U.U.D.M. Project Report 2008:10

Examensarbete i matematik, 30 hpHandledare och examinator: Johan TyskJuni 2008

Department of MathematicsUppsala University

Static hedging of barrier options in discrete and continuous time

Kun Xu

Acknowledgements

I would like to express sincere thanks to my supervisor Prof. Johan Tysk,

whose guidance has been of a great inspiration to me. His useful

suggestions and constant encouragements support me to complete my

work successfully. I also would like to thank Erik Ekstrom for your

Financial Mathematics lectures that gave me the first touch in the area of

financial mathematics. Finally, thanks for my friends and parents as well.

ii

Abstract

In this project, we discuss some hedging strategies which are widely

used in the financial world. We start from the Delta Hedging that is the

most basic tool to hedge risks. However, our aim is to hedge more

complicated options such as barrier options. Since the delta of a barrier

option is extremely large around the barrier region, Delta Hedging does

not work in this case. Then we investigate another hedging technique

called Static Hedging. We describe how to statically hedge a barrier

option both in discrete and in continuous time. In this project, two

different static hedging strategies are considered. One is static hedging

with maturity varying options; the other one is static hedging with strike

varying options. We also show how to implement these hedging

techniques in a practical point of view.

iii

Contents

Chapter 1 What is Hedging? P1

Chapter 2 Delta Hedging P3

2.1 A Simple Discrete Time Example of Delta Hedging P3

2.2 What is behind the Delta Hedging? P8

2.3 Hedging Portfolio in Continuous Time P11

Chapter 3 Static Hedging P16

3.1 Static Hedging with maturity-varying options P17

3.1.1 Intuitive approach towards static hedging P18

3.1.2 Static hedging interpreted by PDE P24

3.1.3 Solving the PDE by finite difference methods P29

3.1.4 Implementation and result P33

3.2 Static Hedging with strike-varying options P36

3.2.1 European security replication P36

3.2.2 Adjusted payoff function P38

3.2.3 Implementation P41

Chapter 4 Summary P45

References P46

iv

Chapter 1 What is Hedging?

In finance, hedging is the process of reducing or offsetting the risks that

either arise in the course of normal business operations or are

associated with investments. Hedging is one of the most important uses

of financial markets, and is an essential part of the modern industrial

activity. Minimizing exposure to an unwanted business risk while still

allowing the business to profit from an investment activity, this is the

main job of hedging.

The best way to understand hedging is to think of it as insurance; by

paying a fixed amount of money (a premium) you can protect yourself

against some possible losses, such as losses due to fire, theft, or even

adverse price changes.

More general hedging case can arise in the following way. Imagine a

large jewelry maker. This maker will purchase gold and other ingredients

and transform these ingredients into beautiful jewelries, such as bracelet,

necklace, ring and so on. Suppose the jewelry maker wins a contract to

supply a large quantity of necklaces to another company over the next

year at a fixed price. The maker is happy to win the contract, but now

faces risk with respect to the gold prices. The maker will not immediately

purchase all the gold needed to satisfy the contract, but will instead

purchase gold as needed during the year. Therefore, if the price of gold

increases during the year, the maker will be forced to pay more to satisfy

the needs of the contract, and hence it will have a lower profit. In some

sense, the maker is dominated by the gold market. If the gold price goes

up, the maker will make less profit, perhaps even losing money on the

contract. On the other hand, if the gold price goes down, the maker will

1

earn even more money than anticipated.

The maker is in the jewelry making business, not in the gold speculation

business. He wants to eliminate the risk associated with gold costs and

concentrate on making. He can do this by obtaining an appropriate

number of gold futures contracts in the futures market. Such a contract

has a small initial cash outlay and at a set future date gives a profit (or

loss) equal to the amount that gold prices have changed since holding

the contract, which means if the price of gold should go up, the value of a

gold futures contract will go up too by a somewhat comparable amount.

Hence, the net effect to the jewelry maker, the profit from the gold futures

contracts together with the change in the cost of gold, is nearly zero.

