Simulation Approaches to Delta Hedging in the Black-Scholes Model · 2017-10-28 · Simulation Approaches to Delta Hedging in the Black-Scholes Model CARL JOHAN KRISTOFFER FÜRST

Simulation Approaches to Delta Hedging in the Black-Scholes Model

C A R L J O H A N K R I S T O F F E R F Ü R S T

Master of Science Thesis Stockholm, Sweden 2012

Simulation Approaches to Delta Hedging in the Black-Scholes Model

C A R L J O H A N K R I S T O F F E R F Ü R S T

Master’s Thesis in Numerical Analysis (30 ECTS credits) at the School of Vehicle Engineering Royal Institute of Technology year 2012 Supervisor at CSC was Anders Szepessy Examiner was Michael Hanke TRITA-CSC-E 2012:006 ISRN-KTH/CSC/E--12/006--SE ISSN-1653-5715 Royal Institute of Technology School of Computer Science and Communication KTH CSC SE-100 44 Stockholm, Sweden URL: www.kth.se/csc

AbstractA delta neutral portfolio in the Black-Scholes model is analysed usingboth simulated and historical data. An in-depth analysis is conducted inorder to investigate what happens when the hedging interval is discreteand when the market maker does not know the true volatility. Basedon simulated data, the risk measured by Expected Shortfall decreasesthe higher the volatility is for a sold call option. This is not the casein reality, where the risk is minimized at about 60-80% of the impliedvolatility depending on the time to maturity.

The so called volatility skew observed in the implied volatilities ofthe market can either remain steady or move as the spot price of theunderlying asset changes. This skew is in contradiction with the Black-Scholes model assumptions. Therefore, what will theoretically happenif it remains frozen in place or moves cannot be analysed in that model.Instead, it is empirically tested how this skew should behave as the mar-ket moves to yield the lowest risk. It turns out that it differs dependingon the time to maturity of the option. An option which has a long timeto maturity seems to generate the lowest risk with a skew that movesin alignment with the spot price. In the case of decreasing time to ma-turity, a skew should remain stable and not move to yield the lowestpossible risk.

While most assumptions of the Black-Scholes model does not con-form with reality, it still gives a reasonable result in this study. Theuncertainty of the input parameters seems to have a greater effect onthe returns than the shortcomings of the model does.

ReferatSimuleringsmetoder för deltahedging i Black-Scholes

modell

Genom både simulerad och historisk data analyseras en deltaneutralportfölj i Black-Scholes modell. En närmare undersökning görs av vadsom händer när hedgingintervallet är diskret och när den sanna volati-liten inte är känd. Expected Shortfalls mätmetod visar att i fallet medsimulerad data minskar risken för en såld köpoption ju högre volatilitetsom används. Detta är inte fallet i verkligheten, där risken minimerasvid ungefär 60% av den implicita volatiliteten. Risken ökar sedan i taktmed att den använda volatiliteten ökar.

Om volatiliteten för en serie optioner med samma parametrar, bort-sett från strike-nivå, beräknas och illustreras som en funktion av strike-nivå, så syns en kurva som liknar ett andragradspolynom. Denna kurvakan antingen vara still i takt med att priset hos den underliggande till-gången förändras, eller följa med. Denna kurva motsäger Black-Scholesmodellen som antar att volatiliteten är densamma oavsett stike-nivå,vilket innebär att effekten av vad som händer när denna kurva antingenföljer marknaden eller ligger still inte kan studeras teoretiskt. Iställettestas empiriskt vad som ger den lägsta risken. Det visar sig att förlånga löptider så skall denna kurva följa med priset på den underlig-gande tillgången för att minimera risken. Ju kortare löptiden blir, jumindre skall kurvan röra sig för att hålla risken så låg som möjligt.

Trots att de flesta antagandena bakom Black-Scholes modell intestämmer överens med verkligen så ger den ett rimligt resultat. Osäker-heten i indataparametrarna verkar ha en större effekt på avkastningenän vad bristerna i modellen har.

Acknowledgements

I would like to thank my supervisor at KTH, Anders Szepessy, as well as my super-visor at Handelsbanken Capital Markets, Mattias Karlsson. Furthermore, I wouldlike to thank Love Lindholm, PhD student at KTH, and Frederik Lange, MSc stu-dent at SSE, for valuable ideas and suggestions. Finally, I address a special thankyou to Robert Andersson, Handelsbanken Capital Markets, for contributing withvaluable market experience.

Stockholm, March 2012Kristoffer Fürst

Contents

1 Introduction 11.1 A Benchmark Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 A Perfect World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

I Theory 5

2 Stochastic Processes 72.1 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 The Wiener Process . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 The Monte-Carlo Method . . . . . . . . . . . . . . . . . . . . 8

2.2 Itô’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Options and Hedging 113.1 Absence of Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 The Value of an Option . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Self-Financing Portfolio . . . . . . . . . . . . . . . . . . . . . 133.2.2 Black-Scholes Equation . . . . . . . . . . . . . . . . . . . . . 133.2.3 Delta Hedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.4 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.1 Distribution of PnL . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Changed Volatility in Delta Hedge . . . . . . . . . . . . . . . . . . . 173.5 Sticky Strike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Utility Functions 214.1 Loss functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Empirical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4 Expected Shortfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

II Results 25

5 Simulated Data 275.1 Changed Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Hedge Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Historical Data 336.1 Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 Change in Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.3 Hedge Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.4 The Impact of Sticky Strike . . . . . . . . . . . . . . . . . . . . . . . 376.5 Distribution of ES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7 Discussion 417.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.3 Further investigations . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Bibliography 43

Appendices 45

A Software Framework 45

B Additional Figures for 3-month Option 49

C Additional Figures for 3-year Option 63

Symbol Explanations

Symbol Explanationatm At-the-money.B(t) Value of a risk-free asset at time t.C(t, S(t)) Value of a (European) call option at time t.ESp Expected Shortfall at level p.f(t, S(t)) The value of an option at time t.F Cumulative density function of a probability distribution.FFF Name of made-up stock used as benchmark case in this

study.hi(t) Number of assets of type i in a portfolio at time t.K Strike price of an option.L Loss functions.ln() Natural logarithm.n Number of times a portfolio is hedged during its existence.N(a, b) Normal distribution with mean a and standard deviation b.p Probability (p ∈ [0, 1]).P (E) Probability that the event E occurs.PnL Profit and Loss (e.g. daily returns of a portfolio).r Annualized continuously compounding risk-free interest

rate.S(t) Spot price of an asset at time t.s Spot price of asset S at time t0.stdDev Standard deviation.∆t Time from ti−1 to ti, i.e. ti − ti−1.t0 = 0 Time today.T − t Time to maturity of an option (in years).tPL Total Profit and Loss (cumulative PnL).Vh(t) Value of a portfolio at time t.VaRp Value-at-Risk at level p.W (t) The value of a Wiener process at time t.X Vector of PnL values for a given portfolio.

Symbol Explanation∆ = ∂f

∂S Delta, the derivative of an option price with respect to theprice of the underlying asset.

µ Drift of an asset, i.e. how much the market expects the assetto increase on average in value per year. The continuouslycompounding drift is used in this study.

ν = ∂V∂σ Vega, the derivative of an option price with respect to the

volatility of the underlying asset.Φ The cumulative density function of the standard normal dis-

tribution.σ Volatility of an asset, defined as the standard deviation of

the asset’s yearly logarithmic returns.

Chapter 1

Introduction

1.1 A Benchmark CaseIn this report, delta hedging in the Black-Scholes model of an at-the-money [7,p. 154] European call option [11, p. 4,48] with FFF1 stock as underlying asset willbe used as a benchmark case. The concepts are briefly explained below

• European call option. Gives its owner the right to buy the underlying assetfor a predetermined price at a predetermined date.

• Underlying asset. The asset on which an option is based. In case of aEuropean call option, it is the asset which the buyer of the option has theoption to buy at maturity.

• Delta hedging. Makes the portfolio immune to price changes in the under-lying asset, i.e. the derivative of the portfolio with respect to the underlyingasset is zero.

• Strike price. The price at which the owner of a European option can buythe underlying asset at maturity.

• Forward price. The price agreed upon today of an asset, with delivery inthe future. In the case of no dividends, it is the same as the spot price of astock.

• At-the-money (atm). An option is atm if the strike price of the optionequals the forward price of the underlying asset with delivery at maturity.

• Black-Scholes model. A model of reality that for example assumes thatthe underlying asset follows a geometric Brownian motion which is a specificstochastic process.

1This name is made up. Another underlying asset was used in reality, but due to secrecy thename of this asset cannot be disclosed.

1

CHAPTER 1. INTRODUCTION

The portfolio considered is

Vh(t) = −C(t, S(t)) + hS(t)S(t) + hB(t)B(t) (1.1)

where C(t, S(t)) is the value of the European call option at time t. The value of thestock FFF at time t is denoted by S(t) and the number of such stocks owned at timet is denoted by hS(t). The amount of money put in a bank account is representedby hB(t)B(t). The theory behind how this portfolio arises and why, is explained inChapter 3.

