Pricing of European Options with Subjective Probability
Ambiguity aversion in the options market during the European sovereign debt crisis Simon Edvinsson
Spring 2016
Abstract
This essay develops an option pricing formula where the market participants are assumed to not follow a uniform approach with respect to uncertainty that arises under extreme market events. By using a continuous Choquet random walk for modeling asset dynamics, as well as including marginal utility, an op- tion price kernel is obtained- this is opposed to the unique price that arises in the standard MMBS framework. By numerically backsolving for the ambiguity parameter, the impact of investor ambiguity aversion can be estimated from observed market option prices. This method is applied on European call op- tions where the underlying assets are various European bank equities observed during the European sovereign debt crisis from late 2009 through early 2011.
Sammanfattning
Denna uppsats utvecklar en optionprissättningsformel som tar hänsyn till så
väl varierande marginalnyttor hos marknadsaktörerena för att inkorporera os-
äkerhetsaversion i prissättning av optioner, som användandet av en kontinuerlig
Choquet random walk för att modellera processen av de underliggande tillgång-
arna. Detta ger upphov till en stokastisk diskonteringsfaktor istället för ett
entydigt bestämt optionspris som erhålls med standard MMBS. Vi använder
denna formel och löser den numeriskt för att estimera osäkerhetsaversion hos
investerare. Denna metod appliceras sedan på Europeiska call optioner med
diverese bankaktier som underliggande tillgång under skuldkrisen i eurozonen
från slutet av 2009 till början av 2011.
Acknowledgement
First of all I would like to thank Tarik Driouchi at King´s college London
for providing generous guidance with non-trivial questions regarding the model
framework used in the essay. I would also like to thank Niklas Hanes at Umeå
university for the help that he generously have provided throughout my studies
in economics. A special thanks is due to my examinators Gauthier Lanot and
Carl Lönnbark, for providing both expertise and excellent remarks. Further-
more, I would like to thank my opponent Reuben Umoh Nyong for insightful
remarks. I owe thanks to Ebba Abelsson and Sarah Pontén who dutifully read
the essay and corrected the inevitable mistakes that I had done. It is needless to
say that all remaining errors are my own. Finally, I would like to thank Tobias
Hedlund, without whom this essay would never have been written.
Contents
1 Introduction 1
1.1 Background . . . . 1
1.2 Problem definition . . . . 2
2 Choquet Ambiguity 5 2.1 Capacities and Choquet Integral . . . . 5
2.2 Choquet Expected Utility . . . . 6
2.3 Choquet Random Walk . . . . 8
3 Ambiguous Black-Scholes derivation 11 3.1 Ambiguous option price . . . . 11
3.2 Unit root non-stationary models . . . . 14
4 Pricing under ambiguity 17 4.1 Option pricing formula evaluation . . . . 17
4.2 Empirical evaluation . . . . 21
4.3 Arima evaluation . . . . 26
5 Conclusion 29 A Derivation of Greeks under uncertainty 35 B Empirical results 39 B.1 Banco Santander . . . . 39
B.2 Deutsche Bank . . . . 43
B.3 HSBC . . . . 47
B.4 Royal Bank of Scotland . . . . 51
C Matlab code 55
1 Introduction
It is widely accepted that the modern option pricing theory was created by Black and Scholes (1973), Merton (1973), and Margrabe (1978) where the latter extended the result to the pricing of options with stochastic strikes. The Margrabe–Merton–Black–Scholes formula (MMBS) is standard in the pricing of options and a common concept in the derivatives markets as well as in the academy. However, it is not common practice among traders to use the MMBS unaltered, due to model assumptions and its incompatibility with real world dynamics (see e.g. Haug and Taleb (2009)).
An important consequence of the MMBS models popularity is the possibility to construct replicating portfolios with dynamic hedging by estimation of the MMBS partial derivatives, or Greeks as they are referred to with standard option market nomenclature. In risk-neutral valu- ation theory, this eliminates all arbitrage possibilities in the market by removing the stochastics of the portfolio such that it can be viewed as a risk-free instrument (see e.g. Björk (2004); Taleb (1997)).
There are, however, numerous examples where the use of MMBS-style dynamic hedging has led to terminal or near-terminal losses after periods of smooth return. A noteworthy example is the hedge fund Long Term Capital Management that experienced terminal losses during the Russian sovereign default (see Edwards (1999)). Besides the empirical performance of the MMBS model, critique has arisen regarding the underlying assumptions of the model both as being too strong and as a reason for market-participant blow-ups (see e.g. Mandelbrot and Taylor (1967); Haug and Taleb (2011)).
1.1 Background
One assumption of the MMBS model is that every future event is a priori known in the sense that it can be assigned a probability in advance. Following the definition as made by Knight (1921), uncertainty is immeasurable, whereas risk is susceptible to some sort of measurement.
The critics of the MMBS argues that the model suffers from misvaluation issues due to its inability to make the distinction between risk and uncertainty, which it solves by omitting the latter from the pricing (see Haug and Taleb (2011)). The investors and market participants have been shown to price options differently, as a consequence of underlying preferences and aversion towards uncertainty. During extreme market conditions and during the presence of never before seen events, e.g. financial crises, sovereign defaults, market crashes, the uncertainty aversion of the investors have been shown to impact the option pricing (see e.g. Smith and McCardle (1998); Nau (2006)).
A consequence of the omission of uncertainty from the MMBS framework is the assumption of a distribution of the market behaviour. As shown by Haug and Taleb (2011), this assumption holds for mild randomness, but exposes a dynamically hedged portfolio to near-fatal losses in the presence of extreme events. It is well known that the distribution of prices for financial markets are fat-tailed, which makes volatile markets a more common occurrence than anticipated by the MMBS model (see Mandelbrot and Taylor (1967)). These volatile markets are accompanied with increased market uncertainty, which have been shown to split the uniform behaviour of investors that can be observed in normal market conditions (see Consigli (2002)). As noted by Keynes (1921), in periods where the uncertainty regarding the states of nature have become apparent to the investors, the option pricing is subjected to their heterogeneous beliefs and irrational behaviour. The ambiguity preferences and ignorance of the investors is therefore a driving force during abnormal market movements (see Ellsberg (1961)).
