Barrier options

1. Barrier options Agela Baze Erjona Mimini

2. Barrier Options • Barrier options are the most popular exotic options(traded OTC in the late ‘60). They are an extension of standard options. • The pay-off of the option depends on the whole path followed by the underlying. • The contract fixes a specified level of the underlying price called barrier. The option pay-off is conditional on the underlying price process hitting the barrier. • Barrier options can be classified according to: • Pay-off type: • Call & Put • Knocking type: • Knock-IN: The option comes INTO life and can be excercised only if the underlying’s price touches the barrier within expiration. • Knock-OUT: The option comes OUT of life and cannot be excercised further if the underlying’s price touches the barrier within expiration. • Barrier type: • UP: The barrier is set above the underlying spot price and is an upper bound. • DOWN: The barrier is set below the underlying spot price and is a lower bound. • Relation between standard options and barrier options: V standard option(t,S)=V(t,S)knock-out+V(t,S)knock-in IN-OUT PARITY

3. Basic Barrier Options • Call: • Up-and-in • Up-and-out • Down-and-in • Down-and-out • Put: • Up-and-in • Up-and-out • Down-and-in • Down-and-out Based on these classifications 8 different basic types of barrier options can be obtained starting from the plain vanilla Call and Put.

4. Barrier Options: Down-and-out Call • It is an european call option which stops to exist if the (is knocked out) if the underlying touches the barrier Hmin before expiration. For the option to have an initial value Hmin < S0. • Pay-off of the Down-and-out Call • (ST-K)+ , if St>Hmin for all t<=T; • 0 , if St<=Hmin for at least one t<=T. • When St does not hit the barrier we are in the situation of a Plain Vanilla Call.

5. Barrier Options: Down-and-out Call • Consider the case in which Hmin<K:

6. Barrier Options: Down-and-out Call Pricing • First analytical formula wasproposedin the 1970s. • There are several methods to price a Barrier Option: •  Static replicating portfolio of vanilla options (priced with B&S) • Law of the maximum (or minimum) of the underlying Closed form solution; • PDE approach; • Monte Carlo simulation pricing; • Lattice evolution of the underlying approach.

7. Barrier Options: Down-and-out Call PDE Pricing • All kinds of barrier options must satisfy the same Black&Scholes equation: • Barrier Option pricing is a terminal boundary problem of the Black&Scholes which compared to the vanilla pricing has one more boundary (here SB denotes the barrier) and a corrisponding boundary condition • The pricing PDE of the down-and-out call option is : • After some manipulations we obtain the following relation:

8. Why buying a barrier option? • Buying a barrier option because: • They may more closely match your beliefs about the market behavior • Lower cost with respect to the standard options (call or put) • You can hedge better with barrier options in some cases. Knock out + knock in = vanilla option. -- >cheaper They help traders who place directional bets enhance their leverage and investors who accept to keep some residual risk on their books reduce their hedging costs.

9. Who buys/sells them? 1) Suppose an investor whose expectations about the behavior of the market are that it will increase / highly increase. • He can buy a down and out call: • If the market increases he can have unlimited profits, as in the case of a standard call option. • Worthless when the price goes down more than H • He could also sell a down and in call with spot price Sthat activates only if the market price falls below H. In this case, he gains premiums from an unlikely market scenario expectation i.e a deep fall of the market.

10. Who buys/sells them? 2) The other two scenarios would be: • Sells an up and in call who expects that the market will decrease or that will remain stable (and so will do the underlying stock). The investor that is interested in buying this type of option expects that the market will increase. • Buys an up and out call the investor that expects a small increase or a rather stable market. Let’s suppose that the spot price is 100 and the barrier is H= 150. The option knockes out only if the price goes higher than 150 Same reasoning in the case of a Barrier put option

11. Lower cost with respect to the standard options • You pay only for scenarios you think are more probable to happen. Example: Knockout options won’t pay off if the stock crosses the knockout barrier. Cheaper than an otherwise identical option without the knockout feature. If you think the chance of knockout is small, you can take advantage of the lower premium and get the same benefits. Or, you can even pay more premium to receive a cash rebate if the option is knocked out Knock out = vanilla option - knock in You avoid to pay for the cases in which the price may have huge changes as a deep fall and then increasing again. Once it is out, it will remain so. So reduces the premium.

12. In some cases you can hedge better with barrier options • Let’s suppose an investor has a stock in his portfolio and his expectations are that the market will decrease. Thus he could sell a call option with strike K, let’s say = 80 and he would not loose from the market fall. But it could happen also the opposite situation in which the market goes up and he might have huge losses. If he sells an up and out call with barrier H= 110, he could get protected also from an increase in the market price. If the spot price of the stock gets higher than 110, the option sold is worthless, without a rebate, and the investor has gained the premium + the stock with a higher market value! On the other side, he would limit his losses because of the lower cost of the barrier call (he must buy it before selling) and because of a market increase. • Example: A global bank might determine that the spread between borrowing and lending rates of a foreign operation is contingent on the foreign currency remaining within pre specified barriers. In this case, a barrier option written on the currency can be used to protect the profitability of these operations in the event that the currency exits these bands.

13. Hedging Barrier options Hedging barrier options is much more complicated than standard plain vanilla because they combine characteristics of the plain vanilla and of the barriers. One can distinguish mainly two types of hedging: • Dynamic hedging - constant rebalancing of the portfolio (eg. Black-Scholes extension of the European options) If the barrier is hit before maturity, the barrier option value at that time is known to be zero, or the price of the corresponding European option otherwise. Thus it is sufficient to hedge the option before hitting the barrier. Prices of barrier options are smooth functions of the underlying asset price in the active region so Ito lemma can be applied up to the stopping time: Hold ∂f (t, St)/ ∂S shares of the underlying asset for t< TH. • Static hedging - “buy and hold” strategy (eg. proof of the put-call parity) 1.Buy an option portfolio with K and T which partially replicates the payoff at the expiry. 2. Make the value of your portfolio equal to 0 at the barrier value by buying and selling call options with strike equal to the barrier H, so to set the value of the portfolio equal to 0 at the boundary. Is not hit: Same payoff with the vanilla option The barrier Is hit: Option value = 0, thus portfolio value must be 0 for St = H

14. Sensitivity- delta of a 1 yeardown and out call 1.Delta of barrier options can differ significatively w.r.t standard options (∆ < 0 or ∆ >1) 2.A barrier option values can actually decrease with increasing volatility (you are long vol w.r.t strike, but short vol w.r.t barrier) * For high values of the spot, the delta behaves as if it was a standard option. ** With a St > H just above the barrier, delta increases rapidly *** It declines as the price declines (but remains positive) and becomes zero when St = H because the option is worthless. ****The down-and-out call is always worth less than the corresponding standard call, but approaches the same value at very high stock prices

15. Sensitivity – Gamma of a 1 yeardown and out call For stock prices near the barrier, anincrease in volatility would actually make knockoutfor the down and out option more likely, and so decrease overall option value.

16. Barrier Options: Perspective • Tailor made options which at present come with different variations from the standard ones: • Rebate (in case the option is knocked out); • Double Barrier(Double-no-touch,China 2011); • Discrete Barrier(Monitored at discrete intervals); • Window Barriers(Time ranges to monitor the price); • Parisian Barrier (Price can spend a certain period out); • Fluffy Barriers(Depth of barrier crossing); • Useful in foreign exchange exposure managing.