Modelling bid- and ask prices of financial derivatives

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Modelling Bid- and Ask Prices of

Financial Derivatives

Jack Schilder 6074510

Bachelorthesis Actuarial Sciences

Faculty of Economics and Business Administration University of Amsterdam

Thesis Supervisor Prof. Dr. R.J.A. Laeven

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Chapter 1 Introduction

Financial markets offer many opportunities for investors. These investors typically concern themselves with finding a satisfying balance between the rewards and risks associated with an investment. To quantify this balance, a variety of different performance measures have been called into life. For an investor to accept or reject a proposed opportunity ultimately depends on the asset prices as established by the market. In classical financial economics, markets are seen as a passive counterparty, accepting any amount and any direction of the trade of a financial asset at the going market price. When the asset is priced correctly according to economic theory, the asset market is said to be efficient. No opportunity for profit in excess of opportunity cosst should exist. Conditions for a unique price depend on the existence, and immediate execution of, arbitrage opportunities and on the availability of trading counterparties. For this to happen, markets need to be liquid.

This view is known as the Law of One Price, which is prevalent in most of the classical asset-pricing models. However, various financial derivatives are being traded in different markets and against different prices. In cases like this, efficient market conditions are often not satisfied. Furthermore, these tend to be illiquid markets, where illiquidity can be thought of as the inability of the market to reach a unique price. When investors lack sufficient

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information to act upon, or when there are too few counterparties interested in trading the derivatives, market illiquidity arises.

In such situations, the Law of One Price fails to hold, and the market is said to be incomplete. Eberlein, Gehrig and Madan (2012) describe incom-pleteness as the holding of residual risk by buyers and/or sellers in a market, which cannot be eliminated by even the best hedge. Under such circum-stances, one can no longer expect to buy from and deliver to the market at the same price. As illiquid and incomplete markets are regularly encountered in practice, one could argue that models based on the Law of One Price are unrealistic, as they are based on conditions that are not met. It is for this reason that Cherny and Madan (2010) abandon the Law of One Price and opt for an extension to a Law of Two Prices. In their view the market is still a counterparty, but when markets are illiquid, their model allows for prices to depend on the direction of the trade. Thus, investors are faced with an ask price when they want to buy from the market, and a bid price when they deliver a financial derivative to the market. The difference between these two prices is known as the bid-ask spread. This spread not only measures the illiquidity of a market, but also the costs for an investor to maintain or discard a position.

There is a variety of different theories trying to model this bid-ask spread, of which the following are mentioned in Cherny and Madan (2010): Some approaches try to determine bid and ask prices that model the effects of informed traders on market makers (Copeland and Galai (1983), Glosten and Milgrom (1985)), while others make the addition of taking into account the costs liquidity providers face concerning order processing and inventory (Ho and Stoll (1981)). Decomposing the spread into order processing, inventory and adverse selection components is considered by Huang and Stoll (1997), and Constantinides (1986) and Lo, Mamaysky and Wang (2004) go by yet another approach by introducing transaction costs, addressing the costs of

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trading in liquid markets.

There exists a large segment of financial markets which are not liquid themselves, yet they create financial products using the markets that are liquid for hedging risks. In these markets, one typically observes two dif-ferent prices for buying and selling and this difference can be quite large at times. The above approaches for modelling the bid-ask spread may have little connection with this difference in bid- and ask prices, as they generally only apply to liquid markets. To model the bid-ask spread in illiquid markets Cherny and Madan (2010) build upon studies by Bernardo and Ledoit (2000) and Carr, Geman and Madan (2001), among others, from which they develop general closed-form expressions for bid- and ask prices, particularly for put-and call options. They see the bid-ask spread as a holding cost. Counter-parties willing to trade are not readily available in an illiquid market, which makes it impossible to have trading in both directions at one going market price.

One may now see the bid-ask spread as a reflection of the cost of holding residual risk, as in Eberlein, Madan and Schoutens (2011). This is where a rather new theory known as conic finance comes into play. This theory allows for bid- and ask prices of financial derivatives to depend on the direc-tion of the trade, and it tries to model these prices by applying the nodirec-tion of acceptability of cash flows to a financial market. The market is assumed to require a minimum level of acceptability for a position to be marketable. Since a market maker would want to out-do competitors, it will set the lowest possible ask price and the highest possible bid price, such that maintaining a short position and a long position in a derivative, respectively, is still ac-ceptable. As a consequence, the bid-ask spread becomes smaller and the risk of a position is minimized. One could see this spread as the cost of unwind-ing a position. The liquidity of a market is closely related to the bid-ask spread and in general, highly liquid financial derivatives have a small spread,

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whereas illiquid assets have a large spread.

The main goal of this thesis is to study the theory of conic finance, and the possibilities it offers for pricing financial derivatives. Based on this the-ory, in particular the closed-form formulas for bid- and ask prices, I seek to analyze past prices of European put- and call options using empirical data. For this analysis, I will focus on the period ranging from 2008 to 2010 in which, as of this writing, the latest financial crises took place. The global financial crisis and the sovereign debt crisis (which in fact is still ongoing) resulted in negative economic growth and worldwide financial trouble in the years that followed September 2008. Based on the bid-ask spreads of certain derivatives, one may be able to quantify the effect of a financial crisis on in-vestors confidence, which in turn affects the liquidity of the financial markets these investors trade in. Then, by modelling the bid- and ask prices using conic finance, one could get more realistic estimations of current and future prices.

In the next chapter, I will give a more detailed overview of the theory behind conic finance. Of importance are the concepts of acceptability indices and distortion functions. With these, closed-form expressions for bid- and ask prices of options can be derived. In the chapter that follows, the theory is used to analyze the data, the results of which will be summarized in the conclusion.

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Chapter 2 Conic Finance Theory

As mentioned in the introduction, conic finance is a new theory for derivative pricing that allows prices to depend on the direction of the trade. In this section, I will introduce the concepts of acceptability indices and distortion functions, and use these to derive expressions for the theoretical bid- and ask prices.

2.1 Theory of Acceptability Indices

Conic finance theory is based on the use of acceptability indices to mea-sure an investment’s performance. This theory provides a new approach for deciding whether or not to accept opportunities, which is an intermediate between expected utility theory and arbitrage pricing theory. Expected util-ity maximization is a powerful tool with strong theoretical appeal, yet it has limited use in practice, for it requires quite an amount of specific inputs, which are difficult to formulate and which also tend to differ over time and across assets.Furthermore, it needs to be consistent with the Von Neumann-Morgenstern axioms.

