Gap Premium Pricing in Leveraged Exchange Traded Notes

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IN

DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS

STOCKHOLM SWEDEN 2016,

Gap Premium Pricing in

Leveraged Exchange Traded Notes

AXEL BROSTRÖM

RICHARD KRISTIANSSON

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Gap Premium Pricing in Leveraged Exchange Traded Notes

A X E L B R O S T R Ö M R I C H A R D K R I S T I A N S S O N

Degree Project in Applied Mathematics and Industrial Economics (15 credits) Degree Progr. in Industrial Engineering and Management (300 credits)

Royal Institute of Technology year 2016 Supervisors at KTH: Thomas Önskog, Jonatan Freilich

Examiner: Henrik Hult

TRITA-MAT-K 2016:07 ISRN-KTH/MAT/K--16/07--SE

Royal Institute of Technology SCI School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

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Abstract

Exchange traded notes have recently seen a surge in popularity. Investors are pouring cash into these debt securities that track various underlying assets.

Riskier leveraged exchange traded notes replicate the daily return of the underlying asset multiplied by a lever. Investors pay various fees for holding these securities, many of which are obscure and hidden. The study has researched what hedging fees underwriter should charge investors. The study has produced two models for pricing these hedging fees. One using insurance pricing based on historical losses and another using arbitrage pricing based on the Black- Scholes model. The models have been used to price leveraged exchange traded notes with a lever of 15 tracking the OMXS30 as the underlying asset, showing that all underwriters but one are within a reasonable price range. A discussion concerning investors risk taking from a behavioral finance perspective and it’s connection to leveraged exchange traded notes is also included.

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Sammanfattning

Det har skett en stor ökning i handeln av börshandlade certifikat de senaste åren.

Investerare köper certifikaten från en bank och certifikaten följer utvecklingen av en underliggande tillgång. Högriskcertifikat ger investeraren exponering mot mer risk genom att applicera en hävstång på investeringen. De betalar alltså ut den dagliga avkastningen av den underliggande multiplicerat med en hävstång.

När investerare investerar i högriskcertifikat betalar de många olika avgifter, många av dem dolda och svåra att beräkna. Särskilt de avgifter som banker tar för att täcka sina hedging-kostnader har undersökts i detta arbete. Undersöknin- gen har lett till två modeller. Den första prissätter avgiften som en försäkring där historiska förluster används och den andra använder arbitrage argument baserat på Black-Scholes-modellen. Modellerna har använts för att prissätta hedging-kostnaderna för högriskcertifikat med hävstång X15 och OMX30 som underliggande tillgång. De visar att alla banker utom en ligger inom ett rim- ligt intervall med sina avgifter. En diskussion om vilka risker investerare tar med perspektiv från beteendeekonomin och deras koppling till högriskcertifikat avslutar arbetet.

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List of Tables

1.1 Prices and returns of a Bull ETN X2 . . . 3

1.2 Prices and returns of a Bull ETN X15 . . . 3

4.1 Results of distribution fitting 1986-2015 . . . 19

4.2 Results of distribution fitting 1998-2001 . . . 22

4.3 Results of distribution fitting 2008 . . . 25

4.4 Summary of Bull Gap Premiums . . . 29

4.5 Summary of Bear Gap Premiums . . . 29

5.1 Bull & Bear figures in the Swedish market . . . 31

List of Figures

4.1 Probability density functions 1986-2015 . . . 20

4.2 Cumulative distribution functions 1986-2015 . . . 21

4.3 Probability density functions 1998-2001 . . . 23

4.4 Cumulative distribution functions 1998-2001 . . . 24

4.5 Probability density functions 2008 . . . 26

4.6 Cumulative distribution functions 2008 . . . 27

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Part I

ETN Pricing

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Chapter 1

Introduction

1.1 Background

1.1.1 ETNs

Exchange traded notes (ETNs) have recently experienced a surge in popularity all over the world. [1] ETNs are unsecured senior debt securities often issued by large financial institutions. An ETN is tied to an underlying asset, often a stock, index, or commodity and tracks the performance of the underlying asset. [2]

As mentioned ETNs are debt securities and therefore ETNs do not own any of the underlying asset they are tracking. ETNs are backed solely by the credit of the issuer. The ETNs that will be considered in this study are settled on a daily basis, meaning that the ETN tracks the daily return of the underlying asset. The returns of an ETN will be clarified in tables 1.1 and 1.2. ETNs are Bull or Bear products which gives the investor the possibility to profit from both positive or negative returns on the underlying asset. A Bull ETN corresponds to a long position and thus gives the investor positive returns if the underlying has positive returns. A Bear ETN corresponds to a short position and thus gives the investor positive returns if the underlying asset has negative returns.

The use of leverage by the underwriter of an ETN creates a leveraged ETN.

They are identical to ETNs with the exception of the fact that the daily returns of the tracked underlying asset are multiplied by a lever. There are many different levers in ETNs, ranging from X2 and upwards. This means that the leveraged ETN will replicate levered daily returns of the underlying. To clarify the situation the tables 1.1 and 1.2 will show daily returns of the underlying and the daily returns of both an ETN with X2 leverage and X15 leverage. These tables do not account for any fees or transaction costs that are incurred.

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Table 1.1: Prices and returns of a Bull ETN X2 Day Price of Return of Price of Return of

underlying underlying ETN X2 ETN X2

Day 1 SEK 100.00 0% SEK 100.00 0%

Day 2 SEK 101.00 1% SEK 102.00 2%

Day 3 SEK 99.99 -1% SEK 99.96 -2%

Day 4 SEK 101.99 2% SEK 103.96 4%

Day 5 SEK 99.95 -2% SEK 99.80 -4%

The effects of movements of the underlying asset on a Bull ETN with leverage X2

Table 1.2: Prices and returns of a Bull ETN X15 Day Price of Return of Price of Return of

underlying underlying ETN X15 ETN X15

Day 1 SEK 100.00 0% SEK 100.00 0%

Day 2 SEK 101.00 1% SEK 115.00 15%

Day 3 SEK 99.99 -1% SEK 97.75 -15%

Day 4 SEK 101.99 2% SEK 127.08 30%

Day 5 SEK 99.95 -2% SEK 88.96 -30%

The effects of movements of the underlying asset on a Bull ETN with leverage X15

1.1.2 Hedging

Hedging is used to reduced risk. For underwriters of ETNs this means buying or selling the underlying to replicate potential cash flows to the investor. Since the ETNs do not own a position in any underlying assets the underwriter of an ETN could choose to leave the position naked and not hedge the potential cash flows to investors. This leaves the underwriter with large amounts of risks, and on the Swedish market most underwriters fully hedge their ETNs to minimize their risk. By being fully hedged the underwriter will be able to recreate the cash flows of the ETN. This means that the underwriter will neither profit nor lose from movements of the underlying asset.

