COMPOUND OPTIONS. Veronica Marchetti Martina Tognaccini. Compound Option. Description of the main features. Pricing Models. Greeks. Applications. Advantages and Disadvantages. What is a Compound Option ?. It is a particular kind of Exotic Options.
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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.
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:
2stikeprices: K1(compound) and K2 (underlying)
2expirationdates: T1 (compound)and T2(underlying) withT1< T2
Itgivesanevaluation formula in closedformassumingConstantVolatiliy
Numericalmethodbased on 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
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
Using the Put – Call parity relation we can compute the price of the compound options which involve puts.
Monte Carlo Simulations (1)
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
3-Finally, byimposing the payoffof the CompoundOption, itfinds the pricewhichbecomes more reliablewith the useofvariancereductionmethodse.g (AntitheticMethod)
1-Monte Carlo Method
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.
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.
The sensitivity of delta to changes in the price of the underlying option.
The sensitivity to changes in the price of the underlying option.
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.
1- Minimize the degreeofriskofaninvestment
Speculation on volatilityofvolatilitybylooking at the price ofanoption in the future