

International Journal of Analysis and Applications

Volume 19, Number 1 (2021), 29-46

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-29

ON MODELLING AND PRICING WEATHER DERIVATIVES DRIVEN BY

NONLINEAR BROWNIAN MOTION

JAVED HUSSAIN∗, PERVEZ ALI

Department of Mathematics, Sukkur IBA University, Sindh, Pakistan

∗Corresponding author: javed.brohi@iba-suk.edu.pk

Abstract. In this paper, our focus is to derive the estimates satisfied by the risk-neutral prices of a class

of weather derivatives, contingent upon temperature which satisfies G-stochastic differential equation driven

by nonlinear G-Brownian motion.

1. Introduction

In this work, we aim to focus on the weather derivatives, HDD Call, and CDD call, where the underlying

temperature is driven by a version of nonlinear Brownian motion, known as G- Brownian motion. Robust

finance is one of the emerging areas of modern finance, where the focus is on developing the risk management

models, where the underlying asset is driven by uncertain volatility. Classically, in the most risk management

model/ financial asset pricing theory, the volatility is either assumed to be constant, deterministic or in case

it is taken stochastic, it is driven by linear noise such as the Wiener process. In all of these cases, models

suffer from reversal disadvantages such as mispricing of financial assets. One very interesting proposal

was given by Levy in [3, 1995], to take the volatility to be uncertain i.e. lie in a closed interval, this

study was a good start but suffered from the problem of risk management tools such as options were not

dynamically priced. The solution to this problem came from Peng in [24, 2007], where he introduced the

motion of the probability space with independent nonlinear expectation known as G-Expectation. This

Received September 6th, 2020; accepted November 3rd, 2020; published November 24th, 2020.

2010 Mathematics Subject Classification. 60J65, 60J70.

Key words and phrases. financial derivatives; weather contracts; nonlinear Brownian motion.

©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

29

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-29


Int. J. Anal. Appl. 19 (1) (2021) 30

was the point of inception of G-Stochastic Calculus and G-measure-theoretic probability. In [24, 2007]

and [25, 2008], Peng introduced this nonlinear version of probability and stochastics, more precisely Peng

introduced G-Conditional expectations,G-normal distributions which can be treated as the solution of fully

nonlinear G-Heat equation. Peng also introduced the notion of G-Brownian motion, G-Stochastic integrals

and their function spaces, G-Martingale, G-Sub & Super Martingales, G-Ito process, G-stochastic differential

equations i.e. SDE driven by G-Brownian motion and multi-dimensional G-Ito formulas. Zhao in [33, 2016]

numerically simulated G-normal distribution and G-Brownian motion. In [27, 2019], Peng introduced the

Feynman Kac formula which is pivotal in solving nonlinear probabilistic motions in G-framework. Using

all these tools in G-framework, Xu in [31, 2010] and [32, 2011] studied the EU call option and Girsanov

theorem in G-framework. Recently Julian in [15, 2020] has developed the theory of interest rate derivative

in G-framework.

The weather puts impacts all kinds of business activities directly or indirectly, so naturally, it is significant

to make good predictions about the weather. Production, transport, and preservation of agriculture crops;

energy production, distribution (cf. [1, 2002]) and consumption; Chain of supermarkets, tourism, and leisure

industries are directly influenced by the weather. Indeed, the key factors involved in weather that must be

taken care of, include a variation of temperature, wind, rainfall, humidity level, snowfall, etc. In particular,

we will focus on modeling the dynamics of temperature. The weather derivatives are becoming more and

more popular due to several reasons. For instance, in the energy market, the energy producers are facing

the challenges, firstly, as the energy sector is getting less and less regulated, and secondly since there is

a positive correlation between the weather and the demand for energy (cf. [29, 2017]), hence, the prices

of the energy is not the hands of energy produces. Indeed this creates competition in the energy market,

among the producers, and therefore the energy companies are interested to hedge their risk by entering

into financial contracts based in the weather. The weather derivative market is not very liquid, it looks

like many companies have not yet define hedging policies or even figured out themselves to weather risk.

If actors outside the energy sector take interest in the weather derivatives market there will be exponential

growth potential. There are some barriers that must be removed if the market is to grow. for example, the

quality and cost of weather data. Companies which want to analyze their performance against historical

weather data, they should often buy information from the national meteorological offices and that is quite

expensive. And the main issue is whether the gained information about the weather is good to rely on or

not. The weather derivatives are a relatively new way to hedge the risk associated with weather, they were

first introduced by Marginson in [20, 2000]. This gives rise to interest in carefully studying the seasonal

weather events El Niño event (cf. Trenberth [30, 1997]), and make a good prediction about the weather

so that seasonal risks can be hedged. After that, the weather derivative market expanded rapidly and

contracts have been traded over-the-counter(’OTC’) as individually negotiated contracts, the primary user


Int. J. Anal. Appl. 19 (1) (2021) 31

of weather derivatives were the energy sectors. Now the first market namely, Chicago Mercantile Exchange

(’CME’) started an electronic marketplace for weather derivatives in September 1999. The founder of CME

is Aquila Energy, Koch Energy Trading, Southern Energy, Enron, and Castle bridge weather markets, All

were active in the OTC market for weather derivatives. For more on weather derivatives we refer to recent

works [18, 2011] [22, 2013], [4, 2013], [23, 2013], [21, 2019] and [28, 2020].

