CC BY
22ORIGINAL PAPER
Doi:10.26794/2308-944X-2021-9-3-52-76 JEL C02, G13
Proximity of Bachelier and Samuelson Models for Different Metrics
Sergey Smirnova, Dmitry Sotnikovb
Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia aoRCID: https://orcid.org/0000-0002-1650-222X aScopus Author id: 57211114685 bORCID: https://orcid.org/0000-0003-4731-7841
ABSTRACT
This paper proposes a method of comparing the prices of European options, based on the use of probabilistic metrics, with respect to two models of price dynamics: Bachelier and Samuelson. In contrast to other studies on the subject, we consider two classes of options: European options with a Lipschitz continuous payout function and European options with a bounded payout function. For these classes, the following suitable probability metrics are chosen: the Fortet-Maurier metric, the total variation metric, and the Kolmogorov metric. it is proved that their computation can be reduced to computation of the Lambert in case of the Fortet-Mourier metric, and to the solution of a nonlinear equation in other cases. A statistical estimation of the model parameters in the modern oil market gives the order of magnitude of the error, including the magnitude of sensitivity of the option price, to the change in the volatility.
Keywords: Bachelier model; Samuelson model; option pricing; probabilistic metrics; sensitivity; volatility
For citation: Smirnov, S., & Sotnikov, D. (2021). Proximity of Bachelier and Samuelson Models for Different Metrics. Review of Business and Economics Studies, 9(3),52-76. DOI: 10.26794/2308-944X-2021-9-3-52-76
Близость моделей Башелье и Самуэльсона для различных метрик
Сергей Смирнова, Дмитрий Сотниковь
'Факультет ВМК, МГУ имени М. В. Ломоносова ьФакультет ВМК, МГУ имени М. В. Ломоносова
АННОТАЦИЯ
В статье представлен метод сравнения цен европейских опционов, основанный на использовании вероятностных метрик, применительно к двум моделям динамики цен - Башелье и Самуэльсона. В отличие от других работ на данную тему, рассматриваются классы опционов, а именно европейские опционы с функцией выплат, удовлетворяющих условию Липшица, а также европейские опционы с ограниченной функцией выплат. Для данных классов выбираются подходящие вероятностные метрики: метрика Форте-Мурье, метрика полной вариации и метрика Колмогорова. Мы доказали, что их вычисление сводится к вычислению W -функции Ламберта в случае метрики Форте-Мурье и к решению некоторого нелинейного уравнения в остальных случаях. Статистическая оценка параметров моделей на современном нефтяном рынке указывает на порядок величины погрешности, включая величину чувствительности цены опциона к изменению показателя волатильности.
Ключевые слова: модель Башелье; модель Самуэльсона; ценообразование опционов; вероятностные метрики; чувствительность; волатильность
© Sergey Smirnov, Dmitry Sotnikov, 2021
52
1 Introduction Description of Models and Motivation for the Study
In this study, the simplest continuous-time financial market models are considered. The movement of prices (Xt )t e[0T ] of an asset in the market is described in the framework of the Bachelier model (Bachelier, 1900), using the stochastic Brownian motion process:
XB = X0 (1 + at + oBWt),t e[0,T],#(1)
where (Wt)te[0T] is the Wiener process, aeR,aB >0.
The model proposed by Samuelson1 (1965) uses geometric (economic) Brownian motion to describe the price dynamics:
XS = X0exp[yt + asWt ],t e[0,T],#(2)
where y e R, aS > 0.
In both models, the volatilities aB and cS are chosen so that they have the dimension [time]-1/2 and the linear trend a and exponential trend y have the dimension [time]-1.
Hereafter, the prices considered are assumed to be discounted, which is equivalent to a zero risk-free interest rate.
The Black-Scholes (1973) and Merton (1973) option pricing model is based on the Samuelson model (describing price dynamics in the market) and is the most popular in practice. Similarly, for the options on futures Black's (1976) pricing model is based on Samuelson's model.
Bachelier (1900) not only described the dynamics of prices but also built a model of option pricing. However, Samuelson (1965) noted that the stock prices should not be negative; thus, Bachelier's model has not been widely used in practice. Nevertheless, for short-term options, the Bachelier model can better fit the real market data than the Black-Scholes-Samuelson model (e.g., Versluis (2006)). Note that the Bachelier model and its modifications have been applied to modern works on mathematical finance. For example, the Bachelier model and its modification with an absorption screen was used by Glazyrina and Melnikov (2020) for pricing life insurance policies with an invested stock index option, and Melnikov and Wan (2021) compared this model with the Bachelier and Samuelson models.
An unprecedented event occurred on April 20, 2020, when West Texas Intermediate (WTI) crude oil futures prices (the benchmark for U.S. crude oil prices) reached negative levels (see CFTC Interim Staff Report, Trading in NYMEX WTI Crude Oil Futures Contract Leading up to, on, and around April 20, 2020). Fuel supply has far exceeded the demand due to the coronavirus pandemic. Due to overproduction, the storage tanks were so full that it would have been difficult to find room for new oil if the future contracts had been brought to delivery. Because the May contract expired on April 21, market participants with long positions did not want to take delivery of oil (which no one needed at that point in time) and incur storage costs and opted to lock in such large losses by entering into offset deals that the prices turned negative. As of April 22, 2020, the Chicago Mercantile Exchange (CME) switched to the Bachelier pricing model for the options on futures for several energy commodities2 to account for the possibility of negative prices.
In this regard, it is interesting to compare the prices of derivative financial instruments obtained using the above-described models. Schachermayer and Teichmann (2005) proved the following estimation for the price difference of a call option "at the money" with an expiration at the moment T:
0<CB -CS
B S i2V2n
(aVT )3.
Here, aB = aS =a and CB,CS denote the option prices in the Bachelier and Samuelson models, respectively. Both processes (1) and (2) are diffusion processes; thus, the Bachelier and Samuelson models are clearly close in case of small (and equal) values of integral volatility <B VT = aS 4T = o4T. Meanwhile, the Samuelson model is close3 to the Bachelier model with
a linear trend y +—52.
2
Grunspan (2011) obtained an asymptotic relation between implicit volatilities for normal and lognormal models at T ^ 0 and compared the sensitivities (greeks) for call options. The differences in option pricing obtained using the Bachelier and Samuelson models are detailed in Thomson (2016).
Another question is for what values of <B VT and aS-\T models can be considered close? We
are interested in the problem of comparing the prices of a European option with an arbitrary payoff function f (•) that belongs to a specific class of functions and depends only on the Xj of the underlying asset at the time of expiration J . For each of the models (1) and (2), there exists a single equivalent risk-neutral (martingale) measure. The option price P (f ,T) with payout function f (•) and time to expiration J is determined as the mathematical expectation relative to the corresponding risk-neutral measure4:
P (f JT ) = E*f (Xj).
The processes given by relations (1) and (2) are martingales if and only if
_2
a = 0, y = -2- .# (3)
Therefore, the difference between the option prices PB (f ,T) and PS (f ,T) in the Bachelier and Samuelson models can be expressed as follows:
PB (f ,T) -Ps (f ,T) = Ef (XBT)-Ef (XST),#(4)
where the process parameters are chosen according to (3).