The underwriters profit from other fees which will be discussed in section 1.1.4. However, to hedge ETNs underwriters use the investor’s money along with loans to purchase a position in the underlying asset. There are two methods for hedging the risk, static hedging and dynamic hedging.

Static Hedging

To create a static hedge for a leveraged ETN the underwriter would follow the following process; use the investor’s money, along with loaned money to acquire a position in the underlying corresponding to the amount of leverage the ETN has.

A Bull ETN with leverage X15 can be used to clarify the hedging process.

For every dollar the investor invests in the leveraged ETN the underwriter loans

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14 dollars, all of this capital is then used to purchase the underlying asset. Thus the underwriter of the ETN recreates the payoff of the underlying asset with X15 lever without paying out of their own pocket. At the end of the day the position is liquidated and the loan payed back as part of the daily rebalancing.

Dynamic Hedging

A dynamic hedge is a hedge which is continuously rebalanced according to the movement of the underlying asset from the start of the contract until maturity.

This is the opposite of the static hedge. In dynamic hedging, the hedge portfolio is constantly being adjusted, weighting up or down the position in the underlying and loaning money. This dynamic hedge will replicate the payoff of a leveraged ETN. This is the style of hedging used when calculating replicating portfolios in arbitrage pricing. Dynamic hedging is the basis for the Black-Scholes differential equation. [3]

1.1.3 Barrier Level

All ETNs have a so called barrier level. If the underlying asset’s daily return passes the barrier level the investors position is worthless and the underwriter closes the position. The barrier level as a multiple of the underlying assets price at time 0 is found by the following formula for Bull ETNs.

Barrier level_Bull= B_Bull= 1 − 1 lever And for Bear ETNs.

Barrier level_Bear = B_Bear= 1 + 1 lever

It is easily seen that the barrier level for a Bull ETN occurs when the underlying moves in a negative direction when the investor has bet on a positive move and vice versa for a Bear ETN. If the underlying moves past the barrier level and the underwriter has used static hedging, they will lose loaned money that was used to create the hedge.

Again a Bull ETN with leverage X15 can be used to clarify. If the underlying has a daily return of -10% the ETN will have a return of -150%. The investor will not owe the underwriter the remaining 50% but his investment will have become worthless. The investors investment along with the loaned money was used to create a hedge and the underlying asset has passed the barrier level.

Using the earlier equation we see.

100% − 10% = 90% < 100% − 1

15 = 93.33%

This means that the underwriter has lost 15 · 3.33% = 50% of the original investment since the underlying asset fell 3.33% past the barrier level. The underwriter owns 15 times the amount the investor invested but since the investors amount covers the loss until the barrier level is passed the underwriter has only lost 3.33% on each position. The investors position in this case is worthless and the position is closed.

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Since static hedging is more practical, it is preferred by many underwriters.

This exposes the underwriter to the risk of the underlying moving past the barrier level. If the underwriter uses dynamic hedging this risk is avoided. The dynamic hedge portfolio means that the portfolio can only become worthless, the underwriter will never need to pay out of pocket.

1.1.4 ETN Fees

Underwriters charge certain fees from investors who have purchased ETNs in order to cover their costs. The standard fees are as follows.

• Administration fee

• Interest fee

• Gap premium

The administration fee is a fee paid to the underwriter for creating and admin- istrating the ETN. The interest fee is for interest on the loans used to create the payoff of the ETN. The gap premium is the fee paid by the investor for the risk that the underwriter takes while creating the ETN. In the case that the underlying passes the barrier level the investor’s position will be worthless but the underwriter using static hedging will start losing money on their hedge positions bought with loaned money as explained earlier, a risk compensated by the gap premium. The administration fees and interest rates are quite similar between all underwriters on the Swedish market but the gap premium varies.

Although the fees investors pay are paid on a daily basis they are often quoted in annual percentage rates (APR). This means that the daily fee is calculated in the following way.

Daily f ee = Quoted AP R 365

1.2 Purpose and Aim

In the ETN market there are usually no to very little transaction costs such as brokerage fees, instead underwriters profit from the other fees they charge that have been named in section 1.1.4. These fees are often found across multiple pages in the prospectus of the ETN. This makes the average investor unlikely to even consider them, much less reflect about their size or impact on the investment. As has been mentioned the administration fees and interest rates are similar across the Swedish market but there is large variation in the gap premium. This gives the potential model two purposes. Firstly, for investors who can compare this result with the gap premium they are paying to see if they are overpaying. Secondly, underwriters can compare their gap premiums with the model and see if they are undercharging their customers for the risk they are taking.

The aim of this study is to provide a model for pricing the gap premium and thereby giving both investors and underwriters a starting point for discussions of gap premiums in leveraged ETNs. By the authors’ knowledge no similar studies of pricing the gap premium have been done.

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1.3 Scope

As has been mentioned leveraged ETNs are available on stock exchanges all around the world with many different underlying assets being tracked. To an- alyze them all would be impossibly time consuming. Instead models will be created which can be applied to ETNs with varying underlying assets with any amount of leverage. The models will be based upon leveraged ETNs tracking the OMXS30 index with leverage X15. The high leverage allows for many in- stances of passing the barrier levels and using the OMXS30 is especially relevant for the Swedish market. The mathematics are identical for other ETNs and the same methods may be applied elsewhere.

1.4 Research Question

The question this study will answer is what the theoretically correct gap premium underwriters should charge investors is. To answer this question the study will provide a model to price the gap premium for an ETN tracking the OMXS30 with leverage X15.

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Chapter 2

Mathematical Theory

2.1 Maximum Likelihood Estimation

Maximum likelihood estimation is a method for estimating the parameters of a statistical model given certain data obtained from an unknown statistical model.

The method was popularized by Ronald Fisher in the 1920s and is today one of the most widely used methods for fitting statistical models to data. Its prevalence is mainly due to the fact that it can be applied to a wide range of statistical models (continuous, discrete, categorical, censored, truncated, etc.), where other methods do not provide a satisfactorily method of estimation. [4]

The principles of the method are that the true model is the model that makes the observed data most likely to occur, which means that one should seek the value of the parameters that maximizes the likelihood function (2.1). [5] The likelihood function represents the probability of the outcomes (xi) given the parameter(s) (θ).