We now give a brief organization of the paper. Section 2 comprises a very brief introduction to G-Stochastic

Calculus. Section 3 gives the reader a detailed account of the weather derivatives. Section 4 comprises of

our key results or estimates satisfied by the risk-neutral prices of weather derivatives (HDD call and CDD

Call), where the underlying non-trade able asset temperature is driven by G-Brownian motion.

2. Preliminaries

2.1. Sublinear Expectation. Linear expectation E satisfies the following relations for random variables X

and Y;

(2.1) E[aX] = aE[X], ∀ a ∈ R, and E[X + Y ] = E[X] + E[Y ].

A sub-linear expectation ÊG satisfies the following weaker condition;

(2.2) ÊG[aX] = aÊG[X] for a > 0, ÊG[X + Y ] ≤ ÊG[X] + ÊG[Y ].

The sublinear expectation ÊG follows the monotonicity property same as a linear expectation: if X ≥ Y then

ÊG[X] ≥ ÊG[Y ]. The sublinear expectation is very important in volatility uncertainty, through sublinear

expectation the people are doing work in super-hedging, super-pricing (cf. [3, 1995] and [19, 1995]) and

measures of risk in finance which caused great attention in finance since the fundamental work of [2, 1999].

In Sublinear expectation(non-linear expectation) space, One can introduce the distributions, of random

variables, like distribution of a single variable, joint distribution, marginal distributions, etc. One still

can show the independence and identically distributed random variables but in the sub-linear expectation

if X is independent to Y it does not directly imply that Y is independent to X. We can still prove

some important theorems in sublinear expectation theory, The law of large numbers, and the central limit

theorem, see [24, 2007]. One can see the G-expectation in [26, 2010] and G-martingales in [17, 2008], also the

Girsanov theorem under G-framework in [32, 2011] is now available. G – Brownian motion has a very rich

and interesting new structure that generalizes the classical structure without triviality. We can determine

the related stochastic calculus, particularly the G – Itô integrals (see [16, 1987]).

A short introduction of sublinear expectation and some definitions are the part of this section.

Definition 2.1. (Sublinear Expectation [27, 2019]) Let χ be the linear subspace of real valued and bounded

functions on Ω and a functional ÊG[.] on χ satisfying the following properties is called sublinear expectation.


Int. J. Anal. Appl. 19 (1) (2021) 32

1. Monotonicity:

X ≥ Y =⇒ ÊG[X] ≥ ÊG[Y ].

2. Constant preserving:

ÊG[c] = c, ∀ c ∈ R.

3. Sub-additivity: For every X,Y ∈ χ

ÊG[X + Y ] ≤ ÊG[X] + ÊG[Y ].

4. Positive homogeneity:

ÊG[λX] = λÊG[X], ∀ λ ≥ 0.

From the above properties(2.1) and(2.1) one can also show the convexity

ÊG[αX + (1 −α)Y ] ≤ αÊG[X] + (1 −α)ÊG[Y ], ∀ α ∈ [0, 1]

Definition 2.2. [27, 2019] Let ÊG,1 and ÊG,2 be sublinear expectations if,

ÊG,1[X] − ÊG,1[Y ] ≤ ÊG,2[X −Y ], ∀ X,Y ∈ χ.

The ÊG,2 is dominated by ÊG,2. From (2.1) of Definition 2.1 a nonlinear expectation is dominated by itself

and the strongest nonlinear expectation on χ is

ÊG,∞[X] := sup
ω∈Ω

X(ω).

every sub linear expectation is dominated by the strongest sublinear expectation.

We will denote Pf set of of all finitely additive probability measures on (ω,F).

Definition 2.3. (Distribution in linear expectation) [27, 2019] Let P ∈ Pf and let X : Ω 7→ R be a F-

measurable function such that |X(ω)| < ∞ for every ω. The distribution of random variable X on (Ω,F,P)

is defined as,

(2.3) FX[ϕ] = EP [ϕ(X)] : ϕ ∈ L∞(R,B(R)) 7→ R.

Here equation (2.3) can be written as

(2.4) FX[ϕ] =

∫
R
ϕ(x)FX(dx).


Int. J. Anal. Appl. 19 (1) (2021) 33

Definition 2.4. [27, 2019] The distributions under nonlinear expectations, is defined as, let the random vari-

able X = (X1,X2, . . . ,X3) be a n-dimensional random vector on a nonlinear expectation space (Ω1, H1,ÊG)

(2.5) F̂X[ϕ] := ÊG[ϕ(X)] : ϕ ∈ Cl.Lip(Rn) 7→ (−∞,∞).

where Cl.Lip(Rn) is the space of Lipschitz continuous functions. the following triple

(Rn , Cl.Lip(Rn) , F̂X[.]) is called sublinear expectation space. F̂X is the distribution of X.

