CC BY
2where g(r, x) is given by
g(r,x) = xa{n-j)ea{n-j)(rf-q-pasav~ay2)rEod (1 + rj)NTI{z<d2>j}\Ft
n> 0
Since Z ~ N(0, 1), we obtain
g(T, x) = xa(n-j ) ea(n—j)(rf-q-p°s av—aS/2)r
/do j -t
-oo V2n
n>0
= xa(n-j) ea(n-j){rf—q—pasav — (1—a(n—j))aS/2}tgAr
m J_ co y^f
X
n0
Applying the substituting v = z + — j)as^/r and setting
di,j ■= d2,j + a(n - j)asVr In (5f( 1 + r^/ii» + a (rf - q - pastry - <4/2) r ,
=--+ a(n "3)asV? (3-n)
= In (Sf(l + rjr/Kf) + a (rf - q - pasW - (1 + cx{n - j)/2) 4/2) r
aaS\fr
we get
g(T1x)=xa{n~j)ea{n~j)^rf~q~pasav~{l~a{n~j))as/'AreXr ^T ~ fdl' ] c~l'2/2dz
\n ^ * J—oo a/27T
«>0 «/—^
n>0 n (3.12)
= i){r/—po-^crv' —(1—J))O-|/2}T- ^ (KT-t)) .y
n> 0
From (3.11) and (3.12), (3.8) becomes
n / \
Cq = V0e(X—Td)(T—t)Y ( ) (-K/)1,j Sta(n—j)ea(n—j){rf —q—pasav — (l—a(n—j))aS/2}r 3=0 ^ j
n> 0
where
ln (Sf (1 + n)n / K/) + a (r/ - q - pas ay - (1 + a(n - j)/2) a|/2) t
di,j =
aasy/r
The diagram below illustrates the Quanto premium evolution with jumps concerning maturity time T and the strike price K/.
Using the same dataset as before, with rd = 0.5, V0 = 30, a = 5,r/ = 0.01, q = 0.1, p = 0.01, as = 0.3, and av = 0.2. Additionally, setting n = 5, A = 5,n = 6 and N = 5, Fig. 2 depicts the progression of Quanto option prices with jumps. It's noticeable that the Quanto option value exhibits an upward trend concerning both variables, maturity time and strike price.
Figure 2. Powered power Quanto option call with jumps plotted against maturity time and strike price.
4. Conclusion
Quanto options are crucial tools for managing risk in the foreign exchange market. Determining their fair prices without arbitrage opportunities is essential. In this study, we have developed formulas to find the no-arbitrage prices for powered Quanto options. We considered scenarios where the underlying currencies follow Brownian motion and Brownian motion with jumps. We supported our theoretical framework with numerical simulations and results. We hope this research will inspire further exploration and interest in pricing exotic options.
REFERENCES
1. Black F., Scholes M. The pricing of options and corporate liabilities. J. Political Economy, 1973. Vol. 81, No. 3. P. 637-654. URL: http://www.jstor.org/stable/1831029
2. Capinski M., Kopp E., Traple J. Stochastic Calculus for Finance. Cambridge, UK: Cambridge University Press, 2012. 186 p.
3. Derman E., Kani I. Stochastic implied trees: arbitrage pricing with stochastic term and strike structure of volatility. Int. J. Theor. Appl. Finance, 1998. Vol. 1, No. 1. P. 61-110. DOI: 10.1142/s0219024998000059
4. Dupire B. Pricing with a smile. Risk, 1994. Vol. 7. P. 18-20.
5. Lee Y., Yoo H.-S., Lee J. Pricing formula for power Quanto options with each type of payoffs at maturity. Global J. Pure Appl. Math., 2017. Vol. 13, No. 9. P. 6695-6702.