The estimate for the right part of (4) can be obtained by calculating the distance in the Fortet-Mourier metric between the distributions of random variables X^,Xs in case of Lipschitz continuity of the payoff function f (•). If the payout function is discontinuous but bounded (e.g., as in the case of a binary option), the total variation metric can be used for the estimation. However, the Kolmogorov metric can also be used to compare the binary option prices; the closeness of distributions under the total variation metric is a very strong assumption, and hence, the corresponding estimate is rougher (but applicable to a broader class of payout functions).
To compare the Bachelier and Samuelson models, it is interesting to find the optimal relation between the volatilities aB, aT . Optimality is understood as the dependence between these indicators that arises when minimizing the distance between XB and XT in (one or another) probability metric d (•,•).
In this paper, the Fortet-Mourier metric between random variables XB and XT is calcu-
lated and the formulae for the total variation metric and Kolmogorov metric are obtained. The dependence of volatilities that minimizes the Fortet-Mourier metric between XB and Xs . Using the probability metrics, the estimates for (4) are obtained to analyze the effect of model choice on option price. By constructing confidence intervals for volatilities in the oil market for standard and binary call and put options, we evaluate the error resulting from the approximate measurement of the volatility.
Notation and Definitions
Let S be a metric space with metric d(•,•) and let us denote by M(T) the set of all signed measures on S and V(T)c M(T) as the set of all probability measures on T equipped with Borel a -algebra.
Definition 1. Let us define a semi-norm in the space Lip(T) of the Lipschitz continuous on T functions as follows:
ii |f(x)-f(y)| r-= suP ,f (•) e Lip(T).
x,y
d ( x, y )
Definition 2. In the space B (T) of bounded measurable functions on T, let us define the norm
|| f iB = sup| f (x)|, f (•) e B(T).
x eT
Definition 3. For T = M in the space Tt (M) of piecewise constant functions with finite number of jumps A1v..,Am, we define a semi-norm
st = £|A f O St (M).
j=1
The introduced semi-norm is a norm in space
Tt (M) / M.
Definition 4. By the coupling of two random variables X u Y , we c} ll5 a j air (X ',Y') for which the following is true X = X, Y=Y. For the monotone coupling of real random variables X u Y with distribution functions FX (•), Fy (•), we call a pair of
(F- (U),F- (U)),U ~ U(0,1),
where FX is the distribution function of a random variable X , which is defined as
Fx (x) = P(X < x),
and F-1 is the generalized inverse function of the monotonically non-decreasing left-continuous function, defined via the relation6
F-1 (y) = inf {x e R: F (x) > y } = = sup{x e R : F(x) < y},y e (0,1).
Let 8(y) be a metric in the space of random variables taking values in S , defined on pairs of (X ,Y) of random variables, with a common probability space.
Definition 5. The minimal metric with respect to 8(y) is the metric
8 (X, Y) = inf |8( X', Y'): X" X ,Y=Y j.
Note that 8(,) is therefore a metric in the
space of distributions and does not depend on
the joint distribution of X and Y .
Let T be a set of measurable functions
f: S ^ R . Then, for each signed measure | on
S such that J\f\\d||<~ for all f e T , the folS
lowing semi-norm can be defined:
Il M II* = sup
f ^
Jfd M
Denote MT = {|eM (S) :|| |||T <^}.
Definition 6. We can say that on MT the dual semimetric if
dT (|,v) =|| v ||T .
In particular, for the probabilistic measures -t = Mt n- (S),
dT (X, Y) = sup |Ef (X) — Ef (Y)|.
f eT
Let (S, B) be a measurable space.
Definition 7. The total variation norm for a signed measure | is defined as
1111 Tv = sup
jJfd|: f eB(S),| | f ||B< 1j.
Definition 8. A total variation metric is a probability metric
djv (Q1Q2 )=||Q —Q21 Tv .
If distributions Q1,Q2 are absolutely continuous with respect to the measure | and have Ra-don-Nikodym densities p1 (), p2 (), then
dTV (Q1,Q2 ) = J|P1 (x) — P2 (x)|l(dx) =
S
= 2J( px (x) — P2 (x ))+|(dx ),#(5)
S
where a + = max (a,0).
Definition 9. If S = R, then the Kolmogorov metric7 is
dK (X,Y) = supIFX (x) — Fy (x)|.
x eR
Definition 10. The Fortet-Mourier metric8 is the probability metric
dFM (X,Y)= sup |Ef (X) — Ef (Y)|.
| |f | Lip <1
There is also an equivalent representation of this metric:
^FM (X,Y) =
min
Ed (XY') : X-X,Y =Y L#(6)
The proof of equivalence of the definitions can be found in Rachev, Klebanov, Stoyanov, and Fabozzi (2013).
It has been shown (e.g., Bogachev (2007)) that in case of S = R, the minimum value in (6) is attained on the monotone coupling
(Fx"1 (U),Fy~' (U)),U ~ U(0,1).
Remark 1. The Fortet-Mourier metric allows one to derive an upper estimate of (4) in the case of Lipschitz continuity of f ( ), for example, if f ( ) is piecewise linear (which corresponds to the portfolio of call and put options). It is also possible to estimate (4) by using the total variation metric if the function f () is bounded. Even if the payout function is neither Lipschitz continuous nor bounded (e. g., if it corresponds to a portfolio of binary and call options), it can most likely be represented as a sum of ones, as in practice, the payout functions usually do not grow faster than linear ones. The Kolmogorov metric provides a more accurate estimate than the total variation metric; however, it is only applicable to piecewise constant payout functions corresponding to a portfolio composed of binary options
Definition 11. Lambert W function is a complex-valued function W: C ^ C, defined as a solu-
tion of the equation z = W (z)
W (z )
z e<
W () cannot be expressed in elementary functions. We are only interested in its two branches, W0 (z), W_1 (z), at z e (-e_1,0) (Fig. 1), which correspond to the real solutions of the equation
xe = zz
(-e -'.0).
The definition and notation are taken from Corless, Gonnet, Hare, Jeffrey, and Knuth (1996).
i 0 -1
S
7 -2 ä
5 -3 à
-4 -5 -6
-e"1
N Wo(z)
\ s, •s.
'—. N. N. N „ W-i(z)
\ N N
S \ \
\ \
-0.3
-0.2
-0.1
0.0
0.1
0.2
Figure 1. Real-valued branches of Lambert W -function
2 Main Results
Let us show how one can obtain the estimates for (4) by using the introduced probability metrics. Let, as mentioned above, PB (f,T),Ps (f,T) stand Source: The authors. for the prices of European options with payoff function f (•) and time to expiration T in the
Bachelier and Samuelson models, respectively. Then, the following estimates are true: If f (-)e Lip (M), then
\Pb (f ,T)-PT (f ,T)|< i i flLp dFM
(Xf, XT ).#(7) If f Qe B(M), then |Pb (f ,T)-Ps (f ,T)| < i i f i B dw (Xf,XT).#(8)
If f (•) e Tt (M), then |Pb (f,T)-Ps (f ,T )| <i i f i Tt dK (XB, XT ).#(9)
Indeed, the price of a European option is defined in the Bachelier and Samuelson models as a mathematical expectation of the payout function relative to the risk-neutral measure:
PB (f ,T) = Ef (XB),Ps (f ,T) = Ef (XT),
c2
where the processes Xf,Xs are martingales, i.e., a = 0,y = —s .
