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Prague University of Economics and Business

Faculty of Economics

Major specialisation: Economic Analysis

Did the Covid19 Increased the Risk Aversion of Investors? Evidence from

Czech Exchange Rate

Diploma Thesis

Author: Mgr. Martin ˇ Casta

Supervisor: Ing. Aleˇs Marˇs´ al MA. Ph.D

Year: 2020

I hereby declare on my word of honour that I have written the diploma thesis independently, using the listed literature.

In PRAGUE date 15.12.2020 Martin ˇCasta

Abstract: The main goal of this study is to obtain expectations regarding the future development of the exchange rate using derived option probabilities and based on them to calculate the perception of risk by investors. More specifically, this study deals with the application of the Ross recovery theorem in the FOREX market using the CZK/EUR exchange rate. From a theoretical point of view, I offer an expression of the Ross recovery theorem using different Numeraire and I also propose a novel approach to the calculation of the implied risk premium.

The results show that both subjective and risk-neutral densities are not unbiased estimators of a future exchange rate. However, the results show a significant increase in the implied risk aversion during the pandemic, which is primarily driven by foreign influences from the perspective of the Czech Republic.

Keywords: Covid19, Risk aversion, Risk-neutral density, Ross Recovery theo- rem

JEL Classification: G13, G14, G15

Abstrakt: Hlavn´ım c´ılem t´eto pr´ace je z´ıskat trˇzn´ı oˇcek´av´an´ı o budouc´ım v´yvoji mˇenov´eho kurzu pomoc´ı pravdˇepodobnost´ı z´ısknan´ych z opˇcn´ıch n´astroj˚u a na jejich z´akladˇe spoˇc´ıst vn´ım´an´ı rizika investory. Konkr´etnˇeji se tato pr´ace zab´yv´a aplikac´ı Rosovy vˇety ohlednˇe moˇznosti extrakce subjetivn´ıch pravdˇepodbnost´ı na FOREX trhu s vyuˇzit´ım smˇenn´eho kurzu CZK/EUR. V teoretick´e ˇc´asti nab´ız´ım vyj´adˇren´ı Rossovy vˇety pomoc´ı rozd´ıln´ych Num´eraire a navrhuji nov´y pˇr´ıstup k v´ypoˇctu implicitn´ı rizikov´e pr´emie. V´ysledky pr´ace ukazuj´ı, ˇze subjektivn´ı a rizikovˇe neutr´aln´ı hustoty nejsou objektivn´ımi odhady budouc´ıho smˇenn´eho kurzu. V´ysledky vˇsak ukazuj´ı v´yznamn´y n´ar˚ust implikovan´e averze k riziku bˇehem pandemie, kter´y je z pohledu ˇCesk´e republiky prim´arnˇe taˇzen zahraniˇcn´ımi vlivy.

Kl´ıˇcov´a slova: Covid19, Rizikov´a averze, Rizikovˇe-neutr´aln´ı pravdˇepodobnost, Rossova vˇeta

JEL klasifikace: G13, G14, G15

I would like to express my sincere gratitude to my supervisor for beneficial advice and comments provided while writing the diploma thesis.

Contents

Introduction 3

1 Risk neutral and subjective densities 5

1.1 Financial markets and risk neutral densities . . . 5

1.2 Ross recovery theorem . . . 10

1.3 Stochastic Discount Factor in FOREX market . . . 14

1.4 Literature review . . . 16

2 Methodology 19 2.1 FX option price . . . 19

2.2 Greeks . . . 23

2.3 Vanna Volga method . . . 26

2.4 Empirical estimation of Ross recovery theorem . . . 29

2.5 Probability Integral Transform (PIT) . . . 32

2.6 Local projections . . . 34

3 Empirical estimation 35 3.1 Data . . . 35

3.2 Risk Neutral Densities - Case of Czech Republic . . . 36

3.3 Subjective Densities - Case of the Czech Republic . . . 43

4 Narrative analysis of risk aversion and exchange rate develop- ment 49 4.1 Regression analysis . . . 49

4.2 Local projection . . . 52

Conclusion 56

A Malz Method 59

B Determination of the strike price of FX option 60

C Option strategies 61

D Non-negative least squares 63

Bibliography 64

List of Figures 68

List of Tables 69

Introduction

In this master thesis, I study how Covid-19 changed the behavior of investors, more specifically how the pandemic increased their risk aversion. I use option pricing methods to obtain probability density functions of exchange rate move- ments and based on them to quantify the changes in the risk aversion of investors.

The work, therefore, focuses on one of the central issues of economics and finance, which is the market ability to predict future developments and its perception of risk. By directly using financial market instruments, we can obtain real-time in- formation on a daily basis about how the participants perceive market risks and the probability of further development.

Financial markets experienced a turbulent period during this time period, where after a period of extreme decline, there was a period of stabilization or even growth in certain markets. However, all this development happened during the ongoing pandemic and the question is, to what extent do these movements in the financial markets really reflect fundamental changes in the real economy, and to what extent they do express the mood of investors. For this reason, one of the main goals of this study is to shed some light on the variability of risk aversion.

The theoretical part deals, in general, with the factors influencing the devel- opment of the exchange rate and the asset pricing of financial instruments by risk-averse investors. Special attention is then paid to FX option instruments, which contain a risk-neutral density that can be extracted. It can be proven that the risk-neutral density corresponds to the second derivative with respect to the strike price (Breeden and Litzenberger, 1978). Furthermore, this thesis deals with methods that can be used to obtain the subjective probabilities of market investors from these risk-neutral probabilities, i.e. in general the question of risk aversion and pricing kernel as is presented in Ross (2011). I offer an expression of the Ross recovery theorem using different currencies and I also propose a novel approach to the calculation of the implied risk premium, which allows us to show the development of investors’ risk premiums over time, including the development in the current period of the Covid19 pandemic.