There are many examples of business risks that can be reduced by

hedging. And there are many ways that hedging can be carried out

through futures contacts, options, and other financial derivatives. Indeed,

the major use of these financial instruments is for hedging, instead of

speculation. We will explain some of the hedging techniques in this

paper. First of all, Delta Hedging is the most widely used tool in the

process of hedging.

2

Chapter 2 Delta Hedging

2.1 A Simple Discrete Time Example of Delta Hedging

Before starting Delta Hedging, it is worthy to discuss the binomial tree,

which is a useful and very popular technique for pricing a stock option.

This is a diagram that represents different possible paths that might be

followed by the stock price over the life of the option.

An option is the right, but not the obligation, to buy or sell an asset under

specified terms. Usually there are a specified price and a specified

period of time over which the option is valid. An option that gives the right

to purchase something is called a call option, whereas an option that

gives the right to sell something is called a put option.

Suppose we now hold a one-year maturity European put option with

strike price 125. And now the stock price is 100. Besides this, we also

need to know the risk free rate r=6%, and volatility v=20%, which is a

measure of the uncertainty of the return realized on the stock. In order to

construct a portfolio with this put option and some shares of stock to

hedge the risk, let us consider a 2-period binomial model [1], the figure is

presented in the following, which means the stock price would move

either up or down at the end of each period.

3

The parameters are:

T = 1, K = 125, v = 0.2, S0 = 100, r = 0.06, nSteps = 2

And the formulas are:

Time in difference in the binomial tree: nSteps

Tdt =

Growth by interest rate: dtreR *=

Stock moving up in the binomial tree: dtveup *=

Stock moving down in the binomial tree: up

down 1=

Value of Q (martingale probability for moving up): ( )

downupdowneQ

dtdivr

up −−

== *

Value of 1-Q (martingale probability for moving down): updown QQ −= 1

Discounting factor: R

df 1=

Based on the binomial model, we can calculate the put option price in

the following way

( )downdownupup QValueQValuedfValue ∗+∗∗=

This formula can be interpreted as stating that the put option value is

found by taking the expected value of the options using the probability

, and then discounting this value according to the risk-free rate. The

probability is therefore a risk-neutral probability. This procedure of

valuation is the concept of risk-neutral pricing.

Q

Q

Running by Matlab[2], we get the Stock Path and Put Value, respectively.

4

Our work is how to hedge this put option’s price risk by using the

underlying stock. At this stage, we will define a parameter called Delta,

which plays an important role in hedging option risk. The Delta of a stock

option is the ratio of the change in the price of the stock option to the

change in the price of the underlying stock, which is denoted by the

Greek letter . In other words, ∆

SP

pricestockinChangepriceoptioninChange

∆∆

==∆______

By definition, Matlab shows us the result of Delta as following

Now we are trying to create a risk-free position. How can we do that?

Note that we already have (long position) one put option worth 20.0901.

One answer is to replicate a short put option position.

At the initial node in Delta matrix, we have ∆ = -0.8496. This means

that, in order to replicate a long (+1) put option, we should sell 0.8496

5

stocks and lend the proceeds to the money market.

By contrast, if we need to create a short put option, we should borrow

from the risk-free money market and buy 0.8496 shares of stocks. By the

perspective of trading, this is a good strategy we should adopt.

Explanation works as the following.

We start with one put option worth 20.0901, and buy 0.8496 stocks. The

stock price at the initial node is 100, so we have to borrow

0.8496x100=84.96. Our portfolio value is Quantity Value

Put Option 1 20.0901 Stock 0.8496 84.96 Cash -84.96 -84.96

Total 20.0901

Notice that the value of our stocks cancels the borrowings, so that our

portfolio value equals the value of the put option we start with.