2
Then,
Pb ( f,T )-Ps ( f,T )|= E ( f ( XB )-f ( XST )
1. In case of Lipschitz continuity of f (•),
|Pb (f,T)-Ps (f,T)|<i i fi \Lip sup Eg (x£ )-Eg (xt ) =i i fi \Lpdm (Xf, XT)
Mhp <1
2. If f (•) is bounded, then
|PB ( f,T )-PT ( f,T )|= jf ( x )( PxBf (x )-pxs ( x ))dx
M
< i i f i if j|px? (x)- pxs (x)|dx = i i f i ib dTv (XB,XT).
M
Here, pxb (•),pXs (•) denote the densities of random variables Xf,XT
3. The function f ( • ) e St (R) can be represented as
m
f (XT ) = f (H+Xf- (XT ), fj (x ) = A j I x >Kj .
j=1
For each function, fj ( ) it is true that
PB (fj,T) — PS (f,j )| = |A j\FlS (Kj) — FXB (Kj )<IA j\dK (XB.XS ).
j=1
dK ( XT , XT ) =
\PB (f ,T)-Ps (f ,T (fj T)-Ps (fj ,T):
j=i
=i fi \stdK ( xb , xs ).
Note 2: If the payout function can be represented as
f ( • ) = fi ( • ) + f2 ( • ) + fs ( ), fx ( • ) e Lip (R), f (• ) e B (R), f3 ( • ) e St (R ),# (10)
then
\PB (f, T)-Ps (f, t)| < i i fi i \LipdFM (XB, XS )+i \f21 \Bdw (XB, XS )+i i fs i \stdK (XB, XS ).# (11)
The representation (10) is obviously not unique. Moreover, f3 ( • ) is unnecessary as soon as any piecewise constant function with a finite number of jumps is bounded. Nevertheless, a proper choice of functions fi ( • ),f2 ( • ) и f3 ( • ) in expansion (10) can significantly improve the estimate (11).
The following statements provide methods of calculation of the metrics appearing in (7)-(9). Finding dFM (XB,X'S) is reduced to the calculation of the metric between random variables ~ N a2) and n ~ СЯ (ц2, a2) that have normal and lognormal distributions. The value of this metric is given by the following theorem.
Theorem 1
Let N a2 ),(ц2, a2 )• Then, under the condition ln
с \ a
vai у
+M2--2 Ml +1 < 0,# (* ), a
1
the metric can be found with the formula
dFM (£, n) = |! (2 [o(*2 ) — )] —i) + 2c1 (^ ) — ^(k2 ))
+
+exp
M2 +-
(l-2[ф(*2 -a2)-Ф(к, -a2)]),
# (12)
where 0( • ) is a cumulative distribution function of the standard normal distribution, • ) is the density of the standard normal distribution, and k1 and k2 are equal to
t, = --L Wo
k2 = -ÜL--L W-,
—- exp
v a, с
M2 —- Ml
ai a 2
—-exp
V a,
M2
-Ml
# (13)
If condition (*) is not satisfied, then
dFM (t n) = -M, + exP
M2 +-
.# (14)
Corollary 1. When trends and volatilities are chosen such that processes (1) and (2) are martingales (i. e., relation (3) is satisfied), the formula for the metric between distributions of the random variables X f, X T can be expressed as
dFM (Xf,XT ) = 2Xo(|>(*2)-0(£i)]-[o(k2-a2)-0(k -a2)] -[^(k2 )-^(ki )]),# (15)
—
1 1 r о - \
^0 a 2 a 2 a 2
-- -- - ^exp 2 2
a1 a2 V a1 2 a1 _ y
1 1 / 2 \
W-1 a2 a 2 a2
-- -- —2exp
a1 a 2 V a1 2 a1_ y
where a1 = aBV7, a2 = as V7 denote the integral volatilities, and k1,k2 are calculated as follows:
# (16)
The following theorem answers the question about the optimal relation between oB and aS minimize the Fortet-Mourier metric in the risk-neutral case. Theorem 2
For fixed a2, the minimum of expression (15) is attained at
a2-\/i—i
3-a2
a, =
1 " -2
+ ln 1 +
vi—e
2
For fixed a1, the minimum in (15) is attained at a2, which is a solution of the equation k1 + k2 = 2a
2
The calculation of the total variation metric and the Kolmogorov metric between Xf and XtT can
where k1,k2 are determined from (16). calc
be reduced to solving a nonlinear equation. This result is formulated in Theorem 3.
Theorem 3 2
L 5~N(^1,aj2),(m-2,a2), and =1,=—Then,
dry (5, n) = 2 [(F5( x1)-Fn(x1 )) + (F5( x3)-F„( x3 ^-^(x,)-F^(x2 ))],# (17) dK (5, n)=max| f*( x)-FJ xi )| ,# (18)
where x1 < x2 < x3 are the roots of the equation (x2 - 2x) - (^)2 (lnx)2 - 3a2lnx +1 -^ - 2a2lnf a.] = 0,# (19)
Va1 У
and the cumulative distribution functions have the form F\ (x) = O Corollary 2. According to Definitions 8 and 9,
г \
x
v ai У
Л( x ) = ф
ln ( x
lx >0'
dTV ( XT ' XT ) = dTV
( vT VT \ Xt -Л t
V Xo Xo y
,dK ( xT , xT ) = dK
( vT VT \ Лт Л t
v X0 X0 y
2
YB Ys
^^ T r ( 2 \ ^^ t r
In the risk-neutral case, —— M(1.,asT ),—— CM
An Ar
-aST. a ST
L0 ^0 V
Theorem 3 by taking into account that a1 = aB\/T, a2 = a^-v/T-
• and the metrics are calculated by
i- uB
3 Proofs of Theorems
Proof of Theorem 1
The cumulative distribution functions of t, - are
Fç( x ) = 0
t \ x
V ai y
Fn( x ) = ®
ln ( x )-^2
lx >0"
Then, their inverse functions can be expressed as
F-1 (u) = ^ + aiO-1 (u), F-1 (u) = e +a2°-1 (u).
As the minimum in (6) is attained on the monotone coupling, we obtain
dm (t-) = E|(^1 + a1Z)-e+a2Z|,Z = O-1 (U) ~ N(0,1).
The expectation is considered here with respect to the measure PZ induced by a random variable Z . Let us divide the space of elementary events into three disjoint sets:
D1 = {rn: ^ + a1Z >e ^+a2Z },
D2 = {rn: ^ + a1Z <e ^+a2Z },
D3 = {rn: ^ + a1Z = e ^+a2Z }.
As P(D3 ) = 0, P(D ^D2 ) = 1holds, and therefore,
dFM (t,-) = E[(^1 + a1Z)-e+a2Z]IA +E[e+a2Z-(^ + axZ)]Id2.
By definition, the set D1 is either empty or comprises those rn for which Z e(k1,k2) for some real k1,k2 as the graph of a linear function can lie above the graph of an exponent only within a finite interval.