Remark: This is a special case. If C(t, S(t)) is replaced by f(t, S(t)) and S(t)represents the underlying asset of the option f(t, S(t)), then this portfolio is therisk-free portfolio in the Black-Scholes model for any option.

1.2 Problem FormulationMain question: Is delta hedging in the Black-Scholes model reasonable to use inpractice?

A portfolio consisting of an option, its underlying asset and a risk-free asset (forexample a bank account) is constructed. Theoretical concepts for what happenswhen

• the market maker does not know the true volatility of the underlying assetand

• the time period between the hedges varies,

are derived. The theoretical results are first tested on simulated data and afterwardson historical data. In the case of historical data, the concept of

• different levels of sticky strikes

is analysed. Sticky strike determines how the volatility should behave as the price ofthe underlying asset changes. The Black-Scholes model assumes that the volatility isconstant regardless of strike price and over the life of the option. As a consequence,the concept of different levels of sticky strikes is only evaluated by using historicaldata.

To evaluate the results, three utility functions are used. These utility functionsare the standard deviation, the Value-at-Risk and the Expected Shortfall. All threefunctions constitute three different measures of risk.

The concepts mentioned above are explained in more detail throughout Part I.

1.3 A Perfect WorldIn a simulated world when assumptions hold true, a hedging strategy should workperfectly. However, the theory considered in this study is based on continuous time

2

1.4. REALITY

but in a simulation, discrete time has to be used. The portfolio will thus changecharacteristics between the discrete time points and a (hopefully small) error willemerge. This error is called a Profit and Loss (PnL) of the portfolio. The expectedvalue of this PnL should still be zero and have a smaller variance than if the hedgingwas not done.

1.4 RealityAs it turns out, most of the assumptions about market behaviour made in the cele-brated Black-Scholes model seems to be wrong. However, this does not by definitionmake it a bad approximation of reality. In case the Black-Scholes model does notapproximate reality accurately enough, what assumptions need to be relaxed orweakened for the model to become accurate enough? This is not an easy questionto answer. It might not even be possible to come up with a good answer, becausethe only thing known about the market is how it functions today and how it hasfunctioned in the past. What if the market behaves the way it does because of themodels? The interesting aspect here is that it might not matter. As long as thetrue model remains unknown to us and all black swans keep hiding, there is no needto know the truth.

3

Part I

Theory

5

Chapter 2

Stochastic Processes

To price any financial contract who’s value depends on future unknown events, amodel for these events is needed. For example, no one knows what the value of astock will be in three months from now. Consider an investor who wants to buy acall option. How can the price of this option be calculated if no one knows whatthe value of the stock will be in 3 months? A stochastic process is one way to solvethis problem. For the price of the contract to be completely fair, this process needsto model reality perfectly. Whether or not this is true for a given process cannotbe tested fully, since historical data is the only way to do this and one cannot becertain that the future will behave as the past.

2.1 Geometric Brownian MotionThe process of choice in this study is the simplest possible model. It is called ageometric Brownian motion (GBM) [3, p. 33]. Its random component is a one-dimensional Wiener process [3, p. 12-13]. The GBM states that the value of theunderlying asset (denoted by S(t)) evolves as

dS(t) = µS(t)dt+ σS(t)dW (t) (2.1)where µ is the drift of the asset, σ is the volatility of the asset and W is a Wienerprocess. It can be shown that the GBM has the closed form [3, p. 33]:

S(T ) = e(µ−σ22 )T+σW (T )S(0).

2.1.1 The Wiener ProcessThe noise process in the GBM (Equation 2.1) is represented by a Wiener process.A Wiener Process, W , is a process such that• W (0) = 0,

• the increments W (tn)−W (tn−1), . . . ,W (t1)−W (t0) are independent for anypartition t0 = 0 < t1 < . . . < tn = T ,

7

CHAPTER 2. STOCHASTIC PROCESSES

• the stochastic variable W (t)−W (s) has the normal distribution N(0,√t− s)

for any two time points s, t ∈ [0, T ], s < t,

• W is continuous,

see [2, p. 40]. The existence of the Wiener process can be proven [3, p. 20-22], butsuch a proof is outside the scope of this study.

2.1.2 The Monte-Carlo Method

The Monte-Carlo method is a general method used to estimate the solution to astochastic problem by forming the mean of a number of samples. Employing thismethod, trajectories of the Wiener process can be simulated in the following way:

• define a vector of time points (t0 = 0, t1, . . . , tn = T ),

• generate samples from the normal distribution N(0,√ti − ti−1),

i = 1, 2, . . . , n.

What has been generated is a discrete sample of the Wiener process. Using thispath in Equation 2.1, a sample of the geometric Brownian motion is created. Ifmany such independent paths are generated, they can be used to estimate the fairprice of any claim who’s value depends on the path of the asset price. This is theMonte-Carlo method for options pricing.

As an example, consider a call option. The value of this claim is dependent onthe asset price at time T . 10000 samples of asset trajectories are generated using theGBM. For every sample, the pay-off is calculated at time T (i.e max(S(T )−K, 0).These pay-offs are then discounted by multiplying them with e−rT where r is therisk-free interest rate. The Law of large numbers [1, p. 160] states that the averageof these discounted pay-offs is the fair value of the option at time t0.

There are more efficient ways to calculate the value of a call option than de-scribed above, but it serves as a good example. In this study, the Monte-Carlomethod is used to calculate the value of the Portfolio 1.1 in discrete time. Afterthat, the historical asset prices are assumed to be samples from a GBM, and thesame calculations are carried out based on these samples.

In Figure 2.1, some generated samples of a GBM are shown.

2.2 Itô’s Lemma

Itô’s lemma [2, p. 51] is a very strong result and should, in the author’s opinion, becalled Itô’s theorem rather than lemma. It states that if{

dY (s) = a(s)ds+ b(s)dW (s)Y (0) = Y0

8

2.2. ITÔ’S LEMMA

Figure 2.1. Simulation of daily stock prices for 3 months with volatility σ = 0.3and drift µ = 0.02.

where a(s) and b(s) are adapted processes [2, p. 43], then for a function f(t, Y (t))it holds that

df(t, Y (t)) =(∂f

∂t(t, Y (t)) + a(t)∂f

∂y(t, Y (t)) + b(t)2

2∂2f

∂y2 (t, Y (t)))

dt

+ b(t)∂f∂y

(t, Y (t)) dW (t).

if some conditions hold true1.This result might seem irrelevant for this study, but it will later be shown to be

an extremely important result when it comes to pricing options.

1The conditions that needs to be satisfied for Itô’s lemma to hold are beyond the scope of thisstudy. For the rest of the study, the assumption is that these conditions hold.

9

Chapter 3

Options and Hedging

An option is a type of derivative contract between two parties. One party (theseller) usually has an obligation and the other party (the buyer) has an option toexercise the contract. This could, in case of a European call option [11, p. 4], beto buy an underlying for a pre-decided (strike) price. The underlying could bewhatever product or event, e.g. a car-crash or the weather. In this study financialproducts such as stocks, indices or other options will be considered as underlyingassets.

To value a European call option within the Black-Scholes model [11, p. 49 &sec. 6.2], six parameters are required:

• S, the current price of the underlying asset, also called spot price,

• K, the strike price of the option,

• r, annualized continuously compounding risk-free interest rate,

• T − t, time to expiration (maturity) of the option, measured in years,

• σ, annualized volatility of the underlying asset price, given as a positive num-ber,

• Annualized continuously compounded dividend yield of the underlying asset(optional).

The pay-off at maturity for a call option is C(T, S(T )) = max(S(T ) − K, 0).This is because if the price of the stock is below the strike, then buying the stockfrom the stock exchange would be cheaper than to buy it at K. If, on the otherhand, the price of the stock is above the strike, the stock is bought at the strikeprice and then sold at the market price. This leaves a profit of S(T )−K.

The Black-Scholes model for valuation of options relies on the following assump-tions [7]:

• there are no transaction cost,

11

CHAPTER 3. OPTIONS AND HEDGING

• there are no arbitrage opportunities in the markets (see Section 3.1),

• it is possible to both lend and borrow any amount of money at the risk-freeinterest rate,

• it is possible to buy and sell any amount of the option and/or underlyingasset,

• the underlying asset follows a Geometric Brownian Motion (see Section 2.1).

A discussion about the assumptions made can be found in Section 7.1.

3.1 Absence of ArbitrageRecall that the portfolio considered in this study is defined as

Vh(t) = −f(t, S(t)) + hS(t)S(t) + hB(t)B(t)

where f(t, S(t)) is the value of the option at time t (denoted by C(t, S(t)) in caseof a call option). Then the definition of arbitrage1 [2, p. 27] is

Vh(t0) = 0 (3.1)P (Vh(t) ≥ 0) = 1 (3.2)P (Vh(t) > 0) > 0 (3.3)

where P (E) denotes the probability of the event E. In other words, Equations3.1-3.3 means that a market maker cannot

• invest nothing (see Equation 3.1),

• be sure to avoid a loss (see Equation 3.2),

• with a positive probability make a profit (see Equation 3.3).