L(θ) = f_X₁_,...,X_N(x₁, . . . , x_N|θ) = [if X independent] =

N

Y

i=1

f_X_i(x_i|θ) (2.1)

where x_i is the observation i, N is the number of observations and f_X_i(x) is the probability distribution for the random variable Xi. The parameter that maximizes the likelihood of the sample is referred to as the M aximum Likelihood Estimate denoted as ˆθ.

θ = arg maxˆ

θ

L(θ) (2.2)

Nevertheless, when evaluating it is common practice to use log likelihood (2.3) since the maximizing parameter ˆθ is the same in both cases and log likelihood is easier to handle computationally, given independence of observations of X.

log[L(θ)] = [if X independent] =

N

X

i=1

log[f_X_i(x_i|θ)] (2.3)

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2.1.1 Convergence

Consistency

Given certain criteria the maximum likelihood estimate is consistent, meaning that the estimated parameter ˆθ converges in probability¹ to the true value θ as the amount of data increases (2.4). Such an estimate is referred to as an Asymptotic Unbiased Estimator, an estimate that minimizes the asymptotic variance.

θˆ→ θ as N → ∞^P (2.4)

This implies that the method is most suited for large to infinite samples sizes (N → ∞). For small samples sizes the asymptotic does not hold, hence the method has no optimum properties. [6]

Asymptotic Normality

Maximum likelihood estimates generally exhibit asymptotic normality, meaning that the estimates contain a random error that decreases with a factor 1/N . Which is yet another reason why maximum likelihood estimates do not work well with small samples.

Nevertheless, for this property to hold some criteria must be satisfied. First of all, the estimate cannot lie on the boundary of the sample, suggesting that the probability does not increases with the sample size. This criteria is standard in asymptotic theory and necessary to get a practically useful estimate. It is easiest averted by using large samples. Secondly, data should not depend on unknown parameters or otherwise suffer from selection bias, as this affects the the probability of an estimate [7] causing a high type I error rate.² Thirdly, nuisance parameters must be independent of the sample size. Which means that the errors cannot increase with the number of observations, since it implies that the method will not converge. [8] Fourthly, for the asymptotic to hold information must increase indefinitely with the sample size in cases when the assumption of independent identically distributed observations does not hold.

Expressed in simpler terms, if the increase in sample size is too dependent on the initial sample it brings no additional information and the the asymptotic does not hold. For the error to decrease with the sample size it is quite intuitive more information is needed to distribute the error on. Hence, data should preferably be independent and identically distributed.

Asymptotic Efficiency

Asymptotic efficiency is perhaps the most important justification for maximum likelihood estimation. Implying that the maximum likelihood estimate is an unbiased estimator with the smallest possible variance when the number of observations increases to infinity, and therefore asymptotically efficient and optimal to all other estimators. A fully efficient estimator achieves the lower bound known as the Cramér-Rao lower bound. [9]

1The probability of an unusual outcome becomes smaller and smaller as n → ∞.

2A type I error is the rejection of a true null hypothesis.

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2.1.2 Distribution Fitting Data

Maximum likelihood is effectively used for fitting a distribution to observed data, which means finding the model that most likely gave rise to the observed data. [10] This model is then interpreted as the model that is most likely the true model.

There are two cases when maximum likelihood is used to fit data to a distribution. First, when the distribution is known but the parameters thereof are not. Secondly, when neither the distribution of the data nor the parameters are known. In both cases the parameter needs to be optimized to the distribution.

When the distribution is unknown it is also necessary to iterate over the different distributions. In this case some of the distributions must be able accommodate skewed data since skewness occasionally appears in real data. [11] [12].

2.2 Pricing

2.2.1 Insurance Pricing

Insurance pricing or risk based pricing means that the issuer of the ETN charges the investor for the risk they take on when they issue the ETN. Thus the expected loss for the issuer is the price the investor should pay for the gap premium.

The insurance price of the gap premium is calculated by looking at historical losses. [13]

Mathematically this is done by fitting a distribution to the data and then calculating the expected loss. An easier but less mathematically justified approach is finding the average of the data without finding a distribution. Both methods have problems when the amount of observed data is small but the distribution fitting will have a smoothing effect that the average will not. Insurance pricing can be summarized in the following way.

Insurance f ee = E[Loss] = Z ∞

−∞

h(x)fX(x)dx (2.5) where h(x) represents the loss when the event x occurs and f_X(x) is the probability density function for the random variable X that generates the events x.

2.2.2 Arbitrage pricing

Arbitrage pricing or valuation is a wide spread theory used in pricing models.

Prices are determined in such a manner as to preclude any arbitrage opportunities.

Black-Scholes Model

Black-Scholes model is an arbitrage pricing model for European options³. The model calculates the price as the risk-neutral expected value of the discounted payoff of the option. This is also know as calculating the price under the risk- neutral measure Q, which is not the real world observed probability measure

3Options that can be exercised only at maturity.

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but a probability measure for arbitrage prices. The risk-neutral measure implies that there is a unique arbitrage free price for each asset in the market. [14] The arbitrage free price is realised by using dynamic hedging, a concept also used by Black and Scholes, to define the Black-Scholes equation. [3] Dynamic hedging can be used to hedge gamma (γ) ⁴ and vega (ν) ⁵ positions. The technique differs from static hedging in such manner that it adjusts the hedge as the underlying moves. [15]

Nevertheless, for the the Black-Scholes model to hold some main assumptions and simplifications must be applied, on the underlying asset.

• Interest rate: assumed to be known, riskless and constant, and is presented as a risk-free rate.

• Log-normal distributed returns: an assumption based on the fact that stock prices cannot be negative. The distribution is skewed with a long tail covering the extreme values. This means that the stock price at maturity ST and the stock price at time 0, S0 have the following distribution.

S_T

S0 ∈ e^Z where Z ∈ N ((r −^σ₂²)T, σ√ T ).

• Volatility: assumed to be constant over time.

• No dividend: a simplification, which is easily worked around by subtract- ing the discounted value of the dividend from the stock price.

And on the market:

• Arbitrage free: there are no risk-less arbitrage opportunities.

• Cash: it is possible to borrow and lend any amount, even fractional at risk-free rate.

• Liquidity: it is possible to buy and sell any amount, even fractional, of the underlying without a buy/sell spread.

• No transaction costs or taxes: a necessary assumption for the constant rebalancing in dynamic hedging.

If these assumptions hold the payoff X can be priced as follows.

Πt(X) = e^{−r(T −t)}E^Q[X] (2.6) where Πtis the price at time t of the payoff X that occurs at time T . To find the expected value under the risk neutral probability measure, the following equation is used.

E^Q[X] = Z ∞

−∞

xψ(z)dz (2.7)

where ψ(z) is the probability density function of Z ∈ N ((r−^σ₂²)(T −t), σ√ T − t).

