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Edited by: Valerio Meloni Claudio Morelli Chooser Options Brief description of the Option Pricing Formulae 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-1 Decision Time t T LONG PUT A brief description (2) We can divide them into two categories: 1. 2. 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 pathdependent 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: Chooser simple max C S t , K , T , P ( S t , K , T ), t 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. C . simple max C ( S t , K , T ), [ C ( S t , K , T ) S t e C . simple C ( S t , K , T ) e D (T t ) D (T t ) max( 0 , Ke Ke r (T t ) ( r D )( T t ) ] St ) Simple Chooser We can therefore decompose the payoff in a long call with maturity “T” and a long put with maturity “t” and strike ( r D )( T t ) Ke price The value of a chooser option is then: Chooser simple Se DT N ( d 1 ) Ke rT N ( d 2 ) Se DT N ( e1 ) Ke rT N ( e2 ) Where: d1 ln( S ) ( r D 0 . 5 ) T 2 K T d 2 d1 T e1 ln( S ) ( r D )T 0 . 5 t 2 K t e 2 e1 t Simple Chooser-Greeks C1 C 2 1 C1 C 2 V V C1 V C 2 C C rKe 1 rt 2 C C Kte 1 2 rt Complex Chooser (1) In a Complex Chooser both Strike Price and Expiry Time could be different between the Call and the Put Chooser complex max[ C ( S t , K c , Tc ), P ( S t , K p , T p ), t ] Complex Chooser (2) Payoff at time 0; Profit K1 K2 Complex Chooser (3) Payoff at time t; Profit K1 K2 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 Double Dim dl As Double, d2 As Double, yl As Double, y2 As Double Dim rhol 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 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 Double Dim Sv As Double, ci As Double, Pi As Double, epsilon As Double Dim dc 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$ December February 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) Why not? More expensive than single plain vanilla options position. If you are too risk averse about the future.