F A C U L T Y O F S O C I A L S C I E N C E S

D e p a r t m e n t o f E c o n o m i c s

U n i v e r s i t y o f C o p e n h a g e n

BA-thesis

Sophie Jelstrup

Currency Hedging: Forwards vs. Options Study of currency hedging in Danish pension funds

Supervisor: Peter Norman Sørensen

Curriculum + ECTS points: 2008, 15 ECTS

Date of submission: 23/11/2012

Abstract

The thesis investigates the costs pension funds encounter when hedging currency risk using

options as opposed to forwards. The focus is strictly on Denmark and the Danish market of

banks and pension funds.

It starts by introducing the Danish market of pension funds and banks. The main players in

the Danish bank market are Danske Bank and Nordea Bank. They create a duopoly situation,

resulting in relatively high costs in the forward currency market, which creates inefficiencies

in the market. To illustrate the costs when using forwards a calculation is performed

comparing the implied rate from forwards and the cash rates from swaps. Later options are

investigated as an alternative hedging tool to the costly forwards. When options are described

it includes looking at implied volatilities and the use of the Black-Scholes model which are

also used in a Monte-Carlo simulation. The simulation illustrates how the assets of a

theoretical pension fund evolve together with the currencies in which the assets are placed.

The results indicate that the theoretical pension fund attains the highest value of the balance

when using options, both compared to having no hedge and when using forwards. However,

there is a trade-off between the highest value of the balance and the standard deviation, as the

standard deviation when using the option is larger than when using the forward.

List of Contents

1 Introduction ........................................................................................................................................... - 1 -

1.1 Denmark- Market Description ....................................................................................................... - 2 -

1.1.1 Pension Funds in Denmark .................................................................................................... - 2 -

1.1.2 Banks in Denmark .................................................................................................................. - 3 -

1.2 Instruments and other definitions ................................................................................................ - 4 -

1.2.1 Forward contract ................................................................................................................... - 4 -

1.2.2 Options .................................................................................................................................. - 4 -

1.2.3 Overnight Index Swap (OIS) ................................................................................................... - 4 -

2 Costs from Forwards .............................................................................................................................. - 5 -

3 Stochastic Calculus ................................................................................................................................ - 8 -

3.1 Option pricing- Black-Scholes ...................................................................................................... - 10 -

4 Options ................................................................................................................................................ - 12 -

5 Pension Fund ....................................................................................................................................... - 13 -

5.1 Monte Carlo simulation ............................................................................................................... - 15 -

6 Conclusion and Discussion................................................................................................................... - 19 -

7 List of Literature .................................................................................................................................. - 21 -

8 Appendix .............................................................................................................................................. - 22 -

8.1 Covered interest arbitrage .......................................................................................................... - 22 -

8.2 Cholesky factorization ................................................................................................................. - 22 -

8.3 Delta Hedging .............................................................................................................................. - 23 -

8.4 Simulation of EURDKK ................................................................................................................. - 24 -

8.5 VBA Code ..................................................................................................................................... - 24 -

8.5.1 Cholesky ............................................................................................................................... - 24 -

8.5.2 Correlation matrix ............................................................................................................... - 25 -

8.5.3 Monte Carlo ......................................................................................................................... - 25 -

- 1 -

1 Introduction

Pension funds worldwide are exposed to a number of risks in the process of managing

the assets of their clients. One of these risks arises from the currency exposure as

investments are placed in foreign currency, but the payout to clients is performed in the

local currency. The currency market offers two hedging tools: Forwards and options. In

the Danish pension sector mainly forwards are used as a hedging tool. Another way to

hedge the currency risk would be to use options. Options are more costly to enter than

forwards, but options also have a potential upside and a limited downside. The following

thesis investigates the costs related to using options as opposed to forwards as a means

of hedging currency risk with focus strictly on Denmark and the Danish market of banks

and pension funds.

First the Danish pension- and bank markets are described. It is in these markets that the

demand and supply of the currency hedging tools originate. Next, important instruments

are defined, as they are to be used frequently later in the thesis. Once the markets,

essential to this thesis, have been described and the definitions defined, forwards and

their costs are presented. The costs arising from using forwards is the teaser to

investigating the possibility of an alternative currency hedge using options. Next a

theoretical part involving stochastic calculus is introduced. It is key to understand this

underlying calculus in order to be able to comprehend both Black-Scholes and later the

simulations. Once stochastic calculus including Black-Scholes has been covered, the

thesis describes the market of currency options by pricing the options from implied

volatilities quoted in the market. The last part is a simulation of a hypothetical pension

fund to try to illustrate what costs a pension fund encounter, given the empirical prices

of forwards and options found in the market. The assets of the pension fund are placed

in currencies: DKK, SEK and USD and at the same time in two different volatility type

assets. All this is simulated using Monte-Carlo and then priced to see the effects of using

either no hedge, hedge with forwards or hedge with options.

The purpose and structure of this thesis has now been introduced. The next section

describes the Danish pension- and bank markets.

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1.1 Denmark- Market Description

1.1.1 Pension Funds in Denmark

Pension funds are among the dominating investors on the global market due to the size

of the funds. Depending on the country specific characteristics and legislation one can

usually categorize the pension system into three distinct groups:

1. Public and social pension

2. Labour market pension often related to terms of employment

3. Private and individual pension

The pension group characterisation above is a good description of the Danish pension

system.

The first group is known as a “pay-as-you-go” system and is not a saving, but rather a

redistribution of income between the citizens. It is a system that relies on those who pay

taxes as they are to support those that are outside the labour market, due to for example

age or illness. This type of system has come under strain in the recent years as the share

of the population in the labour market and thereby the supporters are decreasing

compared to those outside needing pension support. This is mainly due to an aging

population where more people are leaving the labour market than entering.

The second group in Denmark is characterised by pensions related to the labour market

where the main players are Arbejdsmarkedets Tillægspension (ATP), Lønmodtagernes

Dyrtidsfond (LD) and labour market pensions negotiated by the labour market. The

amount paid to the individual client at retirement is dependent on the amount saved by

that individual client and on how the investment has been managed in the period of

saving. The second group has a social perspective or element as some redistribution

between the clients is carried out.

The third group are the private and individual pensions and have many of the same

characteristics as the second group. The pension paid out depends on the amount saved

and on the return of the investment in the period of the saving. The third group differs

from the second group as it is private and that it is more flexible and can therefore be

made to fit the individual client better. In contrast the labour market pensions from

group two are standardised and designed to fit the whole population of Denmark (Lage,

2009).

Pension savings in Denmark has increased over the past many years which can be seen

in figure 1.1. Part of the reason for this increase in saving is the increasing focus on

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private saving and that it has become common to have a labour market pension (Lage,

2009).