If the model would permit arbitrage, then any price of a contingent claim wouldbe an acceptable answer. Thus, to make the modelling reasonable this study as-sumes throughout that arbitrage opportunities are absent.

3.2 The Value of an OptionThe aim is to price a contingent claim on the asset S. Denote the value of this claimat time t with underlying asset price S(t) by f(t, S(t)). In the case of a Europeancall option, it holds that f(T, S(T )) = max(S(T )−K, 0) = C(T, S(T )) where T is

1This is just one definition of no arbitrage, there are others and while they are similar not allof them have the same mathematical interpretation.

12

3.2. THE VALUE OF AN OPTION

the time of maturity for the option. In order to price this claim, the price processof the (risky) asset S needs to be defined. This is done as{

dS(t) = µS(t)dt+ σS(t)dW (t), t > t0S(t0) = s

where t0 is the time today. Note that the risky asset behaves according to a GBM(see Section 2.1). Assume also the existence of a risk-free asset (e.g. a bank account)that evolves according to the process{

dB(t) = rB(t)dt, t > t0B(t0) = 1.

The reason for defining a bank account is that a market maker will often wantthe initial investment in the Portfolio 1.1 to be 0. Generally, the option and theunderlying asset will not be worth the same and if the difference is positive, moneyis put in the bank account to adjust for this. In case of a negative discrepancy, thedifference is lent from the bank.

Remark: The money put in the bank account is risk-free and every other risk-free asset needs to have the same dynamics as the bank account in order to rule outarbitrage.

3.2.1 Self-Financing PortfolioThe Portfolio 1.1 is said to be self-financing [2, p. 87] if

dVh(t) = −df(t, S(t)) + hS(t)dS(t) + hB(t)dB(t).

What the self-financing condition means is that no money can be deposited orwithdrawn from the portfolio during its existence. A portfolio that is self-financingand risk-free is expected to increase in value at the rate of the risk-free interest rate(i.e dV = rV dt). If this was not the case, there would be an arbitrage opportunityin the market.

To see the reason to use a self-financing portfolio, consider a portfolio thatconsists of merely a bank account with an initial investment of 0. Each day, another100 SEK is put into the bank account. Thus when evaluating the result over a timeperiod, it seems that the value of this portfolio has increased without risk. Whathave been overlooked is that the investor must have taken money from some otherportfolio and put into this one. This can be adjusted for, but it is a lot easier tojust demand that the portfolio be self-financing.

3.2.2 Black-Scholes EquationApplying Itô’s lemma with Y (t) = S(t) gives

df(t, S(t)) =(∂f

∂t+ µS

∂f

∂S+ σ2S2

2∂2f

∂S2

)dt+ σ

∂f

∂SdW.

13


The dependency notation is omitted in this section to make the equations morereadable.

Now, consider the portfolio Vh(t) = −f + hSS + hBB, where hS(t), hB(t) ∈ R.The self-financing condition implies that dVh = −df+hSdS+hBdB. This, togetherwith Itô’s lemma, implies

dVh = −df + hSdS + hBdB

= −(∂f

∂t+ µS

∂f

∂S+ σ2S2

2∂2f

∂S2

)dt− σS ∂f

∂SdW + hS(µSdt+ σSdW )

+hBrBdt

=(−∂f∂t− µS ∂f

∂S− σ2S2

2∂2f

∂S2 + hSµS + hBrB

)dt+ (hS −

∂f

∂S)σSdW.

For this portfolio to be risk-free, it cannot change its value as any stochastic processevolves but only as time progresses. It is clear that by setting hS = ∂f

∂S , the termpreceding the Wiener process becomes 0. This makes the portfolio risk-free, i.e.

dVh =(−∂f∂t− µS ∂f

∂S− σ2S2

2∂2f

∂S2 + µS∂f

∂S+ hBrB

)dt

=(−∂f∂t− σ2S2

2∂2f

∂S2 + hBrB

)dt. (3.4)

Absence of arbitrage implies

dVh = rV dt= r(−f + hSS + hBB)dt

= r(−f + ∂f

∂SS + hBB)dt. (3.5)

Setting Equation 3.4 equal to Equation 3.5 yields(−∂f∂t− σ2S2

2∂2f

∂S2 + hBrB

)dt = r(−f + ∂f

∂SS + hBB)dt.

After some algebra, the result becomes

⇒ ∂f

∂t+ rS

∂f

∂S+ σ2S2

2∂2f

∂S2 − rf = 0, t0 ≤ t < T

which is the Black-Scholes equation. In case of a European call option, the boundarycondition is f(T, S(T )) = max(S(T )−K, 0). In this case of a European call option,the Black-Scholes equation has a closed form solution given by

14

3.2. THE VALUE OF AN OPTION

f(t, S(t) = C(t, S(t))= Φ(d1)S − Φ(d2)Ke−r(T−t)

d1 =ln(S(t)

K ) + (r + σ2

2 )(T − t)σ√T − t

d2 = d1 − σ√T − t

where Φ is the cumulative density function of the standard normal distribution.

3.2.3 Delta Hedge

A portfolio that is delta-hedged is insensitive to movements in the underlying asset.Mathematically, this means ∂V

∂S = 0. A portfolio that satisfies this is called deltaneutral and ∂V

∂S is called the delta of the portfolio.For the self-financing portfolio Vh(t) = −f(t, S(t)) + hS(t)S(t) + hB(t)B(t), the

delta becomes

∂V

∂S= −∂f

∂S+ hS

∂S

∂S

∣∣∣∣hS= ∂f

∂S

+ hB(t)∂B∂S

= −∂f∂S

+ hS

∣∣∣∣hS= ∂f

∂S

= −∂f∂S

+ ∂f

∂S= 0

Hence, the portfolio is delta neutral. This is expected because the portfolio wouldelse not be insensitive to movements in the underlying asset and thus not be risk-free.

Remark: The delta of the option f is usually denoted by ∆f .

3.2.4 Implied Volatility

Implied volatility can be calculated using the inverse of the Black-Scholes model.The option prices are quoted in the market. The same goes for the time to maturity,the risk-free interest rate, the current spot price of the underlying asset and thedifferent strike prices. This data can be used to calculate the so called impliedvolatility, i.e. the volatility that must be used for the Black-Scholes model to givethe quoted market price of the option.

When simulating data, a constant volatility is used during the life of the option.When using historical data, the volatilities saved in the systems at HandelsbankenCapital Markets are used2. These will typically not be constant.

2How these are set can unfortunately not be disclosed in this study and they are thereforeregarded as approximately the mid of the bid-ask spread volatilities.

15


3.3 Discrete TimeIn practice, a portfolio is hedged in discrete time. This means that a risk-freeportfolio is only instantaneously risk-free, but it has to be handled discretely. Toillustrate this, consider a delta neutral portfolio that at time ti is defined as

Vh(ti) = −f(ti, S(ti)) + ∂f

∂S(ti, S(ti))S(ti) + hB(ti)B(ti),

see [3, p. 39]. One time step later, at ti+1, the value of this portfolio has changed.

Vh(ti+1) = −f(ti+1, S(ti+1)) + ∂f

∂S(ti, S(ti))S(ti+1) +hB(ti)B(ti+1) 6= Vh(ti). (3.6)

For the portfolio to be self-financing, it is rebalanced at time ti+1 as

Vh(ti+1) = −f(ti+1, S(ti+1)) + ∂f

∂S(ti+1, S(ti+1))S(ti+1) + hB(ti+1)B(ti+1) (3.7)

where every parameter is known except hB(ti+1). From Equation 3.6 the value ofVh(ti+1) is known. The parameters B(ti+1), S(ti+1) and ∂f

∂S (ti+1, S(ti+1)) simplyarises by the passage of time. The value of f(ti+1, S(ti+1)) can be calculated fromti+1 and S(ti+1). Remains to calculate is hB(ti+1) which arises simply by insertionof known parameters in Equation 3.7.

Doing this for all time steps ti, i = 0, 1, . . . , n− 1 gives a Profit and Loss (PnL),or return, defined as Xi = V (ti)−V (ti−1), i = 1, 2, . . . , n. When the option reachesmaturity, n hedges have been conducted which gives n PnL values, denoted by

X = (X1, X2, . . . , Xn).

As the time interval between each hedge (the hedge interval) tends to 0, so will thePnL-values [5].

3.3.1 Distribution of PnL

When using discrete hedge in the Black-Scholes model the standard deviation ofthe cumulative PnL (also called total PnL or tPL) is approximated by

stdDevtPL =√π

4 νσ√n, (3.8)

see [5]. Here, ν = ∂V∂σ is the sensitivity of Portfolio 1.1 value with respect to the

volatility of the underlying asset. Using the assumption that everything but thestock price is deterministic and that there is no volatility skew, only

√n is non-

constant. Thus, as the number of hedges double, the standard deviation of the tPLwill only decrease by half. This is empirically examined in Section 6.5.

