Tristan Guillaume
email: tristan.guillaume@u-cergy.fr

Some sequential boundary crossing results for geometric Brownian motion and their applications in financial engineering

This paper provides new explicit results for some boundary crossing distributions in a multidimensional geometric Brownian motion framework when the boundary is a piecewise constant function of time. Among their various possible applications, they enable accurate and efficient analytical valuation of a large number of option contracts traded in the financial markets belonging to the classes of barrier and lookback options.

2 2 2 2 dS t S t dt S t dB t a s   where   1 B t and   2 B t are standard real-valued Brownian motions and    1 2 1.2 , d B B t dt r  Let 1 2 1 2 3 , , , , H H K K K be positive real numbers and     .

Introduction

The joint law of the maximum (or the minimum) of a real-valued Brownian motion and its endpoint over a finite time interval is a central result in the study of Brownian motion, particularly with regard to the many applications of the theory in finance, medical imaging, robotics and biology. It can be obtained as a consequence of the « reflection principle », which derives from the strong Markov property of Brownian motion (Freedman [6]). Application of Girsanov's theorem easily generalizes this seminal result to the case of a geometric Brownian motion (GBM), a frequently encountered diffusion process that is the building block of financial engineering. Alternatively, the law can be derived by a partial differential equation approach, using Kolmogorov's equation for the transition density function of a diffusion process. The distribution of the first passage time by a one-dimensional GBM to a one-sided or a two-sided straight boundary then follows. A few cases where the boundary is curved have been handled (Barba Escriba [START_REF] Escriba | A Stopped Brownian Motion Formula with Two Sloping Line Boundaries[END_REF] ; Salminen [START_REF] Salminen | On the First Hitting Time and Last Exit Time for a Brownian Motion to/from a Moving Boundary[END_REF]; Kunitomo and Ikeda [START_REF] Kunitomo | Pricing Options with Curved Boundaries[END_REF]). For a comprehensive source of formulae, one may refer to Borodin and Salminen [START_REF] Borodin | Handbook of Brownian Motion -Facts and Formulae[END_REF].

Various extensions to these results are often needed to solve practical engineering problems. In particular, one may look for joint distributions of the highest or lowest points hit over several time intervals, and one may deal with a multi-dimensional GBM. There are few known explicit results in these more general settings, partly because of the laborious analytical calculations involved, but also due to numerical obstacles : the rapidly increasing dimension of the boundary crossing problem leads to analytical solutions that are expressed in terms of functions that are hard to compute with accuracy.

As far as one-sided boundaries are concerned, formulae have been published for the joint law of a sequence of maxima or minima of a one-dimensional GBM over several time intervals (Guillaume [START_REF] Guillaume | Analytical Valuation of Options on Joint Minima and Maxima[END_REF]), as well as for the joint law of the maxima or minima of a two-dimensional GBM over one time interval (Iyengar [19] ; He et al. [START_REF] He | Double Lookbacks[END_REF]). A formula for the joint law of the exit times of a onedimensional GBM from two successive two-sided boundaries is also known (Guillaume [START_REF] Guillaume | Step Double Barrier Options[END_REF]).

This paper focuses on a sequence of two one-sided straight boundaries conditional on two correlated GBMs, while the value of a third correlated GBM is taken into account at the endpoint of the time interval. The state space is thus three-dimensional. The choice of this particular distribution is motivated both by its usefulness in financial engineering applications and by the fact that it leads to tractable analytical solutions that can be computed with great accuracy and efficiency.

Section 2 presents the two main formulae of this paper. Section 3 deals with applications of these formulae and discusses their numerical implementation.

Main formulae

This section contains the two main formulae of the paper. Let

                                                                 2 1 1 1 , 2 s m a   2 2 2 2 2 s m a  
The next proposition provides an exact formula for the four above-mentioned cumulative distribution functions.

Proposition 1

Let   .,.,.; , , N q q q denote the joint trivariate cumulative distribution function of three standard normal random variables

1 2 3
, , X X X , where . a b q is the correlation coefficient between a X and b X ,

    , 1, 2, 3 a b  .
Then, for the up-and-up and the down-and-down distributions, we have : ,,,,,H H K K K t t (2.2)

    . 1 2 1 2 3 1 2 P ,
1 1 1 2 2 1 3 2 2 1 1 3 1.2 1.
                                             1 1 1 2 1 2 2 1 3 2 2 2 1.2 1 1 2 2 1 2 2 2 2 3 2 2 1 1 1.2 1.2 2 2 2 2 , , ; 2 exp , , k t t k t k h t t t t h N t t t t m m m m l r l l s s s s m s r r                                                                        1 1 1 1 2 2 1 1 3 2 2 1 1.2 1.2 1 1 2 1 1 1 2 2 1 2 1 1 3 2 1 1 1 1.2 1.2 2 2 2 2 2 , , ; 2 exp , , k h t k t h k t h t t t t t h N t t t t m m m l l r l r s s s s s m s r r                                                                       1 2 1.2 2 2 1 2 2 1 2 1 2 1 1 1 1 2 1 2 2 1 1 1.2 1.2 1 1 2 2 1 1 1 3 3 2 2 2 1 1 1 1.2 1.2 1.2 2 2 1 2 2 2 2 4 2 exp 2 2 2 , , 2 2 ;
                                                                                 
where

1 l  if         . 1 2 1 2 3 1 2 1 2 1 2 3
H H K K K t t H H K K K t t  1 l   if         . 1 2 1 2 3 1 2 1 2 1 2
3 1 2 down-and-down P , , , , , , P , , , , , ,

H H K K K t t H H K K K t t 
And, for the up-and-down and the down-and-up distributions, we have :

    . 1 2 1 2 3 1 2 P , , , , , , H H K K K t t (2.3) 1 1 1 2 2 1 3 2 2 1 1 3 1.2 1.2 1 1 2 1 2 2 2 2 , , ; , , k t k t k t t t N t t t t t m m m l l l r r s s s                                                  1 1 1 2 1 2 2 1 3 2 2 2 1.2 1 1 2 2 1 2 2 2 2 3 2 2 1 1 1.2 1.
                                                                         1 1 1 1 2 2 1 1 3 2 2 1 1.2 1.2 1 1 2 1 1 1 2 2 1 2 1 1 3 2 1 1 1 1.2 1.2 2 2 2 2 2 , , ; 2 exp , , k h t k t h k t h t t t t t h N t t t t m m m l l r l r s s s s s m s r r                                                                           1 2 1.2 2 2 1 2 2 1 2 1 2 1 1 1 1 2 1 2 2 1 1 1.2 1.2 1 1 2 2 1 1 1 3 3 2 2 2 1 1 1 1.2 1.2 1.2 2 2 1 2 2 2 2 4 2 exp 2 2 2 , , 2 2 ;
, ,

h h k h t t k t h t t t N k h t h t t t t t t m m r m s s s s m m m l r l r s s s s m l r r r s s                                                                                    
where

1 l  if         . 1 2 1 2 3 1 2 1 2 1 2 3
H H K K K t t H H K K K t t  1 l   if         . 1 2 1 2 3 1 2 1 2 1 2 3
H H K K K t t H H K K K t t   Corollary of Proposition 1
The four following joint cumulative distribution functions, that will be useful in Section 3, are deduced from Proposition 1 :

1)           1 1 2 1 1 1 1 1 2 1 2 2 2 2 2 3 0 sup , , , sup , t t t t t S t H S t K S t K S t H S t K                               1 2 1 2 1 2 1 2
3 up-and-up up-and-up P , , , , P , , , ,

H H K K H H K K K    2)           1 1 2 1 1 1 1 1 2 1 2 2 2 2 2 3 0 inf , , , inf , t t t t t S t H S t K S t K S t H S t K                             1 2 1 2 1 2 1
H H K K H H K K K    3)           1 1 2 1 1 1 1 1 2 1 2 2 2 2 2 3 0 inf , , , sup , t t t t t S t H S t K S t K S t H S t K                               1 2 1 2
H H K K H H K K K    4)           1 2 1 1 1 1 1 1 2 1 2 2 2 2 2 3 0 sup , , , inf , t t t t t S t H S t K S t K S t H S t K                               1 2 1 2 1 2 1 2
3 up-and-down up-and-down P , , , , P , , , ,

H H K K H H K K K     Proof of Proposition 1 is provided in the Appendix.
The numerical implementation of Proposition 1 and its corollary is easy using Genz's algorithm for the computation of trivariate normal cumulative distribution functions (Genz [START_REF] Genz | Numerical Computation of Rectangular Bivariate and Trivariate Normal and t Probabilities[END_REF]).

In the next Proposition, we introduce a third correlated geometric Brownian motion that will serve as the endpoint of the joint distribution and we show that this can still be analytically valued. Let

    3 , 0 S t
    1 3 1.3 , d B B t dt r  ,    
                1 1 2 1 2 1 2 3 4 1 2 3 up-and-up 1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4 0 1) P , , , , , , , , sup , , , sup ,
,

t t t t t H H K K K K t t t S t H S t K S t K S t H S t K S t K                                         1 1 2 1 2 1 2
                                      1 1 2 1 2 1 2 3 4 1 2 3 down-and-up 1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4 0 3) P , , , , , , , , inf , , , sup ,
,

t t t t t H H K K K K t t t S t H S t K S t K S t H S t K S t K                                         1 2 1 1 2 1 2 3 4 1 2 3 up-and-down 1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4 0 4) P , , , , , , , , sup , , , inf ,
,

t t t t t H H K K K K t t t S t H S t K S t K S t H S t K S t K                        
Let us introduce the following new notations :

  4 4 0 3 ln , K k S t              2 3 3 3 2 s m a  
The next proposition provides an exact formula for the four above-mentioned joint cumulative distribution functions.

