On Restart Option

What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions

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On Restart Option

Takahiko FUJITA

Graduate School of Hitotsubashi University and Kiyoshi Ito’s Gauss Prize

memorial division at Research Insititute for Mathematical Sciences Kyoto

University

Aug 8, 2009


Contents

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1 What is Restart Option?

Restart Option

Reset Opton

Another Exotic Options

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2 Price of Some Restart Options

Option Price at independent exponential time

Price of meander lookback option

Distrbution results


Restart Option

1. Restart Option

Definition of Restart Option

Definition of Restart Option is the following:

Let St be a stock value process.

Let X[t;T ] = f(St, . . . Su(0 < u < T ), . . . ST ) be the payoff

of a derivative with maturity T at t.

We note

X[0;T ] = f(S, . . . , Su(0 < u < T ) . . . ST ) S0 = S

If there exists a sequence of times(restart times) (fixed or

random (usually stopping times)) T1 < T2 < · · ·X[Ti;T ] = f(STi, . . . , STi, Su(Ti < u < T ), ST ), we call it

”Restart Option”.

Remark 1.

For every vanilla option (which payoff is f(ST )) (For

example, Call case f(ST ) = (ST − K)+) is considered as


Restart Option

”Restart Option”. Because, in every time t < T , that payoff

Xt;T = f(ST ) remains unchange.

Remark 2.

In Japan, many children play a ”Sugoroku game (双六)”. In

Sugoroku, players throw two dice and proceed from the

origin (start point)to another place by the sum of two dice in

the board in oredr to reach to the goal. But there are some

special points at which ”go back to start point” is written

and　 if player stops at those special points, they must go

back to the origin and have to restart. Seeing this, restart

option is like that, if something happen, option contract has

to be restarted.

In this sense, we may call ”Restart Option” ” Sugoroku

Option”.

And we note that The situation of ”Go back to the origin” is


Restart Option

very much important in Mathematics especailly Probability

Theory (usaually called ”Renewal”).

Example of Restart Option 1

(Knock Out Barrier Option with Recovery)

Payoff of ”Knock Out Barrier Option with Recovery” is

following.

Take S0 = S > A, B > A, A < K < B

τA = inf{t|St = A} σ1 = inf{t > τA|St = B} Option is

knocked out if it reaches A before the maturity T , But after

τA before the maturity T , if the stock price St reach to B,

option recovers and restarts. We remark that this option

belongs to so called ”Edokko Option”, which saves options

from manipulation.


Restart Option

Example of Restart Option 2

(Meander Option) (by Fujita, Yor (6))

Payoff of ”meander lookback call option”

=maxg(K)T 5u5T (Su − K)+, where x+ = max(x, 0) and

St is a stock value process,

g(K)T = sup{t < T |St = K}.

The financial meaning of meander lookback option is the

following: If we consider a usual lookback option (with

payoff: max05u5T (Su − K)+ the price of this option is

sometimes extremely high. So partial lookback option (with

payoff: maxu2J(Su − K)+ where J ⊂ [0, T ] is considered

and sometimes traded. Meander lookback option is one

example of this partial lookback option.

Prices of derivatives might be unstable when the time is

approaching to the maturity time T and option since if that


Restart Option

option is traded, when the time approches to maturty time

T , this option is not affected by such unstability of the

market.


Reset Opton

Definition of Reset Option

Definition of Reset option is following:

If there exists a sequence of times Ti (reset times)( fixed or

random time (usually stopping time))

X[0;T ] = f(T1, . . . , Ti, . . . , Su(0 5 u 5 T )) .

Example 1 (Barrier Option 1)

Knock Out Barrier Option

Payoff of ”Knock Out Barrier Option ” is

1fiA>T (ST − K)+

reset time T1 = τA = the first hittig time to A

payoff of Knock Out Barrier option changes from

X(= (ST − K, 0)+) to 0 after T1 = τA.

Example 2 (Barrier Option 2)

Knock In Barrier Option


Reset Opton

Payoff of ”Knock In Barrier Option ” is

1fiA<T (ST − K)+


payoff of Knock In Barrier option changes from 0 to

X(= (ST − K, 0)+) after T1 = τA.


The payoff of Knock in Barrier option changes from 0 to

X(= (ST − K, 0)+) after T1 = τA.

Example 3 （Exotic Barrier Option 1)

(Parisian Option) by (Chesney, Jeanblanc, Yor,(2),(8))

If the lengh of an excursion below A straddlig some t < T is

greater than D, the option vanishies, this means:

Payoff of ”Parisian Option ” is 1T1>T (ST − K)+

reset time T1 = inf{s < T |the lengh of excursion below

A straddling s > D} The payoff of option changes from


Reset Opton

X to 0.