Driouchi et al. (2015a) formulated an option pricing formula for European exchange op-
tions by altering the standard option pricing framework of MMBS, as defined by Margrabe (1978), to allow for Choquet based uncertainty. This model incorporates the changes in investor opinions and market overreactions by modeling the dynamics of the underlying assets with Cho- quet Brownian motion (see Kast and Lapied (2010); Kast et al. (2014)), as well as allowing for heterogeneous investor beliefs in order to account for uncertainty and subjective valuation (see Driouchi et al. (2015a)). The Choquet Brownian motion builds on the Choquet Expected Utility model as defined by Schmeidler (1989) and generalized by Gilboa and Schmeidler (1989), which uses a specific sort of non-additive set functions, known as capacities, instead of standard probability measures to allow for ambiguity in the utility valuation. The continuous Choquet Brownian motion, as developed by Kast et al. (2014), relies on the concept of capacities in terms of being consistent with Theorem 2.9. By representing the capacities as weighted probabilities, they develop a general Wiener process such that it allows for the modeling of asset dynamics using a set of ambiguity adjusted Brownian motions.
1.2 Problem definition
This essay considers the pricing of options under uncertainty by building on the ambiguity adjusted Choquet-option pricing formula of Driouchi et al. (2015a). The model they propose is derived as an uncertainty adjusted equivalent of the MMBS formula, with a fixed strike. In a later paper, Driouchi et al. (2015b) develops a put option equivalent of their formula, and estimates its forecasting power through a series of regressions. We will consider the call option equivalent of the ambiguity adjusted MMBS formula proposed by Driouchi et al. (2015a), and give the formulation of both the pricing formula and corresponding Greeks.
The model is evaluated for various ambiguity levels, and the effect of subjective beliefs on the option pricing under various market conditions is studied. Furthermore, as opposed to Driouchi et al. (2015a) whom backsolves the pricing formula and estimates the level of ambiguity in the market for pre-set levels of uncertainty preferences on European exchange options, we estimate directly the optimal level of ambiguity for European call options. Furthermore, the ambiguity attitudes that are extracted from market prices is forecasted using an ARIMA model. As opposed to Driouchi et al. (2015b), who forecasts the ambiguity level using a regression approach, we use a non-parametric approach. It will be shown that the ambiguity parameter can be forecasted, which could be used to more accurately price options in volatile markets using accounting for investors beliefs. Specifically, we limit ourselves to estimate the ambiguity of call options among European banks that were affected by the European sovereign debt crisis that started in late 2009.
There is a rich literature regarding the pricing of options under ambiguity. Some of the related works includes Cherubini (1997) that develop ambiguity adjusted MMBS bounds using fuzzy measures, as well as Han and Zhou (2007) that uses a similar approach. An interesting application of this method is the pricing of contingent claims on illiquid assets, as done by Cheru- bini and Lunga (2001). Worth mentioning is also the works of Liu et al. (2005) that develops a method of option pricing in the presence of rare events. Zhang and Li (2013) develops a option pricing model when the return rate and volatility are ambiguous by using set-valued differential inclusion. Other noteworthy examples are the optimal stopping model for option pricing with multiple priors by Riedel (2009).
The disposition of the essay is as follows. In section 2, the necessary definitions regarding
the non-additive measures and the Choquet integral are made. The necessary background for
the formulation of the continuous Choquet Brownian motion is also presented. In section 3,
the ambiguity adjusted pricing formula for European call options is derived. In section 4, the
behaviour of various ambiguous option prices with regards to changes in the underlying asset and
other variables is considered. Furthermore, the back solved option pricing formula is applied to
observed market option prices, from which investor uncertainty is estimated. Lastly, in section
5 concluding remarks regarding the results are made. In the appendices the explicit calculations
of the Greeks, as well as figures and tables not presented in the main body of the essay, and
commented Matlab code is presented separately.
2 Choquet Ambiguity
This section presents an overview of non-additive measures in the context of utility theory in general, and option pricing in particular. First of all, the basic properties of general set functions and integration of such functions is treated. This set theoretical framework is then applied in the formulation of the Schmeidler (1989) theorem for expected value under uncertainty and its extension by Gilboa and Schmeidler (1989). From there, conditional capacities as defined by Chateauneuf et al. (2001) is used to define a generalized Wiener process that arises from the continuous time Choquet Brownian motion as shown by Kast et al. (2014). For a general overview of the mathematical concepts presented here, the reader is referred to e.g. Rudin (1987) or Cohn (1980). Specifically, for an overview regarding non-additive measures, see Denneberg (1994).
2.1 Capacities and Choquet Integral
Assume a non empty set Ω, and let A denote an algebra of subsets on Ω. Let v : A → R + such v(∅) = 0, then v is a nonnegative real valued set function on A. A separate definition that imposes additional conditions on v is stated here. (see Denneberg (1994); Choquet (1954)).
Definition 2.1. A set function v : A → R + , v(∅) = 0 on the algebra A is said to be monotone; if A ⊆ B implies v(A) ≤ v(B) for all A, B ∈ A,
convex; if A, B ∈ A such that A ∪ B, A ∩ B ∈ A implies v(A ∪ B) + v(A ∩ B) ≥ v(A) + v(B);
finite; if v(A) < ∞.
The convexity condition is also referred to as supermodularity. If the inequality is reversed in the convexity condition, the set function is conversely said to be concave. Without loss of generality, normalize v such that v(Ω) = 1, and if v in addition satisfies monotonicity and finiteness in Definition 2.1 then v is said to be a non-additive probability set function or capacity for short. Furthermore, a normalized finite set function that is both concave and convex is said to be a probability measure.
Definition 2.2. Let Ω be a nonempty set, and A an algebra of subsets on Ω. Denote the set of probability measures as P(Ω, A) on (Ω, A). Let v be a capacity, and define the core of v by
core(v) = {p ∈ P : p(A) ≥ v(A)}
for all A ∈ A.
If a capacity v is a probability measure, it holds by Definition 2.2 that core(v) = {v}, i.e. the core of v only contains v. For a convex capacity v, it follows also that core(v) 6= ∅. So far, the distinction between additive probabilities and capacities has been made clear. The integration of probability measures is standard in probability theory (see Cohn (1980)). However, integration of capacities is not applicable with the standard Riemann integral due to their non-additive properties.
