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We study a triple of stochastic processes: a Wiener process , , its running maxima process , and its running minima process . We derive the analytical formula for the corresponding copula and show that it is supported on the hemicube, a convex hexahedron with seven vertices. As an application, we draw out an analytical formula for pricing of a double barrier option.
The Wiener process is one of the most ubiquitous stochastic processes used to model complex phenomena. One of such areas is financial mathematics, where this process is the foundation for all models of stock, derivatives, or portfolio analysis. But with the development of the market, more complex derivatives have shown up, and a simple approach to the Wiener process was no longer enough. In particular, there was a question as to what is the distribution of the Wiener process adding boundaries on trajectories. It is especially important when one is dealing with barrier options.
The structure of article is as follows. In the first part, there will be provided the main results. We derive the cumulative distribution function and copula for a triple consisting of a Wiener process and its supremum and infimum. The cumulative distribution function is supported on a cone and the copula on a hemicube. Then, we present special cases of this distribution, particularly focusing on marginals. The next part is devoted to proofs of stated theorems. The starting point is the strong Markov property of Wiener processes. First, we study the simple case based on one stopping time. Finally, to the obtain desired results, we move to double stopping time. We provide a recursive dependency, which will lead to the characterization of the cumulative distribution the function and copula in terms of series.
The last parts show applications in financial mathematics, i.e., we calculate the price of a European double-barrier option in the Black–Scholes model. Compare [1,2,3]. Our approach is based on the Girsanov theorem, which implies the existence a probabilistic measure Q under which the increment of the logarithm of a stock price process divided by the volatility follows the Wiener law. The double change of measure, from the risk neutral to Q and back after some integral calculations, significantly simplifies the proof.
2. Main Results
Consider a probability space and a Wiener process with natural filtration with respect to that process, i.e., —where is the Borel -algebra on .
We denote as
and
the associated running maxima and minima processes.
Note that the triple , is a self-similar process:
Let be the cumulative distribution function of the joint distribution , where . As follows from the celebrated Sklar theorem, there exists a copula process, , , such that
For more details about copula theory and some of its applications, we refer to [4,5,6,7,8,9,10]. Since the process is self-similar, the copula of the triple , where , does not depend on t. We will denote it as .
Based on the strong Markov property of a Wiener process and the Sklar theorem, we prove what follows, where denotes the cumulative distribution of the probability law.
Theorem 1.
The joint cumulative distribution function of , where , is of the form
Note that the copula is absolutely continuous with respect to the Lebesgue measure and is supported on the polyhedron
which has seven vertices
six faces, two rectangles, two trapezoids, two triangles, and eleven edges. Such polyhedrons are called hemicubes because they are linearly equivalent to a polyhedron obtained by cutting the unit cube by half by the surface containing two opposite vertices (for example, and ) (Figure 1).
Furthermore, the copula is invariant with respect to the involution induced by the rotation of the unit cube by an angle about the line
For every triple of generators of the copula , the triple is a set of generators of this copula as well. Hence,
The marginal copulas are given by
All three of the above bivariate copulas are absolutely continuous. The first two are supported on trapezoids. The first one is supported on
the second one is supported on
The third one has a full support . The scatterplots of the above copulas are shown on Figure 2, Figure 3 and Figure 4.
Furthermore, the first and second copula are duals of each other: is a survival copula of (and vice versa). As a matter of fact, they are reflections of a copula introduced in [11]. See also [12]. We have
The third one is self-dual with respect to reflection about the line ,
All three marginal copulas are highly dependent.
Proposition 1.
The Spearman ρ of the marginal copulas of the copula is given by
The strong Markov property plays a crucial role in the proof of Theorem 1. We recall the definition.
Definition 1.
Suppose that is a stochastic process on a probability space with natural filtration . Then, for any stopping time τ on Ω, we can define
Then, X is said to have the strong Markov property if, for each finite stopping time τ, we have that for each , is independent of .
It is well known that the Wiener process fulfills the above definition; see, for example, [13] (Section 37, Theorem 37.5), [14] (Section 3.2), or [15] (Theorem 6.5). Furthermore, for a Wiener process , we have
Theorem 2.
Let τ be a finite stopping time and put
Then, is a Brownian motion, and it is independent of .
Based on the fact that is symmetric
we obtain the following version of the reflection principle.
Theorem 3.