In case of D1 = 0, considering that the expectation of the lognormal distribution with parameters
^ 2, a 2 is equal to exp
^2 +-
, we obtain
dFM (Ç,n) = E[e+a2Z - (^ + axZ)] = + exp
^ 2 +"
.#(20)
If D1 = {ffl: Z e (k1,k2)} , then as it is much more convenient to work with D1 than with D2, we eliminate the indicator ID . Using the formula
EX I„ = EX - EX I
D
for X = e^+a2Z - (^ + a1Z), we get
dFM (t n) = -^1 + eXP
^ 2 +"
.2E [e +a2Z-(^ + aiZ )] I ^ (21)
„a,Zu
As PD) = 0(k2)-0(k1), we need to calculate EZID u Me« To find the first moment of a random variable ZID , we find its Laplace transform
1
Wà) = Ee~XZI°i = 1 -P(D) + f exp
V2n J
-Àx - — 2
dx =
= 1 -P (D1 ) + exp
À2
[o(k2 +À)-0(k + à)].
As the first moment exists, it is equal to
EZI Dj =-V(0 ) = $(k )-^(k2 ).
Now, let us find
1 k2 =772^'
^'exP
02 x
dx = exp
[o(k2)-0(ki)].
+
Combining the above formulas, we obtain
dFM (^= (2[0(k2)-0(kj)]-i) + 2Cj |>(k)-^(k2)] «2
+e'2 + 2 (i-2[0(k2-«2)-0(kj-«2)]).
To obtain the final result, it is necessary to calculate ki,k2 and find the conditions under which Di is nonempty. If Di is nonempty, then ki, k2 are the roots of the equation
| + a1x = exp[|2 + 02 x ].# (22)
Now, let us make the variable replacement y = -—(|1 + a1x ), x = -—- — . Then, the equation is
01 0 2
transformed into
--L y = exp
I2-y
ye
^exp
I2-!r
.#(23)
The right-hand side is negative, so (23) has two real solutions (i.e., D1 is nonempty) only in the
case of — exp
> -e 1 (see the definition of the Lambert W). Taking the logarithm of
this inequality, we obtain (*).
If condition (*) is satisfied, the roots of (23) are found using the W function:
yi =Wo y 2 =W-1
/ - \
0 2 0 2
- exp 12
V 01 01 ] /
/ r \
0 2 0 2
exp 12
V 01 01 ] /
By substituting these solutions into the inverse replacement x = -——- , we obtain (13), which
completes the proof of the theorem.
a1 a2
Proof of Corollary 1
If X,Y are random variables, it immediately follows from (6) that
dFM (cX,cY) = |c| dFM (X,Y),c e M.
Thus,
dFM (, Xt ) = X0dFM
1 + aBW ,exp
= X0dFM n).
Here, we designate % = + aBW , n = exP
4+aW
. Clearly,
%~N (^i, a2 ), n ~ LN a2 ),# (24)
at t
where =1, =-~~, aj2 = aB t, a^ = aS1.
Let us show that condition (*) is satidfied. Suppose that for some a1 > 0,a2 > 0 , this is not true. Then, through (14), dm (%, n) = 0 (i.e., % = n ). We obtain the contradiction with (24). Substituting the parameter values into formula (12) of Theorem 1, we obtain (15) and (16).
Proof of Theorem 2
1. Let us fix a2 > 0 and consider an optimization problem
dFM nH ^
From (15) and (16) and the continuous differentiability of W for a1, a2 > 0, the function dFM (%, n) is found to be continuously differentiable with respect to a1 at a1, a2 > 0. Moreover, the values close to zero and a very large value of a1 are not optimal. Hence, the minimum point satisfies the necessary condition
ddFM n) da1
= 0.
Substituting into (21) the martingale values of parameters and differentiating it by a1 using the Leibniz integral rule, we obtain
ddFM n) =-2
da1
= -2-
da1 3 k
exp
-—+a2 Z 22
-1 -a1Z
da
1 k
/ 2 \
exp a 2 —t2-+a2 z 22 -1 -a1z
V /
-Mz )
exp
- + a2Z
-1 -a1z
$(z )dz = 2 J z $(z )dz -
k
1 = 2EZI A= 2 |>(k )-^(k2 )].
Here, the term with substitution is equal to zero, as k1, k2 are the roots of (22). Thus, the point a1 is optimal if and only if
^(k1 ) = ^(k2 |kl| = |k2 |.
Let us show that the case k1 = k2 is impossible. Indeed, if k1 = k2, then from (15), dFM n) = 0 ;
d
that is, . We obtain the contradiction with
m (ii, 02 ), n~cM (12,0^ ). Thus, k2 = -k1. From (16), we obtain
Wo (z ) + W_i (z ) = -28,
. Adding to this equation the definition of the
where we designate 8 = —, z = -—exp
-0l
2 a,
Lambert W function, we obtain the system
Solving it, we determine
Hence, from (16)
Wo (z ) + W_i (z ) = -28 W (z ) W-i (z )
?Wo (z )
3W-i(z )
=z
=z
W0 (z) = -8 + ^82 - (ze8 )2 W-i (z) = -8-^82 - (zes )
k
= -k1 = —a/Η
Let us substitute the determined value of k2 into (22)
1 + \/l - e "a2 = exp
.02+0l 2 a1
fi-
e "°2
From this, we can easily express as
a =
1 _ „2
+ in 1+
2
2. Analogically to the first point, we equate to zero the derivative
ddFM (t n) =-2
da1
3a.
exp
-—+a2 Z 22
"-2
= -2 J(z-a2 ) exp
—z2-+a2 z 22
1 -a1Z
^(z)dz = -2J(z-a2)^(z-a2)dz :
k2-a2
= -2 J ydO(y) = -2|>(ki-a2H(k2-a2)] = 0.
k1 —a2
2
From here, |k1 - a21 = |k2 - a2 . Again, considering the impossibility of case k1 = k2, we obtain
k1 + k2 — 2a 2.
From (5), we obtain
Proof of Theorem 3
dTV (%, n) = 2 J( p( x )-Pn( x )) dx,# (25)
where set A — {x: p^ (x) > pn (x)} — is the union of intervals whose endpoints are the roots of the equation
P^(x) — Pn( x).
This equation has only positive roots as p^ (x) > 0, pn (x) — 0 at x < 0. Let us write it out explicitly and transform it.
1
lnx + ln
exp
f \ a
( x -1)2 2a 2
1
a2 x
exp
(lnx + a22 / 2)2
Va1 y
= (x2 -2x + 1)-^y( 2a21 ; 2a2
2a 2
lnx )2 + a2lnx + 0-
/ \
2a 2 lnx + 2a2ln
Va 1 y
2
= (x2 - 2x )+1 —2 (lnx)2 - a 2lnx -
_2 _2 aia2 ■
4 ■
(x2 - 2x) - ()2 (lnx)2 - 3a2lnx + 1 - ^ - 2a2ln
f \ a
Va1y
= 0.
Let us denote the left part of the equation by h (x) and find the derivatives of this function:
h ' ( x ) = 2 ( x - 1 )-(^)2
a 2 x x
h"(x) = 2-(^)222 -2lnxI 3a2
2lnx 3af
+
Equality h'(x) — 0 is equivalent to 2x(x -1 ) — 2(—)2lnx + 3a 2 , which has exactly two roots for
a
geometric reasons. Hence, the function h (x) has two local extrema on (0, . Let us denote them
by x*, x2* and x* < x2*.
As lim h(x) — -«>, lim h(x) —, the equation h(x) — 0 has (0, at most three roots. As p^ (x) > pn (x) at x < 0 and at x > 0 p^ (x) > pn (x), when h(x) < 0, set A can be represented as
A — (-^x1 x3 ).#(26)
If the equation has less than three roots, consider x2 — x3. Combining (26) with the integral representation of the total variation metric (25), we obtain the required statement.