Definition 2.3. For a set Ω and corresponding algebra A, let B denote the set of all real-valued A-measurable functions X : Ω → R. Given a capacity v, define the functional I : B → R by
I(X) = Z
X dv = Z
Ω
X(ω) dv = Z 0
−∞
(v(X ≥ x) − v(Ω)) dx + Z ∞
0
v(X ≥ x) dx (2.1)
The functional I in Definition 2.3 is commonly referred to as the Choquet integral. Observe that the two integrals in the right hand side of formula (2.1) are Riemann integrals, where (X ≥ x) = {ω ∈ Ω : X(ω) ≥ x} for a level x ∈ R.
Having defined an integral that is capable of integrating non-additive set functions, it is worth considering integration of probability measures using the Choquet integral. It will be clear that under certain conditions using such measures, the classical expected value is obtained. For a function X ∈ B, it follows from Definition 2.3 and X − inf ω X(ω) ≥ 0 that
Z X − inf
ω X(ω) dv =
Z ∞ 0
v X − inf
ω X(ω) ≥ x dx =
Z
X dv − inf
ω X(ω) (2.2)
Example 2.4. Let X ∈ B be a simple function, i.e. its image is a finite set such that X(Ω) ⊆ R.
Let A 1 , . . . , A n denote the minimal elements of A\∅ such that they are pairwise disjoint, and such that the union of all A i is equal to the whole set Ω. A simple function X can be represented as a linear combination of indicator functions where x i = X(A i ) are in a descending order, i.e.
x 1 ≥ x 2 ≥ . . . x n , such that
X =
n
X
i=1
x i χ A
i=
n
X
i=1
(x i − x i+1 )χ S
i(2.3)
where S i = A 1 ∪ · · · ∪ A i for i = 1, . . . , n. Then by Definition 2.3, formula (2.2) and (2.3) it holds that
Z
X dv =
n
X
i=1
(x i − x i+1 )v(S i ) =
n
X
i=1
x i (v(S i ) − v(S i−1 )). (2.4) If v is a probability measure, denote it p for distinction, then p(S i ) − p(S i−1 ) = p(A i ), and the integral in (2.4) can be represented by
Z
X dp =
n
X
i=1
X(A i )p(A i ) (2.5)
which in statistical literature and probability theory is referred to as the expected value (see Example 5.3. in Denneberg Denneberg (1994)).
It should be clear that capacities are a generalization of classical measures, and consequently a generalization of what can be described as objective probability in Bayesian sense, as in the form of equation (2.5). The integration of capacities is referred to as subjective probability when used in the context of probability theory.
2.2 Choquet Expected Utility
Let X denote a set of outcomes, i.e. rewards, prices, consequences. Let p be a probability measure. Then p is said to be a simple probability measure if the set of p(x) where p(x) 6= 0 for x ∈ X is finite, as well as P
x∈X p(x) = 1. Let Y be the set of all simple probability measures on X. Furthermore, let S be a nonempty set containing all possible states of the world, also known as a state space. Let Σ denote an algebra of subsets on S.
Definition 2.5. Let S be a state space, and let Y be a set of simple probability measures. Then
a lottery act is defined as the function f : S → Y .
Lottery acts are referred to as Anscombe-Aumann acts due to their formulation by Anscombe and Aumann (1963). Denote the set of all lottery acts whose range are finite sets with L 0 . The set up as given here will be used in the stating of the CEU theorem. Before the theorem is stated, some definitions regarding ordering of elements is needed (see Fishburn (1970)).
Definition 2.6. A binary relation on a set Y 6= ∅ is a set of ordered pairs (x, y) where x, y ∈ Y . If is a binary relation, it is a subset of the Cartesian product Y × Y , and the notation x y is used to denote (x, y) ∈ .
Definition 2.7. A binary relation < on a set Y is a defined to be a preference order if for all x, y, z ∈ Y it satisfies
asymmetry: x y implies y x;
negative transitivity: x y and y z implies x z.
Consider a binary relation on L 0 , which implies ordering of the lottery acts included in the set. Given such an ordering on L 0 , an equivalent binary relation can be induced on Y . For all y, y 0 ∈ Y and given the binary relation on L 0 it follows that
y y 0 iff f g
where f (s) = y and g(s) = y 0 for all s ∈ S and f, g ∈ L 0 . In other words, given a particular state of the world s ∈ S, the lottery acts with finite ranges are ordered in a similar manner as the simple probabilities. Before the axioms needed to formulate the theorem of Schmeidler are stated, a final definition regarding comonotonicity needs to be made.
Definition 2.8. Two lottery acts f, g : S → Y are said to be comonotonic if there is no pair s 1 , s 2 ∈ S such that f (s 1 ) < f (s 2 ) and g(s 1 ) > g(s 2 ).
All the elementary parts that are needed to proceed with the axioms, which are necessary for the formulation and proof of the Choquet Expected Utility theorem as shown by Schmeidler (1989), has now been made. The axioms are simply stated here, since a detailed exposition regarding implications and details is out of the scope for this essay. See Schmeidler (1989) or Ozaki (2014) for details.
A1. Ordering: The binary relation on L 0 is a preference order.
A2. Comonotonic additivity: for any triplet (f, g, h) ∈ L 0 that are pairwise comonotonic, it follows that f g implies αf + (1 − α)h αg + (1 − α)h for all α ∈ (0, 1);
A3. Continuity: if f g and g h, it holds αf + (1 − α)h g and g βf + (1 − β)h for any α, β ∈ (0, 1) and f, g, h ∈ L 0 ;
A4. Monotonicity: if for all f, g ∈ L 0 and for all fixed s ∈ S it holds that f (s) g(s) implies f g;
A5. Non-degeneracy: if f g for any f, g ∈ L 0 .
Theorem 2.9. A binary relation on L 0 satisfies A1–A5 if and only if there exists a unique capacity v on the state space S with algebra Σ, and an affine function u on Y which is unique up to a positive affine transformation such that
f g iff Z
S
u(f (s)) dv >
Z
S
u(g(s)) dv. (2.6)
Proof. Omitted (see Schmeidler (1989)).
