Vanilla options The payoff of a European (vanilla) option at expiry is ---call ---put where -- underlying asset price at expiry -- strike price The terminal payoff of a European vanilla option only depends on the underlying price at expiry.

Path-dependent options Their payoffs depend on the historical underlying price during the options’ life. Barrier options: Asian options ： Lookback options ：

Barrier options: an example Barrier options are triggered by the action of the underlying hitting a prescribed value at some time before expiry. For example, if the asset remains below a pre-determined barrier price during the whole life of the option, the contract will have a call payoff at expiry (red line). However, should the asset reach this level before expiry, then the option becomes worthless because it has ``knocked out'‘ (blue line).

types of barrier options The out option, that only pays off if a level is not reached. If the barrier is reached then the option is said to have knocked out The in option, that pays off as long as a level is reached before expiry. If the barrier is reached then the option is said to have knocked in.

continued Up option The barrier is above the initial asset value Down option The barrier is below the initial asset value

continued Call/Put up-out-call, down-out-call, up-in-call, down-in-call, up-out-put, down-out-put, up-in-put, down-in-put, European/American

In-out parity In + Out = Vanilla This parity holds for European, not for American

Pricing by Monte-Carlo simulation For example: up-out-call

Pricing in PDE V=V(S,t): the value of barrier option (before the barrier is reached) The details of the barrier feature come in through the specification of the boundary conditions and solution domains.

Boundary and final conditions out in

Solution domains down up

Closed form solution

American out

American in

BTM

continued

Hedging Barrier options may have discontinuous delta at the barrier. So, it is very hard to hedge a barrier option. We refer interested students to Wilmott (1998) and references therein.

Other features Time-dependent barrier level

continued Double barrier options 1. double out 2. double in 3. one in another out

Asian options Payoff types Fixed strike Floating strike

Type of Averaging Payoff types Arithmetic Geometric

Monte-Carlo Simulation V=exp(-r*T)E[final payoff]

Extending Black-Scholes

Arithmetic Asian Compared with the pricing model with an option on two assets

Geometric Asian

Early exercise

Formulas Explicit formulas exist for all European- style geometric Asian options Not for arithmetic Asian options

An example of reduction

Parity relation

Model independent results Put-call parity for vanilla options, multi-asset options The in-out parity for barrier options American call options should never be early exercised if there is no dividend payment.

Model dependent results Put-call symmetry relation Put-call parities for Asian options

Binomial tree method

Lookback options Types of Payoff

Extending the Black-Scholes equations

Another variable

continued

BTM Equivalence of BTM and PDE

Similarity reduction PDE European/American floating strike options European fixed strike options BTM

Exotic options Barrier options Asian options Lookback options Forward start options Shout options Compound options ……

Forward Start Options An option that starts in the future. An example: a forward start call

Pricing

Pricing Model

Shout Options A shout call option: a vanilla option with an extra feature The holder can at any time reset the strike price The strike price becomes the asset price when resetting The action of resetting is called “shouting”.

Two questions How to value a shout option What’s the optimal shouting policy?

Pricing Model When shouting at time t, the option becomes an at-the-money option with Sf(t) Let V(S,t) be the value of the shout option. Then

Continued

Compound Options An option on another option call on call call on put put on call put on put The payoff two strike prices: X, X1 two expiration dates: T<T1

Future Options An option on a future

Pricing Model for Compound Options

Others Parisian options Employee reload options Asian barrier options ……

Concluding remarks(1): closed-form solutions Available (see Kwok (1998) or Hull (1998)) Barrier options Geometric Asian options Lookback options Russian options Forward start options Compound options Not available Arithmetic Asian options Shout options American-style options

Concluding remarks(2): Binomial tree method The BTM can be readily extended to all options discussed in this chapter But it is not easy to implement for Asian options. (see Kwok (1998) for a modified BTM). Similarity reduction Not available for American-style fixed strike options