Proposition 2

Let the real function  

1 2 3 4 1.2 1.3 1.4 2.3 3.4
, , , ; , , , , b b b b q q q q q  , where   , , , , q q q q q is included in

  1,1 
, be defined as follows :

 

1 2 3 4 1.2 1.3 1.4 2.3 3.4
, , , ; , , , , b b b b q q q q q  (2.6)

    1 2 3 1 2 3 2 2 2 1 1.2 2 3 2.3 2 2 2 2 3 2 1 3 2 2 1 3 2 3 1.3 1 4 1.4 1 3.4 1 2 1.3 3 2 1 4 1.3 1 exp 2 2 2 8 1 b b b x x x x x x x x x x b x N dx dx dx q q f f f f p q q q q f                                                                 where   . N
is the standard normal cumulative distribution function and the following definitions apply :

2 2 1 1.2 1 , f q   2 3 2 2.3 1 , f q   2 2 3.4 1.3 1.4 3.4 1 4 1.3 1.4 3.4 1 3 1 , 1 q q q q f q q f     
Then, for the up-and-up and the down-and-down distributions, we have : ,,,,,,,P ,,,,,,,,H ,,,,,,,P ,,,,,,,,H

    . 1 2
q r q r q r q q r                                                                          1 1 1 2 1 2 2 1 3 2 2 2 1.2 1 1 2 1 2 1 2 2 2 2 4 3 3 2 2 1 2 1 1.3 1.2 1.4 1.2 2 3 3 2 2 2 1 2 2 3.4 1.3 1.4 2 1.3 1 2 2 , , ,
2 2 2 2 exp k t t k t k h t t t t t h k t h t t t t t q q q q m m m m l r l l s s s s m m m m l q r q r s s s s s                                                                      1 1 1 1 2 1.2 1.2 1.3 1.2 1.4 1.3 2.3 3.4 2.3 2 3 2 3 ; , , , ,
t t t t t t t t t q r q r q r q q r                                                1 1 1 1 2 2 1 1 3 2 2 1 1.2 1.2 1 1 2 1 1 1 2 2 1 2 1 1 4 3 3 1 1 1 1.4 1.2 1.3 2 3 3 1 1 1 2 1 1 1 3.4 1.3 1.4 2 1.3 1 2 2 2 , , ,
2 2 2 2 exp k h t k t h k t h t t t t t h k t h h h t t t t q q q q m m m l l r l r s s s s s m m l q r q s s s s s                                                                   1 1 1 2 1.2 1.2 1.3 1.2 1.4 1.3 2.3 3.4 2.3 2 3 2 3 ; , , , , t t t t t t t t q r q r q r q q r                                                   1 2 1.2 2 2 1 2 2 1 2 1 2 1 1 1 1 2 1 2 2 1 1 3 2 2 2 1 1.2 1.2 1.2 1 1 2 1 2 1 1 1 2 2 1 2 4 3 3 2 1.4 1.2 3 3 2 4 2 exp 2 2 2 2 2 , , , 2
h h k h t t k t h k h t h t t t t t t k t t m m r m s s s s m m m m l r l r l r s s s s s s m m q r s l                                                                  1 1 2 1 1 2 1 2 1 1 1.2 1.3 1.2 2 2 1 2 2 1 1 1 1 1 2 1.2 1.2 1.3 1.2 1.4 1.3 2.3 3.4 2 2 3 2 3 3.4 1.3 1.4 2 1.3 1 2 ; 2 2 2 2 , , , ,
t h t h h t h t t t t t t t t t t t q q q q s s m r q r s s s s q r q r q r q q r                                                                       .3                                     where 1 l  if         . 1 2 1 2 3 4 1 2 3 1 2 1 2 3 4 1 2 3 up-and-up P ,
H K K K K t t t H H K K K K t t t  1 l   if         . 1 2 1 2 3 4 1 2 3 1 2 1 2 3 4 1 2 3 down-and-down P ,

H K K K K t t t H H K K K K t t t 

And, for the up-and-down and the down-and-up distributions, we have :

    . 1 2 1 2 3 4 1 2 3 P , , , , , , , , H H K K K K t t t (2.8) 1 1 1 2 2 1 3 2 2 4 3 3 1 1 2 1 2 2 3 3 1 1 1 2 1.2 1.2 1.3 1.2 1.4 1.3 2.3 3.4 2.3 2 3 2 3 , , , ; , , , ,
k t k t k t k t t t t t t t t t t t t t m m m m l l l l s s s s q r q r q r q q r                                                                              1 1 1 2 1 2 2 1 3 2 2 2 1.2 1 1 2 1 2 1 2 2 2 2 4 3 3 2 2 1 2 1 1.3 1.2 1.4 1.2 2 3 3 2 2 2 2 2 3.4 1.3 1.4 2 1.3 1 2 2 , , , 2 2 2 2 exp k t t k t k h t t t t t h k t h t t t t q q q q m m m m l r l l s s s s m m m m l q r q r s s s s s                                                                        1 1 1 2 1.2 1.2 1.3 1.2 1.4 1.3 2.3 3.4 2.3 2 3 2 3 ; , , , , t t t t t t t t q r q r q r q q r                                                 1 1 1 1 2 2 1 1 3 2 2 1 1.2 1.2 1 1 2 1 1 1 2 2 1 2 1 1 4 3 3 1 1 1 1.4 1.2 1.3 2 3 3 1 1 1 2 1 1 1 3.4 1.3 1.4 2 1.3 1 2 2 2 , , , 2 2 2 2 exp k h t k t h k t h t t t t t h k t h h h t t t t q q q q m m m l l r l r s s s s s m m l q r q s s s s s                                                                   1 1 1 2 1.2 1.2 1.3 1.2 1.4 1.3 2.3 3.4 2.3 2 3 2 3 ; , , , , t t t t t t t t q r q r q r q q r                                                       1 2 1.2 2 2 1 2 2 1 2 1 2 1 1 1 1 2 1 2 2 1 1 3 2 2 2 1 1.2 1.2 1.2 1 1 2 2 1 1 1 2 2 1 2 4 3 3 2 1.4 1.2 3 3 2 4 2 exp 2 2 2 2 2 , , , 2
h h k h t t k t h k h t h t t t t t k t t t m m r m s s s s m m m m l r l r l r s s s s s s m m q r s l                                                                   1 1 2 1 1 2 1 2 1 1 1.2 1.3 1.2 2 2 1 2 2 1 1 1 1 1 2 1.2 1.2 1
q q q q s s m r q r s s s s q r q r q r q q r                                                                         2.3                                    
where

1 l  if         . 1 2 1 2 3 4 1 2 3 1 2 1 2 3
4 1 2 3 down-and-up P , , , , , , , , P , , , , , , , ,

H H K K K K t t t H H K K K K t t t  1 l   if         . 1 2 1 2 3 4 1 2 3 1 2 1 2 3
4 1 2 3 up-and-down P , , , , , , , , P , , , , , , , ,

H H K K K K t t t H H K K K K t t t   Corollary of Proposition 2
The four following cumulative distribution functions, that will be useful in Section 3, are deduced from Proposition 2 :

1)             1 1 2 1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4 0 sup , , , sup , , t t t t t S t H S t K S t K S t H S t K S t K                                1 2 1 2 3
H H K K K H H K K K K    2)             1 1 2 1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4 0 inf , , , inf , , t t t t t S t H S t K S t K S t H S t K S t K                              1 2 1 2 3
H H K K K H H K K K K    3)             1 1 2 1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4 0 inf , , , sup , , t t t t t S t H S t K S t K S t H S t K S t K                                1 2 1 2 3
H H K K K H H K K K K    4)             1 2 1 1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4 0 sup , , , inf , , t t t t t S t H S t K S t K S t H S t K S t K                                1 2 1 2 3
H H K K K H H K K K K     Proof of Proposition 2 is provided in the Appendix.
Practical use of Proposition 2 and its corollary in engineering applications depends on the accuracy and the efficiency with which the function  can be numerically implemented. This question is dealt with in Section 3.

Applications and numerical implementation

This Section deals with applications of the results provided in Section 1 in financial engineering. Indeed, Proposition 1 and Proposition 2 can be used as the building blocks for the valuation and the risk management of a large number of option contracts. The main class of instruments under consideration will be barrier options. In their standard form, the latter are contracts whereby the holder is entitled (but not obligated) to buy (call option) or sell (put option) an asset at a prespecified future date (the option expiry) at a prespecified price (the strike price), on condition that the asset price has not (knock-out type) or has (knock-in type) crossed a specific upward level (called the up-barrier) or downward level (called the down-barrier) at any given time from valuation date to expiry. In standard option pricing theory, asset prices are modelled as geometric Brownian motions, so that boundary crossing distributions such as those provided in Section 1 are of immediate use in financial engineering.

