The Exchange Option is one of the interesting instruments provided by Margrabe (1978)[1]. Several studies have been carried out on the modeling of this security, which can be cited by El Karoui et al. (1995), Antonelli et al. (2010), Cheang and Chiarella (2011), Kim and Koo (2016), and Wang(2016)[2, 3, 4, 5, 6, 7] . El Karoui et al. obtained a close formula for the Exchange Option using the change of numeraire, and Kim and Koo provided a close formula for the state of the two assets being the same drift using a Mellin transform. Antonelli et al. worked on this option considering the other structure of Volatility. Cheang and Chiarella, as well as Wang, modeled the Jump diffusion for this structure.

The first studies in the context of the Lookback option were Goldman et al. (1979) and Conze and Viswanathan (1991) [8, 9]research, which provided the exact solution for the floating and fix strike model. In this regard, Dai et al. (2004) [10] provided a close formula for quanto Lookback options. He et al. (1998) [11] have priced it using numerical methods and Monte Carlo by obtaining a joint density function. Jeon and Yoon (2016) [12] also got a closed formula for put-back options using the Mellin transform.

As we said, in this paper, we are combining the Exchange option and the Lookback option to provide a new option called the LOEX option, and for the first time in this study, modeling and pricing it. In addition to being a risk hedging instrument, this financial derivative can have another function; due to its structure, we can use it as a market stimulus. For example, this option can be used to move the ask queues of an asset.

To achieve the intended purpose, we will introduce the Exchange and Lookback options in the next section and provide the price equation for each. In Section 3, we will use the mathematical concepts of LOEX option modeling, develop a PDE for its price model, and provide a way to solve the PDE. Finally, in Section 4, we will run the model with MATLAB software, compare it with the Exchange option price, and show the results.

An exchange option is a conditional claim that the holder can convert assets to another asset at strike time or NOT. Assume that C(t, S, S ) is the Exchange option price at time t with the price of assets S and S and the payoff function is as follows:

C(T, S(T), S (T)) = (S(T) − S (T)) . (2) Now, using the Feynman–Kac formula:

@C @C @C 1 @ C @ C 1 2 @ C + rS + rS + S + 1 2SS + S − rC = 0 (3) @t @S @S 2 @S @S@S 2 @S which dt = dwt dwt and its terminal condition is as follows:

C (T, S, S ) = (S − S) . (4)

The closed formula for the Exchange option according to PDE (3) is as follows [5]

C(t, S1, S2) = S1(d) − S2(d − T − t), (5)

where

S1 p ln S2 T −t

d = p + , p T −t 2 = 1 + 2 − 2 1 2, (6)

−0.5x

(x) = e dx.

The Lookback option is an exotic option that depends on the asset price path. In general, the Lookback option is divided into four categories. In this section, we summarize everything and present the price equation. Consider the following symbols:

M(S(t), t) = max S(!),

0 ! t

m(S(t), t) = min S(!).

0 ! t

Now we have Lookback options and their price equations [13]

Floating strike Lookback call option.

payof f ! S(T) − m(S(T), T)

S 1 S 1

ln + r+ (T −t) ln + r− (T −t)

m(S,t) 2 m(S,t) 2

@ p A −r(T −t) @ p A Eq. ! S − m(S, t)e

T −t T −t

m(S,t) 1 m(S,t) 1

ln − r+ (T −t) ln + r− (T −t)

2 S 2 2 S 2

S1 −r(T −t) S1

− @ p A + e @ p A .

2r T −t 2r T −t

Floating strike Lookback put option.

payof f ! M(S(T), T) − S(T)

S 1 S 1

ln + r+ (T −t) ln + r− (T −t)

M(S,t) 2 M(S,t) 2

@ p A −r(T −t) @ p A Eq. ! −S − + M(S, t)e −

T −t T −t

S 1 S 1

ln + r+ (T −t) 2r ln − r− (T −t)

2 M(S,t) 2 2 M(S,t) 2

S S M 2

1 @ A −r(T −t) 1 @ A + p − e 1 p .