(a) Suppose there is a down-tick in period 1: at this node, the stock price

is 86.8123 and the put is worth 34.4933. Since the risk-free interest rate

is 6%, we have to pay interest on the money we borrowed, so the net

position is Quantity Value

Put Option 1 34.4933 Stock 0.8496 73.7557 Cash -84.96 -87.5474

Total 20.7016

Afterwards calculating the nodes after initial down-tick

Now, the binomial tree for put replication indicates that Delta is -1. So we

need to hold one stock when we leave this node. Since we have 0.8496

6

stocks, this means we have to buy (1-0.8496 =) 0.1504 more stocks. The

stock price is 86.8123, so we have to borrow additional money of

(0.1504x86.8123 =) 13.0566. Let us see what has happened when I do

buy the additional stocks. Note that our borrowings are now

(87.5474+13.0566 =) 100.6040

If there is a subsequent down-tick, our portfolio will be Quantity Value

Put Option 1 49.6362 Stock 1 75.3638 Cash -100.6040 -103.6678

Total 21.3322

If there is a subsequent up-tick, our portfolio will be Quantity Value

Put Option 1 25.0000 Stock 1 100.0000 Cash -100.6040 -103.6678

Total 21.3322

(b) Suppose there is an up-tick in period 1: at this node, the stock price is

115.1910 and the put is worth 10.3833. We have to pay interest on the

borrowed money as well, so the net position is Quantity Value

Put Option 1 10.3833 Stock 0.8496 97.8659 Cash -84.96 -87.5474

Total 20.7019

Afterwards calculating the nodes after initial up-tick

Now, the Delta is -0.7648. So we need to reduce our stock holdings. We

start with 0.8496 shares, so we have to sell (0.8496-0.7648 =) 0.0848

shares of stocks. The stock price is 115.1910, so we get

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(0.0848x115.1910 =) 9.7682 from selling the stocks. Our borrowings are

thus reduced to (87.5474-9.7682 =) 77.7792. Let us see what happens

next.

If there is a subsequent down-tick, our portfolio will be Quantity Value

Put Option 1 25.0000 Stock 0.7648 76.4800 Cash -77.7792 -80.1479

Total 21.3321

If there is a subsequent up-tick, our portfolio will be Quantity Value

Put Option 1 0 Stock 0.7648 101.4808 Cash -77.7792 -80.1487

Total 21.3321

Miracles do happen. The result shows us that, whether the stock climbs

or drops, our portfolio behaves like a risk-free bond. And at the end of

second period, our portfolio is worth 21.3321. How is that possible?

2.2 What is behind the Delta Hedging?

When we analyze the Black-Scholes formula [3], we determine the

option price by constructing a riskless portfolio which includes options

and the underlying assets. The return ratio of such a riskless portfolio

should equal to the risk-free interest rate, otherwise the arbitrage

opportunity exists. The reason why we can build such a riskless portfolio

is the price of option is affected by the underlying asset. Therefore, the

profit (loss) from stock would be cancelled out by the option’s loss (profit).

The net effect eliminates all underlying asset price risk from the position.

8

In other words, the example in the previous section is not a miracle; in

fact, it is a kind of technique. This strategy is named as Delta Hedging.

Let us think about a call option. Because a long call option price is

positively correlated with the stock price, the hedging process should

include a long position in one asset and a short position in the other

asset, in order to make them negatively correlated.

If we have stocks and options ( and are the number of

stock shares and options respectively), the value of portfolio V is

Ns Nc Ns Nc

( )ChSNcNcCNsSV +=+=

Where, NcNsh = is the hedge ratio, in order to make the positions

totally hedged, the value of portfolio V should be immune to the changes

of stock price S. In other words, they are independent of each other, so

SV ∂∂ should be zero.

0=⎟⎠⎞

⎜⎝⎛

∂∂

+=∂∂

SChNc

SV

⇒ 0=∂∂

+SCh

Therefore, the hedge ratio is h SC ∂∂− , the value of SC ∂∂ is the

delta, denoted as if it is a call option, and denoted as if it is a

put option. Refer to the Black-Scholes formula, we find that .

So the delta of a call option is between 0 and 1, which indicates that the

number of stock shares we hold is between 0 and 1 while holding one

call option (

c∆ p∆

( )1dNc =∆

cNcNs ∆−= ). And in the portfolio, the number of stocks is less

than the number of options. We also notice there is a negative sign in the

formula of hedge ratio, which means we should hold a long (short)

position in stock while holding a short (long) position in a call option.

9

Suppose we have a call option with delta 4.0=∂∂=∆ SCc , in order to

hedge perfectly, a investor should have a short position in 100 call

options while having a long position in every 40 shares of stocks.