Figure 1.1 Total pension saving in Denmark 1991-2011

Source: (Finansrådet)

The Danish pension market has now been introduced. Next the focus will be on banks in

Denmark.

1.1.2 Banks in Denmark

The statistics for the largest banks in Denmark can be seen in Figure 1.2 below. The

figure shows that there are few, but large banks namely Danske Bank and Nordea Bank.

The characteristic of the Danish bank sector, with few dominant banks, influences the

market dynamics as the banks are the market makers especially in trades involving DKK.

The inefficiency will specifically influence EURDKK instruments such as forwards and

options where EURDKK is the underlying. The impact on pricing of forwards and options

will be explored later.

Figure 1.2 The largest banks in Denmark ultimo 2011

Million DKK Working capital Debit Credit Balance

Danske Bank 1,247,902 730,542 917,201 2,426,689

Nordea Bank Danmark 357,635 315,374 267,010 765,420*

Jyske Bank 176,968 122,953 110,671 270,021

Nykredit Bank 98,249 57,660 77,613 232,316

Sydbank 93,787 74,567 75,827 153,039

FIH Erhvervsbank 60,910 6,755 33,828 85,283

Spar Nord bank 52,399 37,433 37,572 68,822

Arbejdernes Landsbank 30,439 22,933 16,948 34,570

Source: (Finansrådet) *Note: Nordea total assets under management: EURbn 199.8 (Nordea)

Market descriptions of both pension funds and banks in Denmark have been covered.

Next a few important concepts will be explained in order to be able to use the terms later in

the thesis.

- 4 -

1.2 Instruments and other definitions

1.2.1 Forward contract

A forward is an agreement between two parties to exchange an asset at a future point in

time. The agreement is made over-the-counter (OTC henceforth) and can therefore be

customised to meet the specific requirements from the two parties. Only at maturity will

there be an exchange of money, meaning that there is no exchange of money initially

when the parties enter the forward contract (Wilmott (a), 2006a).

1.2.2 Options

Options (calls) are agreements where the buyer has the right, but not the obligation, to

buy an asset at a specified point in the future at a specified strike price. Compared to

forwards where the parties are obliged to carry out the trade, the buyer of an option can

choose not to exercise (Wilmott (a), 2006a). Options, like forwards, are traded OTC.

Common options are either calls or puts which means that one buys or sells the

underlying respectively (Hull, 2012). Whether exercise is possible until termination or

only at termination, depends on whether the option is American or European. This

thesis will only use European options (only exercise at maturity), but American options

could also have been used. There is an initial premium due to the flexibility of the option

where exercise as explained above, is a right, but not an obligation. Once an option has

been bought and all factors agreed upon, the buyer will at maturity exercise the option,

given that the strike price is less than the spot exchange rate. If the strike price is higher,

the buyer will simply not exercise and the buyer has only lost the premium. This shows

that there is an asymmetry of risk as the maximum downside is the premium, whereas

the upside theoretically is indefinite (Taylor, 2003). The size of the premium reflects the

details of the agreement that is everything from the maturity and strike price to the type

of underlying and its volatility. The longer time to maturity and exercise, the higher the

probability of a positive payoff and therefore a higher premium. With currency options

the underlying is the currency.

1.2.3 Overnight Index Swap (OIS)

OIS swaps are OTC derivative instruments. It is an agreement between two

counterparties to exchange cash flows in the future. One commits to paying a fixed rate

and the other to paying a floating rate according to a specified traded overnight index

- 5 -

such as EONIA (Veronesi, 2010). The swap rate is the fixed rate of interest, which gives

the swap contract par value (zero value) (Wilmott (a), 2006a).

The important concepts described above, will be referred to throughout the rest of the

thesis. Next, costs of using forwards will be covered.

2 Costs from Forwards

As described in the introduction the starting point of this thesis is to show that banks

require a large premium when selling forward contracts to pension funds, meaning the

cost of currency hedging for the pension funds are large. The following will try to

quantify the costs of using forwards as a means of hedging the currency risk. This

quantification is performed by looking at the difference between the spreads of the

interest rates from cash and the implied rates from the forwards.

The pension funds in Denmark typically approach the banks every 3 months with the

purpose of hedging using currency forwards, because a large part of the portfolios are in

foreign currency. In Denmark when one looks at the forward market there are

essentially two banks which can trade currency forwards in the size required by pension

funds: Danske Bank and Nordea Bank (As illustrated in Figure 1.2 on page 3, the two

mentioned banks are by far the largest in Denmark). The dominance of Danske Bank and

Nordea Bank in the forward market creates a duopoly situation and an inefficient

market. It means that the costs pension funds encounter when dealing in the forward

market are very large, compared to making the same forward trade in other countries

(where the currencies also are different). This can be illustrated by looking at the

implied DKK rate derived from the forward prices and comparing them to a cash rate

represented by an OIS swap (see section 1.2 on page 4). These two rates should have a

similar curve and not deviate much.

The comparing cash rate is from an OIS swap. The reason that a swap rate is used is that

the floating leg is traded, which means that the markets expectation is included in the

swap rate. The rate is not risk free rate, but only includes a minimum of risk premium.

When comparing the two rates, it is two rates that are comparable and assumed to be

correlated as they both include market expectations. In the example below Eonia is the

swap rate using EONIA, the euro overnight index, as the floating rate, and Cita is the

swap rate, where the floating rate is the CIBOR Tom/Next fixing.

- 6 -

The prices of the forward and swap rates have been retrieved from Bloomberg. All the

prices are mid prices instead of bid/offer in order to simplify the calculations and results

(the conclusion would be the same if the bid/offer had been used). In Figure 2.1 below

the forward and swap rates, Eonia and Cita, are shown for different time horizons (1 to

12 months).

Figure 2.1 Data from Bloomberg

Time Days EURDKK forward outright Eonia Cita

1M 30 7.4578 0.075 -0.013

2M 61 7.4554 0.072 0.011

3M 92 7.4536 0.069 0.024

4M 120 7.4521 0.068 0.039

5M 152 7.4504 0.067 0.048

6M 181 7.4490 0.067 0.061

9M 273 7.4466 0.067 0.070

12M 365 7.4451 0.067 0.078

Source: Bloomberg 04/11-2012

By looking at covered interest arbitrage one can compare spreads and find an arbitrage

opportunity or market inefficiency described above. The covered interest arbitrage

formula looks the following (ACI, 2003)1:

ariab e urren y rate base urren y rate

outright

spot

variab e year

days 2.1

By applying equation 2.1 the interest rate, implied by the forward rate, can be found. In

the equation the outright is the forward rate and the base currency rate is the swap rate.