In the practical part, I estimated the risk-neutral and subjective densities of exchange rate development using numerical and econometric methods. First, the Vanna Volga method is implemented to obtain the price of currency options outside the quoted prices (Castagna and Mercurio, 2007). This is followed by obtaining risk-neutral densities by the Breeden and Litzenberger methodology mentioned above. Then the obtained risk-neutral densities are transformed to subjective density, using Ross recovery theorem (Ross, 2011).

Furthermore, the predictive abilities of the obtained distributions are tested in their entirety using the probability integral transform (Diebold et al., 1998). The estimated results are also discussed in their entirety. I focus on statistics, numer- ical stability, development over time, and the real-world economic determinants of recovered densities. Finally, I briefly examine the impact of monetary policy decisions on risk premium and the quantile and moments obtained from probabil-

ity densities in the event study by using the local projection method and simple regression analysis.

The results show that both subjective and risk-neutral densities are not an un- biased expectation of a future exchange rate. The results also show an increase in risk aversion during the pandemic period. However, this result is primarily driven by foreign influences from the perspective of the Czech Republic.

I divide this thesis into four major chapters as follows: In the first chapter, I focus on a general description of derivative instruments and risk-neutral densities and the asset pricing by risk-averse investors. Furthermore, Breeden and Litzenberger (1978) method that uses option pricing to obtain risk-neutral densities and Ross Recovery theorem are described and derived here. The second chapter elaborates an methodology. I focus on deriving the pricing formula for currency options, I also derive its Greeks. Furthermore, the Vanna Volga method and probability integral transform is derived here. In this part I also focus on the robust meth- ods for the empirical estimation of Ross Recovery theorem. The third chapter presents an empirical estimation, where the risk-neutral and subjective densities are estimated and tested for their predictive power. Furthermore, the last chapter presents the narrative analysis concerning monetary policy decisions and Covid19 cases.

1. Risk neutral and subjective densities

1.1 Financial markets and risk neutral densities

Financial markets can provide useful information about the perception of risk by market subjects. Forward contracts and options are particularly useful tools, where the reason for their usefulness is due to the fact that one of the key factors determining the prices of these derivative contracts is the probability that certain events will occur. However, these contracts also include information on investors’

risk aversion, as investors demand a premium for their uncertainty.

The approach presented below makes it possible to derive the implied risk-neutral and subjective densities of exchange rate developments using option prices. The obtained densities can theoretically allow us to extract market beliefs about fu- ture developments and in principle act as an indicator of tension in the market.

It should be noted that the risk-neutral probabilities are different from subjec- tive (real-world) densities. Hypothetically, if the investors were risk-neutral, the risk-neutral risk densities denote what the expectations about future develop- ments would need to be so that the present prices were identical to those actually observed. Put differently, the shift from subjective densities to the risk-neutral densities kind-of ”replaces” the abandonment of the risk aversion.

Regarding subjective densities, for a long time, it was considered to be an un- solvable problem to separate risk aversion and the subjective probability that we obtain from the prices of derivative contracts from each other. However, a novel solution was offered by Ross (2011), who uses the restriction on functional form of the utility function of a representative investor together with other limitations, to get the desired solution. This work presents an application of the Ross recovery theorem in the FOREX market. Based on his work, I extract the subjective den- sities of individual participants from previously acquired risk-neutral densities.

Furthermore I propose novel approach using a combination of these results to ob- tain a risk premium. Finally I compare statistics constructed from the obtained densities and investigate their predictive abilities.

The most common types of currency derivative are an FX forward and FX swap, which is a contract to buy/sell/exchange a certain asset in the future at a pre- agreed price. In the case of the currency market described here the price of forward contracts (the forward rate) can be understood as the price that the market participant is forced to pay to hedge against currency risk. The basic method for its determination is the theory of interest rate parity, which is based on the condition of the absence of arbitrage, meaning that in the case of identical assets, investors should expect an identical return in any currency. In reality, however, the agreed rate also reflects the risk aversion and numerous other factors, such as liquidity. For this reason, it cannot be said that the observed rate is really

the market prediction of future expected movement in the exchange rate. Only if we assume the validity of both, the theory of covered and also uncovered interest rate parity this would be an unbiased expectation of the future exchange rate.

This, however, cannot be empirically confirmed, and therefore the forward rate cannot be considered an unbiased forecast of the future spot rate. (e.g. Cuestas et al. (2015)

Options, on the other hand, are contracts that give one party the opportunity to sell/buy an asset at a pre-agreed price in the future. Options that give the right to buy a certain asset are called call options, while those that give the right to sell are called put options. Due to the fact that, unlike forward contracts, options are quoted for several different strike prices with the same time to maturity, they give a better overall picture of future developments, as they allow to observe more of the market’s perceived probability distribution. In this thesis, I focus only on the segment of options called European options. A European option is a version of an options contract that limits execution to its expiration date.