It must be emphasized that the market for barrier options is huge. They are the most actively traded class of non-standard options (usually referred to as exotic options). Many variations on the standard barrier option payoff have been designed to match investors' demand more closely. One of them is the partial-time barrier option, which specifies that the barrier level is monitored during only a fraction of the option lifespan. The basic contracts were studied by Heynen and Kat [START_REF] Heynen | Partial Barrier Options[END_REF] and more general contracts were valued by Armstrong [START_REF] Armstrong | Valuation Formulae for Window Barrier Options[END_REF] and Guillaume [START_REF] Guillaume | Window Double Barrier Options[END_REF]. Another variation is the step barrier option (Guillaume [START_REF] Guillaume | Analytical Valuation of Options on Joint Minima and Maxima[END_REF], [START_REF] Guillaume | Step Double Barrier Options[END_REF]), whereby the barrier evolves as a step function of time. It is also common to encounter outside barrier options, whose payoff is a function of two asset prices : one of them is compared with the strike price at expiry to determine the moneyness of the option, while the other one is monitored up until expiry to check whether the barrier level has been crossed. Outside barrier options were originally valued by Heynen and Kat [START_REF] Heynen | Crossing Barriers[END_REF] and they were further studied by Kwok, Wu and Lyu [START_REF] Kwok | Pricing Multi-Asset Options with an External Barrier[END_REF] and Wong and Kwok [START_REF] Wong | Multi-Asset Barrier Options and Occupation Time Derivatives[END_REF].

These different features -partial time barrier, step barrier and outside barrier, often combine to allow for increased flexibility. But then, performing analytical valuation becomes more and more involved and practitioners have to turn to numerical methods that are slow and relatively inaccurate. This is when the results provided in Section 1 become valuable. Indeed, they enable to price a large number of barrier option contracts sharing the three above-mentioned innovative features (partial time, step and outside barrier), i.e. options based on two or even three assets that may knock-out depending on whether the underlying assets move within a sequence of prespecified ranges of prices over all or part of the option lifespan. More precisely, Proposition 1 and its corollary are the building blocks for the valuation of sequential two-asset knock-out calls and puts whose four payoff structures are defined as follows :

1)

        2 1 2 1 2 0 1 1 2 2 sup , sup t t t t t t t t S H S H K S                       
for a two-asset up-and-up knock-out put

(substitute     2 2 t S K   for     2 2 t K S  
for an up-and-up knock-out call)

2)

          2 1 2 1 2 0 1 1 2 2 inf , inf t t t t t t t t S H S H S K          
for a two-asset down-and-down knock-out call or put 3)

        2 1 2 1 2 0 1 2 1 2 sup , inf t t t t t t t t S H S H S K                        
for a two-asset up-and-down knock-out call or put 4)

        2 1 2 1 2 0 1 1 2 2 inf , sup t t t t t t t t S H S H K S                        
for a two-asset down-and-up knock-out put or call where :

-

   is the indicator function -     1 , 0 S t
              0 0 0 1 2 1 2 1 2 2

P , , , , , , H H H H K t t in Proposition 1

By using Corollary of Proposition 1, one can value up-and-up and down-and-up call options, as well as down-and-down and up-and-down put options.



A sketch of proof of Proposition 3 is provided in the Appendix.

Just as Proposition 1 enables to value two-asset knock-out calls and puts, Proposition 2 can be used to value three-asset knock-out calls and puts. More specifically, if

    3 , 0 S t
K S                        
for a three-asset up-and-up knock-out put or call 2)

          3 1 2 1 2 0 1 1 2 3 inf , inf t t t t t t t t S H S H S K          
for a three-asset down-and-down knock-out call or put 3)

        3 1 2 1 2 0 1 2 1 3 sup , inf t t t t t t t t S H S H S K                        
for a three-asset up-and-down knock-out call or put 4)

        3 1 2 1 2 0 1 1 2 3 inf , sup t t t t t t t t S H S H K S                        
for a three-asset down-and-up knock-out put or call

The no-arbitrage prices of the eight types of option under consideration are now provided by the following proposition.

Proposition 4

Let three geometric Brownian motions ,,,,,,,H H H H H K t t ,,,,,,,H

    1 , 0 S t t  ,     2 , 0 S t
                0 0 0 0 1 2 3 1 2 3 1 2 3 1 3 3 3

H H H H K t t t in Proposition 2

By using Corollary of Proposition 2, one can value up-and-up and down-and-up call options, as well as down-and-down and up-and-down put options.



A sketch of proof of Proposition 4 is provided in the Appendix.

It is easy to notice that Proposition 4 nests Proposition 3 under the following three conditions : S as in Proposition 3. This is achieved by implementing Proposition 4 with the following specifications :

1) 1 1 1 2 1.2 r u d s s r    and 2 2 2 2 r u d s    2) take     . 1 2
1)

2 1 1 1 r u d s    and 2 2 1 2 1.2 r u d s s r    2) 3 2 t t  , 3 1 s s  , 1.3 1 r  and 2.3 1.2 r r  3) compute         0 2 1 1 1 1 2 exp exp K rt P S t t P d    
Similarly, Proposition 4 can be used to value sequential two-asset knock-out options with expiry involves three-dimensional numerical integration. Given the fact that the integrand is very smooth, a classical Gauss-Legendre quadrature can be used, along with a simple adaptive rule increasing the number of points in case the required accuracy is not reached after one iteration, starting with an 8point quadrature. To test the efficiency and the accuracy of this numerical implementation, four series of tests have been implemented. In the first three categories of tests, the option parameters are specified so that the option can be valued by a known analytical formula involving numerical integration of lower dimension. More specifically, tests of category 1 reduce a three-asset up-and-up knock-out put to a two-asset up-and-up knock-out put by letting

4 K   in     1 2 1 2 3
4 1 2 3 up-and-up P , , , , , , , , H H K K K K t t t ; the benchmark is then Proposition 3, that involves the computation of trivariate normal cumulative distribution functions. The latter can be obtained with high precision by Genz's algorithm (Genz [7]), as mentioned earlier.

Tests of category 2 reduce the three-asset up-and-up knock-out put to an outside knock-out put by letting

1 1 K H  , 2 H   , 3 K   , 4 K   and 3 2 1 t t t   in     1 2 1 2 3
4 1 2 3 up-and-up P , , , , , , , , H H K K K K t t t ; the benchmark, then, is the formula for an outside barrier option as given by Heynen and Kat [START_REF] Heynen | Crossing Barriers[END_REF], that involves the computation of bivariate normal cumulative distribution functions. Again, the latter can be obtained with high precision using Genz [START_REF] Genz | Numerical Computation of Rectangular Bivariate and Trivariate Normal and t Probabilities[END_REF].

Tests of category 3 reduce the three-asset up-and-up knock-out put to a single asset up-and-out put by

letting 2 H   , 2 1 K K  , 3 K   , 4 K   , 3 2 1 t t t   , 2 1 s s  and 1.2 1 r  in     1 2 1 2 3
4 1 2 3 up-and-up P , , , , , , , , H H K K K K t t t ; the benchmark, then, is the formula for a basic barrier option as given, e.g., by Haug [START_REF] Haug | The Complete Guide to Option Pricing Formulas[END_REF], that involves the computation of univariate normal cumulative distribution functions.

In the last set of tests, the three-asset up-and-up knock-out put cannot be valued as a simpler contract, so that no reduction of the dimension of numerical integration is possible. Our goal, then, is to find a robust numerical valuation method and study its convergence pattern with the results obtained by implementing Proposition 4. Neither using trees nor numerically solving a partial differential equation stand as the most suitable methods in this case, in view of the double difficulty of having three correlated diffusion processes and path-dependent features in the option payoff. On another hand, crude Monte Carlo simulation would be very poor both in terms of accuracy and efficiency, due to the required time discretization. That is why the most suitable numerical method in this case is probably a Brownian Bridge Monte Carlo approximation (Glasserman [START_REF] Glasserman | Monte Carlo Methods in Financial Engineering[END_REF]), as it involves only 6 random draws of the asset prices for each simulation and avoids a discretization bias. This is indeed the method commonly used by market practitioners when they are faced with a pricing problem involving both multi-asset and path-dependent features.

For each category of tests, a total of 10,000 different option prices have been computed with randomly drawn volatility, correlation and expiry parameters. Let us now sum up the main findings of these numerical experiments. First, the average computational time when applying Proposition 4 on an ordinary personal computer (P7350 2GHz) is 0.6 seconds, which is fast. The longest recorded computational time was 1.4 second. Speed could obviously be greatly increased by involving more hardware resources. In absolute terms, Proposition 4 can thus be considered as an efficient way of computing option prices. In relative terms, Proposition 4 is even extremely efficient. Indeed, 10,000,000 simulations are required to obtain a modest 4 10  digits convergence of the Brownian Bridge Monte Carlo approximation with Proposition 4, which amounts to a computational time of over 4 minutes ! Such slowness becomes a serious issue in real time trading or when large portfolios of options are valued. As far as accuracy is concerned, the results obtained using Proposition 4 always matched to at least 8 10  digits the analytical benchmarks in almost all the numerical experiments conducted. The very rare exceptions were observed when the absolute value of one of the correlation coefficients was above 99.7%. It must be noticed that in this couple of cases, the correlation matrix was found to be almost singular. Besides, risky assets with such levels of correlation are never encountered in the markets, so that it does not seem worthwhile to us to search for more complicated integration rules that would be better suited to these negligible subregions of integration. For all practical purposes, the numerical accuracy of our simple implementation of Proposition 4 can thus be deemed as very satisfactory. The results did not vary significantly between tests of categories 1, 2 and 3, which means that an increase in the number of option parameters that are given limit values does not entail instability in the quadrature rule.