The spot is the exchange rate at which one can exchange if one deals today. By

convention, the number of days used in the calculation is 360. The result can be seen in

Figure 2.2 below in column Implied DKK rates:

Figure 2.2 Implied rates

Time Days Implied DKK rates Cash Spread (bp) Implied Rate Differential (bp)

1M 30 -0.21 -8.80 -28.15

2M 61 -0.25 -6.10 -32.44

3M 92 -0.24 -4.50 -30.96

4M 120 -0.23 -2.90 -29.97

5M 152 -0.22 -1.90 -29.06

6M 181 -0.21 -0.60 -28.01

9M 273 -0.16 0.30 -22.90

12M 365 -0.12 1.10 -19.05

Source: Bloomberg 04/11-2012

The cash spread in Figure 2.2 above is the spread or difference between the Cita and

Eonia. Likewise the implied rate differential is the difference between implied DKK and 1 See appendix: Section 8.1 page - 22 -

- 7 -

Eonia. The cash spread and implied rate differential are illustrated in Figure 2.3 below.

The figure shows that the spread with the implied interest rate is much larger than the

spread using the cash rate. The expectation is that these two should be quite similar, but

as can be seen in Figure 2.3 they differ significantly.

Figure 2.3 Interest Rate Differentials

Source: Bloomberg 04/11-2012

The difference between the rates illustrated in Figure 2.3 above can be quantified by

looking at an empirical example. The total pension saving in Denmark is around DKK

3.34 trillion, where it is assumed that 70% is invested in currencies different from DKK.

This means that 70% of the total savings can be hedged. In Figure 2.4 costs of hedging

different levels is illustrated using a 3 month time horizon as this is the time period that

most pension funds in Denmark would hedge.

Figure 2.4 Costs from rate differentials for pension saving in Denmark

Source: Bloomberg 04/11-2012 and own calculation

Figure 2.4 show that the costs related to the forwards are much higher than expected

from the rate from the cash. This is an indication that the market is not efficient due to

the lack of market makers. There are simply too few suppliers and they can therefore set

a price which is higher than the rest of the market indicates it should be at.

-40

-30

-20

-10

0

10

1M 2M 3M 4M 5M 6M 9M 12M

bp

Cash Spread Implied Rate Differential

0

500

1000

1500

2000

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

DK

K M

ill

Hedge ratio

Difference Cash spread Implied Rate Differential

- 8 -

Summoning up: The two market players in the forward market create a duopoly

situation, where the implied rate from the forward is much higher than the rate from the

cash. This inefficiency results in relatively high costs for the pension funds when

hedging currency risk using forwards. Next stochastic calculus is introduced, before

looking at options and later simulating the movements of assets and currencies of the

theoretical pension fund, in order to understand the underlying math.

3 Stochastic Calculus

When the assets of the pension fund are simulated using Monte-Carlo the assets follow a

path. This path is called a Brownian motion. To understand a Brownian motion

stochastic calculus has to be used. Stochastic calculus will also be used in the pricing

model of options (Black-Scholes) which will be described in the next section to come.

When trying to predict future returns of an asset such as a stock or a currency, one is

looking at the expected returns of that asset. The expected return of assets are

characterised by randomness as one cannot predict the future value with certainty. This

means that in order to be able to model expected returns one has to be able to model

stochastic variables.

The starting point of modelling randomness is to look at a random walk which can later

be used to model expected returns. A simple random walk is in discrete time, but

continuous time is more realistic. Moving from discrete to continuous time in a random

walk, means that the size of the time steps goes to zero. It is important that the random

walk stays finite and does not become infinite which is why we need a Brownian motion

as the limiting process. In order to keep the random walk finite when moving to

continuous time careful scaling between the size of the time step and increments has to

be made. This is done by looking at the quadratic variation of the random walk and

making it equal to t: -

. This will be true when the increment scales with

the square root of the time steps: -

. When n approaches

infinity only with this scaling will the random walk stay finite (Wilmott (a), 2006a).

Summoning up: The limiting process of the random walk is called a Brownian motion

and is denoted by t and has the following properties:

- Finiteness: Random walk stays finite as explained above

- Continuity: Moved from discrete to continuous time

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- Markov: The Markov property is when the random walk of a process has no

memory beyond where it is. In the Brownian motion it means that the conditional

distribution of t given information up until t only depends on

- Martinga e: A martinga e is that the best “guess” of the resu t of an out ome

tomorrow is the same result of the outcome today. Today is the best estimate of

tomorrow. In the Brownian motion it means that the conditional expectation of

t is given information t

- Quadratic variation: As explained above the quadratic variation should approach

or be equal to t.

- Normality: i - i is normally distributed over finite time horizons with

mean zero and variance i i (Wilmott (a), 2006a)

Randomness in the form of t has now been introduced through a Brownian motion.

This can be used in stochastic integrals (and naturally also differentials) with random

walks to enable modelling. They will be in the form: d dt d , where the first

term on the right hand side of the equation is something deterministic and the last term

is something random. The functional form of the deterministic and the random part

depends on the model one is looking at, but before looking at specific functional form a

calculus rule will be introduced namely Itô’s Lemma. For functions of random variables

such as t norma ru es of a u us does not app y. Instead the ru e Itô’s Lemma is used.

It is derived through a Taylor approximation and states: d

d

dt (Wilmott (a),

2006a).

There are many different functional forms of the deterministic and random term which

can be chosen. A widely used random walk because of its use in Black-Scholes option

pricing, is a lognormal random walk where the drift, , and randomness is scaled with S

to give the following: d dt d . The integral form of the log(S) is:

t e . Using Itô’s Lemma on s,t use Itô’s Lemma in higher

dimensions as it is a function of both time and the underlying) and the lognormal

random walk one can find:

d

dt

d

dt 3.1

General stochastic calculus has now been introduced. Next it will be described in relation

to the option pricing model Black-Scholes.

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3.1 Option pricing- Black-Scholes

Before going into the simulation and finding costs related to options a more theoretical

approach is introduced. Options as explained in section 1.2 on page 4 have an

asymmetrical payoff. The value of options, s,t , depends on a number of variables:

underlying asset S and its properties, time t, time to expiry T-t and strike price E. For a

call option there will be a positive correlation between the value of the option and its

underlying as the payoff increases with the value of the underlying. The opposite is the

case with the put. One can make a portfolio which looks the following: s,t - .

The portfolio has an option and is short delta of the underlying asset. The underlying is

assumed to follow a random walk: d dt d as presented in section 3 on page 8.

The portfolio can be written as the change over time (from t to ): - d .

Using Itô’s Lemma and inserting from the equation above the portfolio will look the

following:

dt

d

dt

dt

3.2

The equation above contains a deterministic term dt and a random term dS. If one wants

to eliminate risk arising due to the randomness one can set

. Given these terms can

be determined, randomness has been eliminated, which is referred to as delta hedging.