The basic model for determining the price of a European currency option is the generalized Black Scholes model Garman and Kohlhagen (1983). This model assumes a lognormal movement of the underlying asset and its constant volatility over time. As an input for determining the price using this model, 6 parameters are used C(S_t, τ, X, r_t, q_t, σ(t, X)). These factors are: the spot price, the time to maturity, the exercise price, and the risk-free domestic and foreign interest rate. The last necessary parameter is the implied volatilityσ(t, X). However, the value of this parameter does not correspond to the actual observed volatility of the underlying asset. Therefore, the reverse engineering method is used to determine the value of implied volatility, the condition is the equality of the model and market price. Model and market prices also differ from each other for individual strike prices, even if the other parameters remain the same, which means that the only different parameter is the implied volatility. Knowledge of implied volatility makes it possible to assess, how market participants perceive the risk associated with individual options. For this reason, it is typical for the FX options market that prices are quoted in implied volatility terms.

The currency options market is further specific in the fact, that there exists only a limited number of quoted options with different strike prices. (This is due to the fact that most option traders use standardized strategies) However, the standard of most markets is the existence of at least three options with different exercise prices at the same maturity. In the Czech environment, three quotations of European options are used to determine the risk-neutral density and implied volatility. These are at the money (atm_t), 25 delta put, 25 delta call options.

The issue is more complicated by the fact, that instead of quoting the three options, related so-called option strategies are quoted instead. The most often quoted as Risk reversal (rr_t) and butterfly strategy (bf_t). Risk reversal (rr_t) is an option strategy where the investor simultaneously buys an out of money call option and sells an in the money put option. Butterfly (bf_t) is a strategy that usually consists of 4 options, the portfolio is created by buying a call option with a relatively low strike price, buying a call option with a relatively high strike price,

and selling two call options with a strike price that is midway between these two strike prices. The relationship between individual quotations is expressed as follows (For more details, see the appendix):

σ_t^call(∆) =atmt+bft(∆) +rrt(∆)/2 (1.1) σ^put_t (∆) =atm_t+bf_t(∆)−rr_t(∆)/2 (1.2) To determine the price of the option for non-quoted values, it is necessary to use interpolation and extrapolation methods, which lead to the need to make certain assumptions about the form of implied volatility. The calculation methods used in this thesis are Malz (1997) and the Vanna Volga method from Castagna and Mercurio (2007).¹ Malz is a parametric method, that assumes that the implied volatility has the shape of a second-order polynomial. In contrast, the Vanna Volga method is based on the assumption of dynamic hedging and the creation of a locally risk-free portfolio. This makes this method more attractive due to its greater financial justification. For this reason, the Malz method was used only as of the robustness check of the Vanna Volga method (see appendix). Regardless of the method used, using the knowledge of implied volatility, we have defined all possible option prices at different exercise prices, which is also equivalent to the knowledge of risk-neutral probability.² (Blake and Rule, 2015)

The approach presented in Breeden and Litzenberger (1978) is used to determine the risk-neutral density. The approach is based on the simple idea that price always expresses the expected discounted return on a given asset. In the case of a European call option, its price is expressed as follows:

C(X) = e^−rtE^qmax(S_t−X,0) = e^−rt

∫︂ ∞ X

(x−X)f_qdx (1.3) E^q denotes the conditional expectation at time t according to the risk-neutral probability measure q, S_t denotes the price of the asset at maturity, X denotes the strike price, e^−rt represents the discount factor and f_q is the risk neutral density. Assuming continuity and differentiability, a probability density can be obtained using the second derivative with respect to the strike price. In this work, a numerical solution using second differences is used for its simplicity.

f_q =e^rt∂²C(X)

∂X² ≈e^rtC(X+ ∆X)−2C(X) +C(X−∆X)

(∆X)² (1.4)

The calculation for obtaining the probability density can be illustrated as follows.

Using the Leibniz formula, we obtain the first derivative of the price according to the realization price:

∂C(X)

∂X =e^−rt

(︄

−

∫︂ ∞ X

(X−X)f_qdx+ ∂^∫︁_X^∞(x−X)f_qdx

∂X

)︄

(1.5)

1alternatively, the model can be simplified by using forward rate, ie under certain assump- tions the Black76 model can be usedC^Black76(Ft, τ, X, rt, σ(t, X)).

2The given methods were chosen due to the low liquidity of the option market in the Czech Republic, as they require the smallest number of option quotations

∂C(X)

∂X =−e^−rt

∫︂ ∞ X

fqdx=−e^−rt(1−Fq) (1.6) The basic relation between a probability density function and distribution func- tion was used. For the second derivative according to the strike price, we obtain the previously shown result:

∂²C(X)

∂X² =e^−rtf_q (1.7)

An alternative derivation in a discrete setting is based on the existence of a butterfly strategy. Which is a portfolio that consists of 4 options. More precisely, the portfolio is created by buying one call option with a lower strike price, by selling two call options with a higher strike price and by purchasing one call option with an even higher exercise price. The difference between individual strike prices is denoted asX.

st= C(X+ ∆X)−2C(X) +C(X−∆X)

∆X (1.8)

Dividing the whole expression by ∆X, we get the following expression:

s_t

∆X = C(X+ ∆X)−2C(X) +C(X−∆X)

(∆X)² (1.9)

With a decreasing difference between individual strike prices, we obtain the same relationship that was derived in the continuous setting, ie formally if the limit

∆X approaches 0:

∆X→0lim s_t

∆X = lim

∆X→0

C(X+ ∆X)−2C(X) +C(X−∆X)

(∆X)² = ∂²C(X)

∂X² (1.10) The result shows the so called state price constructed from the option prices. In conculsion, This is identical result as before (risk neutral density) apart from the discount factor.