These equations will give a price for the derivative which gives the payoff X.

This price is also the price of the dynamic hedge. This is because the dynamic hedge will recreate the same cash flows as the derivative and since there is no arbitrage the price of identical cash flows must be equal. Thus by pricing the derivative the price of the dynamic hedging portfolio is found as well.

4Rate of change in the sensitivity of the option value with respect to changes in the underlying asset.

5The sensitivity of the option value with respect to the volatility of the underlying asset.

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2.3 Monte Carlo Integrals

Monte Carlo integration is a stochastic non-deterministic method for numerical integration, often used for complex higher-dimension integrals, integrals that cannot be evaluated analytically, or when the integrated domain is complicated.

Monte Carlo integration is carried out by generating random numbers Xifrom the probability density function fX(x) and computing the objective function for each case and then estimating the average ˆµ (2.9). [16] If Y = h(X) then by the Law of the U nconscious Statistician:

E[Y ] = E[h(X)] = Z ∞

−∞

h(x)fX(x)dx (2.8)

ˆ µ = 1

N

X

i=1

h(xi) (2.9)

where xi is an independent sample of the random variable X.

2.3.1 Convergence

Monte Carlo integration relies on the Strong Law of Large N umbers and converges almost surely⁶ to the expected value as the sample size increases, which is also the true value of the integral. [17]

ˆ µ = lim

N →∞

1 N

N

X

i=1

h(xi)^a.s.→ Z ∞

−∞

h(x)fX(x)dx = E[Y ] (2.10)

2.3.2 Error

The standard error σ_µ_ˆ (2.13) of the Monte Carlo integral is proportional to 1/√

N where N is the number of samples. This means that to halve the error the sample size must be quadrupled.

V ar(1 N

N

X

i=1

h(x_i)) = σ²

N, where σ²= V ar(x_i) (2.11)

ˆ

σ²= 1 N − 1

N

X

i=1

(h(x_i) − ˆµ)² (2.12)

σ_µ_ˆ = ˆσ

√

N (2.13)

6As N → ∞ the probability of the estimator attaining the true value is equal to 1.

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Chapter 3

Method and Model

3.1 Data

3.1.1 Insurance Pricing

The insurance pricing model will be created by first fitting a distribution to the daily returns of the OMXS30 and using these to calculate the underwriters expected loss.

Since the study is done on ETNs tracking the OMXS30 the data used are historical prices of the OMXS30. This data has been collected from Bloomberg Terminal. [18]

Since ETNs are rebalanced every day after closing the returns they track are from close price to close price according to the following formula where Ct

denotes the closing price of day t. This daily return is multiplied by the leverage of the ETN to create the return of the ETN. These daily returns are the data points that will be used in the distribution fitting.

Daily return = Ct− Ct−1

C_t−1 (3.1)

Opening and closing prices were collected from the OMXS30 index’s inception on September 30, 1986 up to December 31, 2015. This gives roughly 7300 data points, of which different subsets will be used for different purposes to evaluate the model.

Time periods

Three different time periods will be considered. The first of which is the entire data set from 1986 to 2015. This will give the most amount of data and give a price to compare against. The next time period will be 1998-2001 which contains the Swedish IT-bubble along with the tragic events of 9/11 which caused large amounts of volatility. The third and final time period to be considered is 2008 with the global financial crisis that occurred. These time periods are chosen because of the large amount of volatility during them. This will give a worst- case scenario which it is likely that underwriters would use if pricing the gap premium in this manner.

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3.1.2 Arbitrage Pricing

The data that will be required to price the gap premium using an arbitrage argument are the following, the risk-free rate r, the volatility of the OMXS30 σ, the maturity time T , the barrier level B, and leverage L. In this study the following will be used.

• r = The risk-free rate = 10 year treasury bond (RGKB 1059)

• σ = Volatility = 1 year predicted volatility of the OMXS30

• T = Time to maturity = 1 day

• B = Barrier level = 1 ±_L¹

• L = Leverage = 15

The predicted volatility of the OMXS30 will be found from the V-Lab at the Volatility Institute of the Stern School of Business at NYU. [19] Even if ETNs rebalance every day the 1 year predicted volatility will be used since the fee charged for the gap premium cannot be adjusted on a daily basis.

3.2 Insurance Pricing

3.2.1 Distribution Fitting

To find a suitable distribution to use in the later steps maximum likelihood estimation was used as described in section 2.1.2. The relevant data was used as observations in the likelihood and log likelihood models, (2.1) and (2.3).

Although the observations of daily returns may not be completely independent observations, the assumption is made that enough additional information is provided for the maximum likelihood estimate to converge. Since not only the parameters but also the distribution is unknown, the maximum likelihood estimation is iterated over multiple distributions. For daily returns it is of course a continuous distribution and the following distributions have been considered.

• Beta

• Birnbaum-Saunders

• Exponential

• Extreme value

• Gamma

• Generalized extreme value

• Generalized Pareto

• Inverse Gaussian

• Logistic

• Log-logistic

• Log-normal

• Nakagami

• Normal

• Rayleigh

• Rician

• t Location-Scale

• Weibull

These distributions cover a range of different skews and shapes. The maximum likelihood estimate is found for every distribution and the likelihood of these estimates are then compared.

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3.2.2 Price and Loss Function

As has been mentioned in section 2.2.1 the insurance price of the gap premium should be the expected loss for the underwriter. If the loss the underwriter incurs is denoted as Y and the return of the underlying is denoted as X, the expected loss is as follows.

E[Y_Bull] = E[h(X)_Bull] = Z ∞

−∞

h(x)_Bullf_X(x) dx (3.2)

E[YBear] = E[h(X)Bear] = Z ∞

−∞

h(x)BearfX(x) dx (3.3) where h(x) denotes the loss function and f_X(x) denotes the probability density function of the random variable X which generates daily returns. This probability density function is the result of the earlier maximum likelihood estimation.

The loss function h(x) is the loss the underwriter incurs when the outcome x is observed from the random variable X and L denotes the leverage. The loss is calculated as a positive percentage of the face value.

h(x)_Bull=

(L · (⁻¹_L − x), if x < ⁻¹_L

0 otherwise (3.4)

h(x)Bear =

(L · (x −_L¹), if x > _L¹

0 otherwise (3.5)

If the underlying moves past the barrier level the investors position becomes worthless. The loaned money that has been placed in the underlying to create the static hedge has also decreased in value and therefore to pay back the loan the underwriter must pay out of their own pocket.

3.2.3 Monte Carlo Integration

To price the gap premium the equations (3.2) and (3.3) must be evaluated.