Delta hedging is when eliminating risk using the correlation between two instruments. It

is dynamic and therefore needs to be continuously rebalanced in order to be fully

hedged2, the portfolio looks the following:

dt 3.3

This portfolio is risk free as it only contains a deterministic term. If it is riskless then

arbitrage means that the portfolio should be equal to as this is the return one

would get by placing the equivalent cash amount in a bank gaining the risk free rate.

Setting these two equations equal to each other and using previous equations one ends

at the Black-Scholes equation:

3.4

2 See appendix for further description of delta hedging, section 8.3 page - 23 -

- 11 -

There are many assumptions related to the Black-Scholes equation some of them are the

following:

- Volatility: The volatility, , must be constant or only depend on time. This is a

strong assumption as financial data do not satisfy this.

- Delta hedging: it is assumed that the delta hedging is done continuously, but in

reality this is not possible instead it would be in discrete time. Another thing

related to the hedging is the transaction costs. In reality there is often a bid offer

spread which makes it costly to keep re-hedging.

- Arbitrage: The model assumes no arbitrage, both model dependent and

independent.

The Black-Scholes equation above does not include which is the expected return and is

related to risk preference: the higher the value of the higher the risk aversion of the

investor. is absent from the equation meaning the Black-Scholes equation is

independent of risk preference and risk neutrality of the investors can be assumed (Hull,

2012). Under the assumption of risk neutrality solutions can be found to the differential

equation. This can be done for the call as an example. Consider ma , where

is the expected value in the risk neutral world. The price of the call is then the

discounted value of the option:

3.5

Using this to solve the Black-Scholes differential equation will lead to(Hull, 2012):

3.6

The Black-Scholes equation can be derived for currency options, which give the

following equation (Wilmott (a), 2006a):

r 3.7

This can be solved to give the foreign currency call option (Finance):

3.8

The theoretical background of stochastic calculus with focus on Black-Scholes has now

been covered to enable the pricing of options using empirical data. The following section

will look at the costs of options like section 2 looked at the costs of forwards.

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4 Options

To look at the costs of using currency options the prices of options in the market has to

be found. This is done by ooking at imp ied vo ati ity whi h is how the “va ue” of the

option is quoted. The implied volatility can be inserted into the Black-Scholes formula

(formula 3.8) to find the price of the option. To find the implied volatility quoted by the

market a trading platform such as Bloomberg or Reuters can be used. However, as

mentioned in section 1.2 on page 4, options are traded OTC and specifically the EURDKK

options are traded in a market where there is a duopoly. This results in prices which can

be inconsistent. The source of the implied volatility used in this thesis will be from

Nordea Analytics, which tends to be more accurate than for example Bloomberg.

The implied volatility is a function of ‘Time to Maturity’ and Strike or Delta. These can be

plotted into a 3D graph called an implied volatility surface. In order to bring it to 2D the

maturity will be set to 1 year as this simplifies the Monte-Carlo simulation later. The

data can be seen in Figure 4.1 where maturity is held constant at 1Y.

Figure 4.1 Implied volatility put, 1Y

Delta 10 25 50 75 90

30-08-2012 4.13% 2.05% 1.60% 3.15% 6.27%

31-08-2012 4.13% 2.05% 1.60% 3.15% 6.27%

03-09-2012 4.13% 2.05% 1.60% 3.15% 6.27%

05-09-2012 5.37% 2.70% 2.20% 3.70% 7.32%

06-09-2012 5.37% 2.70% 2.20% 3.70% 7.32%

Source: Nordea Analytics

The volatility smile for the date 06-09-2012 has been drawn in Figure 4.2. It illustrates

the volatility smile of a currency option.

Figure 4.2 Implied volatility versus delta, 1Y, 06-09-2012

Source: Nordea Analytics

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 20 40 60 80 100

Imp

lied

Vo

lati

lity,

%

Delta, %

Out-of-the-money

calls

Out-of-the-money

puts

- 13 -

The smile arises because returns are not log-normally distributed as assumed in Black-

Scholes (mentioned in section 3.1 on page 10). Currencies both jump and have non-

constant volatilities. The graph shows that both out-the-money and in-the-money

options have higher volatilities than at-the-money options (Hull, 2012).

If the volatilities are inserted into the Black-Scholes formula the cost of the option can be

found. The results can be seen in Figure 4.3 below using as input: strike was 7.4515, the

notional 1,000,000 and the day convention is 360.

Figure 4.3 Put option from implied volatilities, 1Y

Delta 10 25 50 75 90

30-08-2012 3,864.18 1,802.62 1,255.73 2,898.26 5,748.18

31-08-2012 3,684.25 1,625.34 1,080.56 2,719.13 5,567.39

03-09-2012 3,713.26 1,653.13 1,107.42 2,747.78 5,596.78

05-09-2012 6,508.16 2,656.67 1,761.94 3,652.39 8,433.58

06-09-2012 6,513.39 2,661.71 1,766.84 3,657.51 8,438.85

Source: Nordea Analytics

These results mean that the pension fund pays for instance 6,513.39 Euro if the trade

was made on 06-09-2012 with a delta of 10%.

This section covered option pricing by looking at implied volatilities supplied by Nordea

Analytics. Next the theoretical pension fund will be introduced in order for the Monte-

Carlo simulation to commence.

5 Pension Fund

In order to investigate the possibility of using options as a hedging tool a hypothetical

pension fund will be created. Pension funds have assets, equity and liabilities in the form

of payout of pensions to their clients. The balance sheet of such a pension fund can be

seen in Figure 5.1. The payouts will typically be in the local currency which requires the

assets to be liquidated and if necessary changed into the local currency.

Figure 5.1 Balance sheet of a pension fund

Assets Liabilities

Bonds Equity

Stocks Guarantees

Real Estate

Other assets

Source: (Lage, 2009)

The hypothetical pension fund will have assets, liabilities and equity as any other

pension fund. The assets will consist of a low volatility asset such as a bond and an asset

with larger volatility such as a stock. This is a simplification made in order to keep it

- 14 -

simple. An underlying assumption when using this simplification is that all assets are

liquid. It can be discussed how liquid for instance real estate is, but in this thesis high

liquidity of the assets will be assumed.

The scenario will be as follows: The pension fund has assets of 50 billion DKK to be

invested. The allocation can be seen in Figure 5.2 below. The spot currency has been

used to find the amount in the respective local currencies.