To close this section, it is worth mentioning that the results provided in Section 1 can also apply to other popular derivatives contracts. Among them is the lookback option, which allows investors to sell at the highest and buy at the lowest over a time interval, thus optimizing market exit and market entry on a given asset. In its basic form, a fixed strike lookback call pays off

  0 sup t T S t K               at expiry
T . Many variations on this standard structure have been designed. One of them is the double lookback call, written on two assets 1 S and 2 S , which provides investors with the following payoff at expiry T :

    1 1 2 2 0 0 sup sup t T t T a S t a S t K                  ,
    1 1 2 1 1 2 2 0 sup sup t t t t t a S t a S t K                   
. Drawing on Proposition 1, the no-arbitrage price of this option is equal to : One can notice that the valuation of sequential double lookback options is only semi-analytical since it still requires numerical double integration. For a given degree of the quadrature rule, it will not be as accurate as the valuation of two-asset and three-asset sequential knock-out options because of the inaccuracies arising from the numerical differentiation of Proposition 1.

            1 2 0 0 2 1 1 1 2 2 2 0 exp exp exp h h rt a S t h a S t h K               2 1

Conclusion

This paper provides new explicit solutions for some sequential boundary crossing problems in a multidimensional geometric Brownian motion framework. Among their various possible applications, they form the basis for accurate and fast valuation of a large number of option contracts traded in the markets. The methods presented here could be used to solve higher-dimensional problems, especially the introduction of a third boundary monitored with respect to the third process in the model.

However, the resulting formulae would become quite cumbersome and, more importantly, the quality of the numerical integration involved should have to be compared with a Brownian Bridge Monte Carlo approximation. Another possible extension would be to study the analytical tractability of introducing curved boundaries, but it should be pointed out that this form of boundary is rarely encountered in the options markets.

Appendix A. Proof of Proposition 1

Proof is outlined for

    1 2 1 2 3
          1 1 2 1 1 1 1 1 2 1 2 2 2 2 2 3 0 sup , , , sup , t t t t t X t h X t k X t k X t h X t k                       (A.1)
where :

            1 2 1 2 0 0 1 2 ln , ln S t S t X t X t S t S t                           For a fixed 0 t  ,   1 X t and
          1 2 3 1 1 2 3 1 2 1 1 1 1 1 2 1 2 0 3 2 1 2 2 2 2 3 sup , , , sup , k k k t t x x x t t t X t h X t dx X t dx dx dx dx X t h X t dx                                     
By conditioning and using the Markov property of processes

      1 2 3 1 2 3 1 1 2 1 2 3 2 3 3 2 1
, ,

k k k x x x P x P x x P x x dx dx dx        (A.3)
where :

      1 1 1 1 1 1 1 1 0 sup , t t P x X t h X t dx                           2 1 2 2 1 2 1 1 1 , P x x X t dx X t dx             1 2 3 2 3 2 2 2 2 3 2 1 2
, sup ,

t t t P x x X t h X t dx X t dx                   
The functions   , P x x , the following formula is used (Guillaume [START_REF] Guillaume | Analytical Valuation of Options on Joint Minima and Maxima[END_REF]) :

      1 2 2 1 2 2 2 2 2 3
, sup ,

t t t X t x X t h X t x                   (A.4) 2
                                   
where   2 .,.; N r is the bivariate standard normal cumulative distribution function with correlation coefficient r . Differentiating (A.4) twice with respect to variables 2 x and 3

x , and applying the definition of conditional probability yields :

            1 exp 2 2 , 2 2
p                                                                                                                         (A.5)
Then, the most cumbersome part of the proof consists in performing the necessary calculations to solve (A.3) in closed form and find the sum of trivariate standard normal cumulative distribution functions given by Proposition 1.

B. Proof of Proposition 2

Proof is outlined for

q the correlation coefficient between a X and b X ,     , 1, 2, 3 a b  . Then, the joint density function of   1 2 3
, , X X X is given by : , , X X X , one also finds that 3 X admits the following representation :

  1 1 2 2 3 3 , , X dx X dx X dx     (B.1)       2 2 1 3 2 1 2 1.2 1 2 2 1.2 1 3 1.3 1 2.3 1 2 1 3 1.2 2 2 1 2 1 1.2 2 3 1 1 exp 2 2 2 2 x x x x x x x q q q q p f f f f f                                             where 2 2 1 1.2 1 f q   is the standard deviation of 2 X conditional on 1 X , and 2.3 1.2 1.3 2.3 1 2 1 q q q q f   is the partial correlation between 2 X and 3 X conditional on 1 X , and 2
2 3 1.2 1.3 2.3 1 1 f q q    is the standard deviation of 3 X conditional on 1 X and on 2 X .  Proof of Lemma 1 is simple. It is known that 2 X is equal in law to 1.2 1 2 1 2 X Y q f  ,
2 1.2 1 3 1.3 1 2.3 1 2 3 1.2 3 3 1.3 1 2.3 1 3 1.2 3 1 2 X X X X Y Y X X Y q q q f q q f f         (B.2) where 3 Y is a standard normal random variable uncorrelated with 1 X and 2 Y Thus, given 1 X and 2 X , 3 X is normally distributed with mean 2 1.2 1 1.3 1 2.3 1 1 2 X X X q q q f   and standard deviation 3 1.2 f
. Hence, the density function of Next, we proceed to outline proof of Proposition 2. By taking the logarithms of the asset prices at the times 1 t , 2 t , 3 t , and integrating with respect to these variables, the sought cumulative distribution function becomes : (B.4)

    2 1 2 1.2 1 3 3 1 1 2 2 3 1.3 1 2.3 1 3 1.2 2 1 1.2 2 3 1 , exp 2 2 x x X dx X dx X dx x x q q q f p f f                                              (B.
            1 1 2 1 2 3 4 1 2 3 4 1 1 1 1 1 2 1 2 0 4 3 2 1 2 2 2 2 3 3 3 4 sup , , , sup , , t t t t t k k k k x x x x X t h X t dx X t dx dx dx dx dx X t h X t dx X t dx                                         where       3 3 0 3 ln S t X t S t             
and the other terms are defined in the proof of Proposition 1. Then, by conditioning and using the Markov property of the processes , , a a a is a sum of three uncorrelated normal random variables, hence it is normally distributed.

      1 1 1 1 1 0 sup , t t P x X t h X t dx                           2 1 2 2 1 2 1 1 1 , P x x X t dx X t dx             1
Therefore, using Frechet's characterization of multivariate normality (Frechet [START_REF] Fréchet | Généralisations de la Loi de Probabilité de Laplace[END_REF]), the joint density function   x x x x t t t j q j j q j q f f s s s p f f

                                      
where the notations are defined as follows :

      q q q r f q q f q q f        

Then, the most cumbersome part of the proof consists in performing the necessary calculations to solve (B.5) in closed form and find the sum of functions given by Proposition 2.

C. Sketch of proof of Proposition 3 and Proposition 4

Once endowed with Proposition 1, proof of Proposition 3 is straightforward, as an application of option pricing theory (Harrison and Pliska [START_REF] Harrison | Martingales and Stochastic Integrals in the Theory of Continuous Trading[END_REF]) and Girsanov's theorem in two dimensions. The substitutions

                       
are independent standard Brownian motions under the new measure.

  overall, this formula alone suffices to compute 40 different types of sequential multi-asset knock-out options. Proposition 4 implies the numerical implementation of Proposition 2. The function  in Proposition 2

  2 X

 2 t are absolutely continuous random variables, hence one can express (A.1) as the following triple integral : (A.2)

3 X

 3 be three standard normal random variables defined on a probability space with measure  . Denote by . a b

3 X conditional on 1 X and 2 X

 312 writes :

  [START_REF] Borodin | Handbook of Brownian Motion -Facts and Formulae[END_REF] which suffices to prove Lemma 1.

  t  andThe no-arbitrage price of these eight different option contracts is now provided by the following proposition.

	1.2 r . Let 1 d and 2 d be two constant payout rates on assets 1 S and 2 S respectively, and r be the
	constant risk-free interest rate. Let 1 u	1 d    r	1 2 1.2 s s r	and	2 u	2 2 s    . 2 r d
	The value, at time	0 t  , of a two-asset sequential knock-out option with strike price K , with 0
	knock-out barriers 1 H and 2 H monitored w.r.t. 1 S over the time interval 	 t t and 2 0 1 , S over the time
	interval 	 t t respectively, and with maturity 1 2 ,	2 t ,	2 t	0   , is given by : 1 t t
				  S t t  2 ,	0		are the price processes of two risky assets
	-K is the strike price and	2 t is the option expiry
	-1 H is a knock-out barrier monitored w.r.t.		  S t t  1 ,	0		over the time interval 	0 1 , t t	
	-2 H is a knock-out barrier monitored w.r.t.		  S t t  2 ,	0		over the time interval 	1 2 , t t	
	Proposition 3
	Let two geometric Brownian motions		  S t t  1 ,	0		and		  S t t  2 ,	0		be the price processes of
	two risky assets with constant volatilities 1 s and 2 s respectively, and constant correlation coefficient

  t  and

											  S t t  3 ,	0		be the
	price processes of three risky assets with constant volatilities 1 s , 2 s and 3 s respectively, and
	constant pairwise correlation coefficients . a b r ,    , a b 	1, 2, 3		, at any fixed time	0 t  . Let 1 d , 2 d
	and 3 d be three constant payout rates on assets 1 S , 2 S and 3 S respectively, and r be the constant
	risk-free interest rate. Let 1 u	1 d    r	1 3 1.3 s s r	, 2 u	2 d    r	2 3 2.3 s s r	and	3 u	2 3    . 3 r d s
	The value, at time 0 t  , of a three-asset sequential knock-out option with strike price K , with 0
	knock-out barriers 1 H and 2 H monitored w.r.t. 1 S over the time interval 	 t t and 2 0 1 , S over the time
	interval 	 t t respectively, with maturity 1 2 ,	3 t ,	t	3	0    , is given by : 2 1 t t t	(3.2)

  where 1 a and 2 a are two positive weights. The valuation of this contract has been studied by He et al.[START_REF] He | Double Lookbacks[END_REF]. A sequential double lookback call is a

	variation	on	the	double	lookback	call.	Its	payoff	at	expiry	2 t writes	:

  where

				2 Y is a
	standard normal random variable uncorrelated with	1 X . By applying Cholesky's decomposition to the
	correlation matrix associated with 	2	3

  derive from the introduction of a new measure, which is equivalent to the risk-neutral measure, under which the market numeraire is asset 2 S . If we Similarly, once endowed with Proposition 2, proof of Proposition 4 is straightforward, as an application of option pricing theory and Girsanov's theorem in three dimensions. The substitutions