Arbitrage pricing relies on the existence of related market prices, and the problem of whether or not to undertake an opportunity is solved when

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the payoffs from the opportunity are spanned by the payoffs from traded assets. The associated present value of these payoffs can be compared with the initial cost of the opportunity.An investor preferring more to less only takes opportunities resulting in a positive net present value (NPV). For all this to work properly, models using arbitrage theory tend to presume market completeness. However, conditions required for models to produce market completeness are generally quite stringent. Furthermore, this theory lacks strong implications to decide between two cash flows that are not arbitrage opportunities, although one cash flow might be more attractive to the investor than the other.

A new approach combining the above theories was introduced in Carr, Geman and Madan (2001). This approach is based on the idea that the set of opportunities is expanded from arbitrage opportunities to a set of acceptable opportunities. These are investment opportunities that are agreeable to a wide variety of reasonable individuals.

2.1.1 Investment Opportunities in Conic Finance

Transactions in finance involve the purchase or sale of an investment oppor-tunity with a stochastic payoff at a certain time in the future. The space of traded cash flows is taken to be the probability space (Ω, F, P ). When mar-kets are complete, which is the best possible case, there is a unique pricing measure Q equivalent to P . Now, for any cash flow C with a nonzero price w, one can form the difference X = C − w, which now has zero price, and one can buy from or sell to the market any amount of this cash flow X. It is possible to buy at a higher, or sell at a lower price. Thus, the set of marketed cash flows has the property that EQ(X) ≥ 0, for all X.

Though it may not be completely realistic, assuming an investment op-portunity has zero-cost does not affect the generality of conic finance theory, since one can borrow the initial premium at the risk-free rate, and pay it

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back once the final payoff of the cash flow is due. When doing calculations based on this theory, one must use appropriate discount factors to account for the time-value of money.

Cherny and Madan (2009) set out to develop a new collection of perfor-mance measures of efficiency for trades or investments. To that end, they first formulate a set of axioms that should be satisfied by such measures, and if this is the case, satisfactory measures are coined ‘indices of acceptability’. The degree of efficiency is measured by the level of acceptability of a cash flow, with arbitrages having an infinite level of acceptability. The level of acceptability then rises with the size of the associated set of measures that evaluate the trade positively. To put it another way, this means that the higher the level of acceptability, the greater the set of consenting measures, which makes it more likely that a trade or investment is viewed positively by investors. The aim is not to measure personal preferences, meaning that an acceptability index might be used to determine the direction of the optimal trade, but it tells nothing about the optimal size.

2.1.2 Axioms for Acceptability Indices

Acceptability indices should satisfy a number of axioms. To formulate these axioms, the financial outcomes of trading are modeled by zero-cost terminal cash flows seen as random variables on a probability space (Ω, F, P ), such in line with the work by Artzner, Delbaen, Eber and Heath (1999). A perfor-mance measure is then defined as a map α from the set of bounded random variables to the extended positive half-line [0, ∞].

Cherny and Madan (2009) propose and define eight properties that an acceptability index should satisfy. The first four properties define an accept-ability index. The other properties are additional ones which allow for further comparisons between acceptability indices. Let α represent the acceptability index, or performance measure. Let X and Y denote the bounded, random

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variables representing zero-price stochastic cash flows.

1. Quasi-Concavity

The set of cashflows acceptable at level γ is defined as:

Aγ = {X : α(X) ≥ γ}, γ ∈ R.

By the quasi-concavity condition, Aγ is a convex set for all γ:

if α(X) ≥ γ and α(Y ) ≥ γ; then α(λX + (1 − λ)Y ) ≥ γ for any λ ∈ [0, 1].

For this property to hold, the set of cash flows acceptable at level γ, denoted by Aγ, is required to be convex.

2. Monotonicity

Monotonocity is a basic acceptability property, which states that in general, more is preferred over less. This means that if X is acceptable at some level and Y dominates X as a random variable, then Y is at least acceptable at the same level. The monotonicity property should be satisfied on any level:

If X ≥ Y , then α(X) ≥ α(Y ), If X ≤ Y , then α(X) ≤ α(Y ).

3. Scale Invariance

Scale invariance requires that the level of acceptability of a cash flow X does not change under scaling. This property, along with the property

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of quasi-concavity makes that the set Aγ is a convex cone. Thus it

requires that

α(λX) = α(X) for all λ > 0.

This property is quite significant as the direction of the trades is of interest, not their scale.

4. Fatou Property

This property guarantees continuity or closure, and is stated as: If Xn is a sequence of random variables such that

|Xn| ≤ 1, α(Xn) ≥ γ and Xn plim

−−→ X, then α(X) ≥ γ.

5. Law Invariance

Law Invariance requires that the index of acceptability depends only on the probability law of the random variable:

If X law= Y , then α(X) = α(Y ),

with X law= Y meaning that X and Y have the same probability distri-bution.

The Law Invariance property is a strong one because it does not take into account the dependence between cash flows and the personal pref-erences of investors. Hence, a law invariant performance measure is un-able to decide between two cash flows with the same distribution func-tion, that are nonetheless evaluated differently by different investors.

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Though it is a practical property, it does ignore a lot of information.

6. Consistency with Second-order Stochastic Dominance

Acceptability indices need to be consistent with expected utility the-ory. So, if E[U (X)] ≤ E[U (Y )] for any increasing concave function U (·), then α(X) ≤ α(Y ).

7. Arbitrage Consistency

The arbitrage consistency property requires that a cashflow with a positive payoff is accepted by everyone. As arbitrages are universally acceptable, the level of acceptability for such outcomes should be set at infnity. For arbitrage consistency it is required that:

X ≥ 0 a.s. if and only if α(X) = ∞.

8. Expectation Consistency

This property makes it so that cashflows with negative expected payoffs have an acceptability level of zero, whereas any cashflow with positive expected payoffs has an acceptability level larger than zero:

If E[X] > 0, then α(X) > 0; If E[X] < 0, then α(X) = 0.