Example 4 (Exotic Barrier option 2)

(Edokko Option)(by Fujita, Miura)(3)

reset time (in Edokko Option case, called caution time(fisrst

yellow card time) is T1 = τA, The payoff of Option contract

changes from (ST − K)+ to Barrier Option with

τAdependent Knock Out events ( for example,

σ1 = inf{t > inf τA| ∫ tfiA

1(`1;A(Su)du = α(T − τA)}.

Example 5 (Exotic Barrier option 3)

(Local time Barrier option(by Fujita, Petit, Yor(4)) If the

lengh of local time (before T ) at A is greater than D, the

option vanishies, this means: a reset time

T1 = inf{s < T |the lengh of local time before s at A > D}The payoff of option changes from X to 0.

remark financial meaning of these exotic barrier options is


Reset Opton

following:

Barrier options are useful and popular derivatives in over

the-counter markets because they are less expensive than

plain vanilla contracts. Usual barrier options are so called

‘ one touch options ’, i.e., the contracts of which are

knocked out when the price of the underlying asset St hits a

prespecified level (Knock Out barrier) from above or below.

In this barrier option, the option writer might see that the

underlying asset approaches the bar and could try to sell the

underlying asset intentionally and escape payment. It might

be unfair that this kind of price manipulation is possible. So

far,‘ Parisian Option ’(Chesney et al., 1997) and

‘ Cumulative Parisian Option ’(Chesney et al., 1997)

’Edokko Option’ (Fujta, Miura 2003), ’Local Time Barrier

Option’ (Fujita, Petit, Yor 2004) are exotic barrier options


Reset Opton

which make this price manipulation difficult.


Another Exotic Options

Example 1, Average Option

The payoff of Average Option changes from (A[0;T ] − K, 0)+

(where A[a;b] = 1b`a

∫ ba Sudu）→)

totA[0;t]+(T`t)A[t;T ]

T

Example 2, Lookback option

the set of reset times={s| max05u5s Su(= Ms) = Ss}The payoff of lookback option changes from f(Ms) → to

f(Ms + (Ms+∆s − Ms))



2.Price of some Meander Options

(2.1)Option Price at independent exponential time

We consider the following Black Scholes Model under the

risk neutral measure Q:

dSt = rStdt + σStdWt, S0 = S

where St is the stock value at time t, r is the risk free rate,

and σ is the volatility.

We get :

St = S exp((r − 1

2σ2)t + σWt)

Then the risk neutral valuation for derivative with payoff Y

at maturity time T gives V0(Y ), the present value of

derivative Y :



V0 = E(e`rTY )

If Y is of the form φ(FT ), instead of fixed time T , it may be

more convenient to work at time θ, an independent

exponential time, because using such θ often makes

expressions simpler than at fixed time T .

There are 2 ways to access such results.

First attitude: a) to obtain the law of Ft;

in fact, very often for this, it is simpler to consider F„,

θ ∼ Exp(λ), and to invert the Laplace transform to get the

law of Ft. Then, compute E(φ(Ft)) for the particular φ of

interest.

b)second attitude: Start directly with

λ

∫ 10

e`–tE(φ(Ft))dt = E(φ(F„))



and invert the Laplace transform.

In fact, there is the commutative diagram :

Law of (F„) −−−→ E(φ(F„))yy

Law of (Ft) −−−→ E(φ(Ft))

which indicates that we may use either route from NW to

SE.

First we consider the case φ = f(ST ) which is only

dependent on the final stock value ST .



C = E(e`r„f(Se(r`12ff2)„+ffW„))

= E(exp(r − 1

2σ2

σW„ − (

1

2(r − 1

2σ2

σ)2 + r)θ)f(SeffW„)) (∵ Cameron-Martin)

= E(exp(−1

2(r

σ+

σ

2)2θ) exp(

r − 12σ2

σW„)f(SeffW„)))

=λ

λ0E(exp(

r − 12σ2

σW„0)f(SeffW„0 ))

(where θ0 ∼ Exp(λ +1

2(r

σ+

σ

2)2) λ0 = λ +

1

2(r

σ+

σ

2)2)

=λ

λ0

∫ 1`1

e( rff`ff

2)xf(Seffx)

√2λ0

2e`p

2–0jxjdx

Generally we get that　E(e`¸„f(W„)) =∫10 e`¸tE(f(Wt))λe`–tdt = –

–+¸E(f(W„0))

where we used that



forθ ∼ Exp(λ), then θ0 ∼ Exp(λ + α).We also used the simple facts

E(e¸W„0 ) = E(E(e¸W„0 ||θ0)) = E(e¸2„0

2 ) = 2–02–0`¸2 =

∫1`1 e¸x

p2–02

e`p

2–0jxjdx then, we get

　

fW„0 (x) =

√2λ0

2e`p

2–0jxj.

In the case of a call option , f(ST ) = (ST − K)+. 　We

want to get the call option price when K = S.



C =λ

λ0

∫ 1logK=S

ff

e( rff`ff

2)x(Seffx − K)

√2λ0

2e`p

2–0jxjdx

=λ

λ0

∫ 1logK=S

ff

S

√2λ0

2e`(p

2–0`ff`( rff`ff

2))xdx

− λ

λ0

∫ 1logK=S

ff

K

√2λ0

2e`(p

2–0`( rff`ff

2))xdx

=λσ√

2λ0((√

2λ0 − rff)2 − ff2

4)S

2–0ff` rff2 +1

2 K`2–0ff

+ rff2 +1

2

We get the usual Black-Scholes formula by inverting the

above with respect to λ.



(2.2) Price of meander lookback option

V0(Meander lookback option up to time θ)

= E(e`r„ maxg(K)„ 5u5„ Seexp((r`1

2ff2)u+ffWu) − K)+).

In the following, we calculate the above in two cases:

a)S 5 K and b)S = K

a) S 5 K



E(e`r„f( maxgK„ 5t5„

St))

= E(e`r„f( maxgK„ 5t5„

St), θ = τK)

+ E(e`r„f( maxgK„ 5t5„

St), θ < τK)

= E(e`rfiK)E(e`r„f( maxgK„ 5t5„

KeffWt+(r`12ff2)t))

(by memoryless property)



= E(e`rfiK)E(e`r„e( rff`ff

2)W„`1

2(( rff`ff

2)2„f( max

gK„ 5t5„KeffWt))

(by Cameron-Martin)

= E(e`rfiK)λ

λ0E(e( r

ff`ff

2)W„0f( max

gK„05t5„0

KeffWt))

=λ

λ0E(e`rfiK)

∫ ∫

A=x=0e( rff`ff

2)xf(KeffA)

∂

∂A(λ

2

1

1 − e`2–A(e`–x − e–x`2–A))dxdA

=λ

λ0E(e`rfiK)

∫ ∫

A=x=0e( rff`ff

2)xf(KeffA)

λ2

4

sinh λx

(sinh λA)2dxdA (∗)



b) S = K

E(e`r„f( maxgK„ 5t5„

St))

= E(e`r„f( maxgK„ 5t5„

St), θ = τK)


St), θ < τK)

= E(e`rfiK)E(e`r„f( maxgK„ 5t5„

KeffWt+(r`12ff2)t))


St)), mingK„ 5t5„

St = K)

= (∗) +λ

λ0

∫ 10

dbf(Seffb)(− ∂

∂b

(e( rff2`1

2) log K

S sinh(b√

2λ + µ2)

sinh((b − 1ff

log KS

)√

2λ + µ2



+e( rff`ff

2)b sinh(− 1

fflog K

S

√2λ + µ2)

sinh((b − 1ff

log KS

)√

2λ + µ2)

For call option i.e. f(x) = (x − K)+, we obtain that by

some elementary calculation,

a) when S 5 K, the price equals:

K8(–+1

2( rff+ff

2)2)

(KS

)rff2`1

2`q

(12` rff2 )2+ 2r

ff2

ζ(3)(2–`ff2–

,–`ff

2` rff

2–,

3–`ff2` rff

2–) (**)

where ζ(3)(A, B, C) :=1∑

l=0

1

(l + A)(l + B)(l + C).

Especially, if σ2 = 2r, the price equalsK

8(–+ff2

2)(KS

)p

2–ff2 ζ(3)(–`ff

2–, 2–`ff

2–, 3–`ff

2–).

b) when S = K, the price equals: (∗∗)+–

–+12( rff+ff

2)2

((1 − (KS

)(rff2`1

2))(S − K)+



σS∫10 effb

(e( rff2`

12 ) log K

S sinh b√

2–+( rff`ff

2)2+e(

rff`ff2 )b sinh(`

r2–+( rff`ff

22 )2

fflog K

S

sinh((b` 1ff

log KS

)√

2–+( rff`ff

2)2)

−1)db).


(2.3) distribution results

see Fujita,T., Yor, M(6) and for related results Bertoin, J.,

Fujita, T., Roynette, B. and Yor, M(1), Fujita,T., Yor, M(5).


Distrbution results

BM P (´ 5 A) RW P (´ < A)

supu5θ

Bu ‰p„ sup

u51

Bu 1` e−λA supu5θ

Zu 1` ¸A

supu5gθ

Bu ‰ pgθ supu51

bu 1` e−2λA supu5gθ

Zu 1` ¸2A

b : brownian bridge(b:b:)

supgθ5u5θ

Bu1

1+e−λAsup

gθ5u5θ

Zu1

1+αA

m : brownianmeander

supu5dθBu 1` 1−e−2λA

2λAsup

u5dθ

Zu 1` 1A

1

α−1−α(1` ¸2A)

supθ5u5dθ

Bu 1` 1−e−λA

2λAsup

θ5u5dθ

Zu 1` 2

α−1−α

1−αA

A

supgθ5u5dθ

Bu1

1−e−2λA` 1

2λAsup

gθ5u5dθ

Zu1

1−α2A ` 1A

1

α−1−α

e : normalized excursion


Distrbution results

where

• for BM, θ ∼ Exp(λ2/2), i.e., its density is f„(x) =1(0;1)(x)–

2

2exp −–2x

2, and P (ε = 1) = P (ε = 0) = 1/2.

• for RW, θ ∼ Geom(1 − q), i.e., P (θ = k) =

(1 − q)qk, (k = 0, 1, 2, . . . ), α = 1`√

1`q2q

.

• for RW,

gt = sup{u 5 t : Zu = 0}, dt = inf{u > t : Zu = 0}.


BM P (· 5 A, Bθ ∈ dx)

P (supu5θ

Bu 5 A, Bθ ∈ dx) (λ2e−λ|x| − λ

2eλxe−2λ max(A,x))dx

P ( supu5gθ

Bu 5 A, Bθ ∈ dx)∗ λ2e−λ|x|(1 − e−2λA)dx

P ( supgθ5u5θ

Bu 5 A, Bθ ∈ dx) λ2

11−e−2λA 1x5A(e−λ|x| − eλx−2λA)dx


P ( supu5dθ

Bu 5 A, Bθ ∈ dx) (1 − x+

A)1x5A

λ2(e−λ|x| − eλx−2λA)dx

P ( supθ5u5dθ

Bu 5 A, Bθ ∈ dx) (1 − x+

A)1x5A

λ2e−λ|x|dx

P ( supgθ5u5dθ

Bu 5 A, Bθ ∈ dx) 1− x+

A

1−e−2λA 1x5Aλ2(e−λ|x| − eλx−2λA)dx

(∗)Note: We see on this line that supu5g„ Bu and B„ are

independent.


Reference

(1) Bertoin, J., Fujita, T., Roynette, B. and Yor, M.: On a

particular class of self-decomposable random variables: The

durations of a Bessel excursion straddling an independent

exponential time, Prob. Math. Stat.,Vol. 26, issue 2, (2006)

315-366.

(2) Chesney, M., Jeanblanc-Picque, M. and Yor, M.

:Brownian excursions and Parisian barrier options. Adv.

Appl. Prob. 29, (1997).

(3)Fuita, T., Miura, R.:Edokko Options:A New Framework of

Barrier Options (with Ryozo Miura) Asia-Pacific Financial

Markets 9(2) (2003) 141-151.

(4)Fujita, T., Petit, F. and Yor, M.: Pricing pathdependent

options in some Black-Scholes market, from the distribution

of homogeneous Brownian functionals, Journal of Applied


Probability Vol 41 No.1 (March 2004) 1-18.

(5) Fujita, T., Yor, M. : On the remarkable distributions of

maxima of some fragments of the standard reflecting random

walk and Brownian Motion, Prob. Math. Stat. Vol.27, issue

1, (2007), 89-104.

(6) Fujita,T., Yor, M : On the one-sided maximum of

brownian and random walk fragments and its applications to

price of meander options (2008), preprint.

(7) Revuz, D., Yor, M. : Continuous Martingales and

Brownian Motion. Springer- Third edition- 2005.

(8) Yor, M., Chesney, M., Geman, H., Jeanblanc-Picque, M.

: Some Combinations of Asian, Parisian and Barrier Options,

Mathematics of Derivative Securities, Edited by M.H.

Dempster and S.R. Pliska, Cambridge University Press

(1997), 61-87.

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