	1 a in Proposition 1 denote by 1 u  and 2 2 a u      1 , 0 B t t  and     2 , 0 B t t  the two correlated Brownian motions driving
		  S t t  1 ,		0			and			  S t t  2 ,	0		under the risk-neutral measure, then the processes
		  B t 1		2 1.2 , t t s r			0		and		  B t 2		2 s	1		2 1.2 r	, t t		0		are independent standard Brownian
	motions under the new measure.
	1 a	1 u  , 2 a		2 u		and 3 a		3 u	in Proposition 2 derive from the introduction of a new measure,
	which is equivalent to the risk-neutral measure, under which the market numeraire is asset 3 S . If we
	denote by			  B t t  ,  1 , 0		  B t t  2 ,	0		and		  B t t  3 ,	0		the three correlated Brownian
	motions driving			  S t t  1 ,	0		,		  S t t  2 ,	0		and		  S t t  3 ,	0		under the risk-neutral
	measure, then the three following processes :
		  B t 1		3 1.3 , t t s r		0	
	      	  B t 2		s	3	2.3 r	1.2 1.3 2 1.2 r r r   1	, t t		0	      
			  B t 3			s	3 t	1			2 1.3 r					2.3 r	1	1.2 1.3 2 1.2 r r r	 2	,	t	0


Tristan Guillaume
email: tristan.guillaume@u-cergy.fr

A Few Insights Into Cliquet Options

Keywords:

This paper deals with a subset of lookback options known as cliquet options.

The latter lock in the best underlying asset price over a number of prespecified dates during the option life. The specific uses of these contracts are analyzed, as well as two different hedging techniques. Closed form valuation formulae are provided under standard assumptions. They are easy to implement, very efficient and accurate compared to Monte Carlo simulation approximations.

Introduction

The term « cliquet option » is ambiguous. It may refer either to a subset of the lookback options characterized by a small number of fixing dates that are not necessarily uniformly spaced, or to portfolios of forward start options. The latter are also known as reset strike options or ratchet options, to avoid confusion. This paper deals with the former lookbacktype contract, that does not raise the same pricing and hedging issues as reset strike options. Cliquet options, in this sense, are heavily traded in the markets, especially as building blocks for many structured products. A lot of capital-guaranteed investments provide a participation in an index based on a percentage of the best recorded index value over a number of contractually prespecified dates. There are also a lot of equity-linked bonds whose rate of interest is based on a similar formula.

Generally speaking, cliquet options are appealing to investors because they inherit, at least partially, the very attractive payoff of lookback options, while rendering it both more affordable and more flexible, thanks to the decrease in the updating frequency of the running extremum of the underlying asset price, as well as to the possible partial and non uniform spanning of the option life. They also have many advantages compared to other options that allow their holders to « lock in » positive intrinsic values prior to expiry, such as ladder or shout options. In particular, unlike ladder options, they do not cap the greatest possible intrinsic value from the start, and they do not require their holders to have precise anticipations on a set of target price increases in the underlying asset.

However, cliquet options raise a number of pricing and hedging issues. Dynamic hedging is costly because of sharp delta increases during the option life and oscillations of gamma between fixing dates. Pure static hedging is not possible. A semi-static strategy can replicate the option payoff but it is difficult to translate it into a reliable fair price for the option because it is heavily exposed to volatility risk. Finally, it is not easy to come up with a convenient analytical solution of the pricing problem because of quickly increasing dimension.

This paper attempts to address these various issues. In Section 2, the specific properties of cliquet options are highlighted, in comparison with alternative contracts traded in the markets such as lookback, ladder and shout options. Then, the difficulties associated with dynamic hedging are briefly analyzed, leading to the presentation of a semi-static rollover strategy to replicate the cliquet option payoff. In Section 3, closed form valuation formulae are provided, first in dimension 3 and 4, then for any number of fixing dates.

The numerical implementation of these formulae is dealt with in the remainder of Section 3, while Section 4 provides a proof of some of the analytical results in Section 3.

Specific properties of cliquet options

The idea of locking-in positive intrinsic values prior to the option's expiry is embedded in a number of European-type financial options. The type of contract that provides this feature to its fullest extent is the lookback option, whose payoff at expiry T writes (for a call with fixed strike) :

  0 sup t T S t K               (2.1)
where  , 0, S t t  is the underlying asset and K is the strike price Instead of taking into account only the terminal asset value, all the prices quoted from time 0 to time T are factored in when computing the option payoff. The running maximum of   S t is continuously and automatically updated. Taking

  0 K S 
, that is, defining the strike price as the value of the underlying at the contract's inception, this contract guarantees its holder that he or she will sell the underlying at the highest in the interval   0,T . Buying at the lowest and selling at the highest, these are the ultimate goals of equity investors. The difficult problem of market timing, whether it be market entry or market exit, is thus automatically solved in an optimal manner by such contracts, that were originally valued by [START_REF] Conze | Path dependent options : the case of lookback options[END_REF].

The hi-lo option takes this idea even further, as it automatically captures the greatest variability of the underlying asset over the option life, yielding at expiry the following difference :

    0 0 sup inf t T t T S t S t                (2.2)
The main flaw of these two types of contracts is quite obvious : they are very expensive, so that option holders are rather unlikely to reach the breakeven point and thus receive a positive return from their investment. One simple way to lower the cost of lookback options consists in defining a discrete updating frequency of the running maximum or minimum. By the way, there can never be truly continuous updating in practice, even for the most liquid traded stocks, because of the minimum time gap between two quotations in the market, which can be very short but never arbitrarily short. It is clear that decreasing the updating frequency to, say, one day (closing market price) or, better, one week, will result in a lower expected value of the option and hence a lower premium.

Such contracts are called discretely monitored lookback options. They preserve the global lookback property since there are still many equally spaced fixing dates, but not to its full extent.

Cliquet options are essentially a form of discretely monitored lookback options, albeit with three differences in practice :

 there are usually few fixing dates  the lookback period may not span the entire option life

          1 2 max , ,..., n S t S t S t K   (2.
3)

It must be pointed out that, in the markets, the term « cliquet », when applied to a call option, is sometimes used interchangeably to refer to a ratchet option, whereby the strike price resets at each fixing date to the prevailing stock price while at the same time locking in the performance of the previous period, thus yielding at expiry the following payoff :

    1 0 max , 0 n i i i S t S t         (2.4)
Compared to payoff (2.3), payoff (2.4) has no fixed strike and adds every positive contribution up until expiry. Ratchet options are simply a portfolio of forward starting options, hence they admit perfect static replication, unlike cliquet options [START_REF] Buetow | Ratchet Options[END_REF][START_REF] Matosek | Hedging cliquet options[END_REF].

There are no restrictions on the location of the fixing dates over   0,T in a cliquet option contract. In particular, they may all be concentrated in a specific subinterval of   0,T and leave large parts of   0,T non monitored. This flexibility is valuable to investors because it enables them to tailor the option contract to their particular anticipations or hedging requirements, while it substantially reduces the option premium by waiving the lookback property when it is not needed.

The specificity of cliquet options is better grasped through a short analysis of their differences with alternative contracts that also embed a lock-in mechanism. For instance, ladder options are contracts whereby the set of the underlying asset price levels that can be locked in before expiry is prespecified at time zero -they are called the « rungs ».

Denoting by L the highest level or rung reached over the option life, a ladder call then provides its holder with the following payoff :

    max , , 0 S T K L K   (2.5)
Ladder options can be valued as combinations of barrier options and vanilla options [START_REF] De Weert | Exotic Options Trading[END_REF].

The rungs may be attained at any time prior to T , but all prices other than the rungs and the terminal value of S will not be taken into account when computing the option payoff.

In other words, the difference with a cliquet option is two-fold : there are no fixing dates and the maximum level that can be locked in is bounded. One advantage, as far as riskaverse investors are concerned, is that if they set the first rung of the ladder close enough to the spot price at the contract's inception, they are very likely to receive a minimum positive payoff, due to the high probability that the first rung will be attained. However, by definition, this minimum positive payoff is low. You need to add more and more rungs if you want the expected payoff to grow. But then, the premium of the ladder option gradually tends to that of a lookback option. If you think that the actual realized volatility of the underlying will be higher than the implied volatility priced by the market, then you might want to elevate the first rung in your ladder option to lower the premium but it is clearly risky to do so.