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2.2 Performance Measures

The eight axioms in the previous section are eight basic economic properties that a performance measure ought to satisfy. The first four properties are what actually defines an acceptability index. Such an index could possess one or more of the remaining properties, which add to the significancy of an acceptability index, and as such can be used to compare different indices. Together, these first four properties also provide a representation of α(X), connecting it to the family of probability measures:

Let L(Ω, F, P ) be the probability space of a bounded random variable X. α(X) is an index of acceptability and satisfies the first four axioms of acceptability if and only if there exists a family of subset Dγ : γ > 0 of P ,

such that:

α(X) = sup{γ ∈ R+ : inf EQ[X] ≥ 0}, (2.2.1) and Dγ : γ > 0 is an increasing family of sets of probability measures.

What this means is that the different risk preferences of different investors are represented by a range of probability measures, contained within the set Dγ. As more investors evaluate the zero cost cash flow X positively, the size

of the set of probability measures in Dγ supporting X increases, resulting in

a higher level of acceptability of the stochastic cash flow X. As such, when all probability measures have a positive expectation of X, the set containing these measures is at its largest possible size, which is represented by the acceptability level of the cash flow X. In other words, α(X) = γ is the highest value that makes the expectation of X positive under all probability measures in Dγ. Using these values of α, one is able to decide between

different investment opportunities.

Two of such performance measures are the Sharpe Ratio and the Gain-Loss Ratio. The first one of these examines the performance of an investment

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based on measuring expected excess return adjusted for volatility, while the second one is the ratio between expected returns and expected losses of an investment. Both however, have been rejected as acceptability indices ac-cording to Cherny and Madan (2009). Not only does the Sharpe Ratio fail to satisfy the properties of monotonicity and consistency with second-order stochastic dominance, but it was shown by Bernardo and Ledoit (2000) that it does not possess the arbitrage consistency property, either. The Gain-Loss Ratio is shown by Cherny and Madan (2009) to be an acceptability index, but does not possess other economic properties. A limitation of the Gain-Loss Ratio is that it treats small losses and large losses symmetrically. Losses are exaggerated up to a finite level, whereas it would be reasonable if large losses were exaggerated up to infinity. This shortcoming makes the Gain-Loss Ratio less suited as an acceptability index, because when deciding whether or not to invest, investors will want to properly quantify extreme monetary movements, large losses in particular.

Cherny and Madan (2009) formulate their own performance measures, some of which don’t and some of which do satisfy the ideal properties of a performance measure previously introduced. For this, they use what is known as ’distortion functions’.

2.3 Distortion Functions

One can look at the cash flow X = C −w again, which is thought of as having a price of zero, and consider this X a random variable. The relevant set of probability measures that value X positively must now be determined. In the previous section, it was determined that Dγ is an increasing set, whose size is

determined by γ. Furthermore, under the assumption of law invariance, one only has to know the cumulative distribution function FX of X to determine

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to represent such a set of probability measures.

Let us define a distortion function as an increasing, concave function Ψ : [0, 1] → [0, 1], with Ψ(0) = 0 and Ψ(1) = 1. For a random variable X with cumulative distribution function FX, the transform

F∗ = Ψ(F (x)) (2.3.1)

defines a distorted probability measure, with Ψ being the distortion func-tion. A function FX to which this distortion function is applied, will be

distorted at a rate γ. Since Ψ is a concave function, higher weights will be assigned to lower outcomes of the random variable X, and vice versa, causing the expectation of X to fall. As such, a larger γ means that this distorted expectation of X will be lower. This expectation is defined as:

EQγ_{(X) =}

Z ∞

−∞

xdΨγ(FX(x)) (2.3.2)

The distortion parameter γ is seen as a measure of market price risk. Using the definition of a distorted expectation, we can now specify the set of probability measures, so that the above representation can be rewritten as an acceptability level as follows:

α(X) = sup{γ ≥ 0 : EQγ_{[X] ≥ 0} = sup{γ ≥ 0 :}

Z ∞

−∞

xdΨγ(FX(x)) ≥ 0}

(2.3.3) This relationship shows that only if EQγ_{(X) ≥ 0 holds, X is in the set of}

cash flows acceptable at a level γ. The level of acceptability of a cash flow can then be thought of as the maximum level of distortion that the cash flow can withstand such that its distorted expectation remains positive. It can be seen as the maximum level of stress that a random cash flow can withstand while remaining attractive to a large range of investors.

Using the theory of conic finance, all stochastic cash flows X with a level of acceptability of at least gamma, are accepted by the market, which is seen

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as a counterparty in transactions. Cherny and Madan (2010) then use a fixed acceptability level γ for a fixed acceptability index α. The ask price aγ(X)

is what the market asks for a trade, i.e. the price an investor has to pay to buy an asset. The residual cash flow a − X must be α-acceptable at level γ, written as aγ(X) − X ∈ Dγ.

From formula (2.3.1), one can derive the theoretical ask price can be derived as follows: α(a − X) ≥ γ ⇐⇒: Z ∞ −∞ xd Ψγ(Fa−X(x)) ≥ 0, (2.3.4) a + Z ∞ −∞ xd Ψγ(F−X(x)) ≥ 0, (2.3.5)

so the minimum value of a leads to an ask price of: aγ(X) = −

Z ∞

−∞

xd Ψγ(F−X(x)). (2.3.6)

The bid price bγ(X) is the price a trader receives for selling an asset to

the market. When the market buys X at price b, X − b must be acceptable at level γ: X − b ∈ Dγ.

Analogously to the above derivation of the theoretical ask price, one de-rives the bid price as:

α(X − b) ≥ γ ⇐⇒: Z ∞ −∞ xd Ψγ(FX−b(x)) ≥ 0 (2.3.7) −b + Z ∞ −∞ xd Ψγ(F−X(x)) ≥ 0, (2.3.8)

So the maximum value of b leads to a bid price of: bγ(X) =

Z ∞

−∞

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Due to the concavity of distortion functions one can be sure that the ask price is always larger than the bid price, which is consistent with both the literature and observations on the financial markets.

2.4 Acceptability Index Measures

This section presents the new acceptability index measures, motivated by the axioms and distortions functions introduced earlier. These acceptability indices are developed from the weighted VAR (Weighted Value at Risk) which has the form:

WVARγ(X) = −

Z

R

xd Ψγ(FX(x)), (2.4.1)

where FX(x) is the cumulative distribution of the random variable X.

This representation of WVAR is very convenient in constructing particular representatives of this class and Cherny and Madan (2009) employ it to con-struct new risk measures. These being: A(cceptability)I(ndex)MIN, AIMAX, AIMAXMIN and AIMINMAX, with a distortion function corresponding to each of them. For illustrative purposes, graphs of both the distortion function and its effect on the cumulative distribution of a standard normal distributed random variable will be included for each new index measure.

