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COMPOUND OPTIONS

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Veronica Marchetti

Martina Tognaccini

- Description of the mainfeatures

- PricingModels

- Greeks

- Applications

- Advantages and Disadvantages

Itis a particularkindofExoticOptions

COMPOUND OPTION AN OPTION ON OPTION

The underlying asset is itself an option

The compoundoptionor frontoption, givesthe holder the right to buy, (in the case ofcall) or sell (in the case of put)the underlying option or back option.

FourkindsofCompoundOptions:

CALL on CALL

CALL on PUT

PUT on PUT

PUT on CALL

The first Option on Optionmodelwaspublishedby Robert Geske in 1977

OtherresearchhasbeencomputedbyRubistein in 1991.

Research on n-foldCompoundOptions (optionmade on n-1 options) byThomassen and Van Wouwe 2002

A compoundoption can haveall the featuresofotheroptions:

- European or American
- Exoticfeatures: e.g. CompoundOption on a BarrierOption

WewillconsideronlyEuropeanCompoundOptionsand eachofthemhas:

2stikeprices: K1(compound) and K2 (underlying)

2expirationdates: T1 (compound)and T2(underlying) withT1< T2

2 optionpremiums

Compound Option: Call on Call

- To evaluate European Compound Options we can follow different approaches.

Itgivesanevaluation formula in closedformassumingConstantVolatiliy

BLACK-SCHOLES

BINOMIAL TREES

Numericalmethod

Numericalmethodbased on simulations

- MONTE CARLO SIMULATIONS

Geske (1977) and Rubistein (1991) were the first to elaborate and implementclosed form formulae on the basis of the Black & Scholes model (1973) to price European–type compound options.

Assuming a risk- neutral world we can discount the expected value of the option at the expiry date with the risk free rate

We assume Constant Volatility

Using the put-callparity relation we can write the closed formula forall the compoundoptions

TwoExpiry date: T1 (for the compoundoption) and T2 (for the underlyingoption) withT1< T2 and with

Two strike prices : C*(or P*) for the CompoundOption and Xfor the UnderlyingOption

To solve the modelwe start from the Call on Calloption at T0

The valueof the CompoundOption (CC) is a functionof the valueof the underlyingoptionwhichis

WedenoteS*the critical stock price abovewhich the compoundoptiongetsexercisedthenitfollowsthat

And the optiongetsexercixedif

- Whereisthe cumulative distributionfunction of the bivariatenormal random variable with correlationcoefficientρ, thatis

and

Comparing the pricing formula for the compound options and the Black&Scholesonewenoticethat the first two terms correspond, while the third term is the strike price of the compound option multiplied by the probability that the price S exceeds at time and at time the exercisepriceX , embodied by the term

BLACK & SCHOLES for a Call

(

COMPOUND OPTION

Using the Put – Call parity relation we can compute the price of the compound options which involve puts.

Four combinations:

- Call on Call. These payout max{C(S,T1) – X1,0} and have a price of
- Call on Put. These payout of max{P(S,T1) – X1, 0} and have a price of

Four combinations:

- Put on Call. These payout of max{X1 - C(S,T1) ,0} and have a price of
- Put on Put. These payout of max{X1- P(S,T1) ,0} and have a price of
- Put on Put. These payout of max{X1- P(S,T1) ,0} and have a price of

Four combinations:

- Put on Call. These payout of max{X1 - C(S,T1) ,0} and have a price of
- Put on Put. These payout of max{X1- P(S,T1) ,0} and have a price of
- Put on Put. These payout of max{X1- P(S,T1) ,0} and have a price of

Monte Carlo Simulations (1)

- Black & Scholes model assumes constant volatility, because of this, the price obtained with this model is generally underestimated.

Compound options are very sensitive to volatility

We need to develop option pricing models which do not underestimate the price and to do this we have to allow the volatility to evolve stochastically

Monte Carlo Simulations

Monte Carlo Simulations (2)

Monte Carlo methods are a class of computational algorithm that rely on repeated random sampling to compute their results. These methods are suited to calculation by programs such as Visual Basic and Matlab.

1-The modelimplements n simulationsofuniformvariableswhichthentransformsintonormalvariables

2-Then itusesthesevariables in orderto simulate

- S with the GeometricBownianMotion

- R with the short rate interest models i.e. CIR or Vasicek

- The Volatilitywith the Hestonmodel

3-Finally, byimposing the payoffof the CompoundOption, itfinds the pricewhichbecomes more reliablewith the useofvariancereductionmethodse.g (AntitheticMethod)

American typeoptions

1-Monte Carlo Method

2-Binomial Tree

- 2-Sparse Grid Approach firstly introduced by Reisinger (2004) to option pricing problems.

3-The Method of Lines

The key idea behind the method of lines is to replace a PDE with an equivalent system of one-dimensional ordinary-diﬀerential equations(ODEs), the solution of which is more readily obtained using numerical techniques.

The combination technique requires the solution of the original equation only on a set of conventional subspaces deﬁned on Cartesian grids and a subsequent extrapolation step.

The Greeks

To compute the Greeks of a compound option we have to know the derivatives of itsvalue V. Althoughwe are notable to determinethisfunctionexplicitlywe can compute itsderivatives by usingimplicitdifferentiation.

Gamma

Delta

The sensitivity of delta to changes in the price of the underlying option.

The sensitivity to changes in the price of the underlying option.

Theta

Rho

The sensitivity to changes in the time until expiration.

The sensitivity to changes in the domestic and foreign interest rate.

The sensitivity of its value to changes in the underlying option volatility.

Delta

Gamma

Theta

Rho

Vega

1- Minimize the degreeofriskofaninvestment

- In the currency market

- In the fixedincome market

2-HedgingStrategies

- Particulalryusedbycorporationswithseveraltypesof business projectswichmay or notmayfailsuchas:
- Tender for a contract
- Overseasacquisitions (tohedge the foreignexchangerisk)

3-Speculative Strategies

Speculation on volatilityofvolatilitybylooking at the price ofanoption in the future

Advantages

- Cheaperthanstraightoptions

- HighlyLeveraged so it can beusedtogainexposureto the underlyingwhilelimitingdownsideto the initial premium

Disadvantage

- Ifbothoptionsgetexercised, the total premium for the CompoundOptionwillbe more expensivethan the premium for a single optionbutif the strategyissuccessful, the differencewillbeabsorbed