To find the Kolmogorov metric, consider the function g(x) — F\ (x)- Fn (x). As lim g(x) — 0 at the point at which the maximum of the modulus is reached, we have the equality g' (x) —(x) - pn (x) — 0. The solutions of this equation are the roots of x1, x2, x3 obtained in (19). Hence,
dK& n)—maxl F(x) - Fn(x t— ma3l F (x)- Fn (x )|.
2
2
/1\ 1 \
1 \ 1 1
1 \ <2 *2* y/
1 1 1
1 J_____
4.0
4 Numerical Analysis Calculation of the Fortet-Mourier Metric
The value of the Fortet-Mourier metric in (12) cannot be expressed in elementary functions. This is an expected result, which naturally arises when dealing with normal and lognormal distributions: the distribution function O(-) appears, for example, in the Black-Scholes formula (Black and Scholes, 1973). However, in (12) the Lambert W, which is much less frequently used function than 0( ). Nevertheless, many mathematical packages allow calculating the value of any of its Figure 2. Function graph h( ) at o1 = o2 =1 branches, which simplifies the numerical cal- Source: The authors. culation of the formula.
Calculation of the Total Variation Metric and the Kolmogorov Metric Let us discuss here the numerical computation of the total variation metric.
Calculation djy n) Using Quadrature Methods
One of the approaches for the calculation of the total variation metric is the calculation (see (5)) of the integral
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
x
zJ( pj( x )- Pn( x ))+ dx
using quadrature methods. As
(#;(* )- Pn(* ))+ ^ #;(* ), and ^ ~ N (|, o1), we will approximate the integral by the proper one
1+5
J(p§ (*) - Pn (*))+ d* - J (p§ (*) - Pn (*))+ d*.
1-5
As for x < 0,
o( x H^H3
1 x dt J
V2n -
x
exp
dt =
V2nx
exp
x
the approximation error does not exceed
1-5
2 J p.. ( x )dx = 20
V G1 y
G, 2
5 V n
2g2
Now, let us estimate the accuracy of the integral calculation
1+5
J (p( *)- Pn( *))+ d*
1-5
by using the trapezoidal method. The integrand function here is not twice continuously differentiable; however, (26) indicates that it has no more than three break points. As the function is zero at each break point, the integration error in the mesh section containing these points does not exceed 3M1h2, where
M1 — max pE (x)- pn (x).
1 x<=(1-8,1+8) I n V
Combining this with the standard estimation for the trapezoidal rule (Samarsky and Gulin, 1989), we obtain
i i h2(28) 2 M < M2 + 3M1h ,
II 12 2 1 '
28
where M is the error incurred in the integration calculations performed on a grid of size N , h — — grid step, and M2 — max |p(x)- p"n (x)|. N
x <(1 81+8) 1 (x
As pE (x) — —Q - considering ^'(x) —-x^(x), we find
a1 I a1 J
Pç (x) = ^
f i \ X -1
V a1 y
' Pç(x ) =--3 ^
f i \ x -1
V a1 y
(X -1)2 x
f i \ X -1
V a1 y
max
max pÇ ( x ) < -j=-_,
x e(1 -8,1+8)1 ÇV ;l V2na3 X e(1-8,1+8)1
Pç'( X )|
V2n(
na.
' 82 1 + a2
V ai y
Using pn(X ) =-^
a 2 X
lnX
V a 2 y
, we can find
a2 + d
Pn (X ) = —XjT ^(d )' PÎ(X ) =
X)f2+3d(X) , (d(X))3-1
X a
where we designate d (x) — ——.
a2
Let us assume that 1 -8 > 0, which will be true in practice as the values of volatilities are usually small. Let us denote
/1. /. „\ I I. /. „\ r\
d * = max d ( X ) = max
X e(1-8,1+8) v '
ln (1 -8)-^2 ln (1 + 8)-^2
Then,
a 2 + d
max pn(X) <- .— 0 0
X e(1-8,1+8)1 nV ^ V2na2(1 -8)2
max |p" (x)| < — 1--
xe(1-8,1+8^ ^ n >/2na2(1 -8)3
f y t* v 3 \
_ 3d (d )3 +1
2 +-+ -—-—
a2 a22 y
Combining the obtained inequalities, we find
M<
283
3^2nN2
/
\
1 +
V ^ y
1282
+
1
a2(1 -8)3
„ 3d (d )3 +1
2+-+ -—-—
. a2 a2 y
+
42kn 2
8 a 2 + d
-+-
a3 a2(1 -8)2
Calculation of dTy n) using the Monte Carlo method
The same integral can be calculated using the Monte Carlo method, as
J( Pj ( x )-Pn ( x ))+ dx = J(1 - Pjxj ) + Pj ( x )dx = E[1 - ,
Pj(j)
where the expectation is taken with respect to the distribution of a random variable ^ ~ p^ (). We simulate the independent random variables X1v..,Xn ~ p^ ( ) and approximate the integral by
1 Y Y, where Y = 2(1 - ^^
)+. The mean-square deviation in this case can be expressed as
E(- ¿Y - dv (j, n))2 = Var -¿Y
i=1
i=1
-1
Numerical Solution of a Nonlinear Equation
Let us now discuss the numerical solution of (19). Consider the case that has exactly three roots (for cases with fewer roots, the algorithm will be similar). As in the proof of Theorem 3,
On
h (x) = (x2 - 2x) - (-Gl)2 (lnx)2 - 3a?lnx +1 -
2 2 f \ G
G1G2
2a>
Va 1 y
h ■ ( x ) = 2 ( x -1)-( -^)22lnx - ^ h"( x ) = 2+( % In^l + M
Equation h"(* ) = 0 has exactly one root, which means that the function h (z) has one inflection point that lies between *1 u *2 (Fig. 2), and therefore, it is concave on 10, *1 ) and convex on (*2,
This ensures that Newton's method for the root *1 with an initial point *{0) such that *{0) < *1 converges to the root. For the same reason, Newton's method will converge to root *3 at the initial point *30) > *3. Root *2 can be localized by the bisection method for [*1,*3 ] and then calculated by Newton's method.
According to Samarsky and Gulin (1989), if h () is twice continuously differentiable in the neighborhood Ur (* *) of root * * of the equation h (* ) = 0, and
Mn
q = -
A0)
2m1
1,m = inf |h'(x)|,M2 = sup |h"(x)|.
x eUr ( x
x eUr (x
Then, Newton's method converges to x *, and
(k) * x ' - x
-
2k-1
x(0)- x *
.# (28)
Thus, for convergence, it is sufficient to assume that in some neighborhood of the root, the second derivative is bounded and the first one is strictly separated from zero.
At * > *10),
2
a
M2 < 2 + 2(—)2 max
a
1 - lnxf
1W
,1
3a2 (x0)2
and at the localization of the root, the minimum of the modulus of the first derivative is attained at one of the segment endpoints, where it can be computed explicitly. Therefore, by partitioning the segment until q < 1, we can achieve a guaranteed rate of convergence (28).