Let τ be a finite stopping time and let a set A be an element of of positive probability on which ,
such that under the condition A, the probability laws of and coincide:
Proof.
We observe that under the condition ,
Next, we compare the conditional distribution functions of the right sides of the above equations. We apply the fact that the random variables and are measurable and
where denotes the cumulative distribution function of a standard normal random variable ().
To conclude the proof, we apply the second point of the definition of the conditional expected value (see [13] §34).
□
3.2. Marginal Distributions
We consider two families of stopping times. One corresponds to an upper threshold, and the second corresponds to a lower one. In more details, for and , we put
Note that for fixed positive t, the event is equivalent to the event :
and the event is equivalent to the event :
We apply the reflection principle for , where . Since
the reflection principle simplifies to
For we obtain
The further assumption implies that the event contains the event
Indeed, for every from the first set
Therefore, for and ,
where denotes the cumulative distribution function of a standard normal random variable ().
Similarly, for , , and , putting , we obtain
The above, together with inequalities
allows us to determine the joint distribution functions for the pairs and .
Theorem 4.
The joint cumulative distribution function of , where , is of the form
The event is contained in the event . Indeed, for every from the first set,
Thus,
Case .
Since ,
Thus,
□
Note that, since the distributions of running maxima and minima are closely related with stopping times
they might be drawn from the formulas describing exit times (from the half-line) provided, for example, in [16] (Theorem 4.4.5).
3.3. The Two-Barrier Stopping Time
In this section, we apply the reflection principle for the two-barrier stopping time
and for the set A, which are equal, respectively, to
Note that the range of consists only of two points y and z.
Lemma 1.
For any
Proof.
Applying twice the reflection principle, we obtain
Since , the event is contained in the event , and
Therefore,
where denotes the distribution function of an random variable:
Similarly, since and , the event is contained in the event , so
Therefore,
In such a way, we obtain the recurrence for :
Since , we may repeat it. After n repetitions, we obtain
Since
and passing to the limit , we obtain
□
Lemma 2.
For any
Proof.
We base our proof on the symmetry of the Wiener process . We put . Then, we have, for ,
Therefore,
Since is as well a Wiener process, we apply Lemma 1 and obtain
□
Together, the above lemmas give us
Corollary 1.
For and ,
where
Furthermore,
and
Proof.
We observe that
Thus, applying Lemmas 1 and 2, we obtain
Since the event implies the event , we have
Therefore, substituting in the above formula, we obtain
This concludes the proof of the first two formulas of the corollary. The third one follows from the symmetry of the Wiener process. We apply the notation from the proof of Lemma 2.
Since is a Wiener process,
Since
the last equality of the corollary is a consequence of the two previous ones. □
Note that, since the Formula (31) is describing an exit time from a segment, it might be obtained as a solution of a partial differential equation (see [16] Theorem 4.4.5).
3.4. Joint Cumulative Distribution
Theorem 6.
The joint cumulative distribution function of , where , is of the form
where
where
and Φ denotes the distribution function of the standard normal probability law N(0, 1).
Note that the distribution of is absolutely continuous with respect to the Lebesgue measure (Section 3.5) and is supported on the set
Furthermore, the marginal distributions of F are given by the following formulas:
We prove Theorem 6 case by case.
Proof.
Case .
Since is almost surely continuous, and , is almost surely positive:
Case and .
Since is almost surely negative,
Following Theorem 4, we obtain
Case and .
Since the event implies the event ,
As above, following Theorem 4, we obtain
Case and .
We observe that
where is the two-barrier stopping time discussed in Section 3.3. Following Corollary 1 and Theorem 4, we obtain
Case and .
Since ,
□
As a corollary of Theorem 6, we prove the 80-year-old characterization of the Wiener process called the “Lévy Triple Law” ([15] Theorem 6.18), which deals with the trajectories lying between the upper barrier u and lower barrier l.
Corollary 2.
For , , and ,
where denotes the density of .
Proof.
We observe that
Hence,
After shifting by one the parameter k in the second sum, we obtain the required Formula (41).
□
3.5. Properties of the Cumulative Distribution Function F and the Marginal Distributions
We start with the absolute continuity of the marginal distributions of random pairs and :
Proposition 2.
The bivariate cumulative distributions and are absolutely continuous with respect to the Lebesgue measure.
Proof.
We start with the function . We show that a function
is a corresponding density. Let denote an integral of g over the set .