Definition 2.5. (Mean and Variance -Uncertainty) [27, 2019] In general, under sublinear expectation mean

and variance are uncertain, in robust statistics ÊG[X] 6= −ÊG[−X] similarly ÊG[X2] 6= −ÊG[−X2], so now

these are four different parameters

µ̄ := ÊG[X], µ̂ := −ÊG[−X], σ̄2 := ÊG[X2], σ̂2 := −ÊG[−X2].

The purpose of this study is to discuss only variance uncertainty if someone is interested to study the mean

uncertainty he/she may (see [5, 2017] and [6, 2002]). So it is assumed that

ÊG[X] = −ÊG[−X].

Proposition 2.1. [27, 2019] Let X,Y ∈ H and ÊG[Y ] = −ÊG[−Y ] or Y has not mean uncertain. Then

ÊG[X + Y ] = ÊG[X] + ÊG[Y ].

Definition 2.6. (Independence in sublinear expectation) [27, 2019] Notion of independence in sublinear

expectation space (Rn , Cl.Lip(Rn) , ÊG) is same as linear expectation space. Let a random vector Y =

(Y1, . . . ,Yn), Yi ∈ H is said to be independent to random vector X = (X1, . . . ,Xm), Xi ∈ H under ÊG[�] if

for every function ϕ ∈ Cl.Lip(Rm ×Rn), the independence can be expressed as,

ÊG[ϕ(X,Y )] = ÊG[ÊG[ϕ(x,Y )]x=X].

2.2. G-normal distribution. In this section, some definitions has been discussed related to normal distri-

bution under sublinear expectation as defined in [24, 2007].

Definition 2.7. G-normal distribution: [27, 2019] A random variable X ∈ H in sublinear expectation

space
(

Ω, H ,ÊG
)

with σ̄2 = ÊG[X2], σ̂2 = −ÊG[−X2] > 0 is called N(0; [σ̂2, σ̄2])-distributed, if for every

Y ∈ H independent to X , Y ∼ X and aX + bY ∼
√
a2 + b2X, ∀ a,b ≥ 0.

From above definition this is clear that ÊG[X] = −ÊG[−X] = 0 hence random variable X has no mean

uncertainty.

The G-normal distribution N(0; [σ̂2, σ̄2]) is generated by the parabolic PDE defined for [0,∞) ×R:

(2.6) ∂tu−G(∂2xxu) = 0,


Int. J. Anal. Appl. 19 (1) (2021) 34

with Cauchy condition u|t=0 = ϕ where G,

(2.7) G(α) :=
1

2
ÊG[X2α] =

1

2
(σ̄2α+ − σ̂2α−),α ∈ R.

is called the generating function.

Where α+ := max 0,α and α− := max 0,−α, the equation (2.6) is called generating heat equation.

The solution of equation (2.6) is defined as

(2.8) u(t,x) := ÊG
[
ϕ
(
x +
√
tX
)]
, (t,x) ∈ [0,∞) ×R.

The G-heat equations can be also written in the form,

(2.9) ∂tu−
1

2

(
σ̄2(∂2xxu)

+ − σ̂2(∂2xxu)
−
)

= 0.

2.3. G-Brownian Motion. [27, 2019] G-Brownian motion with respect to the G-normal distribution in a

sublinear expectation space is defined as,

Definition 2.8. (G-Brownian Motion) A process
(
BGt (ω)

)
t≥0 is called G-Brownian motion, if for n ∈ N ,

t1 < t2 <,.. . ,< tn and B
G
t1
,BGt2, . . . ,B

G
tn
∈ H it satisfies the following conditions,

• BG0 (ω) = 0

• For every t,s ≥ 0, BGt+s −BGt is N(0;
[
σ̂2s, σ̄2s

]
)-distributed and independent to

(
BGt1,B

G
t2
, . . . ,BGtn

)
G is the same as it was defined in section 2.7. It will be used, for the sake of simplicity, without loss of

generality, in this work, let σ̄ = 1 and σ̂ ≤ 1, by this assumption now

(2.10) G(α) :=
1

2

(
α+ − σ̂2α−

)
, α ∈ R

Now BGt ∼N
(
0;
[
σ̂2s,s

])
. The existence of G-Brownian motion has been proven in [24, 2007].

Definition 2.9. (G-Expectation) [27, 2019] Sublinear expectation is also called G-expectation. The canonical

process
(
BGt
)
t≥0 in the sublinear expectation space

(
Ω, H ,ÊG

)
is G-Brownian process.

There are some properties conditional G-expectations which can be helpful in our study, let for any

X,Y ∈ H 0 here H 0 is the used for L0ip(F),

(1) X ≥ Y =⇒ ÊG [X|Ht] ≥ ÊG [Y |Ht]

(2) ÊG [η|Ht] = η, for everyt ∈ [0,∞) and η ∈ H 0t ,

(3) For every X,Y ∈ χ, ÊG
[
X|Ht] − ÊG[Y |Ht

]
≤ ÊG[X −Y |Ht].