In contrast to a ladder option, a cliquet option does not require its holder to have precise anticipations on a set of target price increases in the underlying asset, which is attractive in uncertain markets or when you do not have enough expertise. Also, cliquet options do not involve capping the greatest possible intrinsic value from the start. Such a capping mechanism obviously does not allow to fully exploit market movements. Moreover, unlike ladder-type contracts, cliquet options enable their holders to take advantage of specific time-related views or constraints. For instance, if there is a strong case that the underlying asset price will probably reach its maximum during a given subinterval of   0,T , because of scheduled events such a public announcement liable to provide impetus to the stock price of a company or elections likely to have a major impact on a currency, then the investor had better locate most of the fixing dates in this specific time area. Indeed, the shorter the lock-in period, the larger the drop in the option premium.

Ultimately, you might even be able to generate a very high payoff with a single fixing date prior to expiry, which makes the option very affordable.

If you are uncertain about both the magnitude and the timing of the underlying asset price increases, an alternative strategy is to turn to a shout option. The latter allows you to lock in the intrinsic value of the option at any time prior to expiry -an action referred to as a « shout », thus providing the following payoff at maturity (for a call):

      max , , 0 S T K S t K    (2.6)
where t  is the time chosen to « shout » A shout option is valued much the same way as an American option [START_REF] Windcliff | Shout options: a framework for pricing contracts which can be modified by the investor[END_REF]. This contract is well suited to investors who wish to play a more active role during the option life and who are not satisfied with the fact that a contract would prespecify either the possible lock-in price levels (ladder option) or the times at which the underlying asset price can be locked in (cliquet option). Compared to a ladder option, one advantage is that you are not stuck at a rung : if the underlying asset price reaches the otherwise highest prespecified rung and keeps on rising, you can wait before you shout in order to lock in even higher levels. Also, you can lock in any positive intrinsic value before expiry, including all those below the first rung or between two different rungs of a ladder option. Relative to a cliquet option, the holder of a shout option is free to wait and see how the market moves before deciding when to lock in an underlying asset price. On the downside, this generates uncertainty as you never know when it is optimal to shout, so that you are faced with the possibility of subsequent regrets, whereas a cliquet option provides as many possibilities of locking in asset price levels as there are fixing dates.

Nothing theoretically rules out allowing for several « shouts » during the option life but then the option quickly becomes more expensive than a cliquet option. Also, shout options require steady monitoring and a lot of investors do not want to dedicate too much of their time to manage their option position. Lastly, from the point of view of derivatives sellers, shout options are difficult to hedge because of the unpredictability of the exercice time. For all these reasons, cliquet options are more popular than shout options in the markets.

As far as hedging is concerned, there are differences between the dynamics of the delta of a cliquet call and those of a vanilla call. In general, unless the option is then deeply out-of-the-money, the delta of a cliquet call will significantly increase in the vicinity of a fixing date as a consequence of the lock-in mechanism. Before the first fixing date 1 t , this phenomenon is observed especially when there is little time left before 1 t and when the underlying asset is then trading near the strike. If it is then trading far below the strike, though, the delta of the cliquet call will be quite similar to that of a vanilla call.

If   1

S t K

 , the holder of the cliquet call option is sure to receive at least   1 S t K  at expiry. This is why the value of a cliquet call is less sensitive than that of a vanilla call to a decline in the value of S below the level   1 S t after time 1 t . It will remain sensitive to such in a decline in the value of S , still, as the latter has a negative impact on the probability that S will reach levels above   1 S t at fixing dates subsequent to 1 t . But at least, unlike a vanilla call, a decline in the value of S below the level   1 S t after time 1 t has no negative impact on the probability that the option will end in-the-money at expiry.

In the vicinity of the second fixing date, 2 t , the magnitude of the delta increase will depend on the current and past levels of the underlying. If the difference   The same analysis carries over to any two consecutive fixing dates. Overall, there may be several areas of sharp delta fluctuations during the option life. These fluctuations will be particularly pronounced when the time is near the last fixing date and the underlying asset is trading near the latest locked-in maximum value. Thus, a delta hedging strategy may turn out to be very costly on a long maturity option, as high transaction costs are incurred due to the frequent rebalancing of the hedging portfolio. This issue is compounded when the option is written on an asset with limited liquidity or jumpy returns, as explained by [START_REF] Petrelli | Optimal dynamic hedging of cliquets[END_REF].

This is why, in practice, the sale of a cliquet option may be hedged by implementing a semi-static rollover strategy. Assuming there are three only fixing dates, 1 t , 2 t , 3 t and the option expiry is 3 t , the semi-static hedge of the sale of a cliquet call would proceed as follows :

-at time zero, buy a vanilla call struck at K and expiring at 1 t Thus, the associated skew should be considered as an at-the-money forward skew, while traders blindly pricing these options off the spot volatility surface would use a volatility estimate that would be too low. Finally, when  

-at time 1 t , if   1 S t K  then buy a new call struck at   1 S t
1 S t K  and   2 S t K 
, and when

  1 S t K  and   2 S t K 
, the variation in implied volatility caused by the skew varies according to the relative levels of  

1 S t
and   2 S t as well as on the relative distance of 1 t and 2 t to the option expiry; as the skew is steeper for short term maturities and the skew curve tends to flatten or even have an element of a smile shape for very high strikes, the net effect on implied volatility is likely to be strongly upward when

  1 S t K  and   2 S t K  .
Thus, the skew exposure is quite difficult to manage ; for more material on this important practical issue, the reader is referred to [START_REF] Gatheral | The Volatility Surface[END_REF], who shows that pricing this kind of structure with a local volatility model tends to underestimate the real value, relative to a stochastic volatility model, as it produces forward skews that are too flat.

There are also issues associated with the risk management of sensitivities other than delta and gamma, such as the vega of a cliquet option. The latter is typically higher than the vega of a vanilla option. Indeed, the larger the volatility of the underlying, the greater the probability to reach a new extremum on each fixing date. Moreover, as each new extremum is locked in, the option value is not exposed to the higher chance of adverse price movements. Another difference with the vega of a vanilla option is that the vega of a cliquet option is at its peak when the underlying is trading near the latest extremum that was locked in at one of the previous fixing dates, not when it is trading near the strike price. However, one must bear in mind that the vega of a cliquet option may be a misleading measure of volatility risk because this is a contract whose gamma changes sign and, as a result, vega may be small at precisely those places where sensitivity to actual volatility is very large. This issue is discussed by [START_REF] Wilmott | Cliquet options and volatility models[END_REF], who analyzes the implications of the fact that the point at which gamma changes sign depends on the relative move in S from one fixing to the next.

Valuation formulae and numerical implementation

As pointed out in the previous section, cliquet options may be hedged by a semi-static rollover strategy, but the latter is difficult to value at the time the option is sold.

Therefore, it would be quite convenient to be able to refer to an easily computed noarbitrage price formula, at least as an analytical benchmark to test more general numerical schemes. Unfortunately, cliquet options are not easy to price by analytical methods because of the rapidly increasing dimension of the valuation problem. To put it simply, the more fixing dates, the higher the dimension, as measured by the required order of numerical integration. The main problem associated with increasing dimension is that analytical results rely on functions that are defined as multiple integrals that may not be easily computed or that may not even be known. As a consequence of this dimension issue, the only closed form solution that has been published so far is a formula for a cliquet option with a single fixing date prior to expiry [START_REF] Gray | Reset put options : valuation, risk characteristics and application[END_REF]) in a standard Black-Scholes framework. Other contributions focus on numerical methods, via either Monte Carlo simulation or a partial differential equation solver [START_REF] Windcliff | Numerical methods and volatility models for valuing cliquet options[END_REF]. It must be pointed out that the classical continuity correction of [START_REF] Broadie | Connecting discrete and continuous pathdependent options[END_REF] used to provide an analytical approximation of the price of discretely monitored lookback options is unfortunately quite inaccurate when there are only few fixing dates, especially when the latter are not uniformly spaced over the option life, and it is therefore not suited to cliquet options.

The following Proposition 1 provides a closed form solution for the no-arbitrage value, V , of a cliquet option with expiry 3 t , three fixing dates, 1 t , 2 t , and 3 t , assuming that the underlying asset S is driven by a geometric Brownian motion with constant volatility s , payout rate d and risk-free interest rate r .

The following expressions will be used, with

  4 , , , , , i j m n i j m n     :  1 l  if the option is a call, 1 l   if the option is a put      0 ln / k K S t           2 2 / 2 , / 2 i i i i r t r t m s m s                2 2 , , / 2 , / 2 j i j i i j i j r t t r t t m s m s            , , i i j i i j t t t s s s s           , , , 1 , , ;
,

j i i i n m j j t t t t i j i j i j m n t t t t b r r      
Also, the function

 

1 2 1 2 1 , ,..., ; , ,...,  is given by :

n n n a a a q q q   ,   , 1,1 , i i a i q       ,
        1 2 1 2 2 1 2 1 1 2 1 1 2 1 / 2 2 1 exp 2 2 1 ... ... 2 1 n n n i i i i i n n n n i i x x x a a a x x x dx dx dx q q p q                                    (3.1)
Proposition 1 can now be stated.