Since all points , , and are equal each other, we conclude that is absolutely continuous.
Next, since is a Wiener process, the pairs and have the same distribution
Therefore, is also absolutely continuous. □
Now, we are in position to show the absolute continuity of the cumulative distribution function F of the random triple .
Proposition 3.
The function F is absolutely continuous with respect to the Lebesgue measure. Furthermore,
Proof.
Since the definition of running maxima and minima implies that
we have
We show that the probability that the random triple belongs to the boundary and is equal to 0.
Since the distribution functions of and are continuous, we have
Since the distribution functions of the random pairs and are absolutely continuous,
To conclude the proof, it is enough to observe that the function F is on the set
Hence, it is absolutely continuous with respect to the Lebesgue measure. □
3.6. Proof of Theorem 1
We recall that a copula is the restriction to the unit n-cube of a distribution function whose univariate margins are uniformly distributed on . Specifically, a function is a copula if it has the following properties. For every and such that for , we have the following:
(C1)
;
(C2)
;
(C3)
C is n-nondecreasing, that is, the C-volume of any n-rectangle with lower vertex and upper vertex is non-negative.
We recall that the C-volume is a signed sum of the values of C at the vertices of the n-rectangle,
where , and .
According to the celebrated Sklar’s theorem, the joint distribution function F of any n-tuple of random variables defined on the probability space can be written as a composition of a copula C and the univariate marginals , i.e., for all ,
Moreover, if the distribution functions are continuous, then the copula C is uniquely determined and can be described in terms of the quantile functions, i.e., the generalized inverses of . Since , we obtain
In our case where and for , the quantile functions of , and are given by
To conclude the proof of Theorem 1, we substitute the above quantile functions into Formula (33).
3.7. Properties of the Copula and Its Marginal Copulas
The copulas mentioned in Section 2 regarding the involution of the copula and its marginal copulas follows from the fact that is a Wiener process as well. Hence, the triple has the same distribution as the triple . We put
Obviously, are the generators of the copula
Since
we apply the inclusion/exclusion principle and obtain
The involutions of the marginal copulas are obtained by substitution in the preceding formulas, respectively, where , , or .
Proof of Proposition 1.
We recall that the Spearman of the copula C is given by the formula ([10] Theorem 5.1.6)
Due to the involution (4), the Spearman of the copula is the same as the Spearman of the copula . We observe that
The first two integrals are obvious:
To calculate the third integral, we substitute
We obtain
Note that the product is a density of the three-dimensional standard normal distribution , which is a spherical distribution. Therefore, the above integral can be expressed in terms of the volume of a spherical triangle. We recall that such a volume equals
where are inner angles (in radians) of the triangle (see, for example, [17] Formula (1.12.2) in Section 1.12). Note that the cosine of the inner angle of the intersection of two half-spaces is given by the scalar product of normal vectors:
Finally, we conclude the following:
The case of the copula can be handled in a similar way. We substitute
Next, we observe that for , we have
Note that . Therefore, putting respectively and , we obtain
Finally, we conclude the following:
□
4. Application of Results: Double-Barrier Option Pricing
In this section, we present one of the possible applications of the derived results. We consider probability space and the Black–Scholes–Merton market model [18] or [19] in which the asset price dynamic follows the geometric Brownian motion, i.e.,
where , , and is a Wiener process. Additionally, the bank account is described with the following differential equation:
The solution to this system of equations is well known in the literature, and we present only a final result without proof. The reader can check that
and
indeed satisfy the above stochastic differential equations.
To valuate in this model a derivative with a payoff X, which is measurable ( is the natural filtration of ), we need to calculate the expected value of the form
where is so-called risk neutral measure, i.e., the probability measure equivalent to and described by the following Radon–Nikodym derivative
Note that, for , coincides with a Wiener process in measure.
For example the Black–Scholes formula for the price of an European call option is
Notice that
This means that under the risk neutral measure , the stochastic differential equation of is of the form
We continue to apply the Girsanov theorem. Let be the probability measure given by the Radon–Nikodym derivative of the form
In the measure Q, for , coincides with a Wiener process. The reader is referred to [20] Theorem 8.6.4 for more details. As in Section 2, we will denote by and the running maxima and minima of .
Note that the inverse Radon–Nikodym derivative is given by the formula
Now, we can move on to the valuation of the double-barrier options.