Knowledge of risk-neutral probability fq provides information on investor’s per- ceptions of the distribution of future asset price values in connection with infor- mation on investors’ risk aversion. However, as already mentioned, it should be noted that the implied risk-neutral density assumes risk-neutral investors. How- ever, this does not correspond to reality, where it can be assumed that most investors are risk averse. That is, investors prefer a certain amount to a bet with the same expected value. For this reason, the implied risk-neutral distribution does not correspond to the real perception of the probability by investors. (More precisely, Risk neutral density depends on the shape of the pricing kernel as will be shown later.)

Nevertheless, the knowledge of risk-neutral density (from now on it will abbreviat- ed to RND) can provide interesting information, especially about the market risk perception. It is possible to value any instrument with the same underlying asset through risk-neutral densities, or to derive a risk premium, although it should be

noted that in this case, it is necessary to use more assumptions regarding utility function. For this reason, the information obtained by this method can be used, for example, by central banks to monitor changes in market expectations, or as an indicator for a certain extreme movement of an asset, i.e. as an indicator of market tensions. It is even possible that these obtained probabilities can be used to assess the effectiveness of the monetary policy instruments used. An example could be the credibility of a fixed exchange rate or another type of exchange rate commitment/statement. It can therefore be concluded that it is an important tool in risk management with a long history of use, as will be shown later in this thesis.

1.2 Ross recovery theorem

Let us now focus on the decomposition of the risk-neutral density, which includes both the real expectation of future movements and the risk premium. For a better understanding of both components, we would ideally be interested in the subjective probability densityfp of investors, although it occurs in the valuation, it is modified through the pricing kernel, denoted by m_t+1. In the simple tree model the pricing kernel expresses the ratio of marginal utilities of a representative investor Lucas (1978). Using simple algebraic manipulations, it can be shown that only if the ratio m_t was equal to one, thus the investors would be risk-neutral, then the option price would reflect subjective market expectation. (Formally E^p denotes the conditional expectation at time t according to the probability measurep.)

C(X) =e^−rt

∫︂ ∞ X

(x−X)fq

f_pfpdx=

∫︂ ∞ X

(x−X)u(c_t+1)^′ u(c_t)^′ fpdx

=E^p(m_t+1max(S_t−X,0))

(1.11) Under very strong assumptions, the subjective probabilities of market partici- pants can be extracted from the risk-neutral densities as is noted by Ross (2011) The first assumption of this method is the existence of a discrete and bounded state space Θ, consisting of θ_i, where i ⊂ (1,2...n), the other assumptions are the absence of arbitrage and a complete markets. The next assumption is that the previously defined pricing kernel ism_ttransition independent, and thus there exist a representative investor with a Time Additive Intertemporal Expected Util- ity Theory preferences over consumption. Provided that all conditions are met, the subjective probabilities can be obtained by the Ross recovery theorem. The intuition behind this approach is that the Ross recovery theorem uses a large number of quoted prices across multiple maturities, from which we are able to derive state prices that are not directly observed in reality.

The solution of the Ross recovery theorem can be shown in the following way. Let us assume a previously defined representative investor, who solves the following two period inter-temporal optimization problem, given that that investor is in genericθ_i initial state of the world Backwell (2015):

maxu(c_t,i, c_t+1,i) =u(c_t,i) +δE^pu(c_t+1) =u(c_t,i) +δ

n

∑︂

j=1

u(c_t+1,ij)f_i,j (1.12) Given a budget constraint:

c_t,i+

n

∑︂

j=1

c_t+1,ijp_i,j =w (1.13)

f_i,j denotes the real probability of transition from the generic initial state θ_i to the generic final state θ_j. p_i,j represents the price of the Arrow Debreu security contingent on the initial state θ_i, which pays 1 unit of Numeraire³ if the next state is θ_j, and zero otherwise. Another notation is the following: c_t,i denotes

3Unit of account, for formal treatment see: Geman et al. (1995)

initial consumption at time t and final consumption at time t+1 contingent on the future state of the world is then denoted byc_t+1,ij,δdenotes a discount factor andwis some positive constant representing the wealth of representative investor.

We create a following Lagrangian:

L=u(c_t,i) +δ

n

∑︂

j=1

u(c_t+1,ij)f_i,j +λ_i(c_t,i+

n

∑︂

j=1

c_t+1,ijp_i,j−w) (1.14) This leads to the following first order conditions:

∂L

∂ct,i

=u(c_t,i)^′ =λ_i (1.15)

∂L

∂c_t+1,ij =δfi,ju(c_t+1,ij)^′ =λipi,j (1.16) The fraction between the first order conditions then leads to the following solution for the price of state contingent claim for every state i,j

p_i,j =δu(c_t+1,ij)^′

u(c_t,i)^′ f_i,j =m_i,j,t+1f_i,j (1.17)

In the expression above we have expressed the stochastic discount factor as the following ratio:

m_i,j,t+1 = p_i,j

f_i,j =δu(c_t+1,ij)^′

u(c_t,i)^′ (1.18)

For further use, the price of the risk-free asset must be derived. The necessary condition is that the sum of all Arrow Debrou prices must give the price of a risk- free asset, i.e. an asset that pays 1 unit of numeraire in each state of the world, otherwise, there would exist an arbitrage opportunity. This in connection with the previously derived results ensures the following expression Cochrane (2009):

n

∑︂

j=1

p_i,j = 1

E^pm_i,j,t+1 = 1

1 +r_f =δE^pu(ct+1,ij)^′

u(c_t,i)^′ (1.19) For a better understanding, let us derive the theoretical pricing kernel, assuming that the logarithmic growth of consumption is normally distributed and CRRA utility function of a representative investor:

ln

(︄ct+1,ij

c_t,i

)︄

=N(a, b) (1.20)