6. Teng L., Ehrhardt M., Giinther M. The pricing of Quanto options under dynamic correlation. J. Comput. Appl. Math., 2015. Vol. 275. P. 304-310. DOI: 10.1016/j.cam.2014.07.017
URAL MATHEMATICAL JOURNAL, Vol. 10, No. 1, 2024, pp. 68-75
DOI: 10.15826/umj.2024.1.006
EXTREMAL VALUES ON THE MODIFIED SOMBOR INDEX OF TREES AND UNICYCLIC GRAPHS
Raghavendra H Kashyap1, Yanamandram B. Venkatakrishnan1, Rashad Ismail2, Selvaraj Balachandran1, Hari Naresh Kumar1^
1 School of Arts Sciences Humanities & Education, SASTRA Deemed University,
Thanjavur, India - 613 401
2Department of Mathematics, Faculty of Science and Arts, King Khalid University,
Mahayl Assir 61913, Saudi Arabia
Abstract: Let G = (V, E) be a simple connected graph. The modified Sombor index denoted by mSo(G) is defined as
mSo(G) = J2 -fWTT'
uT^eV du + dv
where dv denotes the degree of vertex v. In this paper we present extremal values of modified Sombor index over the set of trees and unicyclic graphs.
Keywords: Modified Sombor Index, Trees, Unicyclic graphs, Extremal values.
1. Introduction
A topological index is a real number derived from a structure of a graph that is not dependent on the way the vertices are labeled. A wide range of different topological indices have been employed in QSAR (Quantitative Structure - Activity Relationship) and QSPR (Quantitative Structure -Property Relationship) studies. Any topological indices belong to one of the two classes: they are either bond-additive, or distance based. Typical representation of bond-additive indices are two Zagreb indices, Harmonic index and Randic index.
Let G = (V, E) be a simple connected graph. By the open neighborhood of a vertex v of G we mean the set
Ng (v) = {u € V: uv € E}
and by the closed neighborhood,
Ng[v] = Ng(v) U {v}.
The degree dv of a vertex v is the cardinality of its open neighborhood. We denote by Pn and Cn a path and a cycle with n-vertices, respectively. A length of a cycle is the number of edges contained in the cycle. A star of order n > 2, denoted by Sn is a tree with at least n — 1 leaves. A contraction of an edge e = uv is the replacement of u and v with a single vertex such that edges incident to the new vertex are the edges other than e that were incident with u or v and the resulting graph is denoted by G.uv.
Recently, a degree based topological index called the Sombor index was introduced by Ivan Gutman in [4]. It is defined as
SO(G) = Yl
uveE(G)
and further studied in [1-3, 6, 9, 10]. A variant of Sombor index namely, modified Sombor index, denoted by mSo(G), is defined as
mSo{G) = ^ 1
uveE v^r+^l
In [8], a lower bound on a Modified Sombor index of unicyclic graphs with a given diameter is presented. In [7], bounds of modified Sombor index in terms of spectral radius and energy is given. A study on modified Sombor index matrix is done in [11]. An extreme value of the product of the Sombor index and the modified Sombor index is studied in [5].
In [7], to determine the extremal trees, unicyclic graphs, bicyclic graphs with respect to modified Sombor index were proposed. We determine the extremal graphs for the class of trees and unicyclic graphs, which answers the problem posed in [7]. In particular, we show that star and paths are the graphs with minimum and maximum modified Sombor index among all trees, and for unicyclic graphs we show that Un(n — 1, 2, 2) and cycle are the graphs with minimum and maximum modified Sombor index.
2. Graph transformations
To begin with we present some graph transformations which will be useful to determine the extremal trees and unicyclic graphs.
Transformation A (see Fig. 1). Let G be a nontrivial connected graph and u, v E V(G), such that d(v) > 3 in G and P1 : uu1u2 ... ur and P2 : vv\v2 ... vs be two paths in G. Now we denote the graph H obtained from G by concatenating the paths P1 and P2.
Figure 1. Transformation A.
Theorem 1. Let H be the graph obtained from G using Transformation A, then mSo(G) < mSo(H).
Proof. The vertex vi in path P2 is made adjacent to vertex ur. Then mSo(H) = mSo(G) - + + 1 1
V5 a/8 V8 ^/dfh4 i | ^ i
aeN(v)\Vl + da aeN(v)\Vl ~ lY + dl
= mSo(G) - 4= + 4= -
J—+ y _I___
VE Vs v^Tl VVK-1)2+ 4
mSo(H) > mSo(G)
1 2
+
+ y _1___L_
aGN (v)\vi
> ,„So(G) - -L + ^ - -L + -L - -L > mSo{G).