A double-barrier option is a derivative whose payoff is path-dependent. There are few possibilities, like starting between arbitrary values , and the option becomes worthless whenever crosses at least one of these two barriers. To be precise, we restrict ourselves to the double-barrier European call option with the payoff X at T of the form
where is the indicator of the event , and .
First of all, notice that
is equivalent to
Analogously,
is equivalent to
Now, we are ready to move on toward the final price of the derivative.
Theorem 7.
The price of a European double-barrier option maturing at time T with a strike K and barriers satisfying with the payoff (60) of the form
where
Note that since, for k tending to , are converging to 0, for practical purposes, it is enough to consider only a couple of terms with small values. The similar formulas were provided by Haug ([3]) and by Barker ([1]).
Proof.
The price of a European double-barrier option is given by a formula:
where
and
Next, we apply Theorem 6. Basing on the fact that the density of the joint distribution of , , is vanishing outside the set
we obtain
where denotes the density of the normal distribution , and
Note that the formula in the last brackets may be obtained as well from the Lévy Triple Law; see Corollary 2 or [15] Theorem 6.18.
Next, we return to the probability .
Since
we have
To conclude, for the proof, it is enough to put, respectively, or . □
The alternative method of pricing the double-barrier options is via simulations in the risk-neutral measure. In the Table 1, we present the results. The simulation is provided by an approximation of the Wiener process by a Random Walk process with 100,000 steps. We iterated the procedure 1534 times. The analytic formula for the price is approximated by the sum of the seven terms , where .
Author Contributions
Writing—original draft, K.D. and P.J. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Data are contained within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
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Figure 1.
Polyhedron P—the support of the copula.
Figure 1.
Polyhedron P—the support of the copula.
Figure 2.
Copula (scatterplot).
Figure 2.
Copula (scatterplot).
Figure 3.
Copula (scatterplot).
Figure 3.
Copula (scatterplot).
Figure 4.
Copula (scatterplot).
Figure 4.
Copula (scatterplot).
Table 1.
The comparison of simulated (S) and approximated (A) values of the two-barrier call options with varying strike K and risk-free interest rate r (in %). The initial value of the stock, barriers, volatility, and time to maturity are fixed: , , , , and .
Table 1.
The comparison of simulated (S) and approximated (A) values of the two-barrier call options with varying strike K and risk-free interest rate r (in %). The initial value of the stock, barriers, volatility, and time to maturity are fixed: , , , , and .
r
1
3
6
9
12.5
S
A
S
A
S
A
S
A
S
A
80
7.318
7.373
7.434
7.39
7.486
7.412
7.556
7.428
7.507
7.439
81
6.981
7.037
7.095
7.055
7.149
7.078
7.219
7.095
7.178
7.109
82
6.645
6.703
6.757
6.722
6.813
6.745
6.883
6.764
6.849
6.779
83
6.31
6.371
6.42
6.39
6.477
6.415
6.547
6.435
6.521
6.452
84
5.977
6.042
6.086
6.062
6.144
6.088
6.214
6.109
6.195
6.127
85
5.649
5.718
5.755
5.738
5.814
5.765
5.884
5.787
5.871
5.807
86
5.327
5.4
5.432
5.42
5.49
5.447
5.557
5.47
5.55
5.491
87
5.012
5.087
5.113
5.108
5.173
5.135
5.239
5.159
5.233
5.182
88
4.703
4.782
4.803
4.802
4.862
4.83
4.926
4.855
4.925
4.878
89
4.403
4.484
4.501
4.504
4.56
4.533
4.62
4.557
4.622
4.582
90
4.113
4.194
4.207
4.215
4.265
4.243
4.323
4.268
4.326
4.293
91
3.834
3.913
3.924
3.934
3.98
3.962
4.035