1. MINVAR Acceptability Index − AIMIN(X)

The concave distortion function is given by:

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AIMIN is the largest number γ such that the expectation of the mini-mum of γ+1 draws from the distribution of the cash flow is still positive. Figure 2.1 gives its graphical representation. A disadvantage of MIN-VAR is that the maximal weight assigned to large losses is relatively small. As consumers are generally risk-averse, they will weigh large losses more heavily than large gains. This risk measure is too lenient towards large losses, thus inadequately representing rational consumer behavior, which makes it not a good risk measure. MINVAR is rarely used for this reason, as better alternatives are available.

0.2 0.4 0.6 0.8 1.0 v 0.2 0.4 0.6 0.8 1.0 YΓ YΓ_{HvL 1 - H1 - vL}Γ +1 Γ=32 Γ=4 Γ=1 Γ=0 (a) -4 -2 2 4 x 0.2 0.4 0.6 0.8 1.0 CDF (b)

Figure 2.1: MINVAR on Standard Normal

2. MAXVAR Acceptability Index − AIMAX(X)

The concave distortion function is given by:

Ψγ(v) = vγ+11 , γ ∈ [0, ∞], v ∈ [0, 1] (2.4.3)

This works the same as AIMIN, only now it is the expectation of the maximum of γ + 1 draws from the distribution that we are concerned with. Its effects can be seen in Figure 2.2. MAXVAR is more promising

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than the previous risk measure, yet it still has a potential drawback. It does not discount large gains, which is the reason that this risk measure is rarely used, as well.

0.2 0.4 0.6 0.8 1.0 v 0.2 0.4 0.6 0.8 1.0 YΓ YΓ_{HvL v}Γ +11 Γ=32 Γ=4 Γ=1 Γ=0 (a) -4 -2 2 4 x 0.2 0.4 0.6 0.8 1.0 CDF (b)

Figure 2.2: MAXVAR on Standard Normal

3. MAXMINVAR Acceptability Index − AIMAXMIN(X)

Combining MINVAR and MAXVAR gives the following distortion func-tion:

Ψγ(v) = (1 − (1 − v)γ+1)γ+11 _{, γ ∈ [0, ∞], v ∈ [0, 1]} _(2.4.4)

AIMAXMIN is constructed by first using the MINVAR procedure, and then the MAXVAR procedure to create worst case scenarios. See its effectsin Figure 2.3. This risk measure satisfies all eight axioms for acceptability indices and has neither of the drawbacks of the previous two risk measures.

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0.2 0.4 0.6 0.8 1.0v 0.2 0.4 0.6 0.8 1.0 YΓ YΓ_{HvL I1 - H1 - vL}Γ +1_MΓ +11 Γ=32 Γ=4 Γ=1 Γ=0 (a) -4 -2 2 4 x 0.2 0.4 0.6 0.8 1.0 CDF (b)

Figure 2.3: MAXMINVAR on Standard Normal

4. MINMAXVAR Acceptability Index − AIMINMAX(X)

Combining MAXVAR and MINVAR gives the concave distortion func-tion:

Ψγ(v) = (1 − (1 − vγ+11 ))γ+1, γ ∈ [0, ∞], v ∈ [0, 1] (2.4.5)

AIMINMAX is constructed by first using the MAXVAR procedure, and then the MINVAR procedure to create worst case scenarios. Figure 2.4 shows its effect on a Standard Normal distribution. Like MAXMIN-VAR, this risk measure satisfies all eight axioms for acceptability indices and has essentially no drawbacks. MINMAXVAR and MAXMINVAR have shown to give comparable results in practice, though differences are noticeable when comparing their graphical representations.

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0.2 0.4 0.6 0.8 1.0 v 0.2 0.4 0.6 0.8 1.0 YΓ YΓ_{HvL 1 - 1 - v}Γ +11 Γ +1 Γ=32 Γ=4 Γ=1 Γ=0 (a) -4 -2 2 4 x 0.2 0.4 0.6 0.8 1.0 CDF (b)

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Chapter 3 Application of Conic Finance to

Pricing European Options

In this chapter, the theoretical framework of conic finance as established in the previous chapter, is used to derive explicit expressions for bid- and ask prices of European call- and put options. Distribution functions of options are specified, as there exist straightforward expressions for the stochastic payoffs of such options. With those distribution functions, it is possible to apply the expressions for bid- and ask prices as derived in the foregoing chapter and then calibrate the different stress levels, as indicated by the parameter γ. The movement of γ is then observed over time, with special attention given to the period known as the global financial crisis. From the theory it follows that arbitrage opportunities should be excluded from the data set, as these would yield infinite levels of γ.

3.1 Black-Scholes in Conic Finance

A paper concerning options pricing was published by Black and Scholes back in 1973. That same year, Merton (1973) extended their work and introduced the Black-Scholes model, which is a mathematical model of a financial

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mar-ket for specific financial assets. From this model follows the Black Scholes equation, which gives a theoretical estimation of the prices of European put-and call options. This price is influenced by the remaining time until the expiration date of the option and by the value of the underlying asset. The main idea behind the Black-Scholes formula is to eliminate risk by hedging the options. One method of doing this is by both buying a call and selling a put, thus simulating a long position in the underlying asset of the options.

3.2 Bid- and Ask Prices with Black-Scholes

The Black-Scholes equation gives explicit prices for European put- and call options. Let S be the random variable at time T of an underlying asset. Then the price at time T of a call option C is given by (S − K)+ and for a put option P it is given by (K − S)+_{, where K is the strike price. If F}

S

denotes the cumulative distribution function of S, and one was to consider an acceptability index α based on a family of concave distortions, then the bid- and ask prices for C and P are given by:

aγ(C) = Z ∞ K Ψγ(1 − FS(x))dx (3.2.1) bγ(C) = Z ∞ K (1 − Ψγ(FS(x)))dx (3.2.2) aγ(P ) = Z K 0 Ψγ(FS(x))dx (3.2.3) bγ(P ) = Z K 0 (1 − Ψγ(1 − FS(x)))dx (3.2.4)

Derivations of these are found in Appendix A.1. To calculate the prices of put- and call options, the distribution function of the underlying asset must be known. In the Black-Scholes model, among others, prices are assumed to be lognormally distributed. Building upon this assumption, I derive the function of ST, using the Black-Scholes model. In this model, stock prices

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are assumed to follow a geometric Brownian motion:

dSt= µStdt + σStdW

where µ and σ are constants denoting the drift parameter and the volatil-ity parameter, respectively. W is a standard Brownian motion. Strepresents

the evolution of the stock price over time. In the Black-Scholes model, the principle of risk-neutral valuation is used, denoted as a risk-neutral measure Q. Under this principle, investors preferences, their risk-averseness in partic-ular, does not influence the variables used in the model. From It¯o’s Lemma, it follows that under this risk-meutral measure Q, conditional on S0, the

stock price at time T follows a lognormal distribution, written as:

ST S0 ∼ Lognormal((r − 1 2)T, σ 2_{T ),} or ST ∼ Lognormal((r −1₂)T + ln S0, σ2T ).