(4) ÊG [ηX] = η+ÊG [X|Ht] + η−ÊG [−X|Ht] , for every η ∈ H 0t .

Sublinear expectation theory also have the Tower property:

(2.11) ÊG
[
ÊG [X|Ht] |Hs

]
= ÊG [X|Ht∧s]


Int. J. Anal. Appl. 19 (1) (2021) 35

and

(2.12) ÊG [X + η|Ht] = ÊG[X] + η, for every t ∈ [0,∞) and η ∈ H 0t

The Proposition 2.1 can be defined similarly in the conditional G-expectation. Some moments of G-

Brownian motion’s increments are, ÊG
[
BGt −BGs |Hs

]
= 0, for every s < t, nth moment of increments is

(2.13) ÊG
[
|BGt −B

G
s |
n|Hs

]
=

1√
2π(t−s)

∫ ∞
−∞
|x|n exp−

x2

2(t−s)
dy,

But,

(2.14) ÊG
[
−|BGt −B

G
s |
n|Hs

]
= −σ̂nÊG

[∣∣BGt −BGs ∣∣n|Hs] .
Now these are the formulas for nth moment, so one can easily calculate the moment which is needed, just

like the classical case.

Definition 2.10. (G-Martingale) [27, 2019] An (Mt)t≥0 process is said to be G-martingale if for every

t ∈ [0,∞) ,Mt ∈ H 0 and for every t ∈ [0, t],

ÊG [Mt|Hs] = Ms.

Similarly, G-submartingale and G-super-martingale are defined as, ÊG [Mt|Hs] ≥ Ms and ÊG [Mt|Hs] ≤ Ms

respectively.

Example 2.1. Processes
(
BGt
)
t≥0 and

(
−BGt

)
t≥0 are G-Martingales and

((
BGt
)2)

t≥0
is G-submartingale.

In classical Brownian motion the quadratic variation of Brownian motion is a deterministic function but,

in G-Brownian motion the quadratic variation is itself a process. The definition of quadratic variation

(cf. [27, 2019]) is,

(2.15) 〈BG〉t = BGt
2
− 2

∫ t
0

BGs dB
G
s .

One can easily verify that,

ÊG[〈BG〉t −〈BG〉s|Ht] = t−s,(2.16)

ÊG[−(〈BG〉t −〈BG〉s)|Ht] = −σ̂2(t−s).(2.17)

Following some lemmas are being written without proofs, for proofs (see [24, 2007])


Int. J. Anal. Appl. 19 (1) (2021) 36

Lemma 2.1. [27, 2019] a) For every s ≥ 0,
(〈
BG
〉
s+t
−
〈
BG
〉
s

)
t≥0

is independent of Fs. This is the

quadratic variation process of the Brownian motion BGt
s

= BGs+t−BGs , t ≥ 0, i.e.,
〈
BG
〉
s+t
−
〈
BG
〉
s

=
〈
BG

s〉
t
,

Moreover, ÊG
[〈
BG
〉2
t

]
≤ 10t2.

b) For square integrable process (ηt)t≥0 in the sense
∫T

0
ÊG
[
|ηt|

2
]
dt, then

ÊG
[∫ T

0

η(s)dBs

]
= 0 ,(2.18)

ÊG

(∫ T

0

η(s)dBs

)2 ≤ ∫ T
0

ÊG
[
(η(t))2

]
dt,(2.19)

ÊG
[∫ t

s

ηud〈BG〉u|Fs
]
≤

∫ t
s

|ηu|du.(2.20)

The distribution of quadratic variation contains mean and variance uncertainty, see equation (2.16) and

above lemmas.

2.4. Ito formula for G-Brownian motion. Like classical Brownian motion, Itô’s formula and integral

can be defined under G-Brownian motion, that is,

Theorem 2.1. (Ito’s formula for G-Brownian motion) [27, 2019] Let the G-Itô process of X is of form,

Xt = Xs + αtdt + ηtd〈BG〉s)t + βtdBGt .

Then the G-Itô formula of Φ(Xt) is given as,

Φ (Xt) = Φ (Xs) +

∫ t
s

αu∂xΦ (Xu) du +

∫ t
s

βu∂xΦ (Xu) dB
G
u

+

∫ t
s

(
ηu∂xΦ (Xu) +

1

2
β2u∂xxΦ (Xu)

)
d
〈
BG
〉
u
.

(2.21)

This can be proved by using Taylor series and G-Itô table,

dt dBGt d
〈
BG
〉
t

dt 0 0 0

dBGt 0 d
〈
BG
〉
t

0

d
〈
BG
〉
t

0 0 0

2.5. Product formula for G-Ito processes. For the product G-Itô formula, the technique is same as, it

was in case of standard Brownian motion, Let Xt and Yt are two G-Itô processes than product G-Itô formula

is [7, 2014],

(2.22) d(XtYt) = YtdXt + XtdYt + dXtdYt.


