The Effect of Imprecisely Stated
Volatility on Binomial Option
Pricing
Tomáš Tichý1
Department of Finance, Faculty of Economics, VŠB-TU Ostrava, Sokolská 33, 701 21 Ostrava,
Czech Republic. E-mail: [email protected]
Abstract
Volatility is a crucial parameter to put into the option valuation
model. The volatility of returns is not directly observable. In
many cases, we cannot be sure about proper value. In this paper we study binomial model and provide its generalization for
imprecisely stated input parameter of volatility. The model is formulated in two distinct ways: firstly, the volatility is specified by
its extremes; secondly, it is stated as a fuzzy number.
Keywords
Binomial model, option pricing, imprecisely stated volatility, fuzzy
number
1
Introduction
The binomial model was originally presented in 1979 by Cox, Ross and Rubinstein [5]
(henceforth the CRR model) as a simplification to the more complicated Black and Scholes
model [1]. The big advantage of the CRR model is that it allows to valuate not only
European calls and puts but also many types of more or less exotic payoffs and, what is
more important, also American options.
Both models have been initially set into an idealized market model – constant (deterministic) parameters of drift, diffusion and riskless rate, unconstrained liquidity, no
transaction costs. However, the CRR model is very intuitive so we can easy relax many
of these assumptions and provide more general (or relevant) price of the option.
A crucial parameter to price an option is the volatility of underlying asset returns.
However, the volatility is not directly observable. Moreover, it can change in time. Sometimes, we are able to describe this feature by stochastic volatility models or more generally
by stochastic environment models. A very comprehensive source of various continuoustime models is Cont and Tankov [4].
By contrast, it is not an exception, that the volatility can be stated only imprecisely,
either as an interval or as a fuzzy number. Both approaches are very similar. In the first
case we suppose that the volatility cannot abandon the interval. The problem should be
solved as a superreplication problem, that is, setting up the least cost portfolio which will
dominate the respecting payoff under worst-case scenario. The results will differ due to the
1
Substantial part of this research was done under the support provided by GAČR (Czech Science
Foundation – Grantová Agentura České Republiky) within the project No. 402/05/P085.
1
modeled position – considering the nonlinear payoff, the short position indicates highercost portfolio than the long. The second approach is little more generalized, however,
it requires to know more information about the possibility (or degree of confidence) of
particular levels of volatility.
The paper proceed as follows. In the following Section we define the binomial (CRR)
model. All important relations are derived to allow us to apply them in subsequent
sections. In Section 3 we study the case of the volatility specified by its extreme values.
We also refurmulate the CRR equation to get the extreme values of an option. In Section
4 we argue why to apply the fuzzy theory on imprecisely stated volatility, while Section
5 provides illustrative examples.
2
Simple binomial model
Under classical CRR model it is supposed that knowing the present value the price of any
risky asset can take two particular values in the next time moment. Consider one risky
asset, say stock S(t), with price at time zero S0 and one riskless asset B(t), which gains
riskless rate r, i.e B(1) = B(0) · (1 + r).
Under simple binomial model we suppose, that there is one source of uncertainty, say
Z, which value at time one can be described by
½
u(σ) with probability p
(1)
Z=
d(σ) with probability 1 − p.
It implies that the stock price at time one can be formulated in dependency on Z as
½
S0 u(σ) with probability p
S1 (Z) =
(2)
S0 d(σ) with probability 1 − p.
The parameters u(σ) and d(σ) in equation (2) can be interpreted as indices of up or
down movements in the price and are supposed
√to be dependent on the level of volatility
σ. Usually, we suppose, that u = 1/d = exp(σ ∆t).
Alternatively, we can formulate the (discrete-time) returns µ of the asset price conditionally on Z as
½
1 − u with probability p
µ(Z) =
(3)
1 − d with probability 1 − p.
Here, p ≥ 0 is the true market probability from P. Note, that if u is the index of an up
movement, it is higher than d and to the model make sense, the riskless rate (index of
riskless change R = 1 + r to be more exact) must lie between u and d indices. Hence, the
basic market condition is
d 5 1 + r 5 u.
(4)
Suppose for a moment that (4) does not hold – for example, 1 + r ≥ u. This means
that whichever the probabilities of up and down movements are, the return of the risky
asset is no longer higher then the return of the riskless asset. This implies that no one
will intend to invest in the risky asset. Which should decrease the present price.