Note that the affine function u that represents the binary relation in Theorem 2.9 is a von Neumann and Morgenstern (1947) expected utility function, or for the general case an expected utility function as defined by Herstein and Milnor (1953). Theorem 2.9 makes it possible to use capacities to measure subjective probability where the individual probabilities of lottery acts are nonadditive. Note that (2.6) can be formulated by Riemann integrals, as shown in the discussion regarding the properties of the Choquet integral 2.3.
Theorem 2.9 was latter generalized by Gilboa and Schmeidler (1989) where the authors, by imposing a certain kind of weak topology and additional axioms, restated (2.6) such as
f g iff min
Z
S
u(f (s)) dp
> min
Z
S
u(g(s)) dp
(2.7) where p is a probability measure on a convex subset of the set of probability measures, given the set up as in Theorem 2.9 (see Gilboa and Schmeidler (1989)). By this generalization, it is possible to view a capacity v with regards to a preference order on L 0 to represent uncertainty aversion. For a convex capacity, the preference order behaves as uncertainty averse, and con- versely uncertainty seeking for a concave capacity (see Ozaki (2014)). To see the connection between (2.6) and (2.7), observe first that for a convex capacity v it holds that
Z
X dv = min
Z
X dp : p ∈ core(v)
(2.8) for all X ∈ B. Thus, by (2.8) and a convex capacity v, it is clear that (2.6) is equal to (2.7) by
Z
u(f (s)) dv = min
Z
u(f (s)) dp : p ∈ core(v)
. (2.9)
However, the converse case is not necessarily true (see Ozaki (2014)). The CEU is therefore for convex capacities a special case of (2.7), and we can view Gilboa and Schmeidler (1989) formu- lation as an extension that can be used to model subjective probability under both uncertainty aversion as well as uncertainty seeking. For an example of the economic intuition regarding the difference between the CEU and classical expected utility, see Schmeidler (1989) and the solution to the Ellsberg paradox. Simply put, the paradox states that an investor acting under expected utility is indifferent between picking balls from an urn with known distribution and an urn with unknown distribution. The CEU however penalizes the uncertainty from the unknown distribution, which makes the investor prefer the urn with known distribution.
2.3 Choquet Random Walk
Let S be a finite set of outcomes such that S 6= ∅, and let s ∈ S be outcomes. Consider a
binomial tree as in Figure 2.1, where each node is an outcome with an equal assigned likelihood
c ∈ (0, 1). For discrete time points t ∈ [0, T ), each node s t is succeeded by either an up-
movement, s u t+1 , or a down-movement, s u t+1 , and the process {s} t ∈ S is a symmetric random
S
01S
12S
24. . . c
S
23. . . c c
S
11S
22. . . c
S
21. . . c
c
Figure 2.1: A symmetric binomial tree with equal likelihood c for each node.
walk. If the likelihood c is assigned using a probability measure, then we have c = 1/2, and the model becomes the famous discrete time Brownian motion (see Kast et al. (2014)).
Instead of using a probability measure to assign the likelihoods c, Kast and Lapied (2010) proposes the use of a capacity v that is either convex or concave. In Section 2.1 it was stated that capacities summarize preferences regarding uncertainty in the context of Theorem 2.9. The conditional capacity between nodes is constant irrespective of direction, such that v(s t+1 /s t ) = c where c now reflects ambiguity preferences, and the construction is defined to be a discrete time Choquet random walk (see Kast and Lapied (2010), Kast et al. (2014)).
Kast and Lapied (2010) shows that the Choquet random walk is dynamically consistent and that it satisfies the axioms of Theorem 2.9 and its extension formula (2.7). For the purpose of this essay, it suffices for us to state that the Choquet random walk can be extended to the continuous case, and that it converges to a Wiener process with mean m(c) = 2c−1 and variance s 2 (c) = 4c(1 − c) where c ∈ (0, 1), such that
dW t = m dt + s dB t (2.10)
where B t is a standard Wiener process. If the ambiguity preferences shows aversion, that is c < 1/2 which corresponds to a convex conditional capacity, then the mean m < 0 and the variance s 2 < 1 is lower than for the Brownian motion. The converse case is applicable when the capacity is concave (Kast et al., 2014).
Example 2.10. A convex capacity on a finite set of states of nature S is a real-valued function on the subsets of S as described in Section 2.1. We have that
v(A ∪ B) + v(A ∩ B) ≥ v(A) + v(B)
where v is a capacity and A, B ∈ 2 S . We know that v(S) = 1 and therefore v(S) ≥ v(A) + v(A c )
for all A, A c ∈ 2 S . Consider an uncertainty averse investor, then c ≤ 1/2. Assume also that the investor gains one unit of currency in state A, and one unit of currency if A c occurs. Since c ≤ 1/2 the sum of A and A c is lower than one unit with certainty. The investors uncertainty aversion penalizes the expected outcome where uncertainty is present. For full proof of this sublinearity property see Proposition 3 in Kast and Lapied (2010).
In conclusion, by the use of capacities and Theorem 2.9 it is therefore possible to model the
path of i.e. a risky asset with (2.10) given a certain uncertainty preference that can impact the
evolution of the path.
3 Ambiguous Black-Scholes derivation
This section presents an overview of the pricing of options under uncertainty. A new risky asset price dynamic is derived using the Choquet Wiener process (2.10), as defined in the previous section. By applying the adjusted price dynamics and accounting for heterogeneous marginal utilities to the option pricing in a MMBS framework sense, as proposed by Driouchi et al. (2015b), a new price option formula for European call option is introduced. The section is concluded with a description of the ARIMA model approach used to forecast the empirical point estimates of the backsolved pricing formula. The reader is assumed to be familiar with the concept of option pricing under MMBS framework, and basic stochastic differential calculus. See e.g. Björk (2004) for the former, and Kloeden et al. (2012) for the latter.