3.987
4.04
4.012
92
3.568
3.642
3.654
3.662
3.707
3.69
3.757
3.715
3.764
3.741
93
3.311
3.38
3.395
3.4
3.446
3.428
3.492
3.453
3.497
3.478
94
3.065
3.129
3.146
3.148
3.194
3.175
3.24
3.2
3.244
3.225
95
2.828
2.888
2.908
2.907
2.955
2.933
2.998
2.957
3.002
2.982
96
2.601
2.658
2.678
2.676
2.725
2.701
2.768
2.725
2.77
2.749
97
2.388
2.438
2.461
2.455
2.504
2.48
2.546
2.503
2.549
2.527
98
2.183
2.229
2.255
2.246
2.296
2.269
2.334
2.291
2.337
2.315
99
1.988
2.031
2.056
2.047
2.097
2.07
2.136
2.091
2.137
2.113
100
1.806
1.844
1.871
1.859
1.906
1.881
1.945
1.901
1.949
1.923
101
1.633
1.668
1.696
1.682
1.73
1.703
1.763
1.722
1.768
1.742
102
1.47
1.503
1.529
1.516
1.561
1.535
1.595
1.553
1.598
1.573
103
1.318
1.348
1.374
1.361
1.403
1.378
1.434
1.395
1.441
1.414
104
1.176
1.204
1.229
1.215
1.256
1.232
1.284
1.248
1.291
1.265
105
1.045
1.07
1.095
1.08
1.119
1.096
1.146
1.11
1.151
1.126
106
0.923
0.946
0.972
0.956
0.993
0.97
1.017
0.983
1.023
0.998
107
0.813
0.832
0.857
0.84
0.877
0.853
0.9
0.865
0.904
0.879
108
0.712
0.727
0.753
0.735
0.77
0.746
0.792
0.757
0.795
0.769
109
0.621
0.631
0.658
0.638
0.673
0.648
0.693
0.658
0.696
0.669
110
0.54
0.544
0.573
0.55
0.585
0.559
0.604
0.568
0.606
0.578
111
0.465
0.465
0.496
0.471
0.507
0.479
0.524
0.487
0.525
0.495
112
0.395
0.394
0.424
0.399
0.436
0.406
0.452
0.413
0.452
0.421
113
0.338
0.331
0.362
0.335
0.37
0.341
0.387
0.347
0.387
0.354
114
0.285
0.275
0.308
0.279
0.315
0.284
0.325
0.289
0.328
0.295
115
0.237
0.226
0.257
0.229
0.266
0.233
0.275
0.237
0.273
0.242
116
0.195
0.183
0.213
0.185
0.22
0.189
0.23
0.193
0.228
0.197
117
0.159
0.146
0.175
0.148
0.18
0.151
0.188
0.154
0.189
0.157
118
0.127
0.114
0.141
0.116
0.146
0.118
0.152
0.12
0.151
0.123
119
0.097
0.087
0.111
0.089
0.115
0.091
0.121
0.092
0.119
0.095
120
0.073
0.065
0.085
0.066
0.088
0.068
0.093
0.069
0.093
0.071
121
0.054
0.047
0.063
0.048
0.065
0.049
0.069
0.05
0.07
0.051
122
0.038
0.033
0.046
0.034
0.047
0.034
0.049
0.035
0.049
0.036
123
0.025
0.022
0.032
0.022
0.033
0.023
0.034
0.023
0.033
0.024
124
0.014
0.014
0.02
0.014
0.021
0.014
0.023
0.015
0.021
0.015
125
0.007
0.008
0.011
0.008
0.012
0.008
0.014
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Da̧browski, K.; Jaworski, P.
Modelling the Dependence between a Wiener Process and Its Running Maxima and Running Minima Processes. Mathematics2024, 12, 2707.
https://doi.org/10.3390/math12172707
AMA Style
Da̧browski K, Jaworski P.
Modelling the Dependence between a Wiener Process and Its Running Maxima and Running Minima Processes. Mathematics. 2024; 12(17):2707.
https://doi.org/10.3390/math12172707
Chicago/Turabian Style
Da̧browski, Karol, and Piotr Jaworski.
2024. "Modelling the Dependence between a Wiener Process and Its Running Maxima and Running Minima Processes" Mathematics 12, no. 17: 2707.
https://doi.org/10.3390/math12172707
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.
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Da̧browski, K.; Jaworski, P.
Modelling the Dependence between a Wiener Process and Its Running Maxima and Running Minima Processes. Mathematics2024, 12, 2707.
https://doi.org/10.3390/math12172707
AMA Style
Da̧browski K, Jaworski P.
Modelling the Dependence between a Wiener Process and Its Running Maxima and Running Minima Processes. Mathematics. 2024; 12(17):2707.
https://doi.org/10.3390/math12172707
Chicago/Turabian Style
Da̧browski, Karol, and Piotr Jaworski.
2024. "Modelling the Dependence between a Wiener Process and Its Running Maxima and Running Minima Processes" Mathematics 12, no. 17: 2707.
https://doi.org/10.3390/math12172707
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.