Under these assumptions, we would obtain the following form of pricing kernel:

E^pm_i,j,t+1 = δE^pu(c_t+1,ij)^′

u(c_t,i)^′ =δE^p(c_t+1,ij

c_t,i )^ψ =δe^ψa+b2ψ2/2 (1.21) The pricing kernel manifests itself in the form of the risk premium. To express risk premium, we now define a risky asset, the price of this asset is denoted by p_r. The asset generates cash flow R_j,r depending on the different future states of the worldθ_j., and its return is then denoted byr_j,r, if a generic state θ_j occurs:

p_r=E^p(m_i,j,t+1R_r,i) =

n

∑︂

j=1

p_i,jR_r,ij =δE^p(u(c_t+1,ij)^′R_r,i)

u(c_t,i)^′ (1.22)

By using the formula for covariance, we get following the definition of the price of the risky asset. This means that the price of a risk asset is determined by the discounted expected return, with addition of the second part, which represents the covariance of return with a stochastic discount factor.

p_r =E^pR_rE^pm_i,j,t+1+cov(m_i,j,t+1, R_r) = E^pR_r 1 +rf

+cov(m_i,j,t+1, R_r) (1.23) Alternatively we can express risk premium directly, by dividing the whole expres- sion by p_r and through the previously obtained definition of a risk-free asset and some algebraic manipulation we obtain an alternative expression:

1 = E^p(m_i,j,t+1R_r,i)

p_r =E^p(m_i,j,t+1(1 +r_r,i)) =δE^p(u(c_t+1,ij)^′(1 +r_r,i))

u(c_t,i)^′ (1.24) We can again use the definition of covariance and rewrite the expression accord- ingly:

1 = E^pm_i,j,t+1E^p(1 +r_r,i) +cov(m_i,j,t+1, r_r,j,t+1) (1.25) This leads to the following solution of the risky asset return and risk premium, which is denoted by rp, and is obtained by subtracting risky asset return and risk-free return:

E^pr_r,i =r_f −

(︄cov(m_i,j,t+1, r_r,j,t+1) m_i,j,t+1

)︄

(1.26) rp=

(︄cov(m_i,j,t+1, r_r,j,t+1) m_i,j,t+1

)︄

(1.27) Finally, according to Backwell (2015) we assume an equilibrium condition on consumption, that ensures that the consumption is assumed to be dictated only by the state, not by time or the previous state. In other words, for everyi, j, the requirement is:

ct,j =ct+1,ji=ci (1.28)

For further use in the Ross recovery theorem, we will use the first derived form of Euler equation. The above optimal solution for all i, j can be further simplified using the equilibrium condition of time homogeneity of consumption:

p_i,j =δu(c_j)^′

u(c_i)^′f_i,j =m_i,jf_i,j (1.29) Since this applies to alli, j in θ, the result can be rewritten in matrix form as:

P =δD⁻¹F D (1.30)

P is a n×n matrix of Arrow Debrou prices, F is a n×n matrix of subjective probabilities of investors andD is ann×n diagonal matrix of marginal utilities, which consists ofd_i on the diagonal and zero outside the diagonal. Using matrix algebra in combination with the knowledge that the matrix F is a stochastic matrix, ie the sum of individual rows must be one orF1=1, where 1 is the vector of the ones. Furthermore, due to the Perron Frobenius theorem, which guarantees the existence of a unique real eigenvalue, the problem is reduced to finding the

largest eigenvalue and its corresponding eigenvector. The calculation is as follows, where we defined D⁻¹1 =z.

F = 1

δDP D⁻¹ (1.31)

P D⁻¹1 = δD⁻¹1 (1.32)

P z =δz (1.33)

The critique of Ross Recovery theorem from a theoretical point of view is pre- sented by Boroviˇcka et al. (2016), the authors deconstruct the pricing kernel into a permanent and transitive component. Thus, only if the pricing kernel has the Ross stated form then the subjective probabilities are really extracted. The decomposition of the pricing kernel according to Boroviˇcka et al. (2016), can be written in a simplified form according to Jackwerth and Menner (2018) as follows:

p_i,j =m_i,jp_i,j =m_i,j f_i,j

f_i,j^truef_i,j^true=m_i,jm^permanent_i,j f_i,j^true (1.34) m^permanent_i,j denotes a permanent component of the pricing kernel and is expressed as the ratio of recovered probabilities by real probabilities. Boroviˇcka et al. (2016) interpret the changes in this component as permanent shocks to the economy. For this reason, the Ross recovery theorem is valid only if m^permanent_i,j is equal to 1.

Authors further suggest that in reality m^permanent_i,j is highly variable over time.

However, even in their work, the resulting decomposition is meaningful and the authors describe the obtained value as the market perception of a long-run risk.

1.3 Stochastic Discount Factor in FOREX mar- ket

So far, we have dealt with the general valuation of a risky financial instrument.