3.1 Ambiguous option price
Consider a European call option. It can only be exercised at the time of maturity T , with payoff max{S T − K, 0}.
where S T is the price of the underlying risky asset at T , and K is the strike price. In other words, at the time of maturity the option pays out the positive difference between S and K, or it pays out nothing, which ever is higher. Denote the price of European call option at time t with C t , with corresponding present value defined as
C t (S t , K) = E[e −r(T −t) max{S t − K, 0}]
with risk free interest rate r. In other words, the present value of C t is the discounted payoff which depends on the evolution of the risky asset S. When the risky asset S is assumed to follow a Wiener process, its path evolution is described by
dS t = S t µ dt + S t σ dB t . (3.1) Assuming (3.1) for the dynamics of dS, the price of C t is calculated using the MMBS formula for call options (see Björk (2004)). The price of C t takes the form of a replicating portfolio, specifically
C t (S t , K) = S t N (d 1 ) − Ke r(T −t) N (d 2 ) where
d 1 = ln S K
t+ 1 2 σ 2 (T − t) σ √
T − t , d 2 = d 1 − σ √ T − t.
Here N (·) denotes the standard normal cumulative density function and σ denotes volatility.
Now, assume that the process of the risky asset S can be modeled using the continuous Choquet random walk (2.10) as proposed by Driouchi et al. (2015a). Therefore, by substituting dW in (3.1) with the Choquet Wiener process (2.10) equivalent, the continuous dynamics of dS becomes
dS t = S t (µ + mσ) dt + S t sσ dB (3.2)
with m(c) = 2c − 1 and s 2 (c) = 4c(1 − c) for c ∈ (0, 1). Equation (3.2) implies that investors
heterogeneous beliefs are accounted for by the behavioral ambiguity drift m and volatility mod-
erating parameter s. The capacity parameter c, which could be interpreted as judgment, is used
to alter the predicted price evolution of the asset price. By the definition of the Choquet Wiener
process it is clear that the evolution of the risky asset is dependent on the uncertainty prefer-
ences regarding to both drift and diffusion. This is a result of the investors subjective volatility
measurement under uncertainty, as well as heterogeneous discounting. In the proceeding calcu- lations the n-dimensional It¯ o formula for stochastic differential equations is needed (See Kloeden et al. (2012)).
Definition 3.1. Given a n-dimensional process
dX t = µ t dt + σ t dB t ,
where µ denotes the drift, σ the diffusion and dB is a Brownian motion, the n-dimensional It¯ o’s formula is given by
df t (X t ) = ∂f
∂t dt +
n
X
i=1
∂f
∂x i
dX i + 1 2
n
X
i=1 n
X
j=1
∂ 2 f
∂x i ∂x j
dX i dX j
where (dt) 2 = dt dB = dB i dB j = 0 for i, j = 1, . . . , d such that i 6= j, and (dB i ) 2 = dt for i = 1, . . . , d.
Assume that the price of a riskless bond M can be modeled by the deterministic equation dM = M r dt
where r is the instantaneous rate of return. By Definition 3.1 and equation (3.2), the price process of the European call option dC t becomes
d C(S t , t) = ∂C
∂t dt + ∂C
∂S t (S t (µ + mσ) dt + S t sσ dB) + 1
2
∂ 2 C
∂S t 2 (S t sσdB) 2 + (S t (µ + mσ)dt) 2 + (S t sσdB S t (µ + mσ)dt) which simplifies to
d C(S t , t) = ∂C
∂t + S t (µ + mσ) ∂C
∂S t
+ S t 2 (sσ) 2 1 2
∂ 2 C
∂S t 2
dt + S t (sσ) ∂C
∂S t
dB (3.3)
So far, the price dynamics of riskless bonds has been stated, and the dynamics of price for risky assets and call options have been reformulated according to the Choquet Wiener process, following the proposal of Driouchi et al. (2015a). The next step is to identify the market pricing kernel where ξ t is assumed to be the level of marginal utility in the economy at a given time point t (see Driouchi et al. (2015a)). Assume that ξ t follows the process
dξ t = ξ t u(ξ t , S t ) dt + ξ t v(ξ t , S t ) dW t (3.4) where the functions u and v are to be determined. By substituting for dW t in equation (3.4) with (2.10), the marginal utility process becomes
dξ t = ξ(u(ξ, S t ) + mv(ξ, S t )) dt + ξsv(ξ, S t ) dB. (3.5)
By allowing for varying marginal utility with regards to the price processes of the riskless
bond dM and risky asset dS, the corresponding individual marginal utility option price par-
tial derivative equation can be derived. First of all, by using the stochastic differential formula
d XY = X dY + Y dX + dX dY , and dropping the indexes on the functions u, v and the time index for convenience, it holds that the marginal utility adjusted dM process becomes
dξM = ξ dM + M dξ + dM dξ
= ξ(M r dt) + M (ξ(u + mv) dt + ξsv dB) = ξM ((r + u + mv) dt + sv dB) (3.6) Letting the drift term tend to zero, equation (3.6) implies that u = −r − mv, and the function v is left to be determined. Now, by the process d(ξS) it holds in a similar manner that
d ξS = ξ dS + S dξ + dξ dS
= ξ(S(µ + mσ) dt + Ssσ dB) + S(ξ(u + mv) dt + ξsv dB) + ξSs 2 σv dt
= ξS (µ + mσ) − r + s 2 σv) dt + (sσ + sv) dB . By letting the drift term tend to zero, it is clear that v = (r − (µ + mσ))/s 2 σ. The explicit solutions for both function u and v has therefore been found. Equation (3.4) can by this result be rewritten such that
dξ t = ξ t u(ξ t , S t ) dt + ξ t v(ξ t , S t ) dW t
= ξ
−r − m r − (µ + mσ) s 2 σ
dt + ξ r − (µ + mσ) s 2 σ
dB. (3.7) Notice that the ambiguity aversion is endogenous to the process (3.7) through the variables m(t, c) and s(t, c). Furthermore, v and u are not unique, since the Choquet Wiener process are subjected to the individual beliefs of investors. If the market kernel is equal to the marginal utility ξ i , then perfect hedging would be possible and the model reverts to the risk neutral case of MMBS where it is possible to hedge against uncertainty (see Driouchi et al. (2015a)).