Figure 5.2 Asset allocation in hypothetical pension fund

DKK Local currency

Bonds 35,000,000,000

Denmark 10,500,000,000 10,500,000,000

Sweden 12,250,000,000 14,006,650,000

USA 12,250,000,000 2,091,075,000

Stocks 15,000,000,000

Denmark 4,500,000,000 4,500,000,000

Sweden 5,250,000,000 6,002,850,000

USA 5,250,000,000 896,175,000

Total 50,000,000,000

Source: Bloomberg 08-10-2012

The pension fund is denoted in DKK. That is all the pension funds liabilities are in DKK, it

will both receive and pay its clients in DKK. The assets of the pension fund are in a mix of

the three currencies DKK, SEK and USD. When the fund chooses to invest in DKK, SEK

and USD it has to exchange the money needed from domestic, DKK, into the respective

foreign currencies on the spot market to finance the foreign investment. At the same

time the trader will trade a forward or option to hedge the risk from the currency. In this

setup the investment will be placed for 1 year, then liquidated and exchanged back to

DKK. If the foreign currency increases then the asset will increase in value, but the hedge

instrument will be worth a negative amount or nothing. The opposite will happen if the

currency decreases, then the asset decreases in value, but the hedge instrument will

have a positive value. This is the point of the hedge. The instruments used to hedge will

have a positive payoff in case the currency results in lower values of the assets. A

simplification in the simulation to come is that all hedging is done in EURDKK. In reality

the hedge is done through EURDKK and then for instance EURUSD is used afterwards to

go the whole way from DKK to USD. As the focus is on the inefficient Danish market, only

EURDKK is relevant. USD and SEK is still included in the pension fund to create a more

realistic picture, but the hedge is made in EURDKK only.

- 15 -

The allocation from Figure 5.2 can be used to simulate the ending values of both the

bonds and stocks in the three currencies. To find the values of the future value of stocks,

bonds and currencies, a Monte Carlo simulation will be used.

5.1 Monte Carlo simulation

The Monte Carlo simulation can start now. The bond, stock and currencies will be

simulated using the lognormal random walk described in the section 3 on page 8. To

simulate the risk neutral random walk of the asset an approximation has to be found in

order to update the asset price in each discrete time step. There are many ways to do

this approximation, but this thesis will use the Euler method as it is quite simple to

implement. The Euler method simulates the discrete change in the asset by using the

latest value of the asset and drawing the random part from a standardized normal

distribution (Wilmott (b), 2006b):

r t 5.1

Where is drawn from a standardised normal distribution. The Euler method has an

error of t . To generate the random numbers in VBA different methods can be used,

this thesis will use the Box-Muller method (Wilmott (b), 2006b). In order to catch the

correlation between the movements of all of the assets, a correlation matrix has been

constructed and using Cholesky3 the correlation are included in each assets simulation.

This way it is taken into account that the assets have common factors that affects them

in the same or opposite direction. Figure 5.3 below is the correlation matrix of nine

assets. They are all based on 1 year mid data collected on a weekly basis. Had for

example 3 months or 3 years been used instead the correlation matrix would have been

different, but 1 year provide a good indication of the correlation. EURSEK, EURUSD and

EURDKK are all currencies as the names indicate. SEK-, US- and DK-bonds are based on

total return indexes to indicate the total return of the government bonds. Government

bonds are good indicators of very low volatility investments which pension funds have

high allocations in. SEK-, US- and DK-Stocks are based on domestic weighted indexes

(OMX Stockholm (30), S&P 500 (500) and OMX Copenhagen (20) respectively) that are

based on the most frequently traded stocks on the respective domestic stock exchanges.

The correlation matrix of the nine assets described can be seen below.

3 See appendix: Section 8.2 page - 22 -

- 16 -

Figure 5.3 Correlation matrix, 1Y

EURSEK EURUSD EURDKK

SEK

Bond

US

Bond

DK

Bond

SEK

Stock

US

stock

DK

stock

EURSEK 1.00 0.58 -0.61 -0.59 -0.66 -0.69 -0.61 -0.71 -0.78

EURUSD 0.58 1.00 -0.06 -0.82 -0.76 -0.83 0.03 -0.17 -0.35

EURDKK -0.61 -0.06 1.00 0.45 0.52 0.48 0.44 0.66 0.71

SEK Bond -0.59 -0.82 0.45 1.00 0.95 0.95 -0.02 0.31 0.53

US Bond -0.66 -0.76 0.52 0.95 1.00 0.96 0.14 0.47 0.66

DK Bond -0.69 -0.83 0.48 0.95 0.96 1.00 0.17 0.51 0.69

SEK stock -0.61 0.03 0.44 -0.02 0.14 0.17 1.00 0.86 0.76

US stock -0.71 -0.17 0.66 0.31 0.47 0.51 0.86 1.00 0.94

DK stock -0.78 -0.35 0.71 0.53 0.66 0.69 0.76 0.94 1.00

Source: Bloomberg

Besides the correlation matrix, the basic characteristics of each of the assets have to be

determined. This refers to individual characteristics of the assets such as initial value,

the volatility and the drift. The choices can be seen in Figure 5.4 below. Volatility

(realized4) and drift have been found by looking at historical data from 2009 till present.

Figure 5.4 Asset characteristics

EURSEK EURUSD EURDKK

SEK

Bond

US

Bond

DK

Bond

SEK

Stock

US

stock

DK

stock

Stoday 8.34 1.26 7.45 550.92 424.78 501.44 1,043.93 1,406.58 490.16

Vol 6.30% 7.50% 0.50% 0.10% 0.01% 0.01% 5.00% 5.00% 5.00%

Drift -0.02 0.00 0.00 0.06 0.05 0.07 0.05 0.04 0.06

Source: Bloomberg 15/11- 2012

All initial prerequisites have been covered and the simulation in VBA5 can begin. The

simulation gives the value of the assets at the end of the 1 year period. Figure 5.5 below

shows 3 simulations for each of the assets.

Figure 5.5 Asset simulation

Simulation EURSEK EURUSD EURDKK SEK

Bond US

Bond DK

Bond SEK

stock US

stock DK

stock

1 8.31 1.17 7.44 585.73 446.61 537.85 1043.55 1431.13 529.60

2 8.22 1.14 7.46 585.60 446.61 537.84 1035.56 1424.19 514.10

3 8.20 1.20 7.44 585.30 446.57 537.82 1040.20 1402.36 497.81

Source: Bloomberg

The balance of the pension fund can be calculated from the simulations6. The price of the

forwards and options can also be calculated. The key inputs in the calculation are: strike

of 7.45 forward (mid) is 7.4408 and the hedge is performed on a notional of

4,697,040,864.26 EUR. The forward (mid) is the mid of the bid/offer spread indicated in

4 Volatility in the simulations is the realized, whereas it is implied volatility in the option pricing. 5 See appendix: Section 8.5 page - 24 - 6 See appendix for example of simulation of EURDKK section 8.4 page - 24 -

- 17 -

Bloomberg and therefore includes the premiums described in section 2 on page 5. The

notional corresponds to all assets in foreign currency. The calculations of the balance

can be performed with no hedge, hedge from forward and hedge from option. These

three outcomes can be seen in Figure 5.6, Figure 5.7 and Figure 5.8 respectively.

