### Chooser Options

```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.