Let us now turn to the FOREX market and the risk premium associated with it. We begin with a previously derived definition of gross return of a risky asset denominated in domestic currency as in Backus et al. (2001):

1 =E^p(m_i,t+1(1 +r_r,t+1)) (1.35)

Next, we consider an asset denominated in foreign currency. We adopt the same approach as before for pricing this foreign denominated asset. By ∗ we denote foreign variables, i.e. m^∗_i,j denotes foreign pricing kernel and r_r,t+1^∗ foreign gross return:

1 =E^p(m^∗_i,t+1(1 +r_r,t+1^∗ )) (1.36)

By using the no-arbitrage argument we could alternatively convert foreign instru- ment returns into domestic returns and value them using domestic pricing kernel, where EX_t denotes the exchange rate at time t.

1 =E^p(m_i,t+1(1 +r_r,t+1)EX_t+1

EX_t ) (1.37)

This leads to the following equality between pricing kernels and exchange rate development:

m_i,t+1EX_t+1 EXt

=m^∗_i,t+1 (1.38)

This reason, as was stated before, is that if the equality is violated and both currencies are traded, there is obvious arbitrage opportunity, as there would exist excess return by converting to another currency. In conclusion, using the no- arbitrage argument we have derived the equivalence of these two procedures for the valuation of the foreign asset. Building upon this result, the following relation between logarithm of exchange rate developments and pricing kernels is of the form:

E^p(lnEX_t+1)−lnEX_t=E^p(lnm^∗_i,t+1)−E^p(lnm_i,t+1) (1.39) Combining this result with the Ross recovery theorem stated above gives us fol- lowing relation:

p_i,j =m_i,t+1EX_t+1

EX_t f_i,j =m^∗_i,t+1f_i,j (1.40)

By combining this result with the assumption that the exchange rate development can be described by different Markov transition matrix⁴, we obtain an alternative definition of the Ross recovery theorem.

p_i,j =δu(cj)^′

u(c_i)^′f_i,jex_i,j (1.41)

4Formally we must assume the same condition as in the case of consumption regarding equilibrium value.

This can be rewritten in matrix form using the fact, that the product of two stochastic matrices is stochastic matrix, ieFE=EF=Q and the Perron-Frobenius theorem thus can be applied:

P =δD⁻¹F ED =δD⁻¹QD (1.42)

1.4 Literature review

The use of option pricing to determine risk aversion is a relatively novel topic.

However, in many articles, certain parts of this methodology have been used.

The use of the Breeden and Litzenberger (1978) method has become especially popular for determining market expectations in stock markets. For example, the work of Jackwerth and Rubinstein (1996) deals with the S&P 500 index and the authors are particularly interested in the predictive power and probability of tail events implied by the obtained densities. Alonso et al. (2005) then apply the same analysis to the Spanish IBER35 index. Furthermore, the work of Figlewski (2008) deals with the impact of monetary policy decisions of the Fed on the devel- opment of the S&P 500 index through an event study. Other publications focus on the issue of subjective probabilities and try to extract them. These are papers by Ait-Sahalia and Lo (1998) and Jackwerth (2000), Liu et al. (2007), and Rosen- berg and Engle (2000), where all these works assume that the subjective density distribution is equal to the realized historical distribution. The opposite approach is characterized by the follow-up work of Bliss and Panigirtzoglou (2004), where the authors use risk-neutral densities to derive the risk aversion coefficient and to test the predictive power of option data. They assume a specific form of pricing kernel, which is why they are interested in the risk aversion coefficient in CRRA and the exponential utility function (time separable utility function), which this method implies for the S&P500 and FTSE indices. Most similar to this thesis is Hanke et al. (2020), which deals with quantiles of risk-neutral densities of the world stock markets and their response to Covid19 pandemic. The same topic as this thesis is described by Baker et al. (2020), but using different methodology.

All the methods described above use the mix of lognormal distributions or a smoothing spline method to determine the volatility smile shape, ie the risk neu- tral density. In the case of methods dealing with subjective probabilities and risk aversion, the validity of their results is affected by the extent to which the assumptions used are generally true.

Application of the Breeden and Litzenberger methods to non-equity markets is less common, mainly due to the lower number of option quotations, but there is still some interesting work. In the case of currency markets, the most well- known publication is Malz (1997) and Malz (2014), where the author using ad- ditional assumptions is able to interpolate the volatility smile, based on this risk-neutral densities are then obtained. A classical analysis of their statistics and tail event probabilities is then conducted, the author deals with the forward Premium and ICAPM model and shows that moments risk-neutral probabilities have some predictive power. As previously described, the first Malz (1997) as- sumes a second-order polynomial form of implied volatility, whereas the second work uses interpolation through the clamped spline method.

It is also possible to list papers that all use the Malz methodology on a different time periods and currencies. These are for example the work of Cuaresma et al.

(2010), which deals with risk-neutral densities in the case of Central European countries, the authors further construct an early warning indicator based on these

densities and show its predictive power, concluding the fact, that risk-neutral den- sities provide an additional information about exchange rate movements. I would also like to point out the working papers of the Bank of New Zelland, Gereben (2002) and Lewis (2012), first paper deals with the change in the behavior of the New Zealand dollar against the US dollar, forward premium, and also statistics of moments of the implied distribution. A similar analysis is provided by the Turk- ish Central Bank’s working paper Korkmaz et al. (2019). There is also the work of Ba´nbula (2008) on this topic, and the author performs the same analysis on the polish zloty and the euro. Castr´en (2005) examines the risk-neutral density of the exchange rates of the three new EU member states Poland, Czech Republic and Hungary. The author analyzes whether monetary policy decisions affect the moment of the distribution function and finds find that only some of the implied moments on the Polish zloty exchange rate systematically move around policy events.