2
The standard approach to price any derivative asset f is based on the no-arbitrage
condition. Hence, we are trying to construct the replication portfolio H which will replicate the value of f exactly (or perfectly) for all states of the world. For the model (1),
the following equality must hold with probability one:
P [f1 (Z) = H1 (Z)] = 1
(5)
Thus, the value of the replicating portfolio H must be equal to the value of f whichever
the value of Z will be.
Consider now the European option f , whose payoff at maturity is given by Ψ(Z).
Thus, we have a model with one source of uncertainty (Z) and two possible states in the
future at one side and n + 1 independent assets (i.e. n (n = 1) independent risky asset S
+ one riskless B) on the other side. This indicates, that the market is complete, we can
find the unique risk-neutral probabilities Q to get the risk-neutral price of the option f
by appraising the unique replication portfolio H.
Denote the structure of the replication portfolio by H(x, y), where x indicates the
amount invested into B and y into S, both at time zero. Hence
t = 0 : H(x, y) = xB + yS0
and
½
t=1:
Z(1) = u → H(x, y) = xB(1 + r) + yS0 u
Z(1) = d → H(x, y) = xB(1 + r) + yS0 d.
(6)
(7)
We have stated above, that we should be looking for such H that its time one value
will be equal to the option payoff ΨT (Z) regardless the state Z. Therefore,
½
Z(1) = u → Ψ(u) = xB(1 + r) + yS0 u
(8)
t=1:
Z(1) = d → Ψ(d) = xB(1 + r) + yS0 d.
Note, that the maturity time is the only moment when we can uniquely determine
the financial option value respecting its payoff, fT (Z) = ΨT (Z), without considering any
other conditions. It means that (8) results into two equations with two unknowns x and
y. Solving it gets
Ψ(d)u − Ψ(u)d
,
(9)
x=
(1 + r)(u − d)
y=
Ψ(u) − Ψ(d)
.
S(u − d)
(10)
The no-arbitrage condition should imply that if (8) holds then from (6):
t = 0 : f0 = xB + yS0 .
(11)
Thus, putting x and y from (9) and (10) into (11) we get
f0 =
1
· [qΨ(u) + (1 − q)Ψ(d)] .
1+r
(12)
Here, q = (1+r)−d
can be interpreted as the risk-neutral probability of going up (u) and
u−d
(1 − q) as the risk-neutral probability of going down (d). Thus the risk-neutral probability
space is given by
Q = {P r[Z = u] = q, P r[Z = d] = 1 − q} .
(13)
3
Alternatively, respecting the risk-neutral world, we can make the average value of Z to
be riskless to the asset S generate riskless return, thus (1 − q)d = 1 + r = qu.
The extension of the single-period binomial model into the n−period model is straightforward. The risky asset price evolutes according to (2) rewritten into n-period model
Sn = S0 ·
n
Y
Zk .
(14)
k
Similarly, the riskless asset evolution is given by B(n) = B(0) · (1 + r)n .
Knowing the solution of (6) and (7) and applying the backward recursive procedure,
we are still able to recover the option value at time t on the basis of time t + 1 values.
Thus, (12) changes into
ft (St ) =
1
· [qft+1 (St u) + (1 − q)ft+1 (St d)] .
1+r
(15)
Taking these results into account, we can formulate a time zero value of an option with
general (European) payoff Ψ(ST ) as
¶
n µ
X
1
n
f0 =
·
q j (1 − q)n−j Ψ(S0 uj dn−j ).
(16)
n
j
(1 + r) j=0
3
Volatility specified by its extremes
In this section, the true observed volatility is supposed to lie between two extreme
values σ − (minimal√ volatility) and σ + (maximal volatility). Since we suppose,√that
−
u = 1/d = exp(σ
index u to be between
u− = exp(σ
∆t)
√
√
√ ∆t), we get the true
−
+
+
−
+
+
∆t),
the
and u = exp(σ ∆t). However, since σ ≤ σ√ ⇒ exp(−σ ∆t) ≤ exp(−σ
√
−
+
+
−
down-index d must lie between d = exp(−σ ∆t) and d = exp(−σ ∆t).
Suppose again a single period model. If the asset price at time zero is S0 , the price at
time one must be, in dependence of Z, between:
√
√
½
Z(1) = u → S1 (u) ∈ hS0 exp(σ − √
∆t), S0 exp(σ + ∆t)i
√
t=1:
(17)
Z(1) = d → S1 (d) ∈ hS0 exp(−σ + ∆t), S0 exp(−σ − ∆t)i
or
½
t=1:
Z(1) = u → S1 (u) ∈ hS1− (u), S1+ (u)i
Z(1) = d → S1 (d) ∈ hS1− (d), S1+ (d)i
(18)
where S − indicates minimal possible price of S for given interval volatility in dependency
of Z. Note, that e.g. for Z(1) = d, S1− is S1 (σ + ).