Then, the cumulative distribution of ST can be written as:

FST(x) = Φ( ln_Sx 0 − (r − 1 2σ 2_)T σ√T ) (3.2.5)

The bid- and ask prices for the market can now be generated by the distorted expectations approach using a Wang Transform approach on a log-normal distribution of the underlying asset.

This approach yields option prices that are similar to the Black-Scholes model as used in the market, but it is able to explain bid-ask prices at different levels of acceptability. Using the Wang Transform, the distortion function has the following expression:

Ψγ(FX(x)) = Φ(Φ−1(FX(x)) + γ) = Φ(

x − µx+ γσx

σx

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In this equation, µx and σx are the mean and variance of X, γ is the level

of acceptability and FX(x) is the distribution of X. Using this transform on

the Black-Scholes equation, results in:

Ψγ(FST(x)) = Φ( ln_Sx 0 − (r − 1 2σ 2_{)T + γσ}√_T σ√T ) (3.2.7)

With this, I have specified the distorted distribution function for the underlying asset, which can be used to derive theoretical bid- and ask prices of put- and call options written on it. For this, one needs to use equation (3.2.7) together with equations (3.2.1−4). (Derivations of these equations can be found in Appendix A.2) This results in the following theoretical bid-and aks prices for European put- bid-and call options:

Option Price Formula d1 d2

bγ(C) Ste−γσ √ T_Φ(d 1) − Ke−rT(d2) ln(St/K)+(r+0.5σ2)T −γσ √ T σ√T d1− σ √ T aγ(C) Steγσ √ T_Φ(d 1) − Ke−rT(d2) ln(St/K)+(r+0.5σ 2_{)T +γσ}√_T σ√T d1− σ √ T bγ(P ) Ke−rTΦ(d2) − Steγσ √ T_Φ(d 1) ln(K/St)−(r+0.5σ 2_{)T −γσ}√_T σ√T d1+ σ √ T aγ(P ) Ke−rTΦ(d2) − Ste−γσ √ T_Φ(d 1) ln(K/St)−(r+0.5σ2)T +γσ √ T σ√T d1+ σ √ T

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3.3 Analysis

Now that I have derived the closed-form expressions for bid- and ask prices of European put- and call options, I can use empirical data to estimate these prices. For this I will use data on European put- and call options on the S&P 500 Index. This American stock market index is based on the market capitalization of 500 companies having common stock listed on the NYSE or NASDAQ. There is a variety of companies from different sectors represented in this index, which gives it a certain robustness against failures in a single or multiple sectors it includes. The dataset is obtained from DataStream, containing daily bid-ask prices of a European put- and call option. These options both have a strike price of 1100, and their underlying price is established by the S&P 500 Composite Index, which is a measure for the average performance of common stock of the companies in the S&P 500 Index. The duration of the options is approximately 2 years, starting from 31-12-2007, of which I will look at 562 trading days, starting from 13-08-2008. The other variables required by the model are the risk-free rate of return, the implied volatility and the acceptability level γ. Although not the most realistic approach, I assume the first of these, from here on termed r, to be constant over the duration of the options, at five percent. It is possible to give a better estimate of r, by taking the variable rates of U.S. Treasury Bills, which are thought to be risk-free, and scaling these up to the duration of the options. However, a clear method for establishing such a variable rate of return didn’t present itself. Furthermore, the variable r does not influence the theoretical bid- and ask prices too strongly, hence my choice for a constant rate of five percent. Estimation of the other two variables is discussed in more detail in the following sections.

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3.3.1 Implied Volatility

The volatility in the Black-Scholes model is the standard deviation of the underlying asset’s returns, denoted σ. An asset’s volatility is not readily observable, which is why estimations of it must be made. This can be done using the historical volatilities, or by deducing current and future volatility based on what the market is implying, the implied volatility. In general, the Black-Scholes model assumes this volatility is a constant, which simplifies the estimation at the expense of accuracy. This is an often heard point of critisism towards the Black-Scholes model. Especially when dealing with options with a relatively long time until maturity, as is the case for this thesis, assuming σ to be constant can lead to rather large prediction errors. Even more so when one uses the historical volatilities. For this reason, I will work with implied volatility, using a variable σT for estimating the theoretical

bid-and ask prices.

As mentioned, implied volatility is the future standard deviation as im-plied by the market. A way to estimate this is by working back from readily observable market variables. Following this principle, I make use of the standard Black-Scholes equation, the solutions of which I assume to be the observed market prices for the put- and call option. The other available vari-ables are the risk-free rate of return, the underlying- and the strike price, and the time until maturity. The Black-Scholes equation cannot be solved explicitly for σ, so I use the Newton-Raphson method to estimate it. At the core of the algorithm is the following equation:

σn+1 = σn−

BS(σn) − P

ν(σn)

, (3.3.1)

with P the option’s observed market price and ν(σn) = ∂BS_∂σ = SN0(d1) ·

√ T − t.

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volatility using this algorithm, in particular for the call option. This is due to both lack of observed data, and some occasional failures of the Newton-Raphson method to converge to the root. For these dates, I assume the implied volatility has the same value as it does on the most recent date where an estimation was found.

3.3.2 Calibrating Gamma

The final unknown variable that needs to be estimated is the acceptability level γ. For an increasing value of γ the bid price decreases and the ask price increases relative to the Black-Scholes mid-price. As with the implied volatility, this variable cannot be directly observed in the market, and so an alternative method is needed to estimate daily values of γ.