Since the integrals contain probability density functions of continuous distributions these can be difficult to calculate and can instead be solved by Monte Carlo integration as described in section 2.3. By the same section the following equations are acquired.

E[YBull] ≈ 1 N

N

X

i=1

h(xi)Bull (3.6)

E[YBear] ≈ 1 N

N

X

i=1

h(xi)Bear (3.7)

where xi is the i^thsample of the random variable X and the number of observations is N . Taking random samples from the distribution found by maximum likelihood estimation and evaluating equations (3.6) and (3.7) with the samples results in the theoretical price of the gap premium. As mentioned in section 2.3.1 this result will converge almost surely as N → ∞.

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3.3 Arbitrage Pricing

The assumptions and explanations provided in section 2.2.2 will now be applied to leveraged ETNs. First the payoff as a function of the stock price at maturity Φ(ST) is shown, whereafter the price of a leveraged ETN will be derived.

3.3.1 Bull ETNs

The payoff function for the investor investing 1 dollar in a Bull ETN is as follows.

XBull= Φ(ST)Bull= (L ·

S_T−S0

S₀

if ^S_S^T

0 > B

−1 otherwise

(3.8) In this function, ST denotes the underlying asset price at maturity, S0 the underlying asset price at time 0, L denotes the amount of leverage, and B denotes the barrier level. By the earlier assumptions it is known that ^S_S^T

0 ∈ e^Z and Z ∈ N ((r −^σ₂²)T, σ√

T ). Thus the equation can be rewritten.

X_Bull= Φ(S_T)_Bull=

(L · e^Z− 1

if e^Z > B

−1 otherwise (3.9)

If the price of this payoff at time 0 is denoted by Π0(XBull) along with the earlier assumptions the following equation provides a price for the payoff.

Π₀(X_Bull) = e^−rTE^Q[X_Bull] (3.10) Using the assumption that stock returns are log-normally distributed with certain parameters the following equation can be evaluated.

E^Q[XBull] = Z ∞

−∞

xBullfZ(z) dz (3.11) where fZ(Z) is the probability distribution function of a normally distributed variable Z. Now equation (3.10) and (3.11) will be solved analytically. For ease of notation the following variables are introduced, m = (r −^σ₂²)T and s = σ√

T . This means that Z ∈ N (m, s).

E^Q[XBull] = Z ∞

−∞

xBullfZ(z) dz =

Z log(B)

−∞

(−1) 1

√ 2πs²e

−(z−m)2 2s2 dz +

Z ∞ log(B)

L(e^z− 1) 1

√ 2πs²e

−(z−m)2 2s2 dz

=

Z log(B)

−∞

(−1) 1

√

2πs²e^−(z−m)2^2s2 dz + L Z ∞

log(B)

e^z

√

2πs²e^−(z−m)2^2s2 dz

− L Z ∞

log(B)

√ 1 2πs²e

−(z−m)2

2s2 dz (3.12)

The exponent of the second integral will be examined more closely.

z −−(z − m)²

2s² =2zs²− z²+ 2zm − m² 2s²

=−(z − (m + s²))²

2s² + m +s² 2

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This result is inserted into equation (3.12) to give the following.

E^Q[XBull] = −

Z log(B)

−∞

√1

2πs²e^−(z−m)2^2s2 dz + Le^(m+^s2²⁾

Z ∞ log(B)

√ 1

2πs²e−(z−(m+s2 )2

2s2 dz

− L Z ∞

log(B)

√1

2πs²e^−(z−m)2^2s2 dz (3.13)

The first integral is recognized as the cumulative distribution function of a random variable Z ∈ N (m, s). The second integral is recognized as the cumulative distribution function of a random variable Z ∈ N (m + s², s). The third integral is recognized as the cumulative distribution function of a random variable Z ∈ N (m, s). This result is then used to solve equation (3.10).

Π₀(X_Bull) = e^−rTh

−N (d1) + L · e^(m+^s2²⁾N (d₂) − L · N (d₃)i

(3.14)

d1= log(B) − m

s =log(B) − (r −^σ₂²)T σ√

T d2= (m + s²) − log(B)

s = (r +^σ₂²)T − log(B) σ√

T d3= m − log(B)

s = −d1=(r −^σ₂²)T − log(B) σ√

T

where N (x) is the cumulative distribution function for the standard normal distribution. This is the discounted expected payoff under the risk-neutral probability measure Q and since it is known that this is also the value of the replicating portfolio at time 0 this represents the gap premium for a dynamic hedge of a leveraged Bull ETN.

3.3.2 Bear ETNs

The payoff function for the investor investing 1 dollar in a Bear is as follows.

XBear= Φ(ST)Bear= (L ·

S0−ST

S₀

if ^S_S^T

0 < B

−1 otherwise

(3.15)

In this function, ST denotes the underlying asset price at maturity, S0 the underlying asset price at time 0, L denotes the amount of leverage, and B denotes the barrier level. By the earlier assumptions it is known that ^S_S^T

0 ∈ e^Z and Z ∈ N ((r −^σ₂²)T, σ√

T ). Thus the equation can be rewritten.

XBear= Φ(ST)Bear=

(L · 1 − e^Z

if e^Z < B

−1 otherwise (3.16)

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If the price of this payoff at time 0 is denoted by Π₀(X_Bear) along with the earlier assumptions the following equation provides a price for the payoff.

Π0(XBear) = e^−rTE^Q[XBear] (3.17) Using the assumption that stock returns are log-normally distributed with certain parameters the following equation can be evaluated.

E^Q[X_Bear] = Z ∞

−∞

x_Bearf_Z(z) dz (3.18)

where fZ(Z) is the probability distribution function of a normally distributed variable Z. Now equation (3.17) and (3.18) will be solved analytically. For ease of notation the following variables are introduced, m = (r −^σ₂²)T and s = σ√

T . This means that Z ∈ N (m, s).

E^Q[XBear] = Z ∞

−∞

xBearfZ(z) dz

=

Z log(B)

−∞

L(1 − e^z) 1

√

2πs²e^−(z−m)2^2s2 dz − Z ∞

log(B)

√ 1

= L

Z log(B)

−∞

√1

2πs²e^−(z−m)2^2s2 dz − L

Z log(B)

−∞

e^z

√

− Z ∞

log(B)

√1 2πs²e

−(z−m)2

2s2 dz (3.19)

The exponent of the second integral will be examined more closely.

z −−(z − m)²

2s² =2zs²− z²+ 2zm − m² 2s²

=−(z − (m + s²))²

2s² + m +s² 2 This result is inserted into equation (3.19) to give the following.