16

3.4. CHANGED VOLATILITY IN DELTA HEDGE

3.4 Changed Volatility in Delta HedgeAssume that a market maker does not know the true volatility and for example usesthe implied or historical volatility from the market. The market maker will thenmiscalculate the value of the option and the value of its delta (recall that the deltaof an option is ∂f

∂S ). The self-financing portfolio calculated when using the wrongvolatility is denoted by

Vh = −f + ∂f

∂SS + hBB (3.9)

where a bar indicates that the value has been calculated using the wrong volatility.The portfolio in Equation 3.9 will result in a tPL that will generally not have anexpected value of 0. For a self-financing portfolio with a European call option thiswill result in the following (see [10]):

tPL = 12(σ2 − σ2)

∫ T

0er(T−t)

d2f

dS2 (t, S(t))S(t)2dt, (3.10)

where S(t) follows the process defined in Equation 2.1. Note that unless σ = σ,the tPL of Equation 3.10 will not be zero. Also note that if σ > σ, then tPL > 0,which means that an arbitrage opportunity exists. In practice, it should be difficultto find a buyer at this price which stems from an overestimated volatility.

3.5 Sticky StrikeThe market has generally shown a volatility skew [4] since the stock crash of 1987.The skew is usually negative, and typically looks like the skew displayed in Figure3.1. This means the market demands a higher relative premium the lower thestrike is. Important to note is that the option prices are quoted in the marketand e.g. Black-Scholes model is used to implicitly calculate the volatilities (hencethe name implied volatility). The Black-Scholes model assumes a constant volatilityregardless of strike price. It is thus obvious that the empirical data presented in thissection contradicts one of the assumptions of the Black-Scholes model. The hopeis that this violation is mild enough not to affect the hedging strategy too much.By choosing the level of stickyness (explained below), the effect of this violation isminimized.

The hedged portfolio will be affected by whether or not this skew varies withthe changes in the market. If the skew is completely frozen still as the underlyingasset moves, it is referred to as sticky. This is illustrated in Figure 3.2. If, on theother hand, the volatility skew is fully unsticky, the skew will move in line with theunderlying. This can be seen in Figure 3.3. There is also the possibility of havingcases in between, when the skew moves only slightly in the same direction as theunderlying asset. A negative level of unstickyness can also be used. This wouldmean that the skew moves in the opposite direction of the underlying asset.

17


Figure 3.1. Implied volatilities of call options on OMXS30 with maturity 20 April2012. Data retrieved on January 24 2012.

Sungard Front Arena, one of the trading systems used at Handelsbanken CapitalMarkets, uses a notation called float factor. For a fully unsticky skew, the float factoris 1, which means that the skew moves along with the spot price. A float factor of 0implies that the skew is completely sticky. Negative values down to -1 are possible.

Figure 3.2. Illustrates what full sticky strike means.

Remark: In reality, the true volatility is only known for historical data. Impliedvolatility is not necessarily the true volatility (in fact, in historical data it is oftenhigher than the realized volatility). The Black-Scholes model assumes a constantvolatility during the life of the option. This is almost never the case based on

18

3.5. STICKY STRIKE

Figure 3.3. Illustrates what absence of sticky strike means.

historical data, and there is no reason to believe that it should be the case in thefuture. It is also contradictory to the volatility skew, since the Black-Scholes modelassumes the same volatility regardless of strike price. There are other models (e.g.the Hull-White model [6]) that relaxes this assumption.

19

Chapter 4

Utility Functions

Value-at-Risk (VaR) and Expected Shortfall (ES) are two of the most widely usedrisk measures in the industry. Their clear mathematical definitions makes themsuitable utility functions for this study.

The standard deviation is also a popular measure of risk. It is for example oftenused to asses the risk of funds. The standard deviation will be used in this studyin comparison to the more sophisticated risk measures VaR and ES. Note that thestandard deviation is a very poor risk measure since a high standard deviation withan even higher mean could correspond to an asset with almost no risk of decreasingin value.

4.1 Loss functionsIt might be reasonable to assume that the elements of the sample X (see Section3.3) are weakly dependent, close to identically distributed and have distributionalcharacteristics representable for the return of the next time period [8, p. 172].

Loss functions [9, p. 183] are needed to calculate VaR or ES. These are definedas

Li = Xie−r∆t

T

where ∆t is time period the portfolio is planned to be held (e.g. the time until thenext hedge), measured in years. The vector L represents the Li’s sorted from lowestto highest.

4.2 Empirical DataThe empirical VaR and ES are chosen instead of their respective parametric versions[9, chap. 8]. A parametric distribution is often easier to handle than an empiricalone because it is smooth. On the other hand, by trying to fit a parametric distri-bution to an empirical sample, no information can be added. Instead, there is the

21

CHAPTER 4. UTILITY FUNCTIONS

risk of making assumptions during the process that are not true. If the risk of ascenario worse than the worst observed scenario is of interest, it is reasonable try-ing to extrapolate the empirical distribution using a parametric or semi-parametricapproach (e.g. peaks-over-threshold [9, sec. 8.2.2]). In the cases considered in thispaper, no such extreme scenarios will be of interest. Thus, it makes sense to use theempirical versions of VaR and ES instead of some parametric or semi-parametricones.

Figure 4.1. Empirical and fitted Normal distribution, estimating the future dailyreturn of OMXS30.

In Figure 4.1, an illustration of both a parametric and an empirical distributionis shown. The data used are daily returns of the OMXS30 Index from 2010-01-01to 2011-12-31. The returns, denoted here by X0 to avoid confusion, have beencalculated as

X0 = S(ti)− S(ti−1), i = 1, 2, . . . , n

where S(t0) is the value of OMXS30 at closing 2010-01-01. The vector L0 =X0e

−r∆tT now symbolises a predicted return of a portfolio containing one unit of

OMXS30, held for one day. A Normal distribution [9, p. 203] has been fitted tothe empirical data using maximum likelihood estimation [9, sec. 8.1.3] (again, seeFigure 4.1).

4.3 Value-at-RiskValue-at-Risk (VaR) is defined as

VaRp(X) = F−1L (1− p)

[8, p. 144], where FL is simply the probability density function of the loss functions.Empirically, this is calculated as (see [9, p. 183])

VaRp(X) = Lbnpc+1

22

4.4. EXPECTED SHORTFALL

where Lbnpc+1 is the element at position bnpc + 1 of the vector L. In picture 4.2VaR is shown graphically. The marked area in the picture has area p, which meansthat VaR is the p-quantile of the shown probability density function.

Figure 4.2. Graphical illustration of what Value-at-Risk measures.

The main advantage of VaR is its simplicity. VaR is, perhaps because of this,one the most widely used risk measures in practice. Its disadvantages include nonsub-additivity [8, chap. 6] which means that it might not reward diversification witha lower risk. Since it is simply a quantile measurement, it makes it possible to hidea very large loss far out in the tail of the probability density function. For example,an insurance firm that promises to take on the cost of a potential nuclear meltdownin a nuclear plant will have a very large cost in a very unlikely event, and this willgenerally not affect the VaR.

In short, the interpretation is: the higher the VaRp, the lower the risk. In words,VaR denotes the loss at a probability of p.

4.4 Expected Shortfall

Expected Shortfall (ES) is defined as

ESp(X) = 1p

∫ p

0VaRu(X) du

[9, p. 184], which is the average of the VaR up until a specified p. Its empiricalequivalent is

ESp(X) = 1p

∫ p

0Lbnuc+1 du

23

CHAPTER 4. UTILITY FUNCTIONS

[9, p. 185]. In picture 4.3 it is shown graphically what area ES is concerned with.Note that ES is the average of the area in the picture. The marked area is calculatedby integration and then divided by p to get the average.

Figure 4.3. Graphical illustration of what Expected Shortfall (ES) measures. ES is1ptimes the shaded area.

The advantage of ES compared to VaR is that it is sub-additive [8, chap. 6](i.e. it rewards diversification with a lower risk) and takes the entire tail intoaccount. This means that a very large loss far out in the tail will be detected. Onedisadvantage of ES in contrast to VaR is that it needs a larger sample size to yieldthe same accuracy [12]. In addition, it might be more difficult to understand ESthan to understand VaR.

Just as for VaR, a high ES means a low risk. All VaR and ES measures willthroughout the study be based on the p = 0.05 estimate.

24

Part II

Results

25

Chapter 5

Simulated Data

In this chapter an at-the-money European call option will be considered in detail.The movements in the underlying stock are simulated using a Geometric BrownianMotion [2, sec. 5.2]. The rest of the parameters are set to be the following

• K = 1000,

• r = 0.02,

• T = 0.25 (three months),

• S = Ke−rT ,

• σ = 0.3,

• no dividend yield during the life of the option.

Here, 64 time steps are taken because this is approximately the number ofbanking days in three months. As a result of this, the distribution of ES willbe based on the risk of holding the portfolio one day. The effects of weekends areignored. Furthermore, 10000 simulations are carried out to create enough data foraccurate estimation of ES.