□
Transformation B (see Fig. 2). Let G be a nontrivial connected graph and u G V(G), such that d(u) > 3 in G and P : uuiu2 ... ut be the path in G. The H is constructed from G by removing the leaf ut in the path P and attaching it to the vertex u by an edge uut.
Figure 2. Transformation B.
Theorem 2. Let H be the graph obtained from G using transformation B, then mSo(H) < mSo(G).
Proof. Applying Transformation B to graph G, we have
1 1 1 1
mSo(H) = mSo{G) - — - — + — +
75 a/8 V5 y/(du + l)2 + 1
_ y 1 + y _I_
1
= mSo(G)--— H----= + y ---,
V8 VK + 1)2 + 1 amu)W(d*-!)* + <% VdZTdl
mSo(H) < mSo{G) - 4= + ~7= < rnSo(G).
v 8 V17
□
Transformation C (see Fig. 3). Let G be a nontrivial connected graph, uv G E(G) with N(v) H N(u) = 0. We denote the graph H obtained from G.uv and making the vertex v adjacent to u by an edge uv.
Figure 3. Transformation C.
Theorem 3. Let H be the graph obtained from G using transformation C, then mSo(H) < mSo(G).
Proof. From the definition of Transformation C, we have du ,dv > 2. Then
mSo(H) = mSo(G)--, — V —,
E
1
1
+
a aeN (v)\u 1
+ «mvWuMurt y/fa + dv- l)2 + d2 1 y/(dv +~du — l)2 + 1'
Since,
- E
aeN (u)\v
-
a aeN (v)\u 1
+
+ d'i
+
E
a aeN (v)UN (u)\{u,v}
+ du - l)2 + d?a
< 0,
V/rf2Tdf + du - l)2 +1
< 0 for any du,dv > 2.
Thus mSo(H) < mSo(G).
□
Transformation D (see Fig. 4). Let G be a unicyclic graph with cycle of length a, denoted by Ca and u € Ca, such that d(u) = 3 in G and P : uu\u2 .. .ut (t = 2) be the path in G. Let w be the neighbour of u in Ca. The graph H is constructed from G by removing the leaf vt and including it in the cycle Ca between the vertices u, w.
Theorem 4. Let H be the graph obtained from G using transformation D, then mSo(G) < mSo(H) .
Proof. From Transformation D, we have du = 3. Then Case 1: t > 3
mSo(H) = mSo(G) - - + - + + = mSo(G).
V5 Vl3 Vl3 \/8
Case 2: t = 1
mSo(H) = mSo(G)
VlO vT3 y/E
23
+ — > mSo{G).
□
1
1
1
1
1
Figure 4. Transformation D.
Lemma 1. Let G be a unicyclic path with cycle of length n — 2, say Cn-2 and u € Cn-2 with a path uu1u2. Let H be the graph obtained from G by removing the vertices u1 and u2 and included in the cycle Cn-a. Then mSo(G) < mSo(H).
Proof. Let the du = 3 in V(G). Then,
1
7!
mSo(H) = mSo(G)
--%= + > mSo(G).
V8 Vl3 V8
□
Let Un(ni, n2, n3) be the family of n-vertex unicyclic graph obtained from attaching n\ — 2, n2 — 2 and n3 — 2 pendent vertices to the three vertices of a triangle respectively, where n1 + n2 + n3 = n + 3 and n1 > n2 > n3 > 2.
Lemma 2. For any n > 5, n1 + n2 + n3 = n + 3 and n1 > n2 > n3 > 3,
mSo(Un(n — 1, 2, 2)) < mSo(Un(n1, n2, n3)).
Proof. Since n1 > n2 > n3 > 3, we need to prove
mSo(Un(n + 1, n2 — 1, n3)) < mSo(Un(n1, n2, n))
for n2 > 3. Let
/(x) = , x > 3.
