. The barrier is $106. The time to maturity is 1 year. Risk-free rate is 5% with the annual compounding, the stock price goes up or down by 20%.

c) Is the European up-and-in barrier put option priced higher or lower than the same option wihout a barrier?

Answer 1

Answer a)

European barrier options have a vanilla payoff at expiry plus they also have a singleEuropean barrier. For a European knock-out (EKO) barrier option, if spot at maturity is beyond the barrier level, the contract expires worthless despite being in-the-money.

Barrier Price. The price at which a barrier option becomes active or inactive. A knock-in barrier option may not be exercised unless the price of the underlying asset reaches the barrier price, which is stated in the contract. ... In both cases, the barrier price is $40.

Because barrier options have additional conditions built in, they tend to have cheaperpremiums than comparable options with no barriers. ... The lower premium of thebarrier option may make this more appealing than using non-barrier American or European options

A knock-out option in which the barrier is in-the-money with respect to the strike is called a reverse knock-out option. A knock-in option in which the barrier is in-the-money with respect to the strike is called a reverse knock-in option. ... It will still be cheaper than the plain vanilla option, but not by very much.

The difference between a knock-in and knock-out option is that a knock-in option comes into existence only when the underlying security reaches a barrier, while aknock-out option ceases to exist when the underlying security reaches a barrier.

A knock-in option is a latent option contract that begins to function as a normaloption ("knocks in") only once a certain price level is reached before expiration.Knock-ins are a type of barrier option that are classified as either a down-and-in or an up-and-in.

Knock-outs are a CFD trade on an option. They automatically close – or get 'knockedout' – if your provider's underlying market price reaches your knock-out level. They move one-for-one with the underlying market – meaning that for every point the underlying moves, the price of the knock-out moves the same amount.

Answer b)

Sn is the price of the underlying asset. p is the probability of an upwards price movement. Vu is the option value from node upper node at n+1. Vu is the option value from the lower node at n+1.

Pricing a European Call Option Formula

1. d1 = [ln(P0/X) + (r+v2/2)t]/v √t and d2 = d1 – v √t.

2. P0= Price of the underlying security.

3. X= Strike price.

4. N= standard normal cumulative distribution function.

5. r = risk-free rate.

6. v= volatility.

7. t= time until expiry.

A binomial experiment is a statistical experiment that has the following properties: ... Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. The probability of success, denoted by P, is the same on every trial.

To calculate profits or losses on a put option use the following simple formula: Put Option Profit/Loss = Breakeven Point – Stock Price at Expiration

he maximum amount you can lose on your long position is the price paid for the asset. If you establish a covered call position, your maximum loss would be the stock purchase price minus the premium received for selling the call option
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