In Section 3.3, it is explained why the self-financing risk-free portfolio 1.1 willnot be completely risk-free in practice. However, it should carry a lower risk thanjust owning (or selling) the option or the underlying asset. To illustrate this, thedistribution of ES, based on holding an amount of ∂f∂S (ti, S(ti)), i = 0, 1, . . . , n−1 ofthe stock one day is shown in Figure 5.1. The same risk, but for holding the optioninstead of the stock, is shown in Figure 5.2. The distribution of ES differs quitesubstantially between the two assets. Recall that ∂f

∂S (ti, S(ti)), i = 0, 1, . . . , n − 1is the number of stocks that makes the stock and the option change by the sameamount in value for a given movement in the stock price. This is only true incontinuous time, but it seems reasonable to believe that it is almost true in discretetime for small time steps.

27

CHAPTER 5. SIMULATED DATA

Figure 5.1. Simulated distribution of ES, based on holding ∂f∂S

units of the stockfor one day.

Figure 5.2. Simulated distribution of ES, based on holding the option for one day.

The risk of holding the option or stock can be compared to the risk of holdingthe self-financing portfolio, which is depicted in figure 5.3. Clearly, the risk issignificantly reduced compared to the risk of holding nothing but the option or thestock. This outcome is expected and the risk of holding the self-financing portfoliodecreases even further for smaller time steps (recall that the theory states that isshould be risk-free in continuous time).

5.1 Changed Volatility

In this part, the theoretical analysis conducted in Section 3.4 is compared to itsnumerical equivalent. The same setting is used as in Section 5, but now the mar-ket marker does not know the true volatility. The following seven volatilities areconsidered

• σ = 0, 10, too low volatility

28

5.1. CHANGED VOLATILITY

Figure 5.3. Simulated distribution of ES, based on holding the self-financing port-folio for one day.



• σ = 0, 30 = σ, true volatility

• σ = 0, 35, too high volatility



As mentioned in Section 3.4, when σ > σ the tPL will be positive. This canbe observed in Table 5.1. The opposite can also be seen, i.e. that tPL < 0 whenσ < σ. It can be observed that the empirical and theoretical values are very closeto each other.

σ Empirical value (SEK) Theoretical value (SEK)0,10 -55,20 -55,160,20 -27,66 -27,640,25 -13,85 -13,820,30 -0,03 00,35 13,79 13,820,40 27,59 27,630,50 55,12 55,15

Table 5.1. Empirical and theoretical tPL for different values of σ. The true valueis σ = 0.3.

While the average tPL is interesting, the main focus of this report is the risk.Thus, Figure 5.4 shows three risk measures as functions of scale factors. This scalefactor scales the true volatility which is 0.3. For example, a scale factor of 0.5 gives

29

CHAPTER 5. SIMULATED DATA

a volatility of σ = 0.5 ∗ 0.3 = 0.15. This way of displaying the results was chosen toalleviate the comparison of the simulated and historical results.

The standard deviation seen in Figure 5.4 seems to be minimized when thevolatility is overestimated by about 50%. Recall that the theory states that incontinuous time, the portfolio should be risk-free only if the true volatility is used.If the wrong volatility is employed, the continuous time model would introduce arisk factor (see Section 3.2.2). It seems as the risk-minimizing does not happen atthe same volatility in discrete time as in continuous time, according to the standarddeviation.

Value-at-Risk as well as Expected Shortfall are strictly increasing which is notunexpected. Since the mean return of the portfolio increases as the volatility esti-mate increases, these tail concerned functions will move along with the mean.

Figure 5.4. Simulated average of risk measures, based on holding the self-financingportfolio for one day. Volatility scaled by different factors (σ = 0.3).

5.2 Hedge IntervalIn Section 3.3.1 the theoretical effects of the hedge interval are presented. The firstresult (Equation 3.8), is validated through Figure 5.5. Note that n = T

∆t , where nare the number of time steps and ∆t is the hedge interval measured in years.

Value-at-Risk and Expected Shortfall becomes the same when the hedge intervalis greater than three days. This is because there is only one observation in the 5%tail to base the ES-calculation on. The average of this one observation is of coursejust this observation. The estimates above this level will therefore be inaccurate.

Note that the estimate of the standard deviation in Figure 5.5 seems to showa step behaviour after a while. This is because the same number of hedges areconducted for several step sizes. If the sample size is 63 and steps of 14 days are

30

5.2. HEDGE INTERVAL

taken, only 4 data points are used. If steps of 13 days are used, 4 data points areagain available.

Figure 5.5. Simulated average of risk measures, based on different hedging intervals.

31

Chapter 6

Historical Data

With access to historical data, the benchmark case (Portfolio 1.1) can be analysedhistorically. The following parameters are set

• FFF, as the underlying asset,

• K is set to atm, that is the forward price of the FFF stock with delivery atmaturity,

• r is set to the current risk-free interest rate, i.e. the rate to which Handels-banken Capital Markets can borrow or lend risk-free capital,

• S is set to the spot price of the FFF stock,

• T is set to either 90 days or 3 years,

• dividends are paid by the FFF stock during the life of the option. There whereseveral dividends during the simulated period1.

Two different times to maturity are considered. First, a 90-day option is hedgeduntil maturity. This gives approximately 63 hedges, since there are approximately64 banking days in a 90 day period. The 90-day option is sold 2008-01-01. Thismeans that an option with the parameters in the list above is created at 2008-01-01with maturity 2008-03-31. It is hedged until 2008-03-30. Then a new option iscreated 2008-03-31 with maturity 2008-06-29 and so on. The process is repeateduntil 16 time series have been collected.

Secondly, a 3-year option is hedged for 90 days. This means that when thehedging period ends, there will still be approximately 2 years and 9 months untilmaturity. The procedure used for the 90-day option is repeated but now only 13time series are collected. The first time series starts at 2008-07-01, and then 12subsequent 90-day periods are picked2.

1Due to secrecy, the dividend sizes cannot be disclosed. They are modelled as continuous butare discrete in practice.

2The reason for choosing only 13 time series cannot be disclosed due to secrecy.

33

CHAPTER 6. HISTORICAL DATA

6.1 Time SeriesFigure 6.1 shows the tPL of the Portfolio 1.1 as a function of time when maturityof the option is the end of the hedging period. Note that some of the time seriesseems to fade out and remain thereafter more or less flat. These are options thathave gone out-of-the-money. This means that the spot price of the underlying assetis below the strike price. If the distance up to the strike price is quite large andthere is a short time to maturity, the option value will be rather small and so willits delta be. Thus the change in value of the portfolio is expected to be small too,which is observed in Figure 6.1 (and also in Figure 6.2).

Figure 6.1. tPL of the self-financing portfolio 1.1 as a function of time. 16 sub-sequent 90-days (approximately 64 banking days) hedge periods starting 2008-01-01.Daily hedges have been conducted and the float factor is set to 0. Each portfolio ishedged up to maturity, that is maturity is 90 days from the first hedging day.

Now, turn to Figure 6.2. In this figure, the same time series but for optionswith maturity in 3 years are shown. The behaviour of time series that flatten out isnot observed. This is because even if the spot price of the underlying asset wouldfall quite far below the strike price, the time until maturity would still make thepossibility quite large that the option would go into-the-money before expiry. Thus,both the value of the option and its delta will remain at reasonably high levels andkeep the changes in value of the portfolio moderate.

6.2 Change in VolatilityThe volatility is in reality a stochastic process, since a change in price of an optionon the stock exchange can change without the price of the underlying changing.The entire volatility skew moves according to this unknown stochastic process.

Assume that a hedger calculates an implied volatility skew from the market datausing the average of the bid-ask spread. She then believes that the implied volatility

34

6.2. CHANGE IN VOLATILITY

Figure 6.2. tPL of the self-financing portfolio 1.1 as a function of time. 13 sub-sequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01.Daily hedges have been conducted and the float factor is set to 0. Maturity is threeyears days from the first hedging day.

is either higher or lower than the actual realized volatility will be. As a result, shewants to trade at the volatility she believes to be the correct one. To analyse theeffects of this, the entire implied volatility surface have been scaled with differentfactors and the hedge carried out based on these new volatilities.

Figure 6.3 shows risk measures as functions of volatility scale factors for thecase of the 90-day options. Figure 6.4 shows the same but for 3-year options. Thesemeasures are the average measures of the time series from Section 6.1.

The two graphs are similar, however the second derivative of the curves in Figure6.3 with respect to the volatility scale factor seems to be higher in absolute valuethan those in Figure 6.4. The interpretation is that the hedge of the 90-day optionis more sensitive to changes in the volatility than the hedge of the 3-year option;a result that seems logical. Also note that the lowest risk appears to be closerto the implied volatility in case of the 90-day options than for the 3-year options.Assuming that the true volatility would give the lowest risk, this means that themarket has a tendency to overestimate the volatility more for options with longtime until maturity than for those with short time to maturity. This is reasonablesince there is a larger uncertainty about what will happen in the next 3 years thanabout what will happen in the next 90 days.

Note that the difference between the approach in this section and the approachin Section 5.1 is that now not the true volatility but the implicit volatility hasbeen scaled. In fact, the realised volatilities have over the time period 2008-2012generally been slightly lower than the implicit volatilities. There also seems to be ahuge difference in the behaviour of VaR and ES as compared to the simulated case.Instead of increasing they actually decline as the volatility scale factor increases.Clearly, some model assumption is violated. This also illustrates the advantage

35


of VaR and ES as opposed to the standard deviation. The standard deviationbarely notices that the risk behaves differently as compared to in the simulated case(compare Figures 6.3 and 6.4 to Figure 5.4).