If the stock price drops 1, a call option drops 0.4. The loss from 40 stocks

(40x1=40) will be cancelled by the profit of shorting the call options

(0.4x100=40), and vice versa.

A portfolio with “ stocks + (-1) call option” would protect us from the

changes of stock price. The way of designing such a portfolio is called

the Delta Neutral Strategy. Here, -1 means short position, and is

always between 0 and 1. In the delta neutral portfolio, the number of call

options is always more than the number of stock shares.

c∆

c∆

Since the value of is dominated by stock price and it will change in

almost every second, in order to keep a delta neutral portfolio, we need

to regulate the positions by time to time. For instance, right now the

stock price is 100 with , if the stock price is 110 the next day, the

value of might turn to 0.5. Aiming to keep delta neutral, we should

buy 0.1 shares of stock. This process is called the Dynamic Hedging

strategy. When a short position in call option is hedged by a long position

in stocks, if the stock price goes down (up), then

c∆

4.0=∆c

c∆

c∆ turns down (up),

we need to sell (buy) some stocks to balance the hedged delta. This

method can also be summarized as “buy high sell low”, i.e. buy stocks

while stock price climbs and sell stocks while stock price drops.

As we know about the delta hedging based on the call option, the put

option works in a similar way. The delta of a put option is ,

which is between -1 and 0 (

1−∆=∆ cp

( ) 011 <−=∂∂=∆ dNSPp ). The value of the

10

portfolio ( )PhSNpNpPNsSV +=+= , so pNpNsh ∆−== . Since is

negative, if we have a long position in stocks ( ), and worry about

its price declining, we can realize the hedge by holding a long position in

put options ( ). In this case, we will offset the losses in stocks by

gaining profits in put options.

p∆

0>Ns

0>Np

Now is the time to summarize the Delta hedging strategy.

In the case of call option,

Portfolio A: one short call + c∆ long stocks

Portfolio B: one long call + c∆ short stocks

In the case of put option,

Portfolio A: one long put + p∆ long stocks

Portfolio B: one short put + p∆ short stocks

It is important to note that, the delta hedging only works well under the

condition of small changes in the stock prices. If the stock prices change

dramatically, the tangent of the stock-option curve cannot approximate

the option value correctly, which would cause a dangerous error.

2.3 Hedging Portfolio in Continuous Time

In the previous section, we mainly worked on the hedging strategy in a

discontinuous way. In this section, we will talk about the hedging issue in

a deeper point of view, to explore how it works in the continuous time.

We start with the definition of contingent claim. The book by Tomas Bjork

[4] is the main reference for this section. The common factor of these

11

contracts like European call or put options is that they are completely

defined in terms of the underlying asset S, which makes it natural to call

them derivative instrument or contingent claim. The formal definition of a

contingent claim is the following.

Consider a financial market with vector price process . A contingent

claim with date of maturity (exercise date) , also called a -claim, is

any stochastic variable

S

T T

χ . A contingent claim χ is called a simple claim if

it is of the form ( )( TSΦ= )χ . The function Φ is called contract function.

For instance, the European call is a simple contingent claim whose

contract function is given by ( ) [ ]0,max Kxx −=Φ , and the price for such a

claim is notated by ( )χ;tΠ .

Definition of hedging portfolio: We say that a T –claim χ can be

replicated, alternatively that it is reachable or hedgeable, if there exists a

self financing portfolio such that h ( ) χ=TV h . In this case we say that

is a hedge againsth χ . Alternatively, is called a replicating or

hedging portfolio.

h

It is noted that the value of hedging portfolio should be equal to the price

for the claim χ , otherwise the arbitrage opportunity exists. so

( ) ( )χ;___ ttVportfoliohedgingofValue Π==

On the other hand,

( )( ) ( )χ;,_____Pr ttStFttimeatcontractofice Π==

12

Therefore, we get ( ) ( )( )tStFtV ,=

Since is a hedging portfolio, we have to have V dFdV = .