2r T −t 2r S T −t

Fixed strike Lookback call option.

payof f ! M AX(M(S(T), T) − K, 0)

−r(T −t) t T

At the time t, the price would be e E[max(max(M0(S(t), t), Mt (S(T), T)) − K, 0)],

H(t, S; K) M0(S(t), t) K Eq. ! −r(T −t) t t t , e (M0(S(t), t) − K) + H(t, S; M0(S(t), t)) M0(S(t), t) K −2r (10)

S S 2 2r T −t −r(T −t) −r(T −t) 1 r(T −t)

H(t, S; N) = S(d ) − N e (d ) + e e (d ) − 1 d − .

1 2 2r 1 N 1

ln( )+(r+ ) N p 2

d1 = ,

T −t

2 (11)

ln( )+(r− ) N p 2

d2 = .

T −t Fixed strike Lookback put option.

payof f ! M AX(K − m(S(T), T), 0)

−r(T −t) t T

At the time t, the price would be e E[max(K − min(m0(S(t), t), mt (S(T), T)), 0)],

F or K m0(S(t), t)

−2r p S S 2 2r T −t −r(T −t) −r(T −t) 1 r(T −t)

Eq. ! Ke (−d ) − S(−d ) + e −e (−d ) − 1 −d + ,

2 1 2r 1 K 1

F or K m0(S(t), t)

−r(T −t) t

Eq. ! e (K + m0(S(t), t)((d2) − 1)) − S(−d1),

−r(T −t) 1 r(T −t) S 1 2r T −t

+ e e (−d1) + t −d1 + .

2r m (S(t),t) 1

(12) d1 and d2 are the same as (11) formula.

Sometimes, an issue is a problem that we cannot access the close form of problems, so the numerical methods help us change this circumstance. There are a lot of numerical methods that are used for these problems [14] [15]. In this solution section, we consider solving PDE (17) by numerical methods. The solution method used for the PDE (17) is the mesh-free numerical approximation method [16, 17]. Therefore, by property of exchange and lookback options, we consider boundary and initial conditions as follows:

G(0, S, S ) = (S − S ) , G(,0, S ) = 0, G(,S, 0) = S, (19) lim G(, S, S ) = lim S ,

S,S !1 S,S !1

where is a large number.

For applying the meshfree numerical approximation method, suppose the change of assets S and S is in ithe interval [0, Smax] and [0, Smax] respectively, and also discrete the first interval in M points and the second N points so we have the following radial basis function:

G(t, S, S ) = i(t)'(S, S ), (20)

where p '(x, y) = x + y + " , (21) and is a function that depends on time.

We rewrite the PDE’s (17) derivatives as follow:

M N P G ( ) (t, S, S ) = i(t)' ( ) (S, S ) =: ' ( ) ,

S S S

M N P G ( ) ( ) (t, S, S ) = i(t)' ( ) ( ) (S, S ) =: ' ( ) ( ) , (22)

S S S S S S

M N P G (t, S, S ) = _ (t)'(S, S ) =: ', _

where ', ' ( ) , ' ( ) ( ) supposed that are matrix as follow:

S S S

'i,j = '(Si, Sj ), ' ( ) = ' ( ) (Si, S ), (23)

S i,j S j

' ( ) ( ) = ' ( ) ( ) (Si, S ).

S S i,j S S j

Then, we could have the PDE (17) as below:

' _ + L = 0, (24) where L = rS'S + rS 'S + 1S 'SS + 1 2SS 'SS + 2S 'S S − r'. (25)

Using the concept of derivative, suppose the interval of time is [0, T] whereas mentioned, T is strike time and consider K as the number of equal discrete points of this interval with dt difference so

k+1 k

(tk+1) − (tk) − _ = =: (26) dt dt and using the numerical -method, we can rewrite the ODE (24) as follows:

k+1 k

A = B , (27) where A = ' + dtL, B = ' − (1 − )dtL, (28) 1 k K − 1.

1. Using initial condition, obtain , k = 1;

k+1 k

2. Compute A = B ;

3. G = ' ;

4. Apply boundary condition on G;

k+1 −1

5. = ' G, k = k + 1, and if k < K go step 2.