The valuation formula for the European call option can now be stated by using the dynam- ics (3.3) and (3.7), together with obvious drop of notation, it holds that
d(ξC) = ξ dC + C dξ + dξ dC
= ξ ∂C
∂t + S t (µ + mσ) ∂C
∂S t + S 2 t (sσ) 2 1 2
∂ 2 C
∂S t 2
dt + S t (sσ) ∂C
∂S t dB
+ Cξ
−r − m r − (µ + mσ) s 2 σ
dt + r − (µ + mσ) s 2 σ
dB
+ ξ r − (µ + mσ) s 2 σ
S t (sσ) ∂C
∂S t
dt
. (3.8)
By letting the drift term of equation (3.8) tend to zero and after simplifying, we get
∂C
∂t + (r 0 − ε)S t 2 ∂C
∂S t
+ 1
2 (sσ) 2 S 2 ∂ 2 C
∂S t 2 = r 0 S t (3.9)
for
r 0 = r + m r − (µ + mσ) s 2 σ
, ε = − (m + s 2 σ − sσ)((µ + mσ) − r) s 2 σ
.
The solution to the PDE (3.9) with appropriate initial and terminal conditions can be obtained
in the classical MMBS sense by applying the conventional transformation such that the heat
wave representation can be applied. See e.g. the solution as given in Wilmott et al. (1995). The solution for the Choquet adjusted pricing formula for European call options becomes
C t = S t e −εT N (d 1 ) − Ke −r(T −t) N (d 2 ) (3.10) where
d 1 = ln S K
t+ r 0 − ε + 1 2 (sσ) 2 (T − t) sσ √
T − t , d 2 = d 1 − sσ √ T − t.
For the full derivation of the corresponding partial derivatives of formula 3.10, see Appendix A.
Note also that due to the fact that formula (3.10) have a new derivative, the likelihood parameter of the Choquet Wiener process, a new Greek called Omega ∂ c C can be estimated. Omega denotes the sensitivity of the option price with respect to a change in the preferences of uncertainty aversion in the market.
3.2 Unit root non-stationary models
Modeling non-stationary processes is commonly done in finance by the use of autoregressive integrated moving average models (ARIMA). Consider first the general ARMA model, defined as
r t = φ 0 +
p
X
i=1
φ i r t−i + a t −
q
X
i=1
θ i a t−i ,
where {a t } is a series of white noise, p and q are nonnegative integers. Given that a change series c t = y t − y t−1 follows a stationary and invertible ARMA model, then the series y t is said to follow a ARIMA(p,d,q) process, where d is the number of differences (see e.g. Tsay (2005)).
In the estimation of the parameter order for the ARIMA process, a version of the Hyndman and Khandakar algorithm is used. It minimizes the Aikake information criterion for various models (see Hyndman et al. (2007)) chosen from the data such that the number of differences are chosen by repeated KPSS tests. The standard Aikake Information Criteria (AIC) for ARIMA models is defined as
AIC = −2 log(L) + 2(p + q + k + 1) where the adjusted AIC is
AIC c = AIC + 2(p + q + k + 1)(p + q + k + 2) N − p − q − k − 2
where N is the length of the sample, L is the likelihood of the data, p is the order of the autoregressive part, q is the order of the moving average part, and k = 1 if the constant c = 0.
Otherwise the parameter k is equal to zero (Hyndman et al., 2007). The Schwarz Bayesian Information Criteria (BIC) is also used in the validation of the ARIMA model. BIC is defined as
BIC = N log P N
i=1 e 2 i N
!
+ (k + 2) log N
where e are the residuals, and the other parameters are the same as previously defined (see Hyn- dman et al. (2007)).
In testing for unit root, the standard Augmented Dickey Fuller unit root test (ADF) is used (see e.g. Tsay (2005); Hyndman et al. (2007)). Following Tsay (2005), the test that is performed is H 0 : β = 1 versus H a : β < 1 by using the regression
x t = c t + βx t−1 +
p−1
X
i=1
φ i (x t−1 − x t−2 ) + e t
for a deterministic function of time c t , and residual e t . The ADF is then defined as
ADF = β − 1 ˇ std( ˇ β)
In the in-sample forecasting, the 95% confidence intervals of the multi-step forecast for an ARIMA(0,0,q) model is calculated as ˇ r T +k|T ± 1.96 √
ν T +k|T where
ν T +k|T = ˇ σ 2 1 +
k−1
X
i=1
θ 2 i
!
for k = 2, 3, . . . . The formulation of the confidence intervals for the more complex model
ARIMA(p,d,q) is more involved and omitted. See Hyndman et al. (2007) for formulation of such
confidence intervals.
4 Pricing under ambiguity
In this section, the result of the option pricing under ambiguity as by formula (3.10) is presented.
The partial derivatives for various uncertainty levels are also examined and evaluated. The pricing formula is back solved numerically with respect to the uncertainty parameter, which is then used to estimate the level of ambiguity on observed market option prices. The uncertatinty parameter is estimated on continuous at-the-money European call options with various European bank equities as underlying asset during the eurozone debt crisis. Finally, an ARIMA model is fitted to the parameter point estimates and forecasted.
4.1 Option pricing formula evaluation
First of all, consider Figure 4.1. Here, option prices C t using the ambiguity adjusted pricing formula (3.10) for various values of the underlying asset is presented. The calculation uses the parameters r = 0.04, σ = 0.3, K = 100, T = 1 and c ∈ {0.3, 0.5, 0.7}. We observe that for c = 0.5 the MMBS price is retrieved, which is in line with the discussion in previous sections.
It is evident that the various levels of ambiguity gives rise to a pricing kernel that will need to be considered by the market agents, instead of a uniquely defined MMBS price. It can be observed that the uncertainty averse trader choose to price the call option higher with regards to the value of the underlying asset at any given moment.
50 60 70 80 90 100 110 120 130 140 150
0 10 20 30 40 50 60 70 80
St
Callprice
Price of European Call
Figure 4.1: In the figure above, the price of a European Call option is displayed for c ∈ {0.3, 0.5, 0.7}. The blue line is the option price for c = 0.3, the red line is the MMBS equivalent for c = 0.5, and the green line is for c = 0.7.
Consider now the Greeks as they are defined in the context of ambiguity aversion pricing.
Let c ∈ 0.3, 0.5, 0.7, where the MMBS is shown as a special case with c = 0.5. For the calculation of the Greeks, the parameters that are used are σ = 0.3, r = 0.04, K = 100, and t = 100/252.
Let the drift term µ = 0. In Figure 4.2a it can be observed that the option Delta is more sensitive to changes in the underlying asset for the ambiguous averse case, and slower to react for the ambiguity seeking case. This is confirmed by observing the second order partial derivate Gamma in Figure 4.2b where it is clear that the ambiguous averse option price is more sensitive to changes in the underlying asset.
