Chooser Options«As you like it»

Edited by:Valerio Meloni

Claudio Morelli

Chooser Options

Brief description of the OptionPricing FormulaeSensitivities («Greeks»)Some code (VBA)An exampleWhy «chooser»?

A brief descriptionChooser options are exactly what their name suggests:

the holder has the right to “choose”, up to a certain date, whether his/her option is a call or a put.

St Payoff +

LONG CALL

LONG PUTT

tt-1

Decision Time

A brief description (2)We can divide them into two categories:

1. Simple Chooser Option: either the strike of the call and the put or the time to expiry are the same.

2. Complex Chooser Option: the strikes or even the expiry for both call and put are not the same.

The chooser option could be European or American.These kind of options came from the Compound Option family. Are path-dependent options.They have been traded since July of 1990 with the initial contracts traded by Bankers Trust.Could be traded on: stocks, features, indexes, exchange rates,...

Payoffs and prices…

Simple Chooser

This type of chooser gives the holder of the option a choice of either a vanilla call option or a vanilla put at a predetermined time t, where the payoff can be given as:

tTKSPTKSCChooser ttsimple ),,,(,,,max

Simple Chooser (2)Payoff at time 0;

Profit

K

Simple Chooser (3)Payoff at time t;

Profit

K

Simple Chooser (4)Rubinstein (1991) showed how the above payoff

function can be adjusted to give our valuation formula based on the put-call parity relationship.

]),,([),,,(max. )()( tTrtTDtttsimple KeeSTKSCTKSCC

),0max(),,(. ))(()(t

tTDrtTDtsimple SKeeTKSCC

Simple ChooserWe can therefore decompose the payoff in a long

call with maturity “T” and a long put with maturity “t” and strike price  

The value of a chooser option is then:

Where:

)()()()( 2121 eNKeeNSedNKedNSeChooser rTDTrTDTsimple

T

TDrKS

d

)5.0()ln( 2

1

t

tTDrKS

e

2

1

5.0)()ln(

Tdd 12tee 12

))(( tTDrKe

Simple Chooser-Greeks

121 CC

21 CC

21 CC VVV rt

CC rKe21

rt

CC Kte21

Complex Chooser (1)

]),,,(),,,(max[ tTKSPTKSCChooser pptcctcomplex

In a Complex Chooser both Strike Price and Expiry Time could be different between the Call and the Put

Complex Chooser (2)Payoff at time 0;

Profit

K2K1

Complex Chooser (3)Payoff at time t;

Profit

K2K1

Complex Chooser (4)Pricing:

Function ComplexChooser(S As Double, Xc As Double, Xp As Double, _T As Double, Tc As Double, Tp As Double, _r As Double, b As Double, v As Double) As DoubleDim dl As Double, d2 As Double, yl As Double, y2 As DoubleDim rhol As Double, rho2 As Double, i As Doublei = CriticalValueChooser(S, Xc, Xp, T, Tc, Tp, r, b, v)dl = (Log(S/i) + (b + vA2 / 2) * T) / (v * Sqr(T))d2 = dl — v * Sqr(T)yl = (Log(S/Xc) + (b + vA2 / 2) * Tc) / (v * Sqr(Tc))y2 = (Log(S/Xp) + (b + vA2 / 2) * Tp) / (v * Sqr(Tp))rhol = Sqr(T / Tc)rho2 = Sqr(T / Tp)ComplexChooser = S * Exp((b — r) * Tc) * CBND(dl, yl, rhol) _— Xc * Exp(—r * Pc) * CIESND(d2, yl — v * Sqr(Tc), rhol) _—S * Exp((b — r) * Tp) * CEINID(—dl, —y2, rho2) _+ Xp * Exp(—r * Tp) * CMIX—d2, —y2 + v * Sqr(Tp), rho2)End Function

Complex Chooser (5) The critical stock value “I” is found by calling the function

CriticalValueChooser(•) below, which is based on the Newton-Raphson algorithm, where CND() is the cumulative normal distribution function, and

CBND(•) is the cumulative bivariate normal distribution function.

Function CriticalValueChooser(S As Double, Xc As Double, _Xp As Double, T As Double, Tc As Double, Tp As Double, _r As Double, b As Double, v As Double) As DoubleDim Sv As Double, ci As Double, Pi As Double, epsilon As DoubleDim dc As Double, dp As Double, yi As Double, di As DoubleSv = Sci = GBlackScholes("c", Sv, Xc, Tc — T, r, b, v)Pi = GBlackScholes("p", Sv, Xp, Tp — T, r, b, v)dc = GDelta("c" , Sv, Xc, Tc — T, r, b, v)dp = GDelta("p" , Sv, Xp, Tp — T, r, b, v)yi = ci — Pidi = de — dpepsilon = 0.001'Newton —Raphson s kep roses sWhile Abs(yi) > epsilonSv = Sv — yi / dici = GBlackScholes("c", Sv, Xc, Tc — T, r, b, v)Pi = GBlackScholes("p", Sv, Xp, Tp — T, r, b, v)dc = GDelta("c", SY, Xc, Tc — T, r, b, v)dp = GDelta("p", SY, Xp, T.p — T, r, b, v)yi = ci — Pidi = de — dpWendCriticalValueChooser = SvEnd Function

An Example (Simple Chooser) European chooser option:

Underlying= Australian/$ with S0 = 0,6526 A/$In December the investor decides to buy a chooser option with the right to choice on February if the option will be an European plain vanilla put or a call.Investor Buys in Dec the Chooser option:t=0 (Dec), t1= February, strike (K) = 0,65 A/$, T=March.Until t1 the investor can change his choice.Lets suppose that at time t1=Feb he decides to take a long call because S1>K.The payoff of the long call option at time T=March will be:

Max(S2-K;0)

December

February March

0,6526$

0,65$

Payoff=max(0,66-0,65;0)=0,03

0,66$

Why the «chooser»? If you are a speculative investor who wants bet on volatility.

(Riskier than straddle strategy but cheaper)

If you are sure about the “kind” of volatility.(avoid bad volatility in call position )

If you want to choice at a future time.(because your expectation changes)

Why not?More expensive than single plain vanilla options

position.If you are too risk averse about the future.