Results of Numerical Calculations
The results of metric calculation and optimal Figure 3. Contour Lines dFM n) and optimum values
values a*, a2 for the Fortet-Mourier metric at
a,a2 g (0,1 ),||=1,|2 ^a2 are presented in
a* (a 2), a 2 (a1) plots Source: The authors.
Figs. 3 and 4.
The contour lines show that the distances between random variables n tend to zero as a1 ^ 0, a2 ^ 0. This is because of the convergence of distributions n to the Dirac measure as the volatilities tend to zero.
Application of the Estimates to Certain Options
In this section and hereafter, when referring to processes (1) and (2), we imply that they are martingales; that is, (3) is satisfied.
Estimates (7)-(9), as well as the formulas for the metrics, show that the significant parameters determining the difference between the models are the integrated (or cumulative) volatilities, denoted by a1, a 2.
The application of estimates (7)-(9) to some types of options is shown below.
Put and Call Options
The payoff function of a standard call option fC (XT) — (XT - K)+ is Lipschitz continuous with the Lipschitz constant equal to 1. Therefore, from (7),
\Pb (f ,T)-PS (f ,T)| < dFM (XB,XS) — X0dFM fen).
Figure 4. Contour Lines dTV (%,n) and dK (%,n)
Source: The authors.
Let us use the data obtained by Bachelier (1900). Consider an option with the time to exercise equal to one month, for which the integral volatility equals o = o1 = o2 - 0.008. Then, we find
Pb (fc,T)-Ps (fc,T)| < 3.1-10-5x0.#(29)
Exactly the same estimate is true for a put option.
It is also interesting to compare this estimate with that obtained by Schachermayer and Teichmann (2005) for a call option "at the money" (i.e., for K = x0):
0<PB(fc,T)-Ps(fc,T)< x°o
12V2n'
0.30.2 -0.17 o.o-
*
I -o.i-
X
-0.2 --0.3-0.4-
^ ^ ^ ^ ^ ^ ^ ^ ^
Figure 5. Daily price increments
Source: The authors.
For the same value of o on the right-hand side, we get -1.6 -10 x0. Of course, this exceeds the accuracy of (29) by three orders of magnitude; however, the estimation with the Fortet-Mourier metric allows us to work with a very wide class of payoff functions and therefore is a more universal method.
Binary Options
Consider a binary call option with payout function
X >k-
fB ,c ( Xt ) = MIX Then, from (8),
Pb (fB,c,t)-Ps (fB,cT)| < Mdw (XB,XS). Substituting the Bachelier's data, we obtain
Pb (fb,cT)-PB (fB>c,T)
< 6-10-3 M.
As it was noted, the total variation metric provides less accurate but still acceptable estimate. Let us also apply (9):
Pb (fB,c,T)-Pb (fB,cT)| < MdK (XB,XS) -1.6-10-3M.
The Kolmogorov metric gives a more accurate result, which, however, has the same order as that of the total variation metric.
Estimation of Volatility Using the Oil Market Prices
Let us now try to apply the obtained estimates to the current data. For this purpose, it is necessary to evaluate the parameters ob,os of models (1) and (2). Furthermore, we apply statistical estimation methods assuming that the data satisfy the Bachelier model or the Samuelson model. For real market prices, the distribution of their increments or the increments of their logarithms is poorly approximated by the normal distribution and the increments themselves are not independent (e.g., the effect of volatility clusters occurs). These effects are considered using time-series models with conditional heterogeneity (ARCH models) that allow to describe the asset price behavior more precisely. In addition, the processes obtained using these models, with appropriate normalization, converge to diffusion ones (Gourieroux, 1997; Th. 5.15).
It can justify their application to the estimation of parameters of Bachelier and Samuelson models. However, when comparing these models, we are interested in a rough evaluation of the volatility9.
Consider price Xt as the closing price for WisdomTree WTI Crude Oil from January 2017 to November 2018 (Figure 5). Let us consider dimensionless values
X
Yt —~L ,t — 0,1,..., n — 335. X
According to the Bachelier model, the price increments AYt — Yt -Yt_1 can be represented as
Ax, — a + aBAWt,AWt — Wt-W^ ~ M(0,1).
Thus, as the Wiener process increments are independent, we consider {Axt} as a sample of random variables having a normal distribution M (a, a B).
The maximum likelihood estimate ab for the standard deviation from the sample obtained from the Gaussian distribution with two unknown parameters, mathematical expectation and variance, is
aB =
1 n _
- ¿(ay;-ay; )2,
;=1
where
ay, = - ¿ay; .
;=1
This estimate gives an approximate value for the volatility aB ^ 0.0144. In the Samuelson model, the logarithm increments
A(lnY) — Y + asAWt ~ M(y,a2). Estimating the standard deviation similarly, we obtain aS ~ 0.0150.
Let us construct a confidence interval for the obtained estimates with confidence level q. For the sample Z1v..,Zn obtained from normal distribution with two unknown parameters, the mathemat-
2 ^^ (Z Zn) 2 i \ ical expectation ^ and variance a2, the random variable ^-02-has a distribution of %2 (n -1)
i=1
(e.g., DeGroot and Schervish (2011)). Therefore, to estimate the maximum likelihood 'a of the scale parameter a , we have
P
Y1 < «— < Y2 a
= xn-1 (y 2 )-xn-1 (y 1 ) = qs
where %n-1 () denotes the cumulative distribution function for the law %2 (n -1). Let us choose
Y1 =X--1
'1-q ^
v 2 y
^Y1 = %«-1
f1+q A
v 2 y
then the corresponding confidence interval for a is
n ~ in
' Y2' 1
For the confidence level q = 0.99 , we obtain the confidence intervals as follows: oB e [0.0131,0.0160],os e [0.0136,0.0166].#(30)
0.022-
0.020
0.018-1
X >
0 0.016-
0.014-
0.012
The obtained results are consistent with the normalized values of Chicago Board Options Exchange (CBOE) Oil Volatility Index (OVX) over the same period of time (Fig. 6). This index is calculated Source: The authors. similarly to the volatility index (VIX) but uses oil options. The OVX values should be interpreted as implicit volatility (i.e., volatility calculated based on the observed option prices and reflecting appropriate expectations of market volatility behaviour in the next month). By contrast, the estimates derived from the historical data oB, os reflect the value of realized volatility; therefore, the comparison of these values is not entirely correct. Nevertheless, our goal is to only estimate the order of magnitudes oB u os; thus, it is acceptable for a rough evaluation of "engineering character."
Now we apply the estimate (7) to the call option with the time to expiration equal to one month (T = 30) and obtain
-i.O1 a.
^ ^ ^ ^ ^ ^ ^ ^ tP Figure 6. OVX index
5-
4- __
3-
2 ■ * 1->< X?
0-1-2 ■ -3- X0-3\/varXf X0 - 3\/VarXtfl .......
0.0
0.2
0.4
0.6
0.8
1.0
Figure 7. Process trajectories X t ,X t .
Source: The authors.
pB (fc,t)-Ps (fc,t)|< dFM (XB,XS)« 4.7-10-3x0.#(31)
For a binary option with T = 30 and payout M, according to (8),
\Pb (fB,t)-Ps (fB,t)|< Mdw (XB,XS)« 7.9-10-2M.#(32) If we apply (9), we obtain
pB (fB ,T)-Ps (fB ,t )|< MdK (XB, XS)« 2.110-2 M .# (33) Values of integral volatility
Let us find at what values of the integral volatility parameter the processes XB,Xf remain "close" to each other.