Rho and Kappa behaves similarly to the case of the MMBS, where the ambiguity averse case
are more sensitive to changes in the underlying asset for changes in the interest rate and changes
0 20 40 60 80 100 120 140 160 180 200 220 240 0
0.2 0.4 0.6 0.8 1 1.2
St
Delta
(a) Delta.
0 20 40 60 80 100 120 140 160 180 200 220 240
0 0.5 1 1.5 2 2.5 3 3.5 ·10−2
St
Gamma
(b) Gamma.
Figure 4.2: Delta and Gamma for European call options with various ambiguity parameters c ∈ {0.3, 0.5, 0.7} such that the c = 0.3 is the blue line, c = 0.5 which is the MMBS equivalent is the red line, and c = 0.7 is the green line. In subfigure (a) is the Delta shown, and in subfigure (b) we have the corresponding Gamma.
in the strike. Observe that for both the Rho and Kappa, the ambiguity adjusted option price works as bounds with respect to the MMBS equivalent value of c = 0.5. In Figure 4.3a the ambiguity averse investor prices the option higher for an increase in the interest rate for lower values of the underlying asset than more ambiguity seeking values of c. For Kappa in Figure 4.3b an increase in the underlying asset decreases the option value if the strike price is changed by a positive amount.
0 20 40 60 80 100 120 140 160 180 200 220 240
0 10 20 30 40 50 60 70
St
Rho
(a) Rho.
0 20 40 60 80 100 120 140 160 180 200 220 240
−1.2
−1
−0.8
−0.6
−0.4
−0.2 0
St
Kappa
(b) Kappa.
Figure 4.3: Rho and Kappa for European call options with various ambiguity parameters c ∈ {0.3, 0.5, 0.7} such that the c = 0.3 is the blue line, c = 0.5 which is the MMBS equivalent is the red line, and c = 0.7 is the green line. In subfigure (a) is the Rho shown, and in subfigure (b) we have Kappa.
In Figure 4.4 the Vega and Theta are displayed for various ambiguity parameter values. The
behaviour of the Vega in Figure 4.4a is interesting to observe, since the standard MMBS Vega
is positive for an increase in the underlying volatility σ. Here, an increase in the volatility for other ambiguity levels can have a negative impact on the pricing of the option. For the case of Theta in Figure 4.4b it can be observed that an increase in the elapsed time causes a larger negative valuation of the underlying option even for at the money options. As opposed to Vega, the various ambiguity pricing levels can be viewed as bounds around the risk-neutral MMBS price.
0 20 40 60 80 100 120 140 160 180 200 220 240
−40
−30
−20
−10 0 10 20 30 40
St
Vega
(a) Vega.
0 20 40 60 80 100 120 140 160 180 200 220 240
−60
−50
−40
−30
−20
−10 0
St
Theta
(b) Theta.
Figure 4.4: Vega and Theta for European call options with various ambiguity paramaters c ∈ {0.3, 0.5, 0.7} such that the c = 0.3 is the blue line, c = 0.5 which is the MMBS equivalent is the red line, and c = 0.7 is the green line. In subfigure (a) is the Vega shown, and in subfigure (b) we have Theta.
The fact that the ambiguity adjusted Vega behaves so differently from the risk-neutral case is worth examining closer. Due to the transmission of the uncertainty through the Choquet Brownian motion (2.10) which governs the dynamics of the underlying asset. Since the volatility parameter σ is moderated by the ambiguity parameters m and s, the drift term µ + mσ is overstated and the volatility sσ is understated for the case where c > 0.5. Opposite effects are transmitted for the ambiguity averse case where c < 0.5. The effect is transmitted through large changes in the adjusted interest rate r 0 and the ambiguity multiplier ε as uncertainty increases.
To show the effect on the option price by a change in the underlying volatility, as indicated by the Vega, various examples are considered. We will consider the option valuation for at the money options where K = 100, T = 1, and c ∈ {0.3, 0.5, 0.7}. In Figure 4.5a we observe the case where µ = 0.08 and r = 0.04 for various levels of volatility. Assume similar conditions as in Figure 4.5b with the difference that µ = 0.04. The price for the ambiguity averse case needs a higher discount for high µ than for the case where it is lower. It is also worth noting that the relationship between the ambiguity seeking c = 0.7 and the averse c = 0.3 is inverted when µ is lowered.
For comparison purposes, we consider also the case where the interest rate is negative, r =
−0.02. In Figure 4.5c and 4.5d we have the same conditions regarding the at the money options
as above, with shifting levels of µ. It can be observed that the ambiguity seeking option price is
higher than the other cases irregardless of the level of µ for the negative interest rate.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5
10 15 20 25 30 35 40 45
Volatility, σ Optionprice,Ct
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
5 10 15 20 25 30 35 40 45
Volatility, σ Optionprice,Ct
(b)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 10 20 30 40 50 60 70 80 90
Volatility, σ Optionprice,Ct
(c)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 10 20 30 40 50 60
Volatility, σ Optionprice,Ct
(d)
Figure 4.5: Price of a European call option for various ambiguity parameter values c ∈
{0.3, 0.5, 0.7} such that the c = 0.3 is the blue line, the MMBS equivalent with c = 0.5 is
the red line, and c = 0.7 is the green line.
4.2 Empirical evaluation
The role of uncertainty in option pricing will now be empirically estimated. This is done by illustrating how investors beliefs can be directly extracted from the observed market option prices. This is achieved by numerically back solving the pricing equation (3.10) for the likelihood parameter c in (2.10). The parameter is inferred in the same way as one extracts implied volatility. In this essay the method of bisection is used to solve for the likelihood parameter c.
The time around the European sovereign debt crisis in late 2009 through late 2011 creates an interesting period where the market uncertainty is abnormal. The estimation is performed on continuous European at-the-money call options with 1 months maturity with various European bank equities as the underlying asset. The price data is extracted from Thomson Reuters database. The underlying assets that are considered are Royal Bank of Scotland, BNP Paribas, Deutsche Bank, Banco Santander, and HSBC.