Then
f" (x) =
Vx2 + 1
—4x2 -3x + 2 (x2 + l)5/2
<0
implies that f (x + 1) — f (x) is decreasing function for x > 3. Thus
mSo(Un (n1 + 1,n2 — 1,n3)) — mSo(Un (n ,n2 ,n3)) = mSo(Un (n1 + 1,n2 — 1,n3)) — mSo(Un-1 (n1 ,n2 — 1,n3))
—
n2 — 2
-(mSo(Un (m, n2, n3)) — mSo(Un-1 (n1, n2 — 1, n3))) n2 — 3
+
1
1
+
1
VnfTI V(n2 - l)2 + 1 V^lTnf v^n2 + (n2 - l)2 v/n2 + n3 N/(«2 - I)2 + «3
1
-(
n1 — 1
n1 — 2
+
1
V(ni +1)2 +1 v7^7! V^i+^I
+
+ I)2 + (n2 - l)2 s/n\ + (n2 -l)2'
1
1
Since
then
1 1
< 0,
■\Jnf + n\ \Jn\ + (n2 - l)2 1 1
\ l/'i ■ I if. \/{n2 - l)2 +n't
< 0,
11
< 0
V(ni + l)2 + \fn{ + «3 11
< o,
+ I)2 + (n2 ~ l)2 y/rii + (n2 - l)2
mSo(Un(ni + 1,n - 1,n3)) - mSo(Un(ni, U2, n3)) < f (n2) - f (n2 - 1) - (f (ni + 1) - f(ni)) < 0.
□
3. Extremal trees and unicyclic graphs
In this section, we determine the extremal values of the modified Sombor index on the class of trees and unicyclic graphs.
Theorem 5. Let T be a tree with n-vertices, where n > 3. Then
mSo(S,n) < mSo(T) < mSo(P,n).
Proof. By repeated use of the Transformation A, any tree T can be transformed into a path. Thus by Theorem 1, mSo(T) < mSo(Pn).
Now by repeated use of the Transformation C on T, we obtain a star. Thus by Theorem 3, mSo(T) > mSo(Sn). □
Corollary 1. Let T be a tree on n vertices, where n > 3, then
H ~ 1 < mSo(T) < 4= + n " 2
Vn2 - 2n + 2 ' v/5 \/8
Theorem 6. Let G be an unicyclic graph with n-vertices, where n > 4. Then
mSo(Un(n - 1,2,2)) < mSo(G) < mSo(Cn).
P r o o f. By repeated use of the transformation A, any unicyclic graph G can be transformed into a comet. Thus by Theorem 1, mSo(G) < mSo(COn-a,a). Furthermore by using Theorem 4 and Lemma 1, we get mSo(COn-a,a) < mSo(Cn).
Now by repeated use of the Transformation B on G, we obtain a unicyclic graph G' with a cycle and remaining vertices as leaves. Thus by Theorem 2, mSo(G) > mSo(G'). Furthermore repeating the transformation C on G' we get Un(nl,n2,n3). By Theorem 3, mSo(G') > mSo(Un(nl,n2,n3)). Furthermore using Lemma 2, we get mSo(Un(nl,n2,n3)) > mSo(Un(n - 1,2,2)). □
Corollary 2. Let G be an unicyclic graph on n vertices, where n > 4, then
n - 3 2 1 n
+ = + —= < m,So(G) < —=.
Vn2 - 2n + 2 y/n2 - 2n + 5
'Vn-a+l
Vn—a
M (b)
Figure 5. (a) Comet COn-a,a (b) Un(n - 1, 2, 2). 4. Conclusion
Bounds on modified Sombor index in terms of graph parameters are determined and various topological indices are compared with modified Sombor index in [7]. In [7] an open problem was proposed to determine the extremal trees, unicyclic graphs and bicyclic graphs with respect to modified Sombor index. Extremal trees and unicyclic graphs are determined here, which answers a part on the problem.
5. Acknowledgement
The authors are very greatful to all of the referees for their carefully reading and insightful comments.
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