Using the Ito formula (e.g., 0ksendal, 1991), we find that XB,Xf satisfy the stochastic differential equations
dXB = oBX0dWt, dX f =o sXfdW,
where for a small t value, the optimal relation between the volatilities is oB ~os. 70
Let us now calculate the variances:
VarXf = X02af1 ^ VarXf = X0 aB Jt, VarX S = X02 (eaS ' -l),^VarX S = X0)j(eaS ' -1).
The variances and standard deviations depend only on the initial price and integral volatility. Assuming X0 = 1, a = aB = aS =1, let us model both processes (Fig. 7) such that they correspond to the same Wiener process Wt. At t - 0.2, the standard deviations and the processes themselves begin to differ appreciably. This value corresponds to the integral volatility value a4t - 0.45.
For the options considered in the previous section, the integral volatility is approximately equal to aVr - 0.015 V30 - 0.082.
Option Price Sensitivity to Volatility
To validate the above-used estimates (31)-(33), the option price must change insignificantly for small changes in volatility. This requirement is based on the fact that the value a is never exactly known in the model and its estimation leads to an error when calculating the option price. Let us estimate the sensitivity vega (see Hull, 2012)
V
= dP ( f ,T )
da
for standard and binary put and call options.
The price of a standard call option in the Bachelier model is calculated as
Pf (fcTT) = (X0-K)o
'Xp - K -
vab^X0 j
+ aBVTX0V
'Xq - K ^
KaB4TX, j
Its derivation has been provided by Schachermayer and Teichmann (2005). Similarly, the price of a standard put option can be determined:
K - Xn
^bVTXq j
Pb (fpTT) = (K)®
Let us find the vega coefficient for these options:
dp? ( fc TT)
+ a
jVTXq^
k - x0
vab4tx, j
da.
= ( Xp - K
f X0 - k J f Xp - K J +vtxq v f Xp - K J
f a^VTXp j faB4Tx0J
+
+aBVTx0 f
f Xp - K J f Xp - K J = f Xp - K J
la B^TXq J l aB4ÏX0J [aByfTXp j
as ) = - x ). Similarly, for a put option,
dPf (fp,T)
3a,
=x04t$
K - Xp vafVT j
dPf ( fc TT)
3a,
# (34)
In the Samuelson model, the prices of standard put and call options are determined using the Black-Scholes formulas:
ps ( c TT ) = Xp o
ln X0 + 1 aST K 2 s
as4T
Ps ( fp TT ) = -Xp
- K o
ln
Xo. - 1
K 2
a 2T
1 -O
ln X0 + 1 a2sT K 2 s
a
+K
asVT
'ln Xo 1 '
1 -O
K 2
alT
asVT
The derivatives of these quantities obtained by aS are found to coincide. Denoting
ln X0 + 1a2sT ln X0-^T
y+ = K 2- s , y- = - K 2 s
as^T
as^T
let us find
dps ( fc T ) = xM y+)
' lnXp
K , 1
+
2
-Ky_)
' ln X0 , K 1
s
sir 2
- 1VT
.#(35)
For binary call and put options with payout features,
ib,c (xr) = mixt >,, fbp (xt ) = mixt . Accordingly, the price is determined as an expectation with respect to the martingale measure:
Pf ( ff cC ) = ®Bff,c (Xr ) = mPb (Xr > K ) = M
Pf ( ff ,P ) = MO
( r
1 -O
K - Xn
\\
iibJtxq jj
K - Xo ^
Vub
Jtx,
0
Ps (ff,c ) = Esff ,c (Xr ) = MPs (Xr > K ) = MO
Ps (ff ,P ) = MO
'ln K + 1 asr Xp 2 s
a^VT
ln Xo - 1 asr
K 2 s
From this, we find
dPf ( ff ,c TT) dPf ( ff ,p TT)
daf daf
= M V
K - Xp
lafVTx,
dPs ( ff ,c ,T)_ dPs ( ff ,p ,T)
3as
3as
= M V
ln A - 1 air
K 2 s asy[T
\j
k - xo ia|>txf
,#(36)
0
ln
XL K 1
iVT 2
1VT
.#(37)
Let us now estimate the order of the price calculation error that appears due to an inaccurate measure of volatility. This error approximately equals to | VAa , where V is the option vega coefficient and Aa is the volatility measurement error. As options «at the money» have the greatest liquidity, their study is of the greatest interest. Therefore, we further assume that K = X0,T = 30. From (34) and (35) considering confidence intervals (30), we obtain that for the standard options with the confidence probability, equal to 0.99, the error approximation of \PB (fC,T)-PS (fc,T) calculation does not exceed
IT f i ~
J—X0maxIAoJ + VT4 -aS4T
\1 1 V 2
X0max |AaJ ~ 7 -10 3 X0
For binary options with K = X0,T = 30, according to (36) and (37), with confidence probability 0.99, the error approximation does not exceed
1 f 1 -
-MVT4 -asJT
max I AaJ -1.8-10-3 M.
The resulting estimates differ from (31)-(33) by no more than an order of magnitude. Thus, with the estimation methods used, the error associated with an inaccurate measurement of the volatility can make almost the same contribution to the option price as a model change.
In this section, sensitivity estimation is obtained only for the options of a special form. When applying similar methods for classes of functions, the accuracy of the estimation deteriorates considerably. Let us estimate the vega coefficient in the Bachelier model: if we denote p () as the den-
X
sity of the random variable —T , then the price of the European option with payout function f ( )
Xo ^
and time to expiration T can be found as PB (f ,T) = J f (yX0)p(y)dy.
Based on (1) and (3), the function p () can be expressed as p (y ) = —4 y 1
v -1
After changing the variables z = , we obtain
4r
.( f ,T )=J f ((i+JTz ) Xo )—4
t \ z
Va ^ J
dz.
Let us differentiate the integral by parameter aB. The differentiation performed under the integral
is possible for all aB > 0, as, considering aB on each finite interval, the function
V —A
V daBJ
(-) will be
majorized by an integrable function that does not depend on cb .
daB
4
/ \
Vab J
z
+ ~r 4
/ \
VaB J
( f tt )=J f ((i+4Tz ) Xo )
= — J f ((i+JTaBz ) Xo )[-4(z )+z 24(z )] dz.
dz =
# (38)
For a bounded function f (-)eB (M), 3Pb
3a,
-( f T )
<
J[4(z )+z 24(z )] dz = — y f IB .# (39)
For the Lipschitz continuous functions, we will use the inequality
\f (x )|<| f (X0)| + \\ f \\Lip\x - X0|.
Considering that
7 2 7 3 4
I | x | x)dx = ,—, I | x | x)dx = ,—, - V2n J v2n
3PR
da,
( f T )
<
— J(| f ( X0 )|+| If I aB |z|X0 )[^(z )+z 2^(z )] dz
<
< Tx0
# (40)
V2n
0 il./ iLp
According to estimates (39) and (40), as well as the confidence interval (30), the calculation error PB (f ,T) for a standard call (put) option in money with T = 30 does not exceed
6
V3Ômax I AaB IX0 - 2 • 10-2 • X0,
B
and for a binary option with K = X0,T = 30 does not exceed
i- M = 0.22 • M.
a b
The resulting accuracy estimates are inferior to those obtained using the exact representation of the vega coefficient for these options by one or two orders of magnitude, which is expected as a consequence of the universality of the estimates.