Figure 5.6 Balance: No hedge, DKK

Source: Bloomberg and Nordea analytics

Figure 5.7 Balance: With hedge from forward, DKK

Source: Bloomberg, Nordea analytics and own calculation

40.00 50.00 60.00 70.00

Fre

qu

en

cy

DKK Billion

40 50 60 70

Fre

qu

en

cy

DKK Billion

- 18 -

Figure 5.8 Balance: With hedge from option, DKK

Source: Bloomberg, Nordea analytics and own calculation

It can be difficult to see the difference between the three different figures above. The

interpretation is made easier by looking at the moments of the data behind the graphs.

The first four moments of the 1000 simulations graphed in the three figures above are

shown Figure 5.9 below.

Figure 5.9 First four moments from 1000 simulations with/without hedge

No hedge, DKK With hedge from forward, DKK With hedge from option, DKK

Mean 54,594,902,987.07 54,543,220,698.06 54,652,847,817.72

Std. Dev 3,067,874,301.25 3,065,999,478.00 3,067,038,332.71

Skewness 0.10 0.11 0.11

Kurtosis -0.23 -0.24 -0.24

Source: Own calculation

The mean, the first moment, indicates that the hedge with the option is the most

profitable both compared to no hedge and a hedge with the forward. The standard

deviation, the second moment, indicates that the largest variation is when using no

hedge. This is expected as a hedge is intended to decrease fluctuations. The numbers

also indicate that the standard deviation is the lowest with forwards compared to

options. Skewness, the third moment, quantifies the symmetry of the plot. As the values

are all positive it means that all three plots are skewed to the right, meaning that they all

have heavier tails to the right. Kurtosis (excess), the fourth moment, says something

about the shape of the top and the weight of the tails. The values in the figure above are

slightly negative indicating wide middle, but thinner tails (Wikipedia).

The overall conclusion of the moments is that the balance is the largest when using the

hedge from the options. However the standard deviation is also larger with options

compared to forwards. There seems to be a trade off between the higher mean of the

40 50 60 70

Fre

qu

en

cy

DKK Billion

- 19 -

option, but also a higher standard deviation. Next is the final section of the thesis. It will

sum up on the above sections of this thesis as well as conclude and discuss them.

6 Conclusion and Discussion

This thesis sought to illuminate the forward and option currency market in Denmark as

pension funds use these instruments to hedge currency risk when investing in foreign

currency. The thesis started out by describing the Danish market of pension funds and

banks. It illustrated how a few large banks dominate the market which results in a

duopoly situation with Danske Bank and Nordea Bank as the main players. The

inefficient market results in relatively high costs in the forward currency market. This

was shown by comparing the implied rate from the forward and the cash rates from

swaps. The conclusion drawn was that pension funds encounter relatively large costs

when using forwards as a means of hedging the currency risk. This was the motivation

behind looking at a different tool for hedging: Options. Next the thesis described the

options from empirical implied volatilities and used the Black-Scholes to price the

options. At this stage the empirical part of the thesis had been covered and next a

theoretical pension fund was created. A Monte-Carlo simulation was run to illustrate

how the pension funds’ assets evo ved together with the currencies in which the assets

were placed. The pension fund was simplified to only having investments in DKK, SEK

and USD. The hedge calculated was 100% of the notional and an ATM option. The results

indicate that the highest asset value of the balance of the pension fund is attained when

using options both compared to having no hedge and when using forwards. There is

however a trade-off as the standard deviation when using the option was larger than

when using the forward.

One can discuss the validity of the results by looking at the assumptions and

simplifications of the underlying model and simulations. Some main assumptions can be

mentioned. One is the assumption of continuous rebalancing. This is not a possibility in

the real world where rebalancing is carried out in discrete time due to the time issue

and the costs. As mentioned in section 3.1 on page 10 another strong assumption in

Black-Scholes is the assumption of constant volatility. A more realistic assumption in

regards to volatility is the assumption of volatility clustering where volatility is assumed

to differ across time and high volatility periods are often followed by further periods of

- 20 -

high volatility. This means that the results based on Black-Scholes and delta hedging are

based on assumptions that are unrealistic and their validity can therefore be questioned.

Looking at the simplifications in the simulation many are made to simplify both the

simulation and results: The number of and type of assets, their liquidity, the number of

currencies and the number of simulations. Further simulations could have been made

where for instance different hedge ratios were used as well as testing different realized

volatilities and drifts. If these developments had been implemented it could have

resulted in a clearer result. Looking at the thesis other assumptions were also made. The

Danish bank market for forwards and possibly options are described as a duopoly, but a

rea investigation of the two dominating banks’ osts and profits have not been made,

which means that what looks like a duopoly might not necessarily be correct.

If one overlooks the lacks and simplifications of the model and instead assumes that the

indicative results, that options might be profitable to use instead of forwards as a means

of hedging currency risk, are correct. What will this result in? Likely many Danish

pension funds would start to use options more frequently, due to the reduced costs. Will

the implied volatilities remain unchanged in the market? Most likely the implied

volatilities will increase as the market conditions with few dominating banks have not

changed. It will still be Danske Bank and Nordea Bank that are the main market makers.

This could likely result in, as in the forward market, banks demanding very high

premiums. This eliminates the solution to the problem, and instead creates a new in the

form of expensive options.

Where to go from here is an interesting question. No matter the expectations of the

future market behaviour, the indicative results of this thesis should result in further

investigation of alternatives to using forwards. Another possibility to forwards could be

basis-swaps which are another type of instrument that can also be used to hedge

currency risk.

Hedging is a necessary tool for pension funds to manage their risk when investing. No

matter the costs, some kind of risk management in the form of hedging will always exist.

How to avoid the large premium requirements in the Danish market is a never ending

quest.

- 21 -

7 List of Literature

ACI. (2003). Denmark: Markets, International Limited.

Finance. (n.d.). Retrieved October 27, 2012, from finance.bi.no:

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node9.html#SECTION00940000000000000000

Finansrådet. (u.d.). Interesseorganisation for bankerne i Danmark. Hentede 7. October 2012 fra

Finansrådet: http://www.finansraadet.dk/tal--fakta/statistik-og-tal/pensionsopsparing.aspx,

http://www.finansraadet.dk/tal--fakta/statistik-og-tal/de-stoerste-pengeinstitutter.aspx

Hull, J. C. (2012). Options, Futures and Other Financial Derivatives. England: Pearson Education

Limited.

Lage, C. L. (2009). De Danske Rentegarantier- og afledte konsekvenser (Kandidatspeciale).