E^Q[XBear] = L

Z log(B)

−∞

√ 1 2πs²e

−(z−m)2 2s2 dz

− Le^m+^s2²

Z log(B)

−∞

√ 1

2πs²e−(z−(m+s2 )2

2s2 dz

− Z ∞

log(B)

√ 1

2πs²e^−(z−m)2^2s2 dz (3.20)

The first integral is recognized as the cumulative distribution function of a random variable Z ∈ N (m, s). The second integral is recognized as the cumulative distribution function of a random variable Z ∈ N (m + s², s). The third integral is recognized as the cumulative distribution function of a random variable Z ∈ N (m, s). This result is then used to solve equation (3.17).

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Π0(XBear) = e^−rTh

L · N (d1) − L · e^(m+^s2²⁾N (d2) − N (d3)i

(3.21)

d1= log(B) − m

s =log(B) − (r −^σ₂²)T σ√

T d2= log(B) − (m + s²)

s = log(B) − (r +^σ₂²)T σ√

T d3= m − log(B)

s = −d1=(r −^σ₂²)T − log(B) σ√

T

where N (x) is the cumulative distribution function for the standard normal distribution. This is the discounted expected payoff under the risk-neutral probability measure Q and since it is known that this is also the value of the replicating portfolio at time 0 this represents the gap premium for a dynamic hedge of a leveraged Bear ETN.

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Chapter 4

Results

4.1 Insurance Price

4.1.1 1986-2015

In this section daily returns from the inception of the OMXS30 to the end of 2015 have been used.

Distribution

Using the maximum likelihood estimation as described in sections 2.1.2 and 3.2.1, the following results were achieved.

Table 4.1: Results of distribution fitting 1986-2015 Values

Distribution t Location-Scale

µ 0.000393586

σ 0.0094326

ν 3.63164

The fitted distribution and parameters

The parameters are interpreted in the following way, µ is the location parameter and also the expected value if ν > 1. [20] σ is the scale parameter, which must be larger than zero and finally ν which is the shape parameter. The probability density function of the t Location-Scale distribution is as follows.

fX(x) = Γ(^ν+1₂ ) σ√

νπΓ(^ν₂)

"

ν + (^x−µ_σ )² ν

#−(^ν+1₂ )

(4.1) This is the probability distribution function that will be used in the Monte Carlo equations (3.2) and (3.3) and also used to generate random samples to evaluate (3.6) and (3.7). The following figures show the probability density functions of the original data, the fitted t Location-Scale distributions along with the second, third and fourth most likely distributions.

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Figure 4.1: Probability density functions 1986-2015

The original data plotted against the fitted distributions’ density functions, the values on the x-axis correspond to daily returns in percent.

The following figure shows the error between the fitted distributions cumulative distribution functions and the original data as well as the error between them.

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Figure 4.2: Cumulative distribution functions 1986-2015

The original data plotted against the fitted distributions cumulative distribution functions and the error, the values on the x-axis correspond to

daily returns in percent.

This t Location-Scale distribution will be used to generate the random samples used in the Monte Carlo integration.

Monte Carlo Integration

Running the Monte Carlo integration with N = 10 000 000 000 samples gives the following result for the expected loss per day as a percentage of the original investment.

E[YBull] ≈ 0.0005906798556879345 E[Y_Bear] ≈ 0.0006083933869530255

As mentioned in section 1.1.4 fees of ETNs are usually quoted in annual percentage rates (APR). This gives the following gap premiums for Bull and Bear ETNs with leverage X15 on the OMXS30.

Gap premium^1986−2015_Bull = 0.0005906798556879345 · 365 ≈ 21.56%

Gap premium^1986−2015_Bear = 0.0006083933869530255 · 365 ≈ 22.21%

As mentioned in section 2.3.2 the expected error of the Monte Carlo integration method is proportional to 1/√

N which yields the following result.

E[error] ∝ 1

√ = 1

√10 000 000 000 = 0.00001

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4.1.2 1998-2001

In this section daily returns of the OMXS30 from the start of 1998 to the end of 2001 have been used.

Distribution

Table 4.2: Results of distribution fitting 1998-2001 Values

µ 0.000151396

σ 0.014103

ν 7.35421

The parameters are interpreted in the following way, µ is the location parameter and also the expected value if ν > 1. [20] σ is the scale parameter, which must be larger than zero and finally ν which is the shape parameter. The probability density function of the t Location-Scale distribution is the same as earlier. (4.1) This is the probability distribution function that will be used in the Monte Carlo equations (3.2) and (3.3) and also used to generate random samples to evaluate (3.6) and (3.7). The following figures show the probability density functions of the original data, the fitted t Location-Scale distributions along with the second, third and fourth most likely distributions.

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Figure 4.3: Probability density functions 1998-2001

The original data plotted against the fitted distributions’ density functions, the values on the x-axis correspond to daily returns in percent.

The following figure shows the error between the fitted distributions cumulative distribution functions and the original data as well as the error between them.

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Figure 4.4: Cumulative distribution functions 1998-2001

E[YBull] ≈ 0.0001813424575830784 E[Y_Bear] ≈ 0.0001857789823109766

Gap premium^1998−2001_Bull = 0.0001813424575830784 · 365 ≈ 6.62%

Gap premium^1998−2001_Bear = 0.0001857789823109766 · 365 ≈ 6.78%

E[error] ∝ 1

√ = 1

√10 000 000 000 = 0.00001

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4.1.3 2008

In this section daily returns of the OMXS30 from the start of 2008 to the end of 2008 have been used.

Distribution

Table 4.3: Results of distribution fitting 2008 Values

µ -0.00273509

σ 0.0181151

ν 3.68549

The parameters are interpreted in the following way, µ is the location parameter and also the expected value if ν > 1. [20] σ is the scale parameter, which must be larger than zero and finally ν which is the shape parameter. The probability density function of the t Location-Scale distribution is the same as earlier. (4.1) This is the probability distribution function that will be used in the Monte Carlo equations (3.2) and (3.3) and also used to generate random samples to evaluate (3.6) and (3.7). The following figures show the probability density functions of the original data, the fitted t Location-Scale distributions along with the second, third and fourth most likely distributions.

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Figure 4.5: Probability density functions 2008

The original data plotted against the fitted distributions density functions, the values on the x-axis correspond to daily returns in percent.

The following figure shows the error between the fitted distributions’ cumulative distribution functions and the original data as well as the error between them.