Figure 1 indicates the boundaries of up and down movements of asset price S, S0 =
100. The grey lines are the boundaries given by σ + , the black lines are the boundaries
given by σ − .
In order to get the option price for particular σ we can proceed according to (12) since
it is still valid. However, we should be rather interested in boundary values of f , hf − , f + i.
Thus, to get the minimal price f − , we take an infimum over all possible volatilities
f− =
inf
−
σ∈hσ
,σ + i
1
· [qσ Ψ(uσ ) + (1 − qσ )Ψ(dσ )]
1+r
4
(19)
130
120
110
100
90
80
0
0.2
0.4
0.6
0.8
1
Figure 1: Boundary values of S
and, similarly, to get the maximal price f + we take a supremum over all possible volatilities
f+ =
1
· [qσ Ψ(uσ ) + (1 − qσ )Ψ(dσ )] ,
σ∈hσ − ,σ + i 1 + r
sup
(20)
where q, u, and d are functions of σ.
If the option payoff is sufficiently simple, e.g strictly increasing function of volatility,
we can formulate the interval by means of f (σ − ) and f (σ + ). Otherwise, we need to
examine the function for all possible volatilities.
Suppose, that we want to hedge the payoff of the option to the probability of success
be one. However, the volatility is not known in advance and of course, is not tradable. We
can construct the hedging portfolio Π of the option f and suitable replication portfolio
H(S, B). If the underlying asset price volatility can be described only by boundary values,
we should base the portfolio H(S, B) on such σ that it will dominate the payoff of f for
all possible volatilities.
To be more exact, suppose that we want to hedge a short option. Thus
Π0 = H0 − f0
and Π1 = H1 − f1
P r[H1 − f1 ≥ 0] = 1 ⇒
H 1 ≥ f1 .
(21)
If the option payoff is an increasing function of volatility, then the minimal H for which
H1 ≥ f1 will hold, is identical to the option f with σ + . For general payoff function, it is
the case of
H(S, B) ≡ sup f (σ).
(22)
σ∈hσ − ,σ + i
Similarly, the right solution for long option should be
H(S, B) ≡
inf
σ∈hσ − ,σ + i
f (σ).
(23)
The extension into multiperiod models is usually straightforward. We can suppose
that the volatility is the same for all steps. This implies that we extend (19) or (20) in a
way similar to the reformulation of (12) into (16). Thus
¶
n µ
X
1
n
−
·
q j (1 − q)n−j Ψ(S0 uj dn−j )
(24)
f0 = inf
j
σ∈hσ − ,σ + i (1 + r)n
j=0
5
130
120
110
100
90
80
104
102
100
98
96
0
0.2
0.4
0.6
0.8
1
120
115
110
105
100
95
90
85
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
130
120
110
100
90
80
0
0.2 0.4 0.6 0.8
1
Figure 2: Boundary values of S for two-period model
and
f0+
¶
n µ
X
1
n
= sup
·
q j (1 − q)n−j Ψ(S0 uj dn−j ).
n
j
σ∈hσ − ,σ + i (1 + r)
j=0
(25)
However, for some specific payoff functions it can be important to examine particular
combinations of movements, since the volatility can play various role in different periods.
Suppose for example two period model (σ1 , σ2 ), where (·, ·) indicates particular volatilities
for first and second step, respectively. Although for European call payoff it holds with
probability one that f (σ + , σ + ) ≥ f (σ + , σ̃), where σ̃ ∈ hσ − , σ + i, there can be some specific
payoff, for which exists such σ̃ ∈ hσ − , σ + i, that f (σ + , σ + ) < f (σ + , σ̃). Thus, we must
examine all possible combinations.
As an illustration, Figure 2 shows four types of boundaries for two-period model,
−
(σ , σ − ) and (σ + , σ + ) at the top, (σ − , σ − /σ + ) and (σ + , σ + /σ − ) at the bottom. Grey
lines are given by σ + , black lines by σ − .
It is therefore straight forward, that the model stopped to be recombining, since
σ1 6= σ2 ⇒
u1 =
6 1/d2
.
u2 =
6 1/d1
If the option maturity is partioned into many subintervals, so the effort to calculate a
non-recombining tree would be unsatisfactory high, the recombining feature can be put
backed e.g. by changing the time intervals for particular steps. Unfortunately, this will
not work for very broadly specified volatility intervals.
Finally, the maximal option value f + can be calculated as a combination of (15) and
(20):
Ã
!