Int. J. Anal. Appl. 19 (1) (2021) 37

3. The weather derivatives market

3.1. The Weather Derivative Contract. Weather derivatives are structured as futures, swaps, and

put/call options against different underlying weather indices some of them are cooling and heating degree-

days (defined in next section), snowfall and rain. But here we will discuss only underlying index tempera-

ture(degree days indices). We are giving some definitions and terminology. From now we speak only about

the temperature index. Some definitions which we will use in modeling temperature

Definition 3.1. (Temperature) Given a specific weather station, let Tmaxi and T
min
i are the maximal and

minimal temperatures(Celsius) of iTh day. We define temperature of day i as

(3.1) Ti ≡
Tmaxi + T

min
i

2
.

Definition 3.2. (Degree-days) Let Ti denote the temperature on day i. We define the heating degree-days,

HDDi and the cooling degree-days, CDDi, as

(3.2) HDDi ≡ max{18 −Ti, 0}.

and

(3.3) CDDi ≡ max{Ti − 18, 0}.

respectively.

In the above definitions, it can be seen that the HDDs and CDDs for a specific day are just the number

of degrees that the temperature is deviating from a fixed level called reference level. The names cooling and

heating degree-days originate from the US energy sector because if the temperature is below 18◦C people

tend to use more energy to heat their homes, whereas if the temperature is above 18◦C people start to cool

their homes.

Temperature based weather derivatives is based on the accumulation of HDDs or CDDs during a certain

period, like one calendar month or a winter/summer period. mostly the HDD season includes the winter

months from November to March and CDD season is from May to September. April and October are often

referred to as the ’shoulder months’.

3.2. The CME contract. The CME deals with futures based on the CME Degree Day Index, the aggregate

amount of a calendar month’s average HDDs or CDDs, as well as options for those futures. For more than

11 U.S. cities, the CME Degree Day Index is actually listed.

The futures of the HDD and CDD index are agreements to purchase or sell the HDD and CDD index value

at a specific future date. One contract’s notional value is $100 times the Degree Day Index, and the contracts


Int. J. Anal. Appl. 19 (1) (2021) 38

are quoted as HDD and CDD Index points. The futures are cash-settled, meaning that There is index-based

regular labeling with the gain or loss added to the customer’s account

A CME HDD or CDD call option is a contract that offers the owner the right to buy a HDD / CDD futures

contract at a specific price usually called the strike or exercise price, but not the commitment. Analogously,

the HDD / CDD put option grants the owner the right to sell one HDD / CDD futures contract, but not the

obligation. At the CME, the future options are European style which means that they can only be exercised

at the expiration date, Which means they can only be exercised on the expiry date.

3.3. Weather Options. There are several different contracts traded on the OTC market as mentioned

above. The option is the common type of contract. Calls and puts are two main types of options.

• A call option is the right to buy a specific asset for an agreed amount at a fixed time in the future,

as you must pay the premium at the outset of the deal by buying the right to purchase or not.

• A put option is the right to sell an asset at a fixed time in the future for an agreed amount.

Let someone purchase the call option for some fixed strike level then if the number of HDDs for the contract

period is greater than the agreed strike level, the buyer will receive a payout. The size of the payout is

determined by the strike and the tick size. The tick size is the amount of money that the holder of the call

receives for each degree-day above the strike level for the period. Often the option has a cap on the maximum

payout unlike, for example, traditional options on stocks. A generic weather option can be formulated by

specifying the following parameters:

• The contract nature (call or put)

• The contract tenure (e.g. December 2019)

• The underlying index (HDD or CDD)

• An official weather station for temperature data

• The strike level

• The tick size

• The maximum payout (if the option is capped)

The aim of this study is to find the formula for the payout of contracted option, let K is the strike level and

α is the tick size and contract period is m days. Then the number of CDDs and HDDs for m days period

are,

(3.4) Hm =

m∑
i=1

HDDi and Cm =

m∑
i=1

CDDi.

Then the formula for payout of uncapped HDD call can be written as

(3.5) χ = α max{Hm −K, 0}.


Int. J. Anal. Appl. 19 (1) (2021) 39

Formula for payout of uncapped CDD call can be written as

(3.6) χ = α max{Cm −K, 0}.

Payouts of HDD and CDD puts can be defined similarly.

4. Pricing Weather Derivatives through G-Brownian

4.1. Temperature Model under G-framework.

Theorem 4.1. Let the test model for the temperature under G-Brownian motion and from the Girsanov

theorem under G-framwork (cf. [32, 2011]) one can find, the risk neutral measure Q such that,

(4.1) dTt =

(
dTmt
dt

+ a(Tmt −Tt) −η(t) −λσt
)
dt + σtdB

G
t + η(t)d〈B

G〉t.

Here BGt and 〈BG〉t are G-Brownian motion and quadratic variation of G-Brownian motion respectively, η(t)

is any integrable deterministic function. Then solution of (4.12),

Tt = e
−a(t−s)

(
Ts +

∫ t
s

βue
−a(s−u)du +

∫ t
s

σue
−a(s−u)dBGu +

∫ t
s

ηue
−a(s−u)d〈BG〉u

)
.