In other areas of research, we can mention the work of Nitteberg (2011), which deals with commodity derivatives, more specifically the WTI options, the author calculates risk-neutral densities and the usual statistical moments. There are also works that deal with interest rate options, such as Coutant et al. (2001), which deals with the development of PIBOR futures and different methods for extracting risk neutral densities.

The literature on the extraction of subjective densities is quite limited, although ever since his publication, the Ross Recovery Theorem has attracted some other authors that build upon Rosses research. Regarding the theoretical foundations, Rosses work was followed by Carr and Yu (2012), who have shown that the recov- ery theorem can be extended to the continuous state-space setting for bounded diffusions. Walden also showed that the bounded diffusion condition can be re- laxed, and showed a solution for unbounded diffusion. Another paper by Qin and Linetsky (2014) show how to express Ross recovery in the more general state-space of Borel right processes. Backwell (2015) formalized the Recovery Theorem and added the extension, where he relaxed the assumption of state-independent pref- erences. In contrast, Dubynskiy and Goldstein (2013) and Boroviˇcka et al. (2016) present a critique of the recovery theorem, which focuses on the decomposition of the stochastic discount factor and was already presented in more detail before.

Empirical work literature applying the recovery theorem is also quite limited and the results vary considerably. Martin and Ross (2019) use the recovery theo- rem to analyze long-dated bonds and the yield curve movements. Audrino et al.

(2015) use the S&P 500 option prices to investigate whether the recovery yields predictive information from the sample period from 2000 to 2012 they found that market timing strategies based on recovered moments significantly outperform their counterparts based on their risk-neutral densities. Bakshi et al. (2018) also apply the recovery theorem using Treasury bond futures, and their results un- dermine the implications of the recovery theorem. Authors reject the assumption that the Martingale component of the stochastic discount factor is identical to the unity of the Recovery theorem. Jackwerth and Menner (2018) show that recovered physical distributions based on the S&P 500 index are incompatible with future returns and fail to predict them. These negative results are even

stronger when authors add, as they claim, economically reasonable constraints.

Lan (2018) build on the method of Backwell (2015) using an improved strategy for combining option data with the Recovery Theorem to estimate the subjec- tive densities, authors then used real-world data and investigated whether the Recovery Theorem yields predictive information and rejected the hypothesis.

Kiriu and Hibiki (2019), on the other hand, propose a robust method for the Ross Recovery Theorem that uses a prior information and ensures numerical stability. Gagnon et al. (2019) investigate whether moments recovered using recovery theorem perform better to forecast realized volatility. The authors found evidence that these factors significantly improved realized volatility forecasts.

Flint and Mar´e (2018) implement the Ross Recovery Theorem using option prices of stock market in the South Africa. They compute the moments from the implied risk-neutral and real-world implied distributions and use them as signals for asset allocation with some predictive power. Sanford (2018) shows that more robust results can be achieved using a multivariate Markov chain. Appel and Mare (2019) follow up on their work with specific functional forms of pricing kernel and simulated data, where they determine the robustness of the method.

In short, the results are still ambiguous and do not give a clear answer to the practical value of the Ross recovery theorem. For this reason, we believe that further research is needed and we also believe, that further research is needed and the Ross recovery theorem has not yet been applied to the FOREX markets, which may add some new information.

2. Methodology

2.1 FX option price

For the empirical implementation, first, we need to derive the formula for the price of currency options. The basic model for currency option was formulated by Garman and Kohlhagen (1983). The model is essentially a variation of the most well-known European option pricing model developed in the 1970s by Fischer Black, Myron Scholes, and Robert Merton. Let us now show the simplest way to derive the Black Scholes model for currency options using a probabilistic approach as in Lee and Lee (2013). Due to the length of the derivation, I illustrate this derivation for call options only and price of put option can be then obtained using put-call parity.

The underlying asset for currency options is the spot rate of two currencies, denot- ed byS_t. r andq denote risk-free domestic and foreign interest rates respectively.

σ represents the exchange rate volatility and dWt denotes the standard Wiener process. Assume that the issuer of an option and at the same time the owner of a foreign currency receive interest qS_tdt in the time interval dt. The expect- ed return on the option is then reduced by this income. We also assume that the underlying asset is following geometric Brownian motion, implying constant volatility. This can be written in the form of the following stochastic differential equation:

dS_t= (r−q)S_t+σS_tdW_t (2.1) The solution of this equation can be easily shown by applying the Ito’s lemma to lnS_t:

dln(S_t) = dS_t S_t − 1

2 1

S_t²dS_t² = 1

S_t((r−q)S_tdt+σS_tdW_t)− 1

2S_t²σ²S_t²dt (2.2)

∫︂ t 0

dln(S_x) = (r−q)t− 1

2σ²dt+σdW_t (2.3)

The integration of this expression gives us the following equation of geometric Brownian motion.

