PACKAGED OPTIONS

7.2 PACKAGED OPTIONS

Packaged options are varied. We consider first binary options. A payoff for binary options occurs if the value of the underlying asset S(T ) at maturity T is greater than a given strike price K . The amount paid may be constant or a function of the difference S(T ) − K . The price of these options can be calculated easily if risk- neutral pricing is applicable (since, it equals the discounted value of the terminal payoff). When computations are cumbersome, it is still possible to apply stan- dard (Monte Carlo) simulation techniques and calculate the expected discounted payoff (assuming again risk-neutral pricing, for otherwise simulation would be misleading). The variety of options that pay nothing or ‘something’ is large and therefore we can briefly summarize a few:

r Cash or nothing : Pays A if S(T ) > K . r Asset or nothing : Pays S(T ) if S(T ) ≥ K . r Gap : Pays S(T ) − K if S(T ) ≥ K .

r Supershare : Pays S(T ) if K L ≤ S(T ) ≤ K H .

r Switch : Pays a fixed amount for every day in [0,T ] that the stock trades above

a given level K . r Corridor (or range notes) : Pays a fixed amount for every day in [0,T ] that the

stock trades above a level K and below a level L. r Lookback options : Floating-strike lookback options that provide a payout

based on a lookback period (say three months), equalling the difference be- tween the largest value and the current price. There are Min and Max lookback options:

Min : V (T ) = Max [0,S(T ) − S min ]; Max : V (T ) = Max[0,S max − S(T )]

OPTIONS AND PRACTICE

r Asian options : Asian options are calculated by replacing the strike price by the average stock price in the period. Let the average price be:

¯S = 1

(t) dt; t ∈ [0,T ]

Then the value of the call and put of an Asian option is simply:

Put : V (T ) = Max [0, ¯S − S(T )]; Call: V (T ) = Max[0,S(T ) − ¯S] r Exchange : A multi-asset option that provides the option for a juxtaposition of two assets (S 1 , S 2 ) and given by Max [S 2 (T ) − S 1 (T ),0]. Such options can

also be used to construct options on the maximum or minimum of two assets. For example, buying the option to exchange one currency (S 1 ) with another (S 2 ) leads to:

V (T ) = min [S 1 (T ), S 2 (T )] = S 2 − Max [S 2 (T ) − S 1 (T ),0]

V (T ) = max [S 1 (T ), S 2 (T )] = S 1 (T ) + Max [S 2 (T ) − S 1 (T ),0] r Chooser : Provides the option to buy either a call or a put. Explicitly, say that (T 1 , T 2 ) are the maturity dates of call and put options with strikes (K 1 , K 2 ).

Now assume that an option is bought on either of the options with strike T ≤ (T 1 , T 2 ). The payoff at maturity T is then equal to the max of a call

C [S(T ), T 1 −T;K 1 ] and the put P[S(T ), T 2 −T;K 2 ]: Max{C[S(T ), T 1 −T;K 1 ], P[S(T ), T 2 −T;K 2 ]}

r Barrier and other options : Barrier options have a payoff contingent on the underlying assets reaching some specified level before expiry. These options

have knock-in features (namely in barrier) as well as knock-out features (out- barrier). These options are solved in a manner similar to the Black–Scholes equation considered in the previous chapter, except for a specification of boundary conditions at the barriers. We can also consider barrier options with exotic and other features such as options on options, calls on puts, calls on calls, puts on calls etc., as well as calls on forwards and vice versa. These are compound options and are written using both the maturity dates and strike prices for both the assets involved. For example, consider a call option with

maturity date and strike price given by (T 1 , K 1 ). In this case, the payoff of a call on a call with maturity date T and strike K is a compound option given by:

C c (T 1 , K 1 ) = Max{0, C[S(T ), T 1 − T, K 1 ]−K 1 } where C[S(T ), T 1 − T, K 1 ] is the value at time T of a European call option

with maturity T 1 − T and strike price K 1 . By the same token, a compound put option on a call pays at maturity:

P c (T 1 , K 1 ) = Max (0, K − C(S(T ), T 1 − T, K 1 ))

165 Practically, the valuation of such options is straightforward under risk-neutral

COMPOUND OPTIONS AND STOCK OPTIONS

pricing since their value equals their present discounted terminal payoff (at the exercise time).

r Passport options : These are options that make it possible for the investor to engage in short/long (sell/buy) trading of his own choice while the option

writer has the obligation to cover all net losses. For example, if the buyer of the option takes positions at times t i , i = 1, . . . , n − 1, t 0 = 0, t n = T by buying or selling European calls on the stock, then the passport option provides the following payoff at timeT – the option exercise time:

n−1

Max

u i [S(t i +1 ) − S(t i )], 0

i =0

where u i is the number of shares (if bought, it is positive; if sold, it is negative) at time t i and resolved at period t i +1 . In this case, the period profit or loss would

be: [S(t i +1 ) − S(t i )]. Particular characteristic can be added such as the choice of the asset to trade, the number of trades allowed etc. r As you like it options : These options allow the investor to chose after a specified period of time T , whether the option is a call or a put. If the option is European and the call and the put have the same strike price K , then put-call parity can

be used. The value at exercise is Max(c, p) and consists in selecting either the call or the put at the time the option exercise is made. Thus, put-call parity with continuous and compounded discounting and a dividend-paying stock at

a rate of q implies (as we shall see later on): c+e −q(T −t) Max[0, K e −(R f −q)(T −t) − S(t)]

In other words, ‘as you like it options’ consist of a call option with strike K at T and e −q(T −t) put options with a strike of [K e −(R f −q)(T −t) ] at maturity T .

The finance trade and academic literature abounds with options that are tailored to clients’ needs and to the market potential for such options. Therefore, we shall consider a mere few while the motivated reader should consult the numerous references at the end of the previous and the current chapter for further study and references to specific option types.