Moreover, the conditional expectation and conditional variance of Tt can be given as,

ÊG,Q [Tt|Fs] = e−a(t−s)
(
Ts +

∫ t
s

βue
−a(s−u)du + ξη

)
.(4.2)

where ξη := ÊG
(∫ t

s
ηue
−a(s−u)d〈BG〉u|Fs

)
and

V arG,Q [Tt|Fs] ≤ µ̄T (µ̄T + 2µ̂T ) +

e−2a(t−s)


 T 2s + I(t)2β + +ÊG,Q[I(〈BG〉)2η|Fs] + 2TsI(t)β×∫ t

s

(
2Tsηue

−a(s−u) + 2I(t)βηue
−a(s−u) + σ2ue

−2a(s−u)
)
du


 ,

where

I(t)β :=

∫ t
s

βue
−a(s−u)du, I(BG)σ =

∫ t
s

σue
−a(s−u)dBGu ,

I(〈BG〉)η :=
∫ t
s

ηue
−a(s−u)d〈BG〉u.

Proof. Let us start by rewriting the (4.12) in G-Ito form,

dTt =

(
Tmt
dt

+ a(Tmt −Tt) −ηt −λσt
)
dt + σtdB

G
t + ηtd

〈
BG
〉
t
,

=

(
Tmt
dt

+ aTmt −aTt −ηt −λσt
)
dt + σtdB

G
t + ηtd

〈
BG
〉
t
,

=

((
Tmt
dt

+ aTmt −ηt −λσt
)
−aTt

)
dt + σtdB

G
t + ηtd

〈
BG
〉
t
.


Int. J. Anal. Appl. 19 (1) (2021) 40

By setting βt =
Tmt
dt

+ aTmt −ηt −λσt,

dTt = [βt −aTt] dt + σtdBGt + ηtd〈B
G〉t,(4.3)

by solving equation (4.3) through G-Itô formula equation (2.21) and equation (2.22), first solving the homo-

geneous part of equation (4.3), that is

dTt = −aTtdt.(4.4)

Integrating this from s to t,

ln Tt − ln Ts = −a(t−s)

Tt = Tse
−a(t−s)

Set φs,t = Tt −Ts, and solving equation (4.3), by using following transformation.

(4.5) u(t,Tt) = Ttφ
−1
s,t .

Applying the product G-Itô formula (i.e. (2.22)) on equation (4.5) and using the G-Itô table 2.4, one can

get the following results,

d(Ttφ
−1
s,t ) = dTtφ

−1
s,t + Ttdφ

−1
s,t + dTtdφ

−1
s,t

d(Ttφ
−1
s,t ) = ([βt −aTt]dt + σtdB

G
t + ηtd〈B〉t)φ

−1
s,t + aTtφ

−1
s,t + 0

d(Ttφ
−1
s,t ) = (βtdt + σtdB

G
t + ηtd〈B〉t)φ

−1
s,t .(4.6)

Integrating equation (4.6) from s to t and by simplification, the solution of equation (4.3) is,

Tt = e
−a(t−s)

(
Ts +

∫ t
s

βue
−a(s−u)du +

∫ t
s

σue
−a(s−u)dBGu +

∫ t
s

ηue
−a(s−u)d〈BG〉u

)
.(4.7)

The equation (4.2) represents the stochastic model of temperature in G-framework.

The conditional expected value is,

ÊG,Q [Tt|Fs] = ÊG,Q
[
e−a(t−s)

(
Ts +

∫ t
s

βue
−a(s−u)du +

∫ t
s

σue
−a(s−u)dBGu

+

∫ t
s

ηue
−a(s−u)d〈BG〉u

)
|Fs
]

= e−a(t−s)
(
Ts +

∫ t
s

βue
−a(s−u)du

)
+ ÊG,Q

(∫ t
s

e−a(s−u)σudB
G
u

)
+ ÊG,Q

(∫ t
s

e−a(s−u)ηud
〈
BG
〉
u
| Fs

)
(4.8)


Int. J. Anal. Appl. 19 (1) (2021) 41

Set ξη := ÊG
(∫ t

s
ηue
−a(s−u)d〈BG〉u|Fs

)
and using properties of sublinear expectations from section 2, Propo-

sition 2.3 and Lemma 2.1,

ÊG,Q[Tt|Fs] = e−a(t−s)
(
Ts +

∫ t
s

βue
−a(s−u)du + ξη

)
.(4.9)

The conditional variance can be defined same as it was in linear expectation notion. Let µ̄T = ÊG[Tt|Fs],

such that µ̄T ≥ 0 also let µ̂T = ÊG[−Tt|Fs] now variance is

V arG,Q [Tt|Fs] = ÊG,Q[(Tt − µ̄T )
2 |Fs]

= ÊG,Q[T 2t + µ̄
2
T − 2µ̄TTt|Fs]

= µ̄2T + Ê
G,Q[T 2t + 2µ̄T [−Tt]|Fs]