S_t=S₀e^{(r−q)t−12σ2dt+σdWt} (2.4) Note that lnS_t has a normal distribution N(S₀+ (r−q)t− ¹₂σ²dt, σ²t), i.e. lnS_t has a lognormal distribution. The first and second moments of this distribution, therefore, take following form:

E^pS_t=S₀ + (r−q)t− 1

2σ²dt (2.5)

V arS_t=σ²t (2.6)

Using the change of variables technique we can obtain the following probability density function:

f_St(S) = f_lnSt(lnS)

⃓

⃓

⃓

⃓

⃓

dlnS dS

⃓

⃓

⃓

⃓

⃓

(2.7) f_St(S) = 1

Sσ√ 2tπe

lsS−lnS0−(r−q)t+ 1 2σ2dt

2σ2t (2.8)

The price of the call option under the risk neutral measure is given as follows:

C(X) =e^−rtE^qmax(S_t−X,0) (2.9) Which is equivalent to the following notation using indicator function:

C(X) =e^−rtE^qS_t1_St>X −e^−rtE^qX1_St>Xf_St(S)dS (2.10) Let’s break down the expression into individual parts and then solve the first term first:

E^qSt1_St>X =

∫︂ ∞ X

SfSt(S)dS = S Sσ√

2tπe

lnS−lnS0−(r−q)t+ 1 2σ2t

2σ2t (2.11)

Let us denote:

x= lnS−lnS₀−(r−q)t+¹₂σ²t σ√

t

=> dS

dX = 1 Sσ√

t => dS =σ√ te^xσ

√

t+lnS0+(r−q)t−σ²t/2dx (2.12) The limits of integration change as follows: For S = ∞ => x = ∞ and for S =X => x= ^{lnX−lnS0−(r−q)t+}

1 2σ²t σ√

t The expression becomes:

E^qS_t1_St>X =

∫︂ ∞ x

1 σ√

2tπe

lnS−lnS0−(r−q)t+ 12σ2t

2σ2t σ√

te^xσ

√t+lnS0+(r−q)t−σ²t/2dx (2.13)

Which we further simplify to:

E^q S_t 1₍S_t > X ) = S₀e^(r−q)t

∫︂ ∞ x

e^((x−σ

√ t)2

2 dx (2.14)

We use another substitution: z² = (x − σ√

t)². This simplifies the expres- sion and allow us to express the equation as a standard normal distribution N(0,1), the limits of the integral change as follows due to substitution: x =

(^lnX−lnS0−(r−q)t+¹

2σ²)

σ√

t => z = (^lnX−lnS0−(r−q)t−¹

2σt2)

σ√ t

E^qS_t1_St>X =S₀e^(r−q)t

∫︂ ∞ x

e^{( z}

2 σ√ t)

dx (2.15)

By integrating the expression we get the following:

E^qSt1St>X =S₀e^(r−q)t

∫︂ ∞ x

(︄

1−N

(︄lnX−lnS₀−(r−q)t− ¹₂σ²t σ√

t

)︄)︄

(2.16)

Using the symmetry of the normal distribution, that isN(x) = 1−N(−x) leads to the following solution:

E^qS_t1_St>X =S₀e^(r−q)tN

(︄

−lnX−lnS0−(r−q)t−¹₂σ²t σ√

t

)︄

(2.17) We derive the second term in a similar way:

E^qX1_St>X =XE^q1_St>X =XE^q(S_t > X)

=XE^q

(︃

S₀e^(r−q)t− 1

2σ²dt+σW_t)> X

)︃

=XE^q

(︄σW_t

√t > ln(X)−ln(S₀)−(r−q)t+ ¹₂σ²dt

√tσ

)︄

(2.18)

We have used a heuristic relationship that W_t = √

tz, where z is a random variable with a distribution N(0,1) and again using the symmetry of the normal distribution, the expression can be modified as follows:

E^qX1_St>X =XE^q

(︄

1−N

(︄ln(X)−ln(S₀)−(r−q)t+¹₂σ²dt

√tσ

)︄)︄

(2.19)

E^qX1_St>X =XE^q

(︄

−ln(X)−ln(S₀)−(r−q)t+¹₂σ²dt

√tσ

)︄

(2.20) By combining the two terms we obtain the famous Black Scholes equation:

C(X) = e^−qtS_tN

⎛

⎝−ln^(︂_S^X

t

)︂−(r−q+¹₂σ²t σ√

t

⎞

⎠

−Xe^−rtN

⎛

⎝−ln^(︂_S^X

t

)︂−(r−q)t+ ¹₂σ²)t σ√

t

⎞

⎠ (2.21) For simplicity, let us mention the usual notation, where the first term is denoted as d1, and the second term is denoted as d2

d1 =

⎛

⎝

ln^(︂S_X^t)︂+ (r−q+ ¹₂σ²)t σ√

t

⎞

⎠ (2.22)

d2 =

⎛

⎝

ln^(︂S_X^t)︂+ (r−q− ¹₂σ²)t σ√

t

⎞

⎠ (2.23)

We can clearly see that the relationship between d1 and d2 is as follows:

d1 =d2 +σ√

t (2.24)

The simplification proposed in Malz (1997) used in this thesis, which uses the forward price relation: F_t=S_te^(r−q)t. Which is basically the Black 76 model:

C(X) =e^−rt(F_tN

⎛

⎝−ln^(︂X_F

t

)︂− ¹₂σ²t σ√

t

⎞

⎠−XN

⎛

⎝−ln^(︂_F^X

t

)︂+¹₂σ²t σ√

t

⎞

⎠) (2.25)

For obtaining put option prices, we will use simplification using put-call parity.

This is defined as the relationship between the put price and the call option, which have the same underlying asset, ie the same currency, the same exercise price, and the same maturity. Put-call parity then indicates that the value of the long position in the forward contract is equal to the purchase of the call option and the sale of the put option. However, this means that the current values of a given relationship must also be equal to non-arbitration conditions. For the final solution we will use the currency forward equation given above:

C(X)−P (X) =e^−rt(F_t−X) =S_te^−qt−e^−rtX (2.26)