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Figure 4.6: Cumulative distribution functions 2008

E[YBull] ≈ 0.005848689721346 E[Y_Bear] ≈ 0.004841211914925

Gap premium²⁰⁰⁸_Bull= 0.005848689721346 · 365 ≈ 213.48%

Gap premium²⁰⁰⁸_Bear = 0.004841211914925 · 365 ≈ 176.70%

E[error] ∝ 1

√ = 1

√10 000 000 000 = 0.00001

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4.2 Arbitrage Price

4.2.1 Input Data

The following data has been used in the calculation.¹

• r = 0.79%

• σ = 23.15%

• T =₂₅₀¹ (250 trading days per year)

• B = 93.33% (Bull) or 106.67% (Bear)

• L = 15

• m = (r −^σ₂²)T

• s = σ√ T

Applying the parameters to the risk-neutral pricing equations (3.14) and (3.21) gives the following results.

Π0(XBull) = 0.0004740430063210403 (4.2) Gap premiumBull= 0.0004740430063210403 · 365 ≈ 17.30%

Π0(XBear) = −0.0004737438614705452 (4.3) Gap premiumBear= −0.0004737438614705452 · 365 ≈ −17.30%

1Accurate as of 2016-05-05

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4.3 Summary of Results

Table 4.4: Summary of Bull Gap Premiums

Method Gap Premium

Monte Carlo 1986-2015 21.56%

Monte Carlo 1998-2001 6.62%

Monte Carlo 2008 213.48%

Arbitrage pricing 17.30%

Summary of gap premiums for Bull ETNs found in the study quoted in APR

Table 4.5: Summary of Bear Gap Premiums

Method Gap Premium

Monte Carlo 1986-2015 22.21%

Monte Carlo 1998-2001 6.78%

Monte Carlo 2008 176.70%

Arbitrage pricing -17.30%

Summary of gap premiums for Bear ETNs found in the study quoted in APR

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Chapter 5

Discussion and Analysis

5.1 Discussion of Results

The theoretical gap premiums found in the study are all within a reasonable range of values, except for the gap premium for insurance pricing for the turbulent year of 2008 and Bear ETNs when calculated via arbitrage pricing. The arbitrage pricing model for Bear ETNs yields a result of -17.30%. This would mean that the underwriter should pay the investor for holding the ETN, which is of course unreasonable. Although the other fees of the ETN can compensate for this negative fee it is hard to believe that any underwriter would give the investor a discount for holding a Bear ETN. The reason for this result is the slightly positive shift in the log-normal distribution meaning that the expected return of a Bear ETN is negative leading to a negative price leading to a negative fee. Since Bear ETNs pay off when the market has negative returns this small shift is enough to make the hedging cost negative.

The insurance pricing value using Monte Carlo integration for 2008 is unrea- sonably high, 213.48% for Bull- and 176.70% for Bear ETNs. This is because 2008 was a incredibly turbulent and volatile year, by many considered as the worst financial crisis since the Great Depression in the 30’s. [21] Therefore the risks for the underwriters were huge, and thus this result could be seen as the worst case scenario possible. Also one must not forget that this is a theoretical value. In reality underwriters would more likely put their ETNs in "sold out buy back"-status¹and withdrawn the ETNs with large amounts of leverage from the market, since the risk is too large for the underwriter. The financial crisis of 2008 is so extreme that it affects the interval 1986-2015 and influences the gap premium substantially. Thus it is of interest whether there will ever be a crisis like 2008 again but that is different discussion. Another factor that influences the gap premium is the time era and the way the market works, which has changed immensely since OMXS30 was introduced in 1986. When the OMXS30 was introduced in 1986 it was valued at 125 SEK and currently it is above 1 300 SEK with global companies like H&M, thus it can be difficult to compare the returns even if only daily returns are used.

Nevertheless, the insurance pricing model is preferable for two main reasons

1Only the Market Maker can buy the instrument, and other holders of the instrument can only sell to the Market Maker’s price.

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even if the sample size is only 10 billion samples per integral. This is small and the expected error large but due to computational constraints this is the largest sample size that could be achieved in this study. Firstly, most underwriters use static hedging and therefore it is the most relevant pricing model. Secondly, the arbitrage pricing model relies on the Black-Scholes assumptions. These assumptions lead to an underestimation of extreme movements of the underlying, yielding so called tail risk. [22] When pricing the gap premium it is of course these extreme movements when the underlying moves past the barrier levels which are of interest. Further, as the interest rate is very low the volatility becomes very crucial. This means that a small error in volatility could significantly affect the results of the study.

5.2 Comparison with Market

Today underwriters of ETNs expresses their fees in many different ways, mak- ing it hard for investors to see the actual cost of holding the ETN. Therefore SETIPA²have created a comparative figure, called the Bull & Bear figure. The figure expresses the cost of holding the ETN for one year given that the market does not move at all during the year. [23] Thus the figure only takes into account the daily fee the underwriter charges regardless of the movement in the market.

This could be seen as the gap premium plus the interest margin. Since these margins are not available to private investors this figure will be used. The gap premiums are thought to be about 0.5-1% lower as these are reasonable interest margins. Current underwriters of X15 Bull and Bear ETNs on the OMXS30 on Nasdaq Stockholm charge as follows:³

Table 5.1: Bull & Bear figures in the Swedish market Underwriter Short name Bull & Bear figure

Morgan Stanley AVA 36.64%

Nordea N 5.25%

Société Générale SG 5.87%

Vontobel VON 6.39%

Bull & Bear figures for X15 ETNs on the OMXS30 in the Swedish market The first observation to be made is that all underwriters charge the same fee regardless of the direction of the ETN. Even if the study provides different gap premiums for Bull and Bear ETNs the difference is so small that it is realistic to charge the same fee regardless of direction of the ETN. Only for the worst case scenario (2008) is there a significant difference between the theoretical gap premiums for Bull and Bear ETNs. However, this year also differs in the way that the gap premium for a Bull ETN is higher than for an Bear ETN, implying a downwards market trend. Historically the market has gone up, which supports the theoretical gap premiums for the intervals 1986-2015 and 1998-2001 since the Bull premium is slightly smaller than the Bear premium.

However, looking at the most reasonable result both in aspect of choice of model and time period (Monte Carlo 1998-2001) one can conclude that the

2Swedish Exchange Traded Investment Products Association

3From respective final terms

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actual market gap premiums for all underwriters are close to the theoretical gap premium besides Morgan Stanley (Avanza) who charge much more.

5.3 Possible Effects of Other Factors

As seen there is a small difference between the theoretical gap premiums found in the study and those on the Swedish market. This could be due to a number of factors, some of which will be discussed below.

5.3.1 Stock Borrowing Fees

In the model there have been no costs for buying or selling the underlying other than the actual price of the underlying. In the real world this is not the case. Brokerage fees are applied and there may be taxes. One fee is specifically relevant when shorting stocks. When shorting stocks the owner of the stock may charge a borrowing fee. This has resulted in the discontinuation of Bear ETNs on certain very volatile stocks.