¶Y
j
j
n µ
n
n
X
Y
Y
Y
1
n
·
q(σi )
[1 − q(σi )] Ψ S0
f0+ = sup
u(σi )
d(σi ) .
n
j
~
σ ∈V (1 + r)
j=0
i=1
i=j+1
i=1
i=j+1
(26)
6
The minimal value f − is given by relevant infimum. Here, the option value is obtained as a
supremum over admissible sequences of volatilities for particular steps, ~σ = (σ1 , σ2 , ..., σn ),
and V is an admissible set of these vectors (It is given by permutation of volatilities from
the interval hσ − , σ + i).
4
Volatility as a fuzzy number
Since the volatility as an input parameter to the option pricing model is supposed to be
imprecisely stated here, a suitable methodology to appraise any option can be the application of the fuzzy sets theory. This tool for modeling many types of imprecisely stated
problems was firstly proposed by Zadeh [12]. The most important known applications of
fuzzy sets theory in financial engineering were collected in Ribeiro et al. [8].
Within stochastic setting, the volatility can be described by relevant probability distribution function, including the mean and standard deviation. By contrast, within fuzzy
sets theory, the volatility is described by its membership function µ̃. Suppose again,
that the volatility is given neither precisely as a crisp number nor stochastically by its
probability distribution. Everything, what is know about the volatility is that it is somewhere ”around” its long run mean σ̄ and take any value between σ − and σ + . Thus, the
fuzzy-volatility is defined by σ̃ = (σ − , σ̄, σ + ). Here, the membership function of particular
values, i.e. the degree of confidence, is µ̃σ̃ (σ − ) = 0, µ̃σ̃ (σ̄) = 1, and µ̃σ̃ (σ + ) = 0.
More particulary, the membership function µ̃ of fuzzy-variable σ̃ is defined as
µ̃σ̃ (σ) = sup αIσ̃α (σ).
(27)
α∈[0,1]
Here, α denotes an α-cut and allows to define degrees of confidence. The indicator function
Iσ̃α (σ) gets value one if σ ∈ σ̃α and zero otherwise. Hence, if α = 0, σ belongs to hσ − , σ + i,
however, the degree of confidence is zero. Similarly, any σ, which is either higher than σ +
or lower than σ − has zero membership function to the fuzzy number σ̃ = (σ − , σ̄, σ + ). On
the other hand, as the value of particular σ approaches to the σ̄, the membership function,
as well as degree of confidence, increases. Setting the degree of confidence to be one, we
get a volatility whose feature is crisp. Note also, that the membership function can have
various form in dependence on the set of information we have about the characteristic of
true volatility.
Described features of fuzzy numbers indicates, that for particular level of α-cut the
volatility is σ̃(α) = (σ − (α), σ̄, σ + ( α)). Thus, the boundary values depends on the prespecified degree of confidence.
Since the volatility is given by an interval as in Section 3, we should be again interested
in supremum and infimum of all admissible values of an option, see e.g. formulation (20).
The main difference is as follows. In the preceding section we have handled with interval
hσ − , σ + i. By contrast, here we have infinitely many intervals, because each α produces
distinct interval which results into f − and f + .
Therefore, defining the volatility as a fuzzy number, we can recover a set of option
prices (or intervals of option prices). Subsequently, we can determine the degree of fuzzyness and according to the related α assign relevant price. If there are market participants
with different view about the fuzzyness, the relevant price will be, of course, different.
7
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.1
0.2
0.3
0
0.9250.950.975 1 1.0251.051.075
0.4
Figure 3: Volatility levels (on the left) and up and down indices (on the right) as specified
by α-cuts
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.4
0
0.42 0.44 0.46 0.48
0.5
0.5 0.52 0.54 0.56 0.58 0.6
Figure 4: Probability levels (down on the left and up on the right) as specified by α-cuts
5
Illustrative Example
In this section we briefly apply each method in order to price vanilla call option and barrier
(up-and-out) call option, both of European feature. We suppose the initial underlying
asset price to be 100, the time to maturity is one year, the riskless interest rate is 5%
p.a. and the volatility is supposed to be somewhere around 20% p.a. The exercise price
K = 100, the barrier level is U = 120.
Thus, if we think about the volatility as a precise number, it is σ̄ = 0.05. Otherwise,
it is stated by its extremes, σ − = σ̄/4, σ + = 2 σ̄. Consider, that the maturity is
partioned into 24 equidistant intervals. Particular levels of volatility are illustrated at
Figure 3 as given by α-cuts. Similarly, Figure 4 illustrates particular levels of up and
down probabilities, again as given by various α-cuts.