Proposition 1       0 1 2 3 , , , , , , , V S t r K t t t s d (3.2)           1 0 3 1 1 1 1 exp k r t t t S t m l d l s                              1,2 1,3 2 1,2;1,3 1,2 1,3 , ; m m l l r s s                              1,2 1,3 1 3 1 2 1,2;1,3 1 1,2 1,3 exp , ; k rt K m m m l l l l r s s s                                          2,3 0 3 2 2 1 2,3 exp r t t t S t m l d l s                                    1,2 1 2 1 2 1 1 1,2 1 2 1 1,2 , ; k k k m m m m l l b l l s s s s                                                            2,3 3 1 2,3 exp rt K m l l s                                   1,2 1 2 1 2 1 1 1,2 1 2 1 1,2 , ; k k k m m m m l l b l l s s s s                                                                          1,3 2,3 3 0 3 3 1;3 2,3;1,3 3 1,3 2,3 exp , , ; , k t S t m m m l d l l l r r s s s                                      1,3 2,3 3 3 3 1;3 2,3;1,3 3 1,3 2,3 exp , , ; , k rt K m m m l l l l r r s s s                     A proof of
      0 4 1 1 exp r t t t S t l d                           1,2 1,3 1,4 1 1 3 1,2;1,3 1,3;1,4 1 1,2 1,3 1,4 , , ; , k m m m m l l l l r r s s s s                                            4 exp rt K l                         1,2 1,3 1,4 1 1 3 1,2;1,3 1,3;1,4 1 1,2 1,3 1,4 , , ; , k m m m m l l l l r r s s s s                                                0 4 2 2 exp r t t t S t l d                         1,2 1 2 1 2 1 1 1,2 1 2 1 1,2 , ; k k k m m m m l l b l l s s s s                                                                2,3 2,4 2 2,3;2,4 2,3 2,4 , ; m m l l r s s                 4 exp rt K l                       1,2 1 2 1 2 1 1 1,2 1 2 1 1,2 , ; k k k m m m m l l b l l s s s s                                                                2,3 2,4 2 2,3;2,4 2,3 2,4 , ; m m l l r s s                     0 4 3 3 exp r t t t S t l d                           1,3 2,3 3,4 3 3 1 1;3 2,3;1,3 3 1,3 2,3 3,4 , , ; , k m m m m l l l r r l s s s s                            4 exp rt K l                         1,3 2,3 3,4 3 3 1 1;3 2,3;1,3 3 1,3 2,3 3,4 , , ; , k m m m m l l l r r l s s s s                              0 4 exp t S t l d                           1,
                 4 exp rt K l                           1,
                 Proof of
Proposition 2 is similar to that of Proposition 1 and is therefore omitted.

Finally, for a number of fixing dates greater than 4, the previous results can be generalized to produce the more compact Proposition 3.

Proposition 3       0 1, 2 , , , , , ,..., n V S t r K t t t s d (3.4)                       1 0 1 1 1 exp exp exp n n n n i i i i i n n n i i i S t r t t t t K rt d d l l                                                                        
where :

                          1, 2, 1, 1, 2,
i i i i i i i i i i i i i i i i i i i i i k m m m m l l l l s s s s r r r r                                                      1, 2, 1, 1, 2,
i i i i i i i i i i i i i i i i i i i i i k m m m m l l l l s s s s r r r r                                                , 1 , 2 , , 1 , 2 ,
, 1; , 2 , 2; , 3 , 1; , , ,..., ; , ,...,

i i i i i n i i i i i n i n i i i i i i i i i i n i n m m m l l l s s s r r r                               
 Although Proposition 3 nests Proposition 1 and Proposition 2, its terse formulation may render it relatively unclear and ambiguous without the help of the fully expanded Proposition 1 and Proposition 2, and thus it was useful to state the latter. Moreover, Proposition 2 and Proposition 3 include dimension reductions that do not appear in the more compact Proposition 3, and they are therefore more efficient to use when pricing cliquet options with three or four fixing dates.

From a numerical point of view, the valuation formulae presented in this section raise the question of the computation of the function  . As the number of fixing dates increases, so does the dimension of numerical integration. Fortunately, unlike lookback options, most cliquet options traded in the markets have few fixing dates, as mentioned earlier.

Moreover, the dimension of numerical integration can be reduced by using the following identities :

 when there are three fixing dates, the actual numerical dimension of the function 3  can be brought down from 3 to 1 by using :

    2 2 2 1 1 2 3 2 2 3 1 2 3 1 2 2 2 2 1 2 2 exp / 2 , , ; , 2 1 1 b x x b x b x b b b N N dx q q q q p q q                              (3.5)
 when there are four fixing dates, the actual numerical dimension of the function 4  can be reduced by a factor of two by using :

    2 3 2 2 2 2 2 2 3 4 1 2 3 4 1 2 3 2 1 1 2 4 3 2 3 3 2 2 2 3 2 2 1 3 3 2 2 1 exp 2 , , , ; , , 2 1 1 1 b x b x x x x b b b b b x b x x N N dx dx q q
q q q p q q q q q q q

                                                 (3.6)
 when there are five fixing dates, the actual numerical dimension of the function 5  can be brought down from 5 to 3 by using :

  5 5 1 2 3 4 1 2 3 4 , , , , ; , , , b b b b b q q q q      2 3 4 2 2 4 3 2 3 3 2 2 2 2 2 3 2 2 2 2 3 4 3 / 2 2 2 5 1 1 2 4 3 4 4 3 2 3 4 3 2 2 2 3 4 2 2 1 4 3 2 2 1 1 1 exp 2 2 1 1 1 1 x b x x b x b x x x x x b x b x x x N
N dx dx dx q q q q q q q p q q q q q q q q q q q

                                                         (3.7)
 more generally, when there are over five fixing dates, the actual numerical dimension of the function n  can always be reduced by a factor of 2 by using :

  1 2 1 1 2 2 1
, ,..., , ; , ,..., ,

n n n n n b b b b q q q q                 1 1 2 1 1 2 3 1 2 2 1 1 2 1 3 1 1 exp ... 2 2 1 2 1 1 2 ... 1 1 ... n n n n n n n i i n n n n n n b b b x x x x x x x x b x b x N N dx dx dx q q q q q p q q q q                                                                   (3.8)
The identities (3.5) -(3.8) can be obtained using tedious but straightforward algebra and their proof is therefore omitted. The quality of numerical integration using (3.5) -(3. ,2,3 and 4. In terms of efficiency, it always takes less than one second to compute the option prices on an ordinary personal computer, which is very quick in absolute terms and dramatically more efficient than simulation techniques. It must be pointed out that the efficiency gains are even greater when it comes to the computation of the option sensitivities or greeks. Moreover, there is a clear pattern of linear convergence of the Monte Carlo estimates to the analytical values as more and more simulations are performed in all the tested dimensions. This high quality of numerical integration is not surprising, given the smoothness of the integrands in every dimension. Cases of numerical instability might arise only for extremely high values of the correlation coefficients inside the integrands. After a little numerical experiment, it was found that problems were encountered for values above 99.5%. But such values would come up only for very weird contract specifications, such a single fixing date one day before expiry on a two-year option, and these would actually never be met in the markets. Besides, alternative, more sophisticated quadrature rules that handle almost singular points could then be resorted to.

Thus, in moderate dimensions, the analytical formulae provided in this paper give very efficient and accurate numerical results, while the computational time required for a Monte Carlo simulation to produce reasonably precise approximations is clearly not satisfactory for practical purposes. As more and more fixing dates are added, the quality of a plain Gauss-Legendre implementation of (3.8) will deteriorate. One solution, then, would be to implement an adaptive Gauss-Legendre quadrature based, for instance, on a Kronrod rule. There is plenty of scientific software available for that matter so that it is not necessary to know the technical details. Another, more powerful, solution is to notice that the function  that needs to be computed in increasing dimension has the attractive feature that it matches the special structure of Gaussian convolutions handled by the extremely efficient Broadie-Yamamoto algorithm [START_REF] Broadie | A double-exponential fast Gauss transform algorithm for pricing discrete path-dependent options[END_REF]. Using the latter will ensure that the above stated Proposition 3 can be implemented with confidence in high dimensions. For evidence about the mentioned algorithm, the reader is referred to the above cited original paper.  in Proposition 1 is computed using identity (3.5) along with a 16-point Gauss-Legendre quadrature. The reported averages were computed out of a sample of 500 theoretical option values with randomly drawn parameters. The average computational time is measured in seconds. The average difference between the Monte Carlo approximations and the analytical prices is reported as a percentage of the option prices. So is the maximum difference too.  in Proposition 2 is computed using identity (3.6) along with a 16-point Gauss-Legendre quadrature.  in Proposition 3 is computed using identity (3.8) along with a 16-point Gauss-Legendre quadrature.

Appendix : proof of Proposition 1

The dynamics of the underlying asset S under the risk-neutral measure  are driven by the classical geometric Brownian motion :

          dS t r S t dt S t dB t d s    (4.1)
where r is the constant riskless rate, s is the constant volatility of S , d is a constant payout rate on S and   B t is a standard Brownian motion.

Following the risk-neutral valuation method [START_REF] Harrison | Martingales and arbitrage in multiperiod securities markets[END_REF][START_REF] Harrison | Martingales and stochastic integrals in the theory of continuous trading[END_REF], the no-arbitrage value, at time 0 t , of a cliquet call option with three fixing dates 1 t , 2 t , 3 t and expiry 3 t , is given by :

    0 1 2 3 , , , , , , , C S t K r t t t s d             0 3 1 2 3 exp r t S t S t S t K S t                 (4.2)
where   is the expectation operator under the risk-neutral probability measure  .