Figure 6.3. Risk measures of the self-financing portfolio 1.1 as a function of volatilityscale factor. 16 subsequent 90-days (approximately 64 banking days) hedge periodsstarting 2008-01-01. Daily hedges have been conducted and the float factor is set to0. Each portfolio is hedged up to maturity, that is maturity is 90 days from the firsthedging day.

Figure 6.4. Risk measures of the self-financing portfolio 1.1 as a function of volatilityscale factor. 13 subsequent 90-days (approximately 64 banking days) hedge periodsstarting 2008-07-01. Daily hedges have been conducted and the float factor is set to0. Maturity is three years days from the first hedging day.

36

6.3. HEDGE INTERVAL

6.3 Hedge IntervalIn this section, the same procedure is carried out as in the simulated case (seesection 5.2) with discrete time hedge. What can be observed in Figures 6.5 and6.6 is a standard deviation that seems to be in line with the standard deviation inthe simulated case. VaR and ES however shows an odd behaviour as it moves upand down, especially for the 90-day options. This is not surprising at all becausethe number of data points used in the estimation declines as the hedging intervalincreases. Thus, it seems probable that the hedging error behaves in reality in thesame way as in the simulated case.

Figure 6.5. Different risk measures as functions of the hedging interval. 16 subse-quent 90-days (approximately 64 banking days) hedge periods starting 2008-01-01.The float factor is set to 0. Each portfolio is hedged up to maturity, that is maturityis 90 days from the first hedging day.

6.4 The Impact of Sticky StrikeIn this section, the impact of sticky strike will be analysed. The concept, from hereon referred to as float factor, is explained in section 3.5.

In figure 6.7, the three risk measures as functions of the float factor can beobserved for the 90-day option. The same is shown in figure 6.8 for the 3-yearoptions. It seems as the risk measures are second degree polynomial functions, asfunctions of the float factor.

ES is considerably lower than VaR, which means some large risks are probablylocated far out in the tail. This behaviour seems to be fairly constant regardless ofthe float factor. Based on this data, taking all risk measures into consideration, itwould seem that a float factor of about 0.3 would yield the lowest risk in the caseof the 90-day options. For the 3-year options the number would be approximately0.5. That risk-minimizing happens for lower float factor as the time to maturity

37


Figure 6.6. Different risk measures as functions of the hedging interval. 13 subse-quent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01.The float factor is set to 0. Maturity is three years days from the first hedging day.

decreases seems fairly intuitive because the longer the time to maturity, the morecalm the trading in that product would be. The movements are generally slow andcontrolled in these situations. When maturity approaches, the trading will becomemore unstable and hence the atm volatility increase for a downwards move in thespot price and decrease for an upwards move. The volatility will increase more fora downwards move in the asset the lower the float factor is.

Figure 6.7. Risk measures of the self-financing portfolio 1.1 as functions of thefloat factor. 16 subsequent 90-days (approximately 64 banking days) hedge periodsstarting 2008-01-01. Daily hedges have been conducted. Each portfolio is hedged upto maturity, that is maturity is 90 days from the first hedging day.

38

6.5. DISTRIBUTION OF ES

Figure 6.8. Risk measures of the self-financing portfolio 1.1 as functions of thefloat factor. 13 subsequent 90-days (approximately 64 banking days) hedge periodsstarting 2008-07-01. Daily hedges have been conducted. Maturity is three years daysfrom the first hedging day.

6.5 Distribution of ESThe distribution of ES is here presented for the 90-day and 3-year options. Figure 6.9shows about the same as figure 5.1 does but for historical data instead of simulateddata (and for the FFF stock instead of the simulated stock). The distributions donot look alike at all and this might be because the sample size for the historicaldata is too small or because the samples are not drawn from the same distribution.There is no way to know which reason is true (they might even both be true). Notethat the scaling of the x-axes in these two figures should not be compared sincethe FFF stock does not have the same spot value as the simulated stock at thebeginning of each simulation.

Comparing figure 6.9 and 6.10, no real conclusion can be drawn other thanthat the risk seems to be higher for the 3-year option than for the 90-day option.However, the number of sample points are too few to conclude this with certainty.

39


Figure 6.9. Distribution of ES for the self-financing portfolio 1.1. 16 subsequent90-days (approximately 64 banking days) hedge periods with start 2008-01-01. Dailyhedges have been conducted and float factor is set to 0. Hedged up to maturity, thatis maturity is 90 days from the first hedging day.

Figure 6.10. Distribution of ES for the self-financing portfolio 1.1. 13 subsequent90-days (approximately 64 banking days) hedge periods with start 2008-07-01. Dailyhedges have been conducted and float factor is set to 0. Maturity is three years daysfrom the first hedging day.

40

Chapter 7

Discussion

In this section, a general discussion about the results are presented together withthe major conclusions. Furthermore, a discussion about the assumptions madethroughout the study is conducted and further investigations are suggested.

7.1 Assumptions

Several assumptions have been made throughout this study, and as evident fromSections 5 and 6, the impact of these assumptions on the outcomes varies substan-tially. It would be a troublesome task to seek to uncover what assumption made acertain result divert from the expected. Instead, it is generally more reasonable tostart by asking “Is the diversion from the expected significant? If so, will it help tomake the theory conform better with reality?”.

Three main concepts have been analysed in this study; the wrong volatility, thehedging interval and the sticky strike. The assumptions affecting these concepts arediscussed below.

Firstly, what happened when the volatility was scaled? The diversion from theexpected outcome was significant, so this area is something that might be inter-esting to improve within the theory. However, there is no way to know the truevolatility in the market beforehand so it might not matter if the model is improved.The historical or implied volatilities are the tools available for estimation of futurevolatilities and neither of these might of course estimate the future realised volatilityaccurately.

The second question concerns the hedging interval. The historical data differsfrom the expected outcome by a little. However, the result seems slightly unstableand this is of course due to lack of enough empirical data. Taking every 10th timestep for 64 banking days gives only 5 observations to base a risk calculation on.With this in mind, the results are probable to agree with the theory.

Finally, consider the sticky strike. There is no theoretical expectation sincethe theory does not admit a volatility skew (which need not be the case for othermodels). It is possible to reason ad hoc which implies the results found seems

41

CHAPTER 7. DISCUSSION

reasonable. Note nonetheless that the conclusion is in this case only based onhistorical data which might not model the future accurately.

7.2 ConclusionsThe Black-Scholes model makes some bold assumptions about the market behaviourand it seems almost unreasonable to believe that it would provide an accurate modelof reality. However, it turns out that it might be an accurate model after all. Evenif it is possible to find a model with fewer (or more reasonable) assumptions, theuncertainty of the input parameters will still have a unmeasurable effect on the PnL.Hence, it will not be possible to know if the new model is in fact more accuratethan the Black-Scholes model or not.

Performing a discrete hedge changes the behaviour of the portfolio even in thesimulated world, but it seems that for daily hedges this is not a severe problem.For larger time steps than one day, the estimate becomes too uncertain to draw anystrong conclusion.

7.3 Further investigationsAn obvious extension to the historical test is to add transaction costs and onlypreform a rehedge if the portfolio’s value has changed by a predetermined amount.Varying this amount would make it possible to find the optimum point to rehedge theportfolio with respect to costs and the increasing risk if a rehedge is not performed.Another logical extension would be to investigate more assets and more complexportfolios, for example a semi-static hedge of a Barrier option.

42

Bibliography

[1] S.E.Alm, T.Britton,STOKASTIK. Sannolikhetsteori och statistikteori med tillämpningar. 1st editionLiber, 2009.ISBN 978-91-47-05351-3

[2] T.Björk,Arbitrage Theory in Continuous Time. 3rd editionOxford University Press, 2009.ISBN 0 199 57474 X

[3] J.Carlsson, K-S.Moon, A.Szepessy, R.Tempone, G.Zouraris,Stochastic Differential Equations: Models and NumericsPlace: Stockholm, KTH Royal Institute of Technology.Lecture notes, 2010.

[4] E.Derman,Regimes of Volatility. Some Observations on the Variation of S&P 500 ImpliedVolatilities.Place: Goldman Sachs (Quantitative Strategies Research Notes).1999.

[5] E.Derman,Quantitative Insight. Risk - non continuous hedgePlace: Goldman Sachs (Quantitative Strategies Research Notes).1998.

[6] J-P.Fouque, G.Papanicolaou, K.R.Sircar,Derivatives in Financial Markets with Stochastic Volatility. 1st EditionCambridge University Press, 2000.ISBN 0 521 79163 4

[7] J.C.Hull,Options, Futures & Other Derivatives. 4th EditionPrentice-Hall International Inc., 2000.ISBN 0 13 015822 4

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BIBLIOGRAPHY

[8] H.Hult, F.Lindskog, O.Hammarlid, C.J.Rehn,Risk and portfolio analysis -Part I: Principles-Place: Stockholm, KTH Royal Institute of Technology.Draft, Springer, 2011.