In Table 1 the basic statistical properties of the estimates of the ambiguity parameter for the various considered banks are presented. It can be concluded that the distribution for all considered banks are skewed to the left, and has excess kurtosis. The overall mean is lower than that of the MMBS equivalent, even within two standard deviations. In Figure 4.6 the evolution of the estimated ambiguity parameter can be observed in the upper panel, with its corresponding distribution in the lower panel.
Table 1: Summary statistics for the estimates for optimal level of ambiguity parameter c for at-the-money options with various European bank equities as underlying asset.
Summary Statistics
Ticker n Obs. Mean Std Skew Kurt
BNP.PA 521 0.4954 1.285 ∗ 10 −3 −0.5853 0.0941 RBS 521 0.4962 1.314 ∗ 10 −3 −1.1601 1.4640 HSBC 521 0.4926 0.118 ∗ 10 −3 −1.3909 2.2742 SAN 521 0.4973 0.686 ∗ 10 −3 −0.7051 0.4419
DB 521 0.4966 2.279 ∗ 10 −3 −0.5228 −1.2688
Ticker Median Min Max 1. Quartile 3. Quartile
BNP.PA 0.4956 0.4913 0.4963 0.4945 0.4963
RBS 0.4965 0.4900 0.4982 0.4956 0.4972
HSBC 0.4933 0.4812 0.4973 0.4915 0.4944
SAN 0.4974 0.4946 0.4987 0.4969 0.4978
DB 0.4984 0.4902 0.4987 0.4945 0.4985
An augmented Dickey-Fuller (ADF) test is used to asses whether the considered time series
displays the presence of a unit root. The critical values that are used are interpolated from Table
4.2 on p. 103 in Banerjee et al. (1993). The ADF test confirms the presence of a unit root in
the estimates for BNP Paribas, Banco Santander, and Deutsche Bank. Following standard time
series analysis, the first difference estimates are calculated for all time series. Since the ADF as
used here is interpolated, a non significant p-value indicates nonstationarity but is not conclusive
(see Banerjee et al. (1993). Since three of the time series are nonstationary, we can argue that
0.490 0.494 0.498
Time
Estimate of c
jan 2009 jul 2009 jan 2010 jul 2010 jan 2011
Estimate of c
Density
0.490 0.492 0.494 0.496 0.498 0.500
0 100 200 300
Figure 4.6: Estimated optimal level of ambiguity parameter c for at-the-money call options with BNP Paribas equities as underlying asset. In the upper panel the estimates are displayed over time, and in the lower panel the corresponding histogram as well as fitted distribution is given.
estimates of this kind can be differenced for excluding the possibility of nonstationarity among the estimates for the ambiguity parameter.
In Table 3 the basic statistics are given for the difference time series. In Figure 4.7 the first difference series is displayed over time with corresponding distribution for BNP Paribas. See B for corresponding figures for the remaining options.
Table 2: Augmented Dickey-Fuller test for unit root where the critical values are interpolated from Table 4.2 on p. 103 in Banerjee et al. (1993).
A. Dickey-Fuller (ADF) test
Ticker Test statistic Lag-order p-value
BNP.PA −3.077 8 0.1224
RBS −4.465 8 0.01 ∗
HSBC −5.177 8 0.01 ∗
SAN −3.016 8 0.1484
DB −2.284 8 0.4583
To assert whether the first difference time series are stationary, an augmented Dickey Fuller test is applied again, which shows signs of stationarity for time series. It is therefore suitable to fit an ARIMA model to the original time series, by assessing what type of ARMA model is suitable for the first difference time series. By observing the autocorrelation and partial autocorrelation plots in Figure 4.8, for BNP Paribas, the initial candidate for a suitable ARMA model would be with an AR-part with p = 2. The optimal value for the ARIMA corresponding ARIMA model is obtained using the automatic parameter algorithm as proposed by Hyndman et al. (2007).
Table 3: Summary statistics for the first difference estimates for optimal level of ambiguity parameter c for at-the-money options with various European bank equities as underlying asset.
Summary Statistics, first difference
Ticker n Obs. Mean Std Skew Kurt
BNP.PA 521 0.1 ∗ 10 −5 4.61 ∗ 10 −5 −0.0577 0.9620
RBS 521 0.1 ∗ 10 −5 1.11 ∗ 10 −3 −0.2939 1.4844
HSBC 521 0.2 ∗ 10 −5 0.10 ∗ 10 −3 −0.2504 2.5899
SAN 521 0.1 ∗ 10 −5 0.52 ∗ 10 −3 0.1183 1.3627
DB 521 −0.7 ∗ 10 −5 0.97 ∗ 10 −3 0.0401 8.1091
Ticker Median Min Max 1. Quartile 3. Quartile
BNP.PA 0.0001 −0.0042 0.0036 0.559 ∗ 10 −3 0.568 ∗ 10 −3
RBS 0.0001 −0.0043 0.0036 −0.0006 0.0006
HSBC 0.0004 −0.0101 0.0105 −0.0011 0.0013
SAN 0.0002 −0.0021 0.0022 0.227 ∗ 10 −3 0.291 ∗ 10 −3 DB −0.1 ∗ 10 −5 −0.0053 0.0054 −0.165 ∗ 10 −3 0.144 ∗ 10 −3
Table 4: Dickey-Fuller test for unit root of the first difference estimates where the critical values are interpolated from Table 4.2 on p. 103 in Banerjee et al. (1993).
A. Dickey-Fuller (ADF) test
Ticker Test statistic Lag-order p-value
BNP.PA −11.184 8 0.01
RBS −10.505 8 0.01
HSBC −11.48 8 0.01
SAN −10.444 8 0.01
DB −8.8274 8 0.01
−0.0040.000
Time
First difference estimates of c
jan 2009 jul 2009 jan 2010 jul 2010 jan 2011
First differende estimates of c
Density
−0.004 −0.002 0.000 0.002 0.004
0200400
Figure 4.7: First difference estimates of the ambiguity parameter for continuous at-the-money
European call options with BNP Paribas equities as the underlying asset. In the upper panel
the change in the estimates is displayed over time, and in the lower panel the corresponding
histogram and fitted distribution is given.
−0.4−0.20.0
Lag
ACF
0 5 10 15 20 25
−0.4−0.20.0
Lag
Partial ACF
0 5 10 15 20 25