Obviously, dependency of indices and probabilities on volatility level is linear. Thus,
the boundaries are derived from boundary volatilities. Note, that e.g. q(α) is formulated
as follows:
√
(1 + r) − exp(−σ(α) ∆t)
(1 + r) − d(α)
√
√
=
.
(28)
q(α) =
u(α) − d(α)
exp(σ(α) ∆t) − exp(−σ(α) ∆t)
Thus, the results of equation (28) is an interval of probabilities, whose boundaries are
obtained by taking an infimum and supremum over all σ according to particular α-cut.
However, through the linearity it is sufficient to calculate it just for boundary values.
8
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
6
8
10
12
14
16
0
18
0
1
2
3
4
5
6
Figure 5: Fuzzy value of plain vanilla call option and barrier up-and-out call option
Figure 5 illustrates the value of plain vanilla call option (on the left) and up-and-out
call option (on the right) for various levels of volatility.
Look first at vanilla call. Considering crisp level of volatility σ = 0.2 we get the
price slightly above 10. Considering the volatility specified by its boundaries, we get
prices between 5 and 18. Finally, consider the volatility specified as a fuzzy number, as
indicated by Figure 3. Once again, the vanilla call value strictly increases with volatility.
Thus, we can get option prices for particular αcuts evaluating the formula for particular
boundary volatilities.
However, the situation is little different for the up-and-out call option. As the volatility
increases, also the probability of reaching the barrier level rises. Hence, we can see that
the relevant chart is not symmetric. The maximal option price f + is not obtained for
the highest volatility of the particular interval, but as we can see, for the volatility of
approximately 30%, which is σ + (α = 0.4).
6
Conclusions
In this paper we study the option pricing binomial model under the case of imprecise
volatility. In many cases we are not sure what the future volatility can be. Since the
volatility is difficult to measure, it can be specified by its extremes - the minimum and
maximum.
If we are able assign the level of confidence to particular level of volatilities around its
crisp value, we can apply the fuzzy theory to express the intervals of respecting option
price by virtue of particular α-cuts. We have shown, that the relevant chart for plain
vanilla call option has a shape similar to the the one of input volatility. By contrast, the
chart illustrating the prices of barrier call option is cut off from the right at the level of
α = 0.4.
References
[1] BLACK, F., SCHOLES, M. The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81 (May-June 1973), 637–659, 1973.
9
[2] BROADIE, M., DETEMPLE J. B. Option Pricing: Valuation Models and Applications. Managament Science 50 (9), 1145–1177, 2004.
[3] BOYLE, P.P., VORST,T. Option replication in discrete time with transaction costs.
Journal of Finance 47 (1), 271-293, 1992.
[4] CONT, R., TANKOV, P. Financial Modelling with Jump Processes. Chapman &
Hall/CRC press. 2004.
[5] COX, J.C., ROSS, S.A., RUBINSTEIN, M. Option Pricing: A simplified approach.
Journal of Financial Economics 7, 229–263, 1979.
[6] FAMA, E.F. The Behaviour of Stock Market Prices. Journal of Business 38, 34–105,
1965.
[7] KEMNA, A.G.Z., VORST, A.C.F. A pricing method for options based on average
asset values. Journal of Banking and Finance 14, 113–129,1990.
[8] RIBERIO, R.A., ZIMMERMANN, H.-J., YAGER, R.R., KACPRZYK, J. editors.
Soft Computing in Financial Engineering, Physica-Verlag, 1999.
[9] RUBINSTEIN, M. Implied binomial trees, Journal of Finance 69, 771–818, 1994.
[10] STETTNER, L. Option pricing in discrete-time incomplete market models. Mathematical Finance 10 (2), 305-321, 2000.
[11] ZADEH, L.A. Fuzzy sets. Information and Control 8, 338–353, 1965.
[12] ZMEŠKAL, Z., DLUHOŠOVÁ, D., TICHÝ, T. Financial Models, VSB-TU Ostrava,
2004.
10
0.15
0.15
0.125
0.125
0.1
0.1
0.075
0.075
0.05
0.05
0.025
0.025
0
0.15
0.125
0.1
0.075
0.05
0.025
0
50
100
100
150
200
200
300
0
250
0.15
0.125
0.1
0.075
0.05
0.025
0
0
400
50
100 150 200 250 300
100 200 300 400 500 600 700
Figure 6: Approximation of probability distribution function of the underlying asset price
for various levels of α-cuts (α = 1, α = 0.75, α = 0.5, α = 0)
11