[9] H.Hult, F.Lindskog, O.Hammarlid, C.J.Rehn,Risk and portfolio analysis -Part II: Methods-Place: Stockholm, KTH Royal Institute of Technology.Draft, Springer, 2011.

[10] D.Talay,Around Model Risk in FinancePlace: National Institute for Research in Computer Science and Control (IN-RIA).2004.

[11] P.Wilmott, S.Howison, J.Dewynne,The Mathematics of Financial Derivatives. 1st EditionCambridge University Press, 1995.ISBN 0 521 49789 2

[12] Y.Yasuhiro, Y.Toshinao,Comparative Analyses of Expected Shortfall and Value-at-Risk: Their Estima-tion Error, Decomposition, and Optimization.http : //www.imes.boj.or.jp/research/papers/english/me20− 1− 4.pdfPlace: Bank of Japan, Research Division I, Institute for Monetary and EconomicStudies.2002. Retrieved at 2011-11-17.

44

Appendix A

Software Framework

To analyse hedging strategies based on historical data, access to historical data isrequired. A framework has been created for this purpose and its GUI (GraphicalUser Interface) is shown in Figure A.1. A reader not familiar with Sungard FrontArena might not fully understand this section and it is mainly intended for readersat Handelsbanken Capital Markets. The version of Sungard Front Arena Primeused is 3.2.2. The code logs on to the trading system specifying an historical date,which grants access to market data from that date.

Delta hedging is the only implemented strategy, but the program code is de-signed to make extensions to other hedging strategies as simple as possible.

The user has the option to use the data retrieved in the last execution. Thisway, different settings can be analysed without having to retrieve the data again.The reason for this function is the time consuming task of logging on to the systemwith historical dates. One month worth of data takes the application approximatelyone hour to collect and save1.

Several time series are usually of interest and these will be regarded as samplesfrom a GBM (see section 2.1). To get the program to extract several time series theuser starts by specifying the first time period to use (e.g. 2008-01-01 to 2008-03-30,with expiry of the option at 2008-03-31). Then how many days the expiry shouldbe pushed are chosen. There are, for example, 90 days between 2008-01-01 and2008-03-30 so to start the next time series after this period 90 days are chosen.Afterwards, how many times to push the expiry are chosen. If the first time periodstarts at 2008-01-01 then 16 ∗ 90 days later is at the end of 2011 and another 90days time period would land in the future at the time of writing. Hence, to pushexpiry 15 times is maximum in this example (date of writing is February 2012).

The input parameters seen in Figure A.1 are explained below

• Type of strategy: Hedging strategy used, e.g. delta hedging.

• Filter name: The Prime-filter that selects the option to hedge.

1Intel R© Xeon R© CPU W3550 @ 3.07GHz, 3.73 GB of RAM

45

APPENDIX A. SOFTWARE FRAMEWORK

Figure A.1. GUI for program used to access historical data.

• Filter name underlying fwd: The Prime-filter that selects a forward con-tract on the underlying asset. Used to set atm-strike for the option.

• Start date: Historical simulation starts at this date.

• End date: Historical simulation ends at this date.

• Expiry date: Maturity/expiry of the option and the forward contract.

• Prime username: What Prime-username to log on with.

• Prime password: What Prime-password the above user has.

• Prime servername: What Prime-server to log on to, preferably a test-serverto avoid stress on production server.

• Progress: Indicates the progress of the execution.

• Use same data as last simulation (saves computation time): Runsanalysis on the data retrieved in the last program execution.

• Run simulation. Button that executes the specified simulation.

46

• Calendar name, used to identify banking days: E.g. Stockholm tradingcalendar in which the dates when the Stockholm stock exchange is closed arespecified.

• Push expiry for ... days: How many days to wait from start date of onesimulation until the next simulation is started.

• Push expiry ... times: One more simulation than the number specifiedhere are conducted.

• Volatility surface float factor: Sets the float factor (see section 3.5) of thevolatility surface.

• Scale volatility surface with factor: A volatility surface is every volatilitypoint with respect to maturity and strike. The surface is scaled with thisconstant scalar.

• How often to hedge (days): How many days that should pass betweeneach hedge. Only positive integer numbers are acceptable.

• Show in right plot: What figure to show in right window. For example riskmeasures as a function of float factor.

• Risk measure at level (%): VaR and ES will be measured at this proba-bility level.

• Exclude the ...% most extreme values in stat-calc: For example if thehighest and lowest observations out of 100 seems to be outliers, they can beremoved by setting this parameter to 2(%).

• Mean: The mean of the tPLs.

• Additional info to user: Displays information that could be useful to theuser, e.g. a short description of the data collected in the last execution.

47

Appendix B

Additional Figures for 3-month Option

The figures presented here are intended for the interested reader to get a morecomplete view of the dynamics analysed in the study. A risk level of 5% is used inall VaR and ES calculations in this section.

Figure B.1. tPL of self-financing portfolio as a function of time. 16 subsequent90-days (approximately 64 banking days) hedge periods with start 2008-01-01. Dailyhedges have been conducted and float factor is set to 0. Hedged up to maturity, thatis maturity is 90 days from the first hedging day.

49

APPENDIX B. ADDITIONAL FIGURES FOR 3-MONTH OPTION

Figure B.2. tPL of self-financing portfolio as a function of time. 16 subsequent 90-days (approximately 64 banking days) hedge periods with start 2008-01-01. Hedgeshave been conducted every 2nd day and float factor is set to 0. Hedged up to maturity,that is maturity is 90 days from the first hedging day.

Figure B.3. tPL of self-financing portfolio as a function of time. 16 subsequent 90-days (approximately 64 banking days) hedge periods with start 2008-01-01. Hedgeshave been conducted every 3rd day and float factor is set to 0. Hedged up to maturity,that is maturity is 90 days from the first hedging day.

50

Figure B.4. tPL of self-financing portfolio as a function of time. 16 subsequent 90-days (approximately 64 banking days) hedge periods with start 2008-01-01. Hedgeshave been conducted every 5th day and float factor is set to 0. Hedged up to maturity,that is maturity is 90 days from the first hedging day.

Figure B.5. tPL of self-financing portfolio as a function of time. 16 subsequent90-days (approximately 64 banking days) hedge periods with start 2008-01-01. Dailyhedges have been conducted and float factor is set to −1. Hedged up to maturity,that is maturity is 90 days from the first hedging day.

51


Figure B.6. tPL of self-financing portfolio as a function of time. 16 subsequent90-days (approximately 64 banking days) hedge periods with start 2008-01-01. Dailyhedges have been conducted and float factor is set to 1. Hedged up to maturity, thatis maturity is 90 days from the first hedging day.

Figure B.7. tPL of self-financing portfolio as a function of time. 16 subsequent90-days (approximately 64 banking days) hedge periods with start 2008-01-01. Dailyhedges have been conducted and float factor is set to 0. Hedged up to maturity, thatis maturity is 90 days from the first hedging day. The volatility surface has beenscaled by a factor of 0.5.

52

Figure B.8. tPL of self-financing portfolio as a function of time. 16 subsequent90-days (approximately 64 banking days) hedge periods with start 2008-01-01. Dailyhedges have been conducted and float factor is set to 0. Hedged up to maturity, thatis maturity is 90 days from the first hedging day. The volatility surface has beenscaled by a factor of 2.0.

Figure B.9. Risk measures of self-financing portfolio as a function of float factor. 16subsequent 90-days (approximately 64 banking days) hedge periods with start 2008-01-01. Daily hedges have been conducted. Hedged up to maturity, that is maturityis 90 days from the first hedging day.

53


Figure B.10. Risk measures of self-financing portfolio as a function of float factor.16 subsequent 90-days (approximately 64 banking days) hedge periods with start2008-01-01. Hedges have been conducted every 2nd day. Hedged up to maturity,that is maturity is 90 days from the first hedging day.

Figure B.11. Risk measures of self-financing portfolio as a function of float factor.16 subsequent 90-days (approximately 64 banking days) hedge periods with start2008-01-01. Hedges have been conducted every 3rd day. Hedged up to maturity, thatis maturity is 90 days from the first hedging day.

54

Figure B.12. Risk measures of self-financing portfolio as a function of float factor.16 subsequent 90-days (approximately 64 banking days) hedge periods with start2008-01-01. Hedges have been conducted every 5th day. Hedged up to maturity, thatis maturity is 90 days from the first hedging day.

Figure B.13. Risk measures of self-financing portfolio as a function of float factor.16 subsequent 90-days (approximately 64 banking days) hedge periods with start2008-01-01. Daily hedges have been conducted. Hedged up to maturity, that ismaturity is 90 days from the first hedging day. The volatility surface has been scaledby a factor of 0.5.

55


Figure B.14. Risk measures of self-financing portfolio as a function of float factor.16 subsequent 90-days (approximately 64 banking days) hedge periods with start2008-01-01. Daily hedges have been conducted. Hedged up to maturity, that ismaturity is 90 days from the first hedging day. The volatility surface has been scaledby a factor of 2.0.