One way of finding values of γ for this dataset, is to estimate them by calibration. This is done using the total-squared-error criterion, where the squared error for one of the two types of options at date i is the sum of the squared differences between the observed and theoretical bid- and ask prices:

T SE(γ) =X i (bidi− bi,γi) 2_{+ (ask} i− ai,γi) 2 _(3.3.2)

Then, the desired levels of γ are found by minimizing this expression with respect to γ: γ∗ = arg min

γ T SE(γ), with γ ∈ R n

+. This variable can

be estimated for periods of varying sizes: daily, weekly, monthly and so on, were the total number of such periods is denoted by n. For the dataset at hand, only the daily values of γ are of concern.

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3.3.3 The Financial Crisis

With this, all the necessary variables are available, and I can look at how close the modelled price movements approximate the observed price movements of the options. The period that marks the beginning of the financial crisis lends itself quite well for this purpose, as the strong fluctuations in prices give rise to a clear pattern for both options, as displayed in the figures in this section. For the pricing model to hold under such circumstances would be much more significant than the case were estimated prices closely approximate the observed prices in a period with relatively stable markets.

Graphical representations of the estimated bid- and ask prices of put- and call option, respectively, are shown in Figures 3.1 and 3.2 below.

0 50 100 150 200 250 300 350 400 450 13‐8‐2008 13‐9‐2008 13‐10‐2008 13‐11‐2008 13‐12‐2008 13‐1‐2009 13‐2‐2009 13‐3‐2009 13‐4‐2009 13‐5‐2009 B‐S (put) bid put (est.) ask put (est.)

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0 50 100 150 200 250 300 350 13‐8‐2008 13‐9‐2008 13‐10‐2008 13‐11‐2008 13‐12‐2008 13‐1‐2009 13‐2‐2009 13‐3‐2009 13‐4‐2009 13‐5‐2009 B‐S (call) bid call (est.) ask call (est.)

Figure 3.2: Estimated Bid-Ask Prices of the Call Option

The prices move as one would expect during a financial crisis. As the price of the underlying asset decreases, eventually dropping below the strike price of the options, put prices rise sharply, while call prices strongly decrease. When compared to the observed prices, these figures suggest that these are indeed very good approximations. For reference and comparison, additional figures for both observed and estimated prices are included in the appendix. To more specifically express the accuracy of the approximation, I have tried to assign a numerical value to it, using a two-sample t-test. Denoting the estimated bid- and ask prices of respectively the put- and call option as X and the observed bid- and ask prices of the same options as Y , the following t-test can be formulated:

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H0 : µX = µY

H1 : µX 6= µY

The t-statistic for this test is: T = X− bb Y

Sq 1

n(X)+

1 n(Y )

. The test is performed a total of four times, on both the bid price and the ask price of the two options, seperately. The resulting probabilities are then combined into an average for both option types, as summarized in table 3.1:

Option type P (T ≤ t; H0)

Put Option (average) 0,99312013

Call Option (average) 0,75883889

Table 3.2: T-test Results

The resulting values are clearly too high to reject the null-hypothesis. They are in fact of such magnitude, in particular for the put option, that they appear to speak quite favourably for the accuracy of the model.

Another measure of interest is the bid-ask spread, as seen in Figure 3.3. Due to uncertainty and illiquidity in the financial markets during a financial crisis, this spread tends to increase.

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0 2 4 6 8 10 12 14 16 Bid‐Ask Spread (avg.) Bid‐Ask Spread (avg.)

Figure 3.3: Average Bid-Ask spread (Put & Call)

The movements of the bid-ask spread, as seen in the figure, coincide with the events that occured from the latter half of 2008 onward. An increas-ing number of large and important financial institutions encounterd serious trouble, with the interbank lending market worsening and many companies being bailed out or going bankrupt.

Most of these events occured in the last months of 2008, though the bid-ask spread can be seen sparking up in both 2009 and 2010. This can be attributed to the reports of negative economic growth worldwide and in particular to the financial problems slowly culminating in countries such as Greece, Ireland and Spain, marking the beginning of the European sovereign-debt crisis.

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Chapter 4 Conclusion

This thesis began with the introduction of conic finance, a theory meant to combine no-arbitrage theory and expected utility theory. With conic finance, it is possible to define levels of acceptability of an investment’s stochastic cashflow. This allows investors to better distinguish between different in-vestment opportunities, while at the same time helping the market formulate better bid- and ask prices.

To this end, acceptability indices are introduced. When combined with distortion functions, these give expressions for the acceptability level of a stochastic cashflow, which in turn can be used to derive theoretical expres-sions for the bid- and ask prices.

Following this principle, I formulated the theoretical bid- and ask prices of European put- and call options. This was done by extending the standard Black-Scholes model for pricing such options, using the Wang transform to formulate Black-Scholes option prices that take into account the direction of the trade. These theoretical expressions were then used to analyze price movements during the financial crisis. Save for a few instances, the modelled prices were shown to be consistently accurate, when compared to the ob-served prices. When option prices vary based on the direction of the trade, the theory of conic finance should be preferred to comparable theories that

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assume one market price.

One point to note is that the analysis in this thesis was done retrospec-tively. As such, both implied volatility and γ could be estimated using ob-served data. If one were to use this model for predicting future option prices, the accuracy of the modelled prices would most likely be less than seen in this analysis. The variables mentioned would have to be estimated based on the expected values of all other variables used in the extended Black-Scholes model. Future research on this subject will, in all probability, result in dif-ferent methods for finding these variables. Nevertheless, I believe the use of conic finance to be a good and very promising method for modelling option prices.

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Bibliography

[1] Artzner, P., et al., (2000), ’Coherent Measures of Risk’, Mathematical Finance 9, 203−228.

[2] Bernardo, A. & Ledoit, O. (2000), ’Gain, Loss, and Asset Pricing’, Jour-nal of Political Economy 108, 144−172.

[3] Black, F., & Scholes, M. (1973). ’The pricing of Options and Corporate Liabilities’. The Journal of Political Economy 81, 637−659.

[4] Carr, P., Geman, H., & Madan, D.B. (2001). ’Pricing and hedging in incomplete markets’. Journal of Financial Economics 62, 131−167. [5] Cherny, A., & Madan, D.B. (2009). New Measures for Performance

Eval-uation. Review of Financial Studies 22, 2571−2606.

[6] Constantinides, G. & Lapied, A. (1986), ’Capital market equilibrium with transaction costs’, Journal of Political Economy 94, 842−862.

[7] Copeland, T. & D.Galai (1983), ’Information effects on bid ask spread’, Journal of Finance 38, 1457−1469.