Breaking down the various possible outcomes, the right-hand side of (4.2) can be expanded as follows :

                3 0 1 1 2 2 3 3 exp r t S t K I S t K I S t K I S t                    (4.3) with :             1 1 1 2 1 3 , , I S t K S t S t S t S t      (4.4)             2 2 2 1 2 3 , , I S t K S t S t S t S t      (4.5)             3 3 3 1 3 2 , , I S t K S t S t S t S t      (4.6)
where  is the indicator function.

Using Girsanov's theorem, (4.3) becomes :

            1 0 0 1 3 1 1 exp t r t t S t I S t d                    2 0 0 2 3 2 2 exp t r t t S t I S t d             0 3 exp t S t d                3 0 0 3 3 1 2 3 exp I S t rt K I I I S t           (4.7)
where     1 2 ,   and   3  are three measures equivalent to  whose Radon-Nikodym derivatives are given by :

      2 exp , 1, 2, 3 2 i i t i i d t B t i d s s                   (4.8)
where t  is the natural filtration of

            1 1 1 2 1 3 , , I X t k X t X t X t X t      (4.9)             2 2 2 1 2 3 , , I X t k X t X t X t X t      (4.10)             3 3 3 1 3 2 , , I X t k X t X t X t X t      (4.11)
Next, by definition of Brownian motion, the random variables

  1 X t ,         2 1 3 1 , X t X t X t X t   and     3 2 X t X t 
can be written as follows :

  1 1 1 1 X t t t m sf   (4.12)       2 1 2 1 2 2 1 X t X t t t t t m sf      (4.13)         3 1 3 1 2 2 1 3 3 2 X t
        2 2 1 , X t k X t X t                    2 1 2 1 1 , 0 X t k X t k X t X t X t k          
(4.17)

Then, using the independence of the variables

    3 2 X t X t  and     2 1 X t X t  ,
one can obtain :

    0 2 I S t     2 1 1 2 1 1 2 1 1 1 2 2 1
                                                     3
                                                (4.20)
Then, performing the necessary calculations, one can obtain the following result :

            0 3 3 1 3 2 3 1 3 2 3 3 3 3 1 3 1 3 2 , , ; 1 ,
                                       (4.22)

  8) was tested up to six fixing dates. That would be the required dimension for the valuation of a three-year expiry cliquet option with semi-annual observation points or a six-month expiry cliquet option with monthly observation point, for instance. The identities (3.5) -(3.8) were implemented using a simple 16-point Gauss-Legendre quadrature rule. In every dimension, 500 option prices were computed by means of Proposition 1, Proposition 2 and Proposition 3, with randomly drawn parameters. The results were compared with those obtained by a Monte Carlo simulation using antithetic variates and the Mersenne Twister random number generator. The main findings are summarized in Tables 1

  and the time is close to 2 t , then the delta of the cliquet call tends to 1 and the gamma drops to zero. The more volatility around the level

	1 S t		K	is
	positive and S is trading near the level  
	  1 S t between 1 t and 2 t , the more delta oscillates, with gamma reaching a local
	maximum at the level   1 S t just before 2 t .

1 S t little before 2 t , then the delta increase is sharp. If S rises above the level   1 S t

  The same procedure easily extends to a higher number of fixing dates. However, it does not work without costs. First, the ability to implement the hedge depends on the availability of the required options in the market, which may be a problem if the underlying is not liquid enough or if the maturities are non-standard. Even if the required options are traded, there are vanilla option bid-ask spreads that need to be factored in.

	  1 S t		K		and   2 S t			K	, both vanilla options bought at times 1 t and 2 t are struck
	at K . To value the former at time zero, one should look at the 	 t t forward implied 1 2 ,
	volatility point on the volatility surface. It is important to bear in mind that the associated
	skew is the 	 t t one, which is typically steeper than the  1 2 ,	 t t one. The farther 1 0 2 , t is
	from 0 t , the higher the chances that the 	 t t option will be priced at time zero using 1 2 ,
	an inappropriate skew, because there is uncertainty about the level / K S at time 1 t and
	about the variation in the slope of the skew curve between 0 t and 1 t . These problems are
	compounded when it comes to the valuation of the 	1 2 , t t		option in the semi-static
	hedge. Next, when   1 S t		K	and   2 S t		K	, both vanilla options bought at times 1 t
								and expiring at 2 t ; if
	  1 S t		K	then buy a new call struck at K and expiring at 2 t
	-at time 2 t , if   2 S t		K	then buy a new call struck at   2 S t and expiring at 3 t ; if
	  2 S t		K		and   1 S t		K	then buy a new call struck at K and expiring at 3 t ; if
	  2 S t		K	and   1 S t		K	then buy a new call struck at   1 S t and expiring at 3 t

Besides, it is not easy to value the cost of this semi-static rollover strategy at the time the cliquet call is sold. Indeed, it involves purchasing options in the future whose strikes are not known at time zero and depend on future values of S . Numerical methods must be used to approximate the value of these future contracts. Whatever numerical methods are used, the skew must also be taken into account to obtain a reliable price. When and 2

t are struck at a level above K but they are at-the-money at the time of purchase.

  Proposition 1 is provided in the Appendix.

	Next, Proposition 2 gives a closed form solution for the no-arbitrage value, V , of a
	cliquet option with expiry 4 t , and four fixing dates, 1 t , 2 t , 3 t and 4 t , under the same
	assumptions as in Proposition 1.
	Proposition 2
	  0 V S t 	 , , , , , , , , 1 2 3 4 r K t t t t s d			(3.3)

Table 1

 1 Cliquet option values with three non-uniformly spaced fixing dates a

		Average computational	Average difference	Maximum difference
		time	with analytical	with analytical
			formula	formula
	Analytical formula	0.15	0	0
	Monte Carlo	0.82	0.63	0.96
	100,000 simulations
	Monte Carlo	5.58	0.19	0.56
	1,000,000 simulations
	Monte Carlo	52.6	0.025	0.161
	10,000,000
	simulations

a : The analytical formula used is Proposition 1. The function 3

Table 2

 2 Cliquet option with four non-uniformly spaced fixing dates b

		Average computational	Average difference	Maximum difference
		time	with analytical	with analytical
			formula	formula
	Analytical formula	0.26	0	0
	Monte Carlo	1.07	0.45	1.05
	100,000 simulations
	Monte Carlo	7.88	0.17	0.46
	1,000,000 simulations
	Monte Carlo	74.81	0.029	0.213
	10,000,000
	simulations
	b : The analytical formula used is Proposition 2. The function 4

Table 3

 3 Cliquet option with five non-uniformly spaced fixing dates c

		Average computational	Average difference	Maximum difference
		time	with analytical	with analytical
			formula	formula
	Analytical formula	0.38	0	0
	Monte Carlo	1.24	0.51	0.82
	100,000 simulations
	Monte Carlo	9.85	0.22	0.63
	1,000,000 simulations
	Monte Carlo	95.32	0.042	0.176
	10,000,000
	simulations
	c : The analytical formula used is Proposition 3. The function 5  in Proposition 3 is computed
	using identity (3.7) along with a 16-point Gauss-Legendre quadrature.

Table 4

 4 Cliquet option with six non-uniformly spaced fixing dates d

		Average computational	Average difference	Maximum difference
		time	with analytical	with analytical
			formula	formula
	Analytical formula	0.89	0	0
	Monte Carlo	1.45	0.54	0.91
	100,000 simulations
	Monte Carlo	12.16	0.26	0.49
	1,000,000 simulations
	Monte Carlo	123.66	0.038	0.184
	10,000,000
	simulations

d : The analytical formula used is Proposition 3. The function 6

  By the Markov property of Brownian motion, we have :

				x	3			x	3			
		x	1	2    3 x x k    		  1 X t		  2 dx X t 1 ,		  3 dx X t 2 ,		 dx dx dx dx 3 3 2 1
													  1 X t		  2 dx X t 1 ,		  3 dx X t 2 ,		dx	3	
						  1 X t					  2 X t		  1 dx X t 2					  3 X t		  2 dx X t 3	
								2 p	 3 2			1			1 t t 2 1 /		1		2 t t /	3			exp	      		1 2	    	x	1 s 	1 t m 1 t	2      
							x	2		t m	2							1 1 t x	1 t m	2	x	3	t m	3	2 2 t x	t m	2	2
								s	t	2										t	2		s	1 t	s	t	3	t	3	s	t	2
											 2 1				1 t t /	2		 2 1	2 t t /	3	
																									(4.18)
																									1      	m  s	3 t t	 	2 2 t t	    
	Next, the expectation								  0 I S t 3	
																												  0 I S t 3		(4.19)





can be expanded into integral form as follows :

  This completes the proof for a cliquet call option with three fixing dates 1 t , 2 t , 3 t and expiry 3 t . The same method can be applied to value cliquet options with a greater number of fixing dates, by using the following generalization of (4.21) for 4 n :

						I S t										(4.21)
				 	    	k s   t	t m		m s	t t	t t  	m s	t t	t t  		t t	t t	 	t t	    
			  n X t		  n k X t ,			X	    1 , n t X t		X	  2 t	,...,	  n X t		X		1 t  n		
			n	k 1 s	1 t n m , n t n t t	,	m n t n t s	  t n t 2 1 n t t	,	  1 1 t n t n t t	,	  3 2 t n m t ,..., n t t s		  2 2 n t t n t t	,..., n t n t	1 2 m s	  t n n t	n t n t	  1 1	;