In the Black-Scholes model

SdwSdtdS σα += ( )( )St,σσ =

rBdtdB =

We start from dF

( ) ( )22

2

21, dS

SFdS

SFdt

tFStdF

∂∂

+∂∂

+∂∂

=

( ) dtSFSSdwSdt

SFdt

tF

2

222

21

∂∂

++∂∂

+∂∂

= σσα

dwSFSdt

SFS

SFS

tF

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

= σσα 2

222

21

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

= dwF

SFS

dtF

SFS

SFS

tF

Fσσα 2

222

21

Now we consider dV

VofbondsinpartrelativeU B _____=

VofstocksinpartrelativeU S _____=

which indicates . 1=+ SB UU

⎟⎠⎞

⎜⎝⎛ +=

BdBU

SdSUVdV BS

( )( )rdtUdwdtUV BS ++= σα

( )( )dwUdtrUUV SBS σα ++=

13

Since and , we have FV = dFdV =

FSFS

U S∂∂

=σ

σ ⇒ F

SFS

U S∂∂

=

Now we set the drift equal:

FSFS

SFS

tF

rUU BS

2

222

21

∂∂

+∂∂

+∂∂

=+σα

α

⇒ rF

SFS

tF

U B

2

222

21

∂∂

+∂∂

=σ

.

Due to 1=+ SB UU , we have

121

2

222

=∂∂

+∂∂

+∂∂

=+rF

SFrS

rFSFS

tF

UU SB

σ

⇒ rFSFS

SFrS

tF

=∂∂

+∂∂

+∂∂

2

222

21σ

which is the Black-Scholes PDE. This should hold at any possible value

of underlying asset . S

In the explanation above, it is easy to find the weights of stocks and

bonds in the hedging portfolio. The relative weights are:

Stocks: F

SFS

U S∂∂

=

Bonds: F

SFSF

UU SB∂∂

−=−= 1

This also indicates the numbers of stocks and bonds we should invest in

14

the hedging portfolio

Number of bonds : Bh

FSFSF

FBhB ∂

∂−

=∗ ⇒

BSFSF

hB∂∂

−=

Number of stocks Sh

FSFS

FShS ∂

∂

=∗ ⇒

SFhS ∂∂

=

Apparently, SFhS ∂∂

= is the ∆ of the option, which is consistent with

what we discussed in the previous section. Delta indicates how many

stocks we should hold to hedge the option.

So far we have introduced what is going on with the Delta Hedging, from

a simple example to its inside principle, from discontinuous time to

continuous time. Indeed, Delta Hedging is a good tool and widely used

by traders in the real financial world.

15

Chapter 3 Static Hedging

Besides plain vanilla options (standard European call and put), a lot of

exotic options are traded in the market. Barrier options are one of the

most heavily traded exotic derivatives in the OTC market [5], since they

have lower prices than the plain vanilla options. Barrier options are

options where the payoff depends on whether the underlying asset price

reaches a certain level during a certain period of time. They can be

classified as either knock-out options or knock-in options. A knock-out

option ceases to exist when the underlying asset price reaches a certain

barrier, while a knock-in option comes into existence only when the

underlying asset price reaches a barrier. From the bank’s point of view,

an important issue becomes how to hedge these barrier options. Options

traders ordinarily hedge options by the Delta Neutral strategy, by

shorting the dynamic hedging portfolio against a long position in the

option to eliminate all the risk related to stock price movement. This

Delta hedging is also called as dynamic hedging, because we must

continuously adjust the weights in our portfolio according to the formula

as time passes and stock price moves.

Unfortunately, there are some drawbacks with hedging barrier options

under Delta Neutral method. Firstly, due to the character of the barrier

options, when the time is close to maturity T, the value of delta is very

large and changes rapidly especially in the neighborhood of the barrier.

Thus it is very difficult to delta hedge the options while its underlying

asset price is in the barrier region. Secondly, continuous weight

adjustment is impossible, and so traders adjust at discrete intervals. This

would cause small errors that compound over the life of the option, and it

would result in the portfolio’s accuracy problem. Thirdly, there are

16

transaction costs associated with adjusting the portfolio weights which

would grow with frequency of adjustment.