5 Conclusion
The approach based on the use of probability metrics enables the estimation of how much the transition from one model to another affects the price of a European option with a payout function from a certain class (represented as a sum of Lipschitz continuous and bounded functions). This price change can be estimated by using an appropriate probabilistic metric and the norm (or semi-norm) of the payout function in a suitable function space. However, the main factor affecting the value of the estimation is the integral volatility, at a large value of which the Bachelier and Samuelson models, which are essentially arithmetic and geometric random walks, cease to be similar. As expected, the estimates obtained using the Fortet-Mourier metric were the most accurate, whereas the use of the total variation metric and the Kolmogorov metric led to similarly less accurate results. Moreover, the calculation of the latter two metrics was reduced to the numerical solution of the same nonlinear equation describing the points of intersection of normal and lognormal densities.
For the oil market, measures of realized volatility were estimated and confidence intervals were constructed assuming that the models are true. By calculating the sensitivity (vega coefficient) for standard and binary options, the error arising in the estimation of model parameters was found to be comparable to the change in price when the model changed.
Acknowledgments
The authors express their gratitude to I. Vorobieva for valuable comments and advice in the design of this article, as well as to the reviewer for useful comments, which helped to improve the presentation.
References
Bachelier, L. (1900). Théorie de la speculation. Annales Scientifiques de l'École Normale Supérieure, 3(17), 2186.
Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167-179. Black, F., & Scholes, M. S. (1973). The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654.
Bogachev, V. I. (2007). Measure theory. Vol. I, II. Springer-Verlag, Berlin.
Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., & Knuth, D. E. (1996). On the Lambert W function.
Advances in Computational Mathematics, 5(1), 329-359. Glazyrina, A., & Melnikov A. (2020). Bachelier model with stopping time and its insurance application. Insurance: Mathematics and Economics, 93, 156-167. DeGroot, M. H., & Schervish, M. J. (2011). Probability and Statistics. 4th Ed. In Pearson, Delbaen, F., Schachermayer, W. The Mathematics of Arbitrage., Berlin Heidelberg: Springer-Verlag. Gouriéroux, C. (1997). ARCH Models and Financial Applications. Springer Series in Statistics. Grunspan, C. (2011). A note on the equivalence between the normal and the lognormal implied volatility:
A model free approach. Retrieved from https://arxiv.org/pdf/1112.1782.pdf. Hull, J. (2012). Options, Futures, and Other Derivatives. Boston: Prentice Hall.
Kolmogorov, A. (1933). Sulla determinazione empirica di una legge di distribuzione. G. 1st. Ital. Attuari., 4, 83-91.
Melnikov, A., Wan, H. (2021). On modifications of the Bachelier model. Annals of Finance, 17(2), 187-214. Merton, R.C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141-183.
0ksendal, B. (1991). Stochastic Differential Equations. New York: Springer.
Rachev, S.T., Klebanov, L.B., Stoyanov, S.V., & Fabozzi, F.J. (2013). The Methods of Distances in the Theory of
Probability and Statistics. New York: Springer. Samuelson, P.A. (1965). Rational theory of warrant pricing. Industrial Management Review, 6(2), 13-39. Schachermayer, W., & Teichmann, J. (2005). How close are the option pricing formulas of Bachelier and Black-
Merton-Scholes. Mathematical Finance, 18(1), 55-76. Thomson, I.A. (2016). Option Pricing Model: Comparing Louis Bachelier with Black-Scholes Merton. Available at SSRN 2782719.
Versluis, C. (2006). Option pricing: back to the thinking of Bachelier, Appl. Financ. Econ. Lett., 2(3), 205-209. Melnikov, A. V., Volkov, S. V., & Nechaev, M. L., (2001). Mathematics of Financial Liabilities. Moscow: Publishing house of State University — Higher School of Economics. 260 p. (In Russian) Samarsky, A. A., & Gulin, A. V. (1989). Numerical methods. Moscow: Nauka. 432 p. (In Russian) Sverchkov, M. Y., & Smirnov S. N. (1990). Maximum coupling for processes from D[0, »]. Doklady. AN SSSR,
311, 5, 1059-1061; Dokl. Math., 41, 2, 352-354. (In Russian) Shiryaev A. N. (1998). Fundamentals of Stochastic Financial Mathematics. Vol. 2. Theory Moscow: PHASIS. 512 p. (In Russian)
Footnotes
1 Apparently, Samuelson was the first economist to propose this modification of the Bachelier model. Therefore, we use the term "Samuelson's model."
2 For a complete list of contracts, see CME Group Advisory Notice 20-171, 2020.
3 This follows directly from the Ito formula.
4 The assumptions made in Bachelier's thesis (in an informal way) actually mean that the price process is a martingale.
5 The term "coupling" is also used in random process theory in a different sense; see, for example, Sverchkov, and Smirnov (1990).
6 The generalized inverse function defined in this manner is also left-continuous. In this case, the random variable F_1 (U), where U is uniformly distributed on (0,1) random variable, has a distribution function equal to F .
7 This metric forms the basis of the nonparametric criterion of the same name, which is based on the theorem proved by Kolmogorov (1933).
8 Also, Kantorovich metric, Wasserstein metric, and Dudley metric. The variety of names can be explained by many equivalent representations (for details, see Ruschendorf, https://wwwhttps://www.encyclopediaofmath. org/index/index.php?title=Wasserstein_metric=Wasserstein_metric).
9 An exposition of the statistical analysis concerning volatility has been presented by Melnikov, Volkov, and Nechaev (2001), paragraph 4.3. In contrast to this study, we use the maximum likelihood estimation (instead of an unbiased estimation with uniformly minimal variance) for the volatility, as such estimation for bijec-tive transformation of the parameter reduces to this transformation of the parameter estimate. Among other things, this is applicable when determining implicit volatility.
ABOUT THE AUTHORS / ИНФОРМАЦИЯ ОБ АВТОРАХ
Sergey Smirnov — Cand. Sci. of Physical and Mathematical Sciences since 1982, Faculty of Computational Mathematics and Cybernetics, Department of System Analysis, Associate Professor since April 1, 2017. S. N. Smirnov also is Professor at National Research University Higher School of Economics (HSE) e-mail: s.n.smirnov@gmail.com
Сергей Смирнов — кандидат физико-математических наук с 1982 г., Факультет вычислительной математики и кибернетики, кафедра системного анализа, доцент с 1 апреля 2017. С. Н. Смирнов является также профессором Национального исследовательского университета «Высшая школа экономики» (НИУ ВШЭ). Факультет вычислительной математики и кибернетики Московского государственного университета имени М. В. Ломоносова, Москва, Россия
Dmitry Sotnikov — student, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia e-mail: dmitrysotni@gmail.com
Дмитрий Сотников — студент факультета вычислительной математики и кибернетики Московского государственного университета имени М. В. Ломоносова, Москва, Россия
The article was submitted on 17.05.2021, reviewed on 16.07.2021, and accepted for publication on 17.08.2021. The authors read and approved the final version of the manuscript.
Статья поступила в редакцию 17.05.2021; после рецензирования 16.07.2021; принята к публикации 17.08.2021.
Авторы прочитали и одобрили окончательный вариант рукописи.