Figure B.15. Risk measures of self-financing portfolio as a function of volatilityscale factor. 16 subsequent 90-days (approximately 64 banking days) hedge periodswith start 2008-01-01. Daily hedges have been conducted and float factor is set to 0.Hedged up to maturity, that is maturity is 90 days from the first hedging day.

56

Figure B.16. Risk measures of self-financing portfolio as a function of volatilityscale factor. 16 subsequent 90-days (approximately 64 banking days) hedge periodswith start 2008-01-01. Hedges have been conducted every 2nd day and float factor isset to 0. Hedged up to maturity, that is maturity is 90 days from the first hedgingday.

Figure B.17. Risk measures of self-financing portfolio as a function of volatilityscale factor. 16 subsequent 90-days (approximately 64 banking days) hedge periodswith start 2008-01-01. Hedges have been conducted every 3rd day and float factor isset to 0. Hedged up to maturity, that is maturity is 90 days from the first hedgingday.

57


Figure B.18. Risk measures of self-financing portfolio as a function of volatilityscale factor. 16 subsequent 90-days (approximately 64 banking days) hedge periodswith start 2008-01-01. Hedges have been conducted every 5th day and float factor isset to 0. Hedged up to maturity, that is maturity is 90 days from the first hedgingday.

Figure B.19. Risk measures of self-financing portfolio as a function of volatilityscale factor. 16 subsequent 90-days (approximately 64 banking days) hedge periodswith start 2008-01-01. Daily hedges have been conducted and float factor is set to−1. Hedged up to maturity, that is maturity is 90 days from the first hedging day.

58

Figure B.20. Risk measures of self-financing portfolio as a function of volatilityscale factor. 16 subsequent 90-days (approximately 64 banking days) hedge periodswith start 2008-01-01. Daily hedges have been conducted and float factor is set to 1.Hedged up to maturity, that is maturity is 90 days from the first hedging day.

Figure B.21. Different risk measures as functions of the hedging interval. 16 subse-quent 90-days (approximately 64 banking days) hedge periods with start 2008-01-01.Float factor is set to −1. Hedged up to maturity, that is maturity is 90 days fromthe first hedging day.

59


Figure B.22. Different risk measures as functions of the hedging interval. 16 subse-quent 90-days (approximately 64 banking days) hedge periods with start 2008-01-01.Float factor is set to 0. Hedged up to maturity, that is maturity is 90 days from thefirst hedging day.

Figure B.23. Different risk measures as functions of the hedging interval. 16 subse-quent 90-days (approximately 64 banking days) hedge periods with start 2008-01-01.Float factor is set to 1. Hedged up to maturity, that is maturity is 90 days from thefirst hedging day.

60

Figure B.24. Different risk measures as functions of the hedging interval. 16 subse-quent 90-days (approximately 64 banking days) hedge periods with start 2008-01-01.Float factor is set to 0. Hedged up to maturity, that is maturity is 90 days from thefirst hedging day. The volatility surface has been scaled by a factor of 0.5.

Figure B.25. Different risk measures as functions of the hedging interval. 16 subse-quent 90-days (approximately 64 banking days) hedge periods with start 2008-01-01.Float factor is set to 0. Hedged up to maturity, that is maturity is 90 days from thefirst hedging day. The volatility surface has been scaled by a factor of 2.0.

61

Appendix C

Additional Figures for 3-year Option

The figures presented here are intended for the interested reader to get a morecomplete view of the dynamics analysed in the study. A risk level of 5% is used inall VaR and ES calculations in this section.

Figure C.1. tPL of self-financing portfolio as a function of time. 13 subsequent90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Dailyhedges have been conducted and float factor is set to 0. Maturity is three years fromthe first hedging day.

63

APPENDIX C. ADDITIONAL FIGURES FOR 3-YEAR OPTION

Figure C.2. tPL of self-financing portfolio as a function of time. 13 subsequent90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Hedgeshave been conducted every 2nd day and float factor is set to 0. Maturity is threeyears from the first hedging day.

Figure C.3. tPL of self-financing portfolio as a function of time. 13 subsequent90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Hedgeshave been conducted every 3rd day and float factor is set to 0. Maturity is threeyears from the first hedging day.

64

Figure C.4. tPL of self-financing portfolio as a function of time. 13 subsequent90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Hedgeshave been conducted every 5th day and float factor is set to 0. Maturity is threeyears from the first hedging day.

Figure C.5. tPL of self-financing portfolio as a function of time. 13 subsequent90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Dailyhedges have been conducted and float factor is set to −1. Maturity is three yearsfrom the first hedging day.

65


Figure C.6. tPL of self-financing portfolio as a function of time. 13 subsequent90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Dailyhedges have been conducted and float factor is set to 1. Maturity is three years fromthe first hedging day.

Figure C.7. tPL of self-financing portfolio as a function of time. 13 subsequent90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Dailyhedges have been conducted and float factor is set to 0. Maturity is three years fromthe first hedging day. The volatility surface has been scaled by a factor of 0.5.

66

Figure C.8. tPL of self-financing portfolio as a function of time. 13 subsequent90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Dailyhedges have been conducted and float factor is set to 0. Maturity is three years fromthe first hedging day. The volatility surface has been scaled by a factor of 2.0.

Figure C.9. Risk measures of self-financing portfolio as a function of float factor. 13subsequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Daily hedges have been conducted. Maturity is three years from the first hedgingday.

67


Figure C.10. Risk measures of self-financing portfolio as a function of float factor.13 subsequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Hedges have been conducted every 2nd day. Maturity is three years from thefirst hedging day.

Figure C.11. Risk measures of self-financing portfolio as a function of float factor.13 subsequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Hedges have been conducted every 3rd day. Maturity is three years from thefirst hedging day.

68

Figure C.12. Risk measures of self-financing portfolio as a function of float factor.13 subsequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Hedges have been conducted every 5th day. Maturity is three years from thefirst hedging day.

Figure C.13. Risk measures of self-financing portfolio as a function of float factor.13 subsequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Daily hedges have been conducted. Maturity is three years from the firsthedging day. The volatility surface has been scaled by a factor of 0.5.

69


Figure C.14. Risk measures of self-financing portfolio as a function of float factor.13 subsequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01. Daily hedges have been conducted. Maturity is three years from the firsthedging day. The volatility surface has been scaled by a factor of 2.0.

Figure C.15. Risk measures of self-financing portfolio as a function of volatilityscale factor. 13 subsequent 90-days (approximately 64 banking days) hedge periodsstarting 2008-07-01. Daily hedges have been conducted and float factor is set to 0.Maturity is three years from the first hedging day.

70

Figure C.16. Risk measures of self-financing portfolio as a function of volatilityscale factor. 13 subsequent 90-days (approximately 64 banking days) hedge periodsstarting 2008-07-01. Hedges have been conducted every 2nd day and float factor isset to 0. Maturity is three years from the first hedging day.

Figure C.17. Risk measures of self-financing portfolio as a function of volatilityscale factor. 13 subsequent 90-days (approximately 64 banking days) hedge periodsstarting 2008-07-01. Hedges have been conducted every 3rd day and float factor isset to 0. Maturity is three years from the first hedging day.

71


Figure C.18. Risk measures of self-financing portfolio as a function of volatilityscale factor. 13 subsequent 90-days (approximately 64 banking days) hedge periodsstarting 2008-07-01. Hedges have been conducted every 5th day and float factor isset to 0. Maturity is three years from the first hedging day.

Figure C.19. Risk measures of self-financing portfolio as a function of volatilityscale factor. 13 subsequent 90-days (approximately 64 banking days) hedge periodsstarting 2008-07-01. Daily hedges have been conducted and float factor is set to −1.Maturity is three years from the first hedging day.

72

Figure C.20. Risk measures of self-financing portfolio as a function of volatilityscale factor. 13 subsequent 90-days (approximately 64 banking days) hedge periodsstarting 2008-07-01. Daily hedges have been conducted and float factor is set to 1.Maturity is three years from the first hedging day.

Figure C.21. Different risk measures as functions of the hedging interval. 13 sub-sequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01.Float factor is set to −1. Maturity is three years from the first hedging day.

73


Figure C.22. Different risk measures as functions of the hedging interval. 13 sub-sequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01.Float factor is set to 0. Maturity is three years from the first hedging day.

Figure C.23. Different risk measures as functions of the hedging interval. 13 sub-sequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01.Float factor is set to 1. Maturity is three years from the first hedging day.

74

Figure C.24. Different risk measures as functions of the hedging interval. 13 sub-sequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01.Float factor is set to 0. Maturity is three years from the first hedging day. Thevolatility surface has been scaled by a factor of 0.5.

Figure C.25. Different risk measures as functions of the hedging interval. 13 sub-sequent 90-days (approximately 64 banking days) hedge periods starting 2008-07-01.Float factor is set to 0. Maturity is three years from the first hedging day. Thevolatility surface has been scaled by a factor of 2.0.

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Simulation Approaches to Delta Hedging in the Black-Scholes Model · 2017-10-28 · Simulation Approaches to Delta Hedging in the Black-Scholes Model CARL JOHAN KRISTOFFER FÜRST