There is an alternative way to hedge the barrier options, which can also

solve the problems mentioned above. Instead of dynamic delta hedging,

we can use the Static Hedging method. In the static hedging portfolio,

given a particular target option, we can construct a portfolio of standard

options with varying strikes, maturities and fixed weights, which will not

require any further adjustment and will exactly replicate the value of the

target option. In this paper, we discuss two kinds of static hedging

strategy. One is to construct a hedging portfolio with standard options

with different maturities; the other one is to construct a hedging portfolio

with standard options with different strikes.

One very important issue should be clear here, when we talk about a

hedging portfolio, it is the same meaning as a replicating portfolio.

However, when we talk about how to replicate a barrier option, it means

approximately copying a barrier option; while when we hedge a barrier

option, the strategy is to take a negative position of the hedging portfolio

to offset the target barrier option. Therefore, the concept of ‘hedge’ is

negatively correlated with the concept of ‘replicate’.

3.1 Static Hedging with maturity-varying options

This method is to construct a hedging portfolio with standard options with

varying maturities. In general, such portfolio requires an infinite number

of options. Even so, a static hedging portfolio with only several options

can provide adequate replication over a wide range of future market

conditions. By increasing the number of options in the hedging portfolio

we can increase the accuracy of replication. Often, a fairly small portfolio

17

works adequately [6], and limits the transaction costs.

3.1.1 Intuitive approach towards static hedging

This approach is introduced by Derman,E. Ergener.D & Kani.I.[6]. The

method is trying to use standard options to replicate an up-and-out

European call option, described in the following table.

---------------------------------------------------------------------------------- An up-and-out European call option ---------------------------------------------------------------------------------- Stock price: 100 Strike: 100 Barrier: 120 Rebate: 0 Time to expiration: 1 year Dividend yield: 5.0% (annually compounded) Volatility: 25% per year Risk-free rate: 10.0% (annually compounded) ---------------------------------------------------------------------------------- Up-and-Out Call Value: 0.7482 Ordinary Call Value: 11.7320 ---------------------------------------------------------------------------------- Note that, the value of up-and-out call option and the value of ordinary

call option are calculated by Matlab.

In terms of the barrier option, there are two different classes of stock

price scenarios that determine the option’s payoff:

(1) The stock price does not hit the barrier before expiration. In this case,

the up-and-out call behaves totally the same as an ordinary call with

strike 100.

(2) The stock price does hit the barrier before expiration. Then the

up-and-out call is worthless with value of zero.

From a trader’s point of view, a long position in this up-and-out call is

equivalent to owning an ordinary call if the stock never hits the barrier,

18

and owning nothing otherwise. Let’s try to construct a portfolio of

ordinary options which behaves like this.

First we replicate the up-and-out call for scenarios in which the stock

price never reaches the barrier of 120 before expiration. In this case, the

up-and-out call has the same payoff as an ordinary one-year European

call with strike 100. We name this PORTFOLIO 1, as shown in the

following table. It replicates the target up-and-out call for all scenarios

which never hit the barrier before expiration.

PORTFOLIO 1 (the last two columns are option values)

Quantity Type Strike Expiration 1 year before T

Stock at 100

1 year before T

Stock at 120

1 call 100 1 year 11.7320 26.0711

The value of PORTFOLIO 1 at a stock level of 120 is 26.0711, which is a

big difference compared to the zero-value of an up-and-out call on the

barrier. Consequently, its value at a stock level of 100 is 11.7320, which

is also much greater that the value (0.7482) of the up-and-out call.

PORTFOLIO 1 replicates the target option for scenarios of type (1). By

adding a suitable short position into the PORTFOLIO 1, we can attain

the zero value for the hedging portfolio at one definite time on the barrier

at a stock price of 120. Let’s choose to do this at 1 year from expiration,

and aim to cancel the value of 26.0711 at stock price 120. Because the

value of a call with strike 120 and 1 year before T is 14.0784 when it is at

the stock level of 120, we need to short 852.10784.140711.26

= call options to

cancel out the value of 26.0711 to ensure that PORTFOLIO 2 will have

zero value on the 120 barrier. Following table is the PORTFOLIO 2.