The most obvious example is the Swedish stock Fingerprint Cards AB (FING- B.ST). It is an incredibly volatile stock (σ = 54.06%)⁴ and owners of the stock charge high borrowing fees. This has led to the discontinuation of leveraged Bear ETNs on the stock. At the time of writing there are currently no Bear ETNs available for Fingerprint Cards AB. In the case of stocks that are slightly less volatile but the owners still charge borrowing fees underwriters often in- corporate this fee into the gap premium. Thus the gap premium may not only reflect the cost purely for hedging but also related costs.

5.3.2 Liquidity of the Underlying Asset

The liquidity of the underlying asset is one of the main assumptions in the Black-Scholes world and is important for creating the static hedge as well. If the underlying cannot be both bought or sold in any amount, even fractional, then the underwriter assumes more risk than what has been examined in the study. If the liquidity of the underlying asset is poor the underwriter assumes a liquidity risk beyond the other risks involved when creating the hedge. This risk should be included in the gap premium, something that has not been examined in this study. In the case of poor liquidity the gap premium should be higher than the theoretical gap premiums found in the study. There are also other ways for the underwriter to handle poor liquidity in the underlying asset. This includes hedging with positions in highly liquid assets that are strongly correlated with the underlying assets but this also carries other risks.

5.3.3 Profit Margin

Although underwriters may claim that the gap premium should solely represent the hedging costs that are incurred there is a possibility that they may over- charge investors to increase profits from ETNs. The complexity of the investing prospectus where the fees are detailed make this a possibility. An investor looking to understand exactly which fees he is being charged and how they are

4As of 2016-05-20

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calculated must make a significant time investment that most investors are not willing to make. An even greater time investment must be made to compare fees across the market as different underwriters use slightly different standards and calculation methods.

5.3.4 Credit Rating

Credit rating reflects an individuals or institutions credit worthiness, predicting their ability to pay back debt. Thus a downgrade in credit rating will affect the value of a leveraged ETN since ETNs are debt securities. A factor that may affect the gap premium in particular is the amount of risk the underwriter is willing to take. On the Swedish market must underwriters fully hedge their ETN positions and this has been the assumption throughout the study but on other markets this assumption may not hold. Underwriters can choose to take on the risk of potential cash flows and leave their positions naked without a hedge. This leads to more risk but also potential rewards. If the underlying asset moves in the opposite of the ETNs direction the underwriter will profit from the movement of the underlying as well as the fees they charge. This potential reward comes with the risk of having to pay out large amounts to investors when the underlying moves in the same direction as the ETN they have sold. Thus it becomes a question of what amount of risk the underwriter is willing to take and their ability to pay back the debt they owe. This means underwriters willing to take on more risk can charge lower gap premiums and more risk averse underwriters can charge higher gap premiums.

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Chapter 6

Conclusion

The study has found theoretical gap premiums using two different methods with consideration for the underwriters preferred hedging style. This allows discussion internally within the underwriting organization if they are undercharging their investors but also allows investors to have a benchmark to compare against when looking at investing in a leveraged ETN.

The study has used OMXS30 and leverage X15 but the proposed models can be applied in similar fashions to other underlying assets and amounts of leverage. Since leveraged ETNs are fast becoming very popular investments all around the world [1], the models must deal with different circumstances. The arbitrage price based on the Black-Scholes model is of course sensitive to the assumptions within the Black-Scholes world. Thus care must be taken when using the model with different underlying assets. One of the biggest problems is the possibility of so called early exercise. Investors can choose to close their positions while they are ahead or before the underlying falls past the barrier level during trading hours. This is a circumstance that is not covered by the Black- Scholes model. The possibility of early exercise creates an optimal stopping problem of when to close the position. In general there is no analytical closed form solution for pricing options that allow early exercise and therefore it is difficult to analytically price leveraged ETNs that can be exercised at any point during the day.

The insurance pricing model is more robust with respect to different underlying assets but of course relies heavily on the maximum likelihood estimation of the distribution of daily stock returns. One of the problems of using maximum likelihood in this study is the fact that for the purpose of the study only the tails of the distribution are of interest but the distribution is being fitted to all values. This means there could be better models for exploring specifically the tails of the distribution specifically when the stocks move past the barrier levels.

A proposal to model returns more accurately, would be modelling jump- diffusion processes as done by Andersen and Andreasen. [24] This approach models the stock being affected by a standard one-dimensional Brownian motion and a Poisson counting process with a deterministic jump intensity. This leads to a forward partial integro-differential equation that is solved by European call option prices.

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Gap Premium Pricing in Leveraged Exchange Traded Notes

Gap Premium Pricing in

Leveraged Exchange Traded Notes

AXEL BROSTRÖM

RICHARD KRISTIANSSON

Gap Premium Pricing in Leveraged Exchange Traded Notes

A X E L B R O S T R Ö M R I C H A R D K R I S T I A N S S O N

Table of Contents

I ETN Pricing 1

II Risk Taking 35

III References 48

List of Tables

List of Figures

Part I

ETN Pricing

Chapter 1

Introduction

1.1 Background

1.1.1 ETNs

1.1.2 Hedging

1.1.3 Barrier Level

1.1.4 ETN Fees

1.2 Purpose and Aim

1.3 Scope

1.4 Research Question

Chapter 2

Mathematical Theory

2.1 Maximum Likelihood Estimation

2.1.1 Convergence

2.1.2 Distribution Fitting Data

2.2 Pricing

2.2.1 Insurance Pricing

2.2.2 Arbitrage pricing

2.3 Monte Carlo Integrals

2.3.1 Convergence

2.3.2 Error

Chapter 3

Method and Model

3.1 Data

3.1.1 Insurance Pricing

3.1.2 Arbitrage Pricing

3.2 Insurance Pricing

3.2.1 Distribution Fitting

3.2.2 Price and Loss Function

3.2.3 Monte Carlo Integration

3.3 Arbitrage Pricing

3.3.1 Bull ETNs

3.3.2 Bear ETNs

Chapter 4

Results

4.1 Insurance Price

4.1.1 1986-2015

4.1.2 1998-2001

4.1.3 2008

4.2 Arbitrage Price

4.2.1 Input Data

4.3 Summary of Results

Chapter 5

Discussion and Analysis

5.1 Discussion of Results

5.2 Comparison with Market

5.3 Possible Effects of Other Factors

5.3.1 Stock Borrowing Fees

5.3.2 Liquidity of the Underlying Asset

5.3.3 Profit Margin

5.3.4 Credit Rating

Chapter 6

Conclusion