København: Københavns Universitet.

Nordea. (n.d.). Nordea . Retrieved November 19, 2012, from Nordea:

http://www.nordea.com/About+Nordea/Nordea+overview/Facts+and+figures/1081354.html

Riaz, D. (n.d.). Retrieved October 28, 2012, from

http://casm.lums.edu.pk/pdf/Financial%20Math%20Workshop%20Material/DrRiaz-1.pdf

Sercu, P. (2009). International Finance- Theory Into Practice. Princeton: Princeton University Press.

Sundaresan, S. (2009). Fixed Income Markets and Their Derivatives. London: Elsevier Inc.

Taylor, F. (2003). Mastering Foreign Exchange & Currency Options. Great Britain: Pearson Education

Limited.

Veronesi, P. (2010). Fixed Income Securities- Valuation, Risk, and Risk Management. New Jersey: John

Wiley & Sons, Inc.

Wikipedia. (n.d.). Wikipedia. Retrieved November 20, 2012, from Wikipedia:

http://en.wikipedia.org/wiki/Moment_(mathematics)

Wilmott (a), P. (2006a). Paul Wilmott On Quantitative Finance, volume one. England: John Wiley &

Sons Ltd.

Wilmott (b), P. (2006b). Paul Wilmott On Quantitative Finance, volume three. England: John Wiley &

Sons Ltd.

Wilmott, P. (2006c). Paul Wilmott on Quantitative Finance, volume one. England: Wiley & Sons Ltd.

- 22 -

8 Appendix

8.1 Covered interest arbitrage

There is a link between interest rates and forward swaps. A firm needing to finance itself

in one currency can choose to do the financing in another currency if the interest rate

(the cost of finance) is lower and use a forward swap to convert to the first currency.

The result would be a lower cost of finance. This is the idea behind the covered interest

arbitrage.

The formula can be derived from the forward swap:

8.1

This formula should be inverted:

8.2

The forward swap is per definition the difference between the outright and the

spot: . Plotting this into the formula above

will give us the covered interest arbitrage:

ariab e urren y rate base urren y rate

outright

spot

variab e year

days

8.3

8.2 Cholesky factorization

Cholesky factorization is a way to include the correlation between assets when for

example trying to simulate the movements of the assets using Monte- Carlo. Assets are

often affected by the same market factors and their random terms are therefore

correlated:

. The random terms of assets in the Monte- Carlo simulation

should catch this correlation. The Cholesky factorization can be thought of as the square

root of the original matrix: where the positive definite matrix is the

correlation matrix and M is the Cholesky decomposition multiplied by itself transposed.

- 23 -

The matrix M should be multiplied by a vector of random numbers to give a vector of

random numbers where the correlation is included: M . This vector will be used in

the Monte Carlo simulation instead of the random numbers from the Box-Muller.

8.3 Delta Hedging

One way of managing risk is by delta hedging. Delta is one of the Greek letters and is

defined as the rate of change in the option price with respect to the change in the

underlying (Hull, 2012).

8.4

In the currency option the underlying is the currency (Hull, 2012):

e rfT d

e rfT d 8.5

The delta says how much the option changes in value when the currency changes and is

the slope as can be seen in Figure 8.1 below. More specifically delta hedging is a way to

hedge by using the correlation between the option and its underlying (Wilmott (a),

2006a). In order to be delta-neutral the change in the currency should be exactly

counteracted by the change in the value of the option. This way no matter the movement

of the currency, it will be neutralised. It should be noted however that with the

movement in the currency the delta hedge also changes. This means that to remain delta

neutral the portfolio has to be rebalanced continuously (Hull, 2012). In reality this

dynamic hedging is costly and not possible to be do continuously, therefore true delta-

neutrality is not possible.

Figure 8.1 Delta

Source: (Hull, 2012)

- 24 -

8.4 Simulation of EURDKK

Figure 8.2 below is an illustration of EURDKK distribution with 1000 simulations. As it

can be seen the distribution looks like a normal distribution as expected as the random

terms from the Box-Muller method are drawn from a normal distribution.

Figure 8.2 EURDKK distribution from 1000 simulations

Source: Bloomberg

8.5 VBA Code

8.5.1 Cholesky Function choleskys(Sigma As Object)

' copyright PJ Schonbucher

Dim n As Integer

Dim k As Integer

Dim i As Integer

Dim j As Integer

Dim X As Double

Dim a() As Double

Dim M() As Double

n = Sigma.Columns.Count

ReDim a(1 To n, 1 To n)

ReDim M(1 To n, 1 To n)

For i = 1 To n

For j = 1 To n

a(i, j) = Sigma.Cells(i, j).Value

M(i, j) = 0

Next j Next i

For i = 1 To n

For j = i To n

X = a(i, j)

For k = 1 To (i - 1)

X = X - M(i, k) * M(j, k)

Next k

If j = i Then

M(i, i) = Sqr(X)

Else

M(j, i) = X / M(i, i)

End If

Next j Next i

choleskys = M

End Function (Wilmott (b), 2006b) and copyright PJ Schonbucher

7.3 7.35 7.4 7.45 7.5 7.55 7.6

Fre

qu

en

cy

EURDKK

- 25 -

8.5.2 Correlation matrix Sub CholMult()

Dim BM As Variant

Dim BoxM As Double

Dim Chol As Variant

Dim CholMMult As Variant

ReDim BM(1 To 9)

ReDim Chol(1 To 9, 1 To 9)

ReDim CholMMult(1 To 9)

Chol = choleskys(Range("N57:V65"))

For i = 1 To 9

BoxM = Boxmuller

Range("X" & 99 + i).Value = BoxM

BM(i) = BoxM

Next i

For i = 1 To 9

For j = 1 To 9

Range("AA99").Offset(i, j).Value = Chol(i, j)

Next j

Next i

CholMMult = Application.WorksheetFunction.Transpose(WorksheetFunction.MMult(Chol,

Application.WorksheetFunction.Transpose(BM)))

For i = 1 To 9

Range("AL" & 99 + i).Value = CholMMult(i)

Next i

End Sub (Wilmott (b), 2006b)

8.5.3 Monte Carlo Sub BalanceSheet()

ReDim BM(1 To 9)

ReDim Chol(1 To 9, 1 To 9)

ReDim CholMMult(1 To 9)

ReDim Asset(1 To 9)

ReDim InfoMatrix(1 To 6, 1 To 9)

ReDim CostPut(1, 1 To 5)

Dim SToday(9) As Double

Dim Expn(9) As Double

Dim Vol(9) As Double

Dim IntRate(9) As Double

Dim NTS(9) As Double

Dim NPaths(9) As Double

Dim Num As Long

Dim TStep(9) As Double

Dim Drift(9) As Double

Dim SD(9) As Double

Dim DF(9) As Double

Sheets("Correlation").Range("B11:AD6000").ClearContents

StrikeEURDKK = Sheets("correlation").Cells(8, 2)