19

PORTFOLIO 2 (the last two columns are option values)

Quantity Type Strike Expiration 1 year before T

Stock at 100

1 year before T

Stock at 120

1 call 100 1 year 11.7320 26.0711

-1.852 call 120 1 year -8.8559 -26.0711

Net 2.8761 0.0000

PORTFOLIO 2 consists of a single 1-year 100 strike standard call (which

is PORTFOLIO 1) plus a short position in 1.852 1-year calls with strike

120. Why do we use 120 strike call? Because the 120 strike call has no

payoff at expiration below the barrier, it will not damage the replication

for scenarios of type (1) which is already achieved by PORTFOLIO 1.

Any strike greater than 120 would achieve the same goal.

PORTFOLIO 2 replicates the value of the up-and-out call at expiration

below the barrier, and exactly on the 120 barrier at 1-year before

expiration. The table also tells us that, the value of the up-and-out call is

2.8761 when stock is 100, which is still larger than its value of 0.7482 at

the same market level. This difference reflects the fact that, the value of

PORTFOLIO 2 is zero on the barrier only at 1 year before expiration,

whereas the up-and-out call’s value is zero on the barrier at all times.

This idea tells us that, our next step is to find out how to make our

portfolio’s value approximately zero on the barrier at all times. The figure

below shows the value of PORTFOLIO 2 on the 120 barrier at all times

before expiration.

20

PORTFOLIO 3 in the following table illustrates an alternative hedging

portfolio. It adds to PORTFOLIO 1 a short position of one extra option so

as to get zero value for the hedging portfolio at a stock price of 120 with

0.5-year (6 months) to expiration, as well as for all stock prices below the

barrier at expiration.

PORTFOLIO 3 (the last two columns are option values)

Quantity Type Strike Expiration 0.5year before

T Stock at 100

0.5year before T

Stock at 120

1 call 100 1 year 8.0544 23.0179

-2.382 call 120 1 year -4.5444 -23.0179

Net 3.5100 0.0000

In order to cancel the value of 23.0179 at stock price 120, since the

value of a call with strike 120 and 0.5 year before T is 9.6652 when it is

at the stock level of 120, we need to short 382.26652.90179.23

= call options to

cancel out the value of 23.0179 to make sure its zero value on the 120

barrier. The following figure shows the value of PORTFOLIO 3 for stock

21

120 at all times before expiration. We also can see that the replication on

the barrier is good only at the point of 0.5 year (6 months). At all other

times, it fails to match the up-and-out call’s zero payoff.

By adding one more call to PORTFOLIO 3, we can construct a portfolio

to match the zero payoff of the up-and-out call at stock 120 at both 0.5

year and 1 year. This portfolio is PORTFOLIO 4 shown in the following

table.

PORTFOLIO 4 (the last two columns are option values)

Quantity Type Strike Expiration Stock at 120

0.5 year

T Stock at 120

1 year

1 call 100 1 year 23.0179 26.0711

-2.382 call 120 1 year -23.0179 -33.5347

0.772 call 120 0.5 year 0.0000 7.4636

Net 0.0000 0.0000

The value of PORTFOLIO 4 at the barrier at 120 for all times before

expiration is shown in the following figure. We can see that this portfolio

22

has a better match with the zero value of an up-and-out call on the

barrier. For times between 0.5 and 1 year before T, the boundary value

at stock 120 is quite close to zero.

By adding more options to the hedging portfolio, we can match the value

of target option at more points on the barrier. The figure following shows

the value of a portfolio with seven standard options at the stock level of

120 that matches the zero value of up-and-out at the barrier about every

two months (0.167 year). Obviously, the match between the target option

and our hedging portfolio on the barrier is much improved.

In principle, we can match the target option payoff at as many points on

the boundary as we like. The more points we match, the better the

23

hedging is. If we use an infinite number of options in the portfolio, we

could match the payoff everywhere on the boundary. Our hedging

portfolio would have exactly the same value as the target option at all

times and all stock prices, as long as interest rate, volatility and other

parameters stay unchanged. In practice, a small number of options in

the hedging portfolio would meet our requirement.

Recall the binomial tree model mentioned in the previous chapter, option

valuation is calculated by backward induction