Chooser Options « As you like it »

ChooserOptions«Asyoulikeit» Editedby: Valerio Meloni Claudio Morelli

ChooserOptions • Brief description of the Option • PricingFormulae • Sensitivities («Greeks») • Some code (VBA) • An example • Why «chooser»?

A brief description Chooser 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. LONG CALL Payoff + St T t-1 t LONG PUT Decision Time

A brief description (2) We can divide them into two categories: • Simple Chooser Option: either the strike of the call and the put or the time to expiry are the same. • 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…

SimpleChooser • 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:

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.

SimpleChooser • We 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:

SimpleChooser-Greeks

ComplexChooser (1) • In a ComplexChooserboth Strike Price and ExpiryTimecouldbedifferentbetween the Call and the Put

ComplexChooser (2) • Payoff at time 0; Profit K1 K2

ComplexChooser (3) • Payoff at time t; Profit K1 K2

ComplexChooser (4) • Pricing: FunctionComplexChooser(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 Double Dim dl As Double, d2 As Double, yl As Double, y2 As Double Dimrhol As Double, rho2 As Double, i As Double i = 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

ComplexChooser (5) • The critical stock value “I” is found by calling the function CriticalValueChooser(•) below, which is based on the Newton-Raphsonalgorithm, where CND() is the cumulative normal distribution function, and CBND(•) is the cumulative bivariate normal distribution function. FunctionCriticalValueChooser(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 Double Dim Sv As Double, ci As Double, Pi As Double, epsilon As Double Dimdc As Double, dp As Double, yi As Double, di As Double Sv = S ci = 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 — Pi di = de — dp epsilon = 0.001 'Newton —Raphson s kep roses s While Abs(yi) > epsilon Sv = Sv — yi / di ci = 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 — Pi di = de — dp Wend CriticalValueChooser = Sv End 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/$, T=March. Until t1 the investor can change his choice. Lets suppose that at time t1=Feb he decides to take a long call because S1>K. The payoff of the long call option at time T=March will be: Max(S2-K;0) 0,66$ 0,6526$ 0,65$ February December March Payoff=max(0,66-0,65;0)=0,03

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) Whynot? • More expensive than single plain vanilla options position. • If you are too risk averse about the future.

Chooser Options « As you like it »