ForwardMid = Sheets("correlation").Cells(8, 5)

CostPut = Sheets("Correlation").Range("T3:AD3")

HedgeNotional = Sheets("Correlation").Cells(8, 8)

BondsDKK = Sheets("Pension Fund").Cells(21, 4)

BondsSEKDKK = Sheets("Pension Fund").Cells(22, 4)

BondsUSDDKK = Sheets("Pension Fund").Cells(23, 4)

StockDKK = Sheets("Pension Fund").Cells(26, 4)

StockSEKDKK = Sheets("Pension Fund").Cells(27, 4)

StockUSDDKK = Sheets("Pension Fund").Cells(28, 4)

BondsSEKLoc = Sheets("Pension Fund").Cells(22, 5)

BondsUSDLoc = Sheets("Pension Fund").Cells(23, 5)

StockSEKLoc = Sheets("Pension Fund").Cells(27, 5)

StockUSDLoc = Sheets("Pension Fund").Cells(28, 5)

'cholesky

Chol = choleskys(Sheets("Cholesky").Range("N57:V65"))

NoSim = Sheets("Correlation").Cells(9, 2)

- 26 -

For S = 1 To NoSim

'box-muller

For i = 1 To 9

BM(i) = Boxmullers()

Next i

'cholesky and boxmuller

CholMMult = Application.WorksheetFunction.Transpose(WorksheetFunction.MMult(Chol,

Application.WorksheetFunction.Transpose(BM)))

'define variables of assets

InfoMatrix = Sheets("Correlation").Range("B2:J7")

'skal gemmes for alle assets j

For j = 1 To 9

SToday(j) = InfoMatrix(1, j)

Expn(j) = InfoMatrix(2, j)

Vol(j) = InfoMatrix(3, j)

IntRate(j) = InfoMatrix(4, j)

NTS(j) = InfoMatrix(5, j)

NPaths(j) = InfoMatrix(6, j)

TStep(j) = Expn(j) / NTS(j)

Drift(j) = (IntRate(j) - 0.5 * Vol(j) * Vol(j)) * TStep(j)

SD(j) = Vol(j) * Sqr(TStep(j))

DF(j) = Exp(-IntRate(j) * TStep(j))

'simulate asset

' Simulate stock

Asset(j) = SToday(j)

Asset(j) = Asset(j) * (Exp(Drift(j) + SD(j) * CholMMult(j)))

Sheets("Correlation").Cells(S + 10, 1 + j).Value = Asset(j)

Next j

Dim MatrixSum As Variant

ReDim MatrixSum(1, 1 To 4)

MatrixSum(1, 1) = BondsSEKLoc * Sheets("Correlation").Cells(S + 10, 5).Value / SToday(4) + StockSEKLoc *

Sheets("Correlation").Cells(S + 10, 8).Value / SToday(7)

MatrixSum(1, 2) = BondsUSDLoc * Sheets("Correlation").Cells(S + 10, 6).Value / SToday(5) + StockUSDLoc *

Sheets("Correlation").Cells(S + 10, 9).Value / SToday(8)

MatrixSum(1, 3) = BondsDKK * Sheets("Correlation").Cells(S + 10, 7).Value / SToday(6) + StockDKK * Sheets("Correlation").Cells(S +

10, 10).Value / SToday(9)

'MatrixSum(1, 4) = MatrixSum(1, 1) / (Sheets("Correlation").Cells(S + 10, 2).Value) + MatrixSum(1, 2) /

(Sheets("Correlation").Cells(S + 10, 3).Value)

Sheets("Correlation").Cells(S + 10, 17).Value = Payoff((Sheets("Correlation").Cells(S + 10, 4).Value), StrikeEURDKK) * Exp(-

IntRate(3))

Sheets("Correlation").Cells(S + 10, 18).Value = (ForwardMid - Sheets("Correlation").Cells(S + 10, 4).Value) * Exp(-IntRate(3))

'For i = 1 To 5

Cells(S + 10, 19).Value = -CostPut(1, 3) + Payoff((Sheets("Correlation").Cells(S + 10, 4).Value), StrikeEURDKK) * Exp(-IntRate(3))

Cells(S + 10, 21).Value = -CostPut(1, 3) + Payoff((Sheets("Correlation").Cells(S + 10, 4).Value), StrikeEURDKK) * Exp(-IntRate(3)) -

(Sheets("Correlation").Cells(S + 10, 4).Value - ForwardMid) * Exp(-IntRate(3))

'Next i

Balance = MatrixSum(1, 1) / Asset(1) * Asset(3) + MatrixSum(1, 2) / Asset(2) * Asset(3) + MatrixSum(1, 3)

Cells(S + 10, 23).Value = Balance

Cells(S + 10, 24).Value = Balance + HedgeNotional * (ForwardMid - Sheets("Correlation").Cells(S + 10, 4).Value)

Cells(S + 10, 25).Value = Balance + HedgeNotional * (-CostPut(1, 3) + Payoff((Sheets("Correlation").Cells(S + 10, 4).Value),

StrikeEURDKK))

Cells(S + 10, 26).Value = Cells(S + 10, 24).Value - Cells(S + 10, 25).Value

Next S

Sheets("Correlation").Cells(8, 17).Value = Application.WorksheetFunction.Average(Range("Q11:Q5000"))

Sheets("Correlation").Cells(8, 18).Value = Application.WorksheetFunction.Average(Range("R11:R5000"))

'Sheets("Correlation").Cells(8, 20).Value = Application.WorksheetFunction.Average(Range("T11:T5000"))

Sheets("Correlation").Cells(8, 19).Value = Application.WorksheetFunction.Average(Range("S11:S5000"))

Sheets("Correlation").Cells(8, 21).Value = Application.WorksheetFunction.Average(Range("U11:U5000"))

Sheets("Correlation").Cells(8, 23).Value = Application.WorksheetFunction.Average(Range("W11:W5000"))

Sheets("Correlation").Cells(8, 24).Value = Application.WorksheetFunction.Average(Range("X11:X5000"))

Sheets("Correlation").Cells(8, 25).Value = Application.WorksheetFunction.Average(Range("Y11:Y5000"))

Sheets("Correlation").Cells(8, 26).Value = Application.WorksheetFunction.Average(Range("Z11:Z5000"))

Cells(1, 17).Value = Exp(-IntRate(3)) * Application.WorksheetFunction.Average(Range("Q11:Q5000"))

Cells(1, 18).Value = Exp(-IntRate(3)) * Application.WorksheetFunction.Average(Range("R11:R5000"))

End Sub(Wilmott (b), 2006b)