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.

TS

)0,max()( XSXS TT

)( TSX

X

Path-dependent options

Their payoffs depend on the historical underlying price during the options’ life.

Barrier options:

Asian options：

Lookback options：

T

tTT dtST

AXA0

1 where,)(

tTt

TT SMXM

0max where,)(

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 optionThe barrier is above

the initial asset value

Down optionThe barrier is below

the initial asset value

continued

Call/Putup-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

options1. double out2. double in3. 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

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