≤ µ̄2T + Ê
G,Q[T 2t |Fs] + Ê

G,Q[2µ̄T (−Tt)|Fs]

= µ̄2T + Ê
G,Q[T 2t |Fs] + 2µ̄T Ê

G,Q[−Tt|Fs]

= µ̄2T + Ê
G,Q[T 2t |Fs] + 2µ̄T µ̂T

V arG,Q [Tt|Fs] ≤ µ̄T (µ̄T + 2µ̂T ) + ÊG,Q[T 2t |Fs].(4.10)

Let N = µ̄T (µ̄T + 2µ̂T ) + ÊG[T 2t |Fs] and from the fact that V ar [Tt|Fs] ≥ 0 one can write it as Σ̄2 =

V ar [Tt|Fs] ∈ [0,N]. Here ÊG,Q[T 2t |Fs] can be estimated by following approximation of it, but for the sake

of simplicity there are some conventional notations have been defined as, let

I(t)β :=

∫ t
s

βue
−a(s−u)du, I(BG)σ :=

∫ t
s

σue
−a(s−u)dBGu ,

I(〈BG〉)η :=
∫ t
s

ηue
−a(s−u)d〈BG〉u.

Now

ÊG,Q
[
T 2t |Fs

]
= ÊG,Q

[(
e−a(t−s)

(
Ts +

∫ t
s

βue
−a(s−u)du +

∫ t
s

σue
−a(s−u)dBGu

+

∫ t
s

ηue
−a(s−u)d〈BG〉u

))2 ∣∣Fs
]

= e−2a(t−s)ÊG,Q
[
T 2s + I(t)

2
β + I(B

G)2σ + I(〈B
G〉)2η + 2TsI(t)β + 2TsI(B

G)σ

+2TsI(〈BG〉)η + 2I(t)βI(BG)σ + 2I(t)βI(〈BG〉)η + 2I(BG)σI(〈BG〉)η
∣∣Fs] ,

using properties of sublinear expectations from section 2, Proposition 2.3 and Lemma 2.1,

ÊG,Q
[
T 2t |Fs

]
≤ e−2a(t−s)

(
T 2s + I(t)

2
β + 2TsI(t)β + 2TsÊ

G,Q[I(BG)σ|Fs]

+ 2TsÊG,Q[I(〈BG〉)η|Fs] + 2I(t)βÊG,Q[I(BG)σ|Fs] + 2I(t)βÊG,Q[I(〈BG〉)η|Fs]

+ÊG,Q[I(BG)2σ|Fs] + Ê
G,QI(〈BG〉)2η|Fs] + 2Ê

G,Q[I(BG)σI(〈B〉)η|Fs]
)
,


Int. J. Anal. Appl. 19 (1) (2021) 42

Using some properties from [24, 2007], and properties of sublinear expectations from section 2, Proposition

2.3 and Lemma 2.1,

ÊG,Q[T 2t |Fs] ≤ e
−2a(t−s)




T 2s + I(t)
2
β + 2TsI(t)β + 2Ts

∫ t
s
ηue
−a(s−u)du

+2I(t)β
∫ t
s
|ηu|e−a(s−u)du +

∫ t
s
σ2ue
−2a(s−u)du

+ÊG,Q[I(〈BG〉)2η|Fs]


 ,

≤ e−2a(t−s)




T 2s + I(t)
2
β + 2TsI(t)β + 2Ts

∫ t
s
ηue
−a(s−u)du

+2I(t)β
∫ t
s
ηue
−a(s−u)du +

∫ t
s
σ2ue
−2a(s−u)du+

ÊG,Q[I(〈BG〉)2η|Fs]




≤ e−2a(t−s)




T 2s + I(t)
2
β + 2TsI(t)β + 2Ts

∫ t
s
ηue
−a(s−u)du

+2I(t)β
∫ t
s
ηue
−a(s−u)du +

∫ t
s
σ2ue
−2a(s−u)du+

ÊG,Q[I(〈BG〉)2η|Fs]




≤ e−2a(t−s)




T 2s + I(t)
2
β + 2TsI(t)β×∫ t

s

(
2Tsηue

−a(s−u) + 2I(t)βηue
−a(s−u) + σ2ue

−2a(s−u)
)
du

+ÊG,Q[I(〈BG〉)2η|Fs]


 .

(4.11)

Substituting the value of

V arG,Q [Tt|Fs] ≤ µ̄T (µ̄T + 2µ̂T ) + e−2a(t−s) ×


T 2s + I(t)
2
β + 2TsI(t)β×∫ t

s

(
2Tsηue

−a(s−u) + 2I(t)βηue
−a(s−u) + σ2ue

−2a(s−u)
)
du

+ÊG,Q[I(〈BG〉)2η|Fs]


 .

�

4.2. Pricing Weather Derivative.

Theorem 4.2. Suppose that dynamics of temperatures process (Tt)t≥0 satisfies following the G-SDE,

dTt =

(
dTmt
dt

+ a(Tmt −Tt) −η(t) −λσt
)
dt + σtdB

G
t + η(t)d〈B

G〉t.