[8] Eberlein, E., Gehrig, T. & Madan, D. (2012), ’Accounting to Acceptabil-ity: With Applications to the Pricing of One’s Own Credit Risk’, Journal of Risk 15 , 91−120.

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[9] Eberlein, E., Madan, D. B. & Schoutens, W. (2011), ’Capital require-ments, the option surface, market, credit and liquidity risk’, Working Paper, Robert H Smith School of Business.

[10] Glosten, L. & Milgron, P. (1985), ’Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders’, Journal of Financial Economics 14, 71− 100.

[11] Ho, T. & Stoll, H. (1981), ’Optimal Dealer Pricing under Transactions and Return Uncertainity’, Journal of Financial Economics 9, 47−73. [12] Huang, C., Litzenberger, R. (1988). Foundations for Financial

Eco-nomics. North-Holland, NY: Elsevier Science Ltd.

[13] Lo, A., Mamaysky, H., & Wang, J. (2004). ’Asset prices and trading volume under fixed transactions costs’. Journal of Political Economy 112, 1054−1090.

[14] Madan, D.B., & Cherny, A. (2010). ’Markets as a Counterparty: An Introduction to Conic Finance’. International Journal of Theoretical and Applied Finance IJTAF 13, 1149 1177.

[15] Merton, R.C. (1973). ’Theory of Rational Option Pricing’. The Bell Journal of Economics and Management Science 4, 141−183.

[16] Wang, S.S. (2003). ’Equilibrium Pricing Transforms: New Results using Buhlmann’s 1980 Economic Model’. Astin Bulletin 33, 57−73.

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Appendix A

Deriving bid-ask prices

A.1 European put- and call options (conic

price formulas)

A.1.1 Bid price of a call

Consider the bid price of a call option with payoff at maturity C = (S − K)+, and let bγ(c) ≥ 0, then

FC(x) = FS(S − K). (A.1.1) Knowing that: bγ(C) = Z ∞ K xdΨγ(FC(x)), (A.1.2)

This can be rewritten as: bγ(C) =

Z ∞

0

(S − K)dΨγ(FS(x)). (A.1.3)

Write S = x and split (A.1.3) to get: bγ(C) =

RK

0 (x − K)dΨγ(FS(x)) +

R∞

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This first integral equals 0 for a call option, reducing this expression to: bγ(C) =

Z ∞

K

(x − K)dΨγ(FS(x)), (A.1.4)

Which can be integrated by parts, resulting in: (Ψγ(FS(x)) − 1)(x − K) ∞ K + R∞ K (1 − Ψ γ_(F S(x)))dx = Z ∞ K (1 − Ψγ(FS(x)))dx. (A.1.5)

A.1.2 Ask price of a call

As C = (S − K)+, and it must hold that aγ(C) ≥ 0, one can say that:

F−C(x) = FS(−(S − K)) = 1 − FS(K − S). (A.1.6)

The ask price of a call is:

aγ(C) =

Z 0

−∞

xdΨγ(F−C(x)), (A.1.7)

and combining this with (A.1.6) yields:

aγ(C) = − Z 0 −∞ xdΨγ(1 − FS(K − x)) = − Z ∞ 0 xdΨγ(1 − FS(K + x)) = − Z K 0 xdΨγ(1 − FS(K + x)) + Z K ∞ xdΨγ(1 − FS(K + x)) ! .

Seeing that the first integral equals 0, and rewriting the second one gives: −R∞

K (x − K)dΨ

γ_{(1 − F} S(x)),

which after integrating by parts results in: =

Z ∞

K

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A.1.3 Bid price of a put

As P = (K − S)+, one can write:

FP(x) = 1 − FS(K − S). (A.1.9)

The bid price of a put is:

bγ(P ) =

Z ∞

0

xdΨγ(FP(x)), (A.1.10)

letting x = K − S and combining with (A.1.9) this gives: bγ(P ) =

Z ∞

0

(K − S)dΨγ(1 − FS(x)). (A.1.11)

Let random variable S be denoted by x and split the integral:

bγ(P ) = Z ∞ 0 (K − x)dΨγ(1 − FS(x)) = Z K 0 (K − x)dΨγ(1 − FS(x)) + Z ∞ K (K − x)dΨγ(1 − FS(x)) = Z ∞ K (K − x)dΨγ(1 − FS(x)),

because the second integral equals 0 for a put option. Integrating by parts: = −(Ψγ(1 − FS(x)) − 1)(K − S) K 0 + Z K 0 (1 − Ψγ(1 − FS(x)))dx = Z K 0 (1 − Ψγ(1 − FS(x)))dx. (A.1.12)

A.1.4 Ask price of a put

As P = (K − S)+, one can write:

F−P(x) = 1 − FS((−K − S)) = FS(K + S). (A.1.13)

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aγ(P ) = −

Z 0

−∞

xdΨγ(F−P(x)), (A.1.14)

and combined with (A.1.13) it is rewritten as:

aγ(P ) = Z 0 −∞ xdΨγ(FS(K + x)) = − Z ∞ 0 xdΨγ(FS(K − x)) = Z K 0 (K − x)dΨγ(FS(x)) + Z ∞ K (K − x)dΨγ(FS(x))

The second integral equals 0 for a put option. Integrating the first integral by parts:

(Ψγ(FS(x))(x − S)) K 0 + Z K 0 Ψγ(FS(x))dx = Z K 0 Ψγ(FS(x))dx. (A.1.15)

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A.2 Wang-transform bid-ask prices for

European put- and call options

A.2.1 Bid price of a call

bγ(C) = Z ∞ K xdΨγ(FC(x)) = Z ∞ K (x − K)dΨγ(FS(x)) = Z ∞ K (x − K)dΦ ln(x/St) − (r − σ2 2 )T + γρσ √ T σ√T ! = Z ∞ K xdΦ ln(x/St) − (r − σ2 2 )T + γσ √ T σ√T ! − Z ∞ K KdΦ ln(x/St) − (r − σ2 2 )T + γσ √ T σ√T ! . (A.2.1) For a random variable X with CDF FX(x) and X ∼ Lognormal(µ, σ2), it holds

that: Z ∞ K xdF (x) = eµ+12σ 2 Φ µ + σ 2_{− ln K} σ ! . (A.2.2) The first integral of (A.2.1) can then be written as:

eln St+(r−σ2₂ )T −γσ √ T +1₂σ2_T Φ ln St+(r− σ2 2 )T −γσ √ T +σ2_{T −ln K} σ√T ! = SterT −γσ √ T_Φ ln(St/K) + (r + σ2 2 )T − γσ √ T σ√T ! . (A.2.3) As for the second integral of (A.2.1):

Z ∞ K KdΦ ln(x/St) − (r − σ2 2 )T + γσ √ T σ√T ! = K 1 − Φ ln(x/St) − (r − σ2 2 )T + γσ √ T σ√T !! . (A.2.4)

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Putting (A.2.3) and (A.2.4) back into (A.2.1) and simplifying leads to: bγ(C) = Ste−γσ √ T_Φ(d 1) − Ke−rTΦ(d2), (A.2.5) with d1= ln(St/K)+(r+σ2₂ )T −γσ √ T σ√T and d2 = d1− σ √ T .

