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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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.
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Restart Option
option is traded, when the time approches to maturty time
T , this option is not affected by such unstability of the
market.
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Reset Opton
Payoff of ”Knock In Barrier Option ” is
1fiA<T (ST − K)+
reset time T1 = τA = the first hittig time to A
payoff of Knock In Barrier option changes from 0 to
X(= (ST − K, 0)+) after T1 = τA.
reset time T1 = τA = the first hittig time to A
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Reset Opton
which make this price manipulation difficult.
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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))
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Option Price at independent exponential time
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 :
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Option Price at independent exponential time
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„))
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Option Price at independent exponential time
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 .
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Option Price at independent exponential time
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Option Price at independent exponential time
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.
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Option Price at independent exponential time
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.
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Price of meander lookback option
(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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Price of meander lookback option
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)
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Price of meander lookback option
= 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 (∗)
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Price of meander lookback option
b) S = 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))
+ E(e`r„f( maxgK„ 5t5„
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Price of meander lookback option
+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)+
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
Price of meander lookback option
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).
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
(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).
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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}.
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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.
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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
What is Restart Option? Price of Some Restart Options Table 1 : A list of interesting maxima Table 2 : A list of joint distributions
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.