Then the risk-neutral (arbitrage free) price HCt of the uncapped HDD call (described in section 3) satisfies

the following estimate,

HCt ≤ 18mα−e−(tm−t)Kα−
m∑
i=1

α

(
e−a(ti−s)

(
Tti +

∫ ti
s

βue
−a(ti−u)du + ξη(ti)

))
.

(4.12)


Int. J. Anal. Appl. 19 (1) (2021) 43

and the risk-neutral (arbitrage free) price CCt of the uncapped CDD call (described in section 3) satisfies the

following estimate,

CCt ≤ e−(tm−t)Kα + 18mα +
m∑
i=1

(
αe−a(ti−s)

(
Tti +

∫ ti
s

βue
−a(ti−u)duξη(ti)

))
.

(4.13)

where ξη = ÊG,Q
(∫ t

s
ηue
−a(s−u)d〈BG〉u|Fs

)
.

Proof. Let us recall, from section 3, the payout of HDD call with tick size α is

(4.14) χ = α(Hm −K)+,

where Hm =
∑m
i=1 max (18 −Tti ), it is known that Tt is G-Normally distributed but the Hm is not G-

Normally distributed just because there is a maximum in its definition, so just for the simplicity if someone

is interested to find out the explicit formula for option, then it will be possible for winter months and with

an assumption that 18−Tti ≥ 0. Now just to be precise, let for winter months Hm = 18m−
∑m
i=1 Tti , which

is G-normally distributed.

ÊG,Q [Hm|Ft] = ÊG,Q
[

18m−
m∑
i=1

Tti|Ft

]
= ÊG,Q[18m] + ÊG,Q

[
m∑
i=1

(−Tti )|Ft

]

≤ 18m +
m∑
i=1

ÊG,Q [−Tti|Ft] ,(4.15)

Set

µ̄m = ÊG,Q [Hm|Ft] , µ̄m = ÊG,Q [−Hm|Ft] ,

σ̂2m = V ar [Hm|Ft] , σ̂
2
m = −V ar [−Hm|Ft] .

Now from Feynman-Kac Formula (cf. [27])

HCt = ÊG
(
αe−(tm−t)(Hm −K)+|Ft

)
The case when the derivative is in the money i.e. temperature is above 18 for some days, then

HCt = αe
−(tm−t)ÊG,Q ((Hm −K)|Ft) = αe−(tm−t)ÊG,Q (Hm|Ft) −e−(tm−t)K,

≤ 18mα−e−(tm−t)Kα +
m∑
i=1

αÊG,Q[−Tti|Ft]

≤ 18mα−e−(tm−t)Kα−
m∑
i=1

(
αe−a(ti−s)

(
Tti +

∫ ti
s

βue
−a(ti−u)du + ξη(ti)

))
.

(4.16)

Now lets move towards, uncapped CDD call. Recall the payoff of the uncapped CCD call,

(4.17) χ = α(K −Cm)+,


Int. J. Anal. Appl. 19 (1) (2021) 44

where Cm =
∑m
i=1 max (Tti−18), it is known that Tt is G-Normally distributed but the Hm is not G-Normally

distributed, so to find out the explicit formula for option, then it will be possible for summer months there

can be days where the temperature goes above 18 i.e. Tti − 18 ≥ 0. Now just to be precise, let for winter

months Cm =
∑m
i=1 Tti − 18m, which is G-normally distributed.

ÊG,Q [Cm|Ft] = ÊG,Q
[
m∑
i=1

Tti − 18m|Ft

]
= ÊG,Q

[
m∑
i=1

(Tti )|Ft

]
− 18m

≤
m∑
i=1

ÊG,Q[Tti|Ft] − 18m,(4.18)

Now from Feynman-Kac Formula (cf. [27])

Ct = ÊG,Q
(
αe−(tm−t)(K −Cm)+|Ft

)
The case when the derivative is in the money i.e. temperature is above 18 for some days, then

CCt = αe
−(tm−t)ÊG ((K −Cm)|Ft) ,

= e−(tm−t)Kα−αe−(tm−t)ÊG,Q (Cm|Ft) ,

≤ e−(tm−t)Kα + 18mα +
m∑
i=1

αÊG,Q[Tti|Ft]

≤ e−(tm−t)Kα + 18mα +
m∑
i=1

(
αe−a(ti−s)

(
Tti +

∫ ti
s

βue
−a(ti−u)du + ξη(ti)

))
.

(4.19)

�

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	2. Preliminaries
	2.1. Sublinear Expectation
	2.2. G-normal distribution
	2.3. G-Brownian Motion
	2.4. Ito formula for G-Brownian motion
	2.5. Product formula for G-Ito processes

	3. The weather derivatives market
	3.1. The Weather Derivative Contract
	3.2. The CME contract
	3.3. Weather Options

	4. Pricing Weather Derivatives through G-Brownian
	4.1. Temperature Model under G-framework.
	4.2. Pricing Weather Derivative.

	References