A.2.2 Ask price of a call

aγ(C) = − Z ∞ K xdΨγ(F−C(x)) = − Z ∞ K (x − K)dΨγ(1 − FS(x)) = − Z ∞ K (x − K)dΨγ 1 − Φ ln(x/St) − (r − σ2 2 )T σ√T !! = − Z ∞ K (x − K)dΨγΦ ln(x/St) + (r − σ2 2 )T σ√T ! = Z ∞ K (x − K)dΦ ln(x/St) − (r − σ2 2 )T − γσ √ T σ√T ! = Z ∞ K xdΦ ln(x/St) − (r − σ2 2)T − γσ √ T σ√T ! − Z ∞ K KdΦ ln(x/St) − (r − σ2 2 )T − γσ √ T σ√T ! . (A.2.6) Analogous to the derivation of the call-bid-formula, one can rewrite and simplify the two integrals in (A.2.6), resulting in:

aγ(C) = Steγσ √ T_Φ(d 1) − Ke−rTΦ(d2), (A.2.7) with d1= ln(St/K)+(r+σ2₂ )T +γσ √ T σ√T and d2 = d1− σ √ T .

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A.2.3 Bid price of a put

bγ(P ) = Z ∞ 0 xdΨγ(FP(x)) = − Z K 0 (K − x)dΨγ(1 − FS(x)) = − Z K 0 (K − x)dΨγ Φ ln(x/St) + (r − σ2 2 )T σ√T !! = Z K 0 (K − x)dΦ ln(St/x) − (r − σ2 2 )T − γσ √ T σ√T ! = Z K 0 KdΦ ln(St/x) − (r − σ2 2 )T − γσ √ T σ√T ! − Z K 0 xdΦ ln(St/x) − (r − σ2 2 )T − γσ √ T σ√T ! . (A.2.8) Evaluating the first integral in (A.2.8) gives:

Z K 0 KdΦ ln(St/x) − (r − σ2 2 )T − γσ √ T σ√T ! = K " Φ ln(St/x) − (r − σ2 2 )T − γσ √ T σ√T ! K 0 = KΦ ln(St/x) − (r − σ2 2 )T − γσ √ T σ√T ! . (A.2.9) The second integral in (A.2.8) can be rewritten as:

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Z K 0 xdΦ ln(St/x) − (r − σ2 2 )T − γσ √ T σ√T ! = eln St+(r−σ2₂ )T +γσ √ T +1₂σ2_T 1 − Φ ln St+ (r − σ2 2 )T + γσ √ T + σ2T − ln K σ√T !! = SterT +γσ √ T_Φ ln(K/St) − (r + σ2 2 )T − γσ √ T σ√T ! . (A.2.10) Putting (A.2.9) and (A.2.10) back into (A.2.8) and simplifying leads to:

bγ(P ) = Ke−rTΦ(d2) − Steγσ √ T_Φ(d 1), (A.2.11) with d1 = ln(K/St)−(r+σ2₂ )T −γσ √ T σ√T and d2= d1+ σ √ T .

A.2.4 Ask price of a put

aγ(P ) = − Z 0 −∞ xdΨγ(F−P(x)) = Z K 0 (K − x)dΨγ(FS(x)) = Z K 0 (K − x)dΨγ Φ ln(x/St) + (r − σ2 2 )T σ√T !! = Z K 0 KdΦ ln(x/St) − (r − σ2 2 )T + γσ √ T σ√T ! − Z K 0 xdΦ ln(x/St) − (r − σ2 2 )T + γσ √ T σ√T ! . (A.2.12) Evaluating the first integral in (A.2.12) gives:

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Z K 0 KdΦ ln(x/St) − (r − σ2 2 )T + γσ √ T σ√T ! = K " Φ ln(x/St) − (r − σ2 2 )T + γσ √ T σ√T ! K 0 = KΦ ln(x/St) − (r − σ2 2 )T + γσ √ T σ√T ! . (A.2.13) The second integral in (A.2.12) can be rewritten as:

Z K 0 xdΦ ln(x/St) − (r − σ2 2 )T + γσ √ T σ√T ! = eln St+(r−σ2₂ )T −γσ √ T +1₂σ2T _{1 − Φ} ln St+ (r − σ2 2 )T − γσ √ T + σ2T − ln K σ√T !! = SterT −γσ √ T_Φ ln(K/St) − (r + σ2 2 )T + γσ √ T σ√T ! . (A.2.14) Putting (A.2.13) and (A.2.14) back into (A.2.12) and simplifying leads to:

aγ(P ) = Ke−rTΦ(d2) − Ste−γσ √ T_Φ(d 1), (A.2.15) with d1 = ln(K/St)−(r+σ2₂ )T +γσ √ T σ√T and d2= d1+ σ √ T .

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Appendix B

Additional Figures

0 50 100 150 200 250 300 350 400 450 Observed Bid‐ and Ask Prices (Put Option) B‐S (put) bid put (obs.) ask put (obs.) Figure B.1

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0 50 100 150 200 250 300 350 400 450 Estimated Bid‐ and Ask Prices (Put Option) B‐S (put) bid put (est.) ask put (est.) Figure B.2 0 50 100 150 200 250 300 350 Observed Bid‐ and Ask Prices (Call Option) B‐S (call) bid call (obs.) ask call (obs.) Figure B.3

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0 50 100 150 200 250 300 350 Estimated Bid‐ and Ask Prices (Call Option) B‐S (call) bid call (est.) ask call (est.) Figure B.4

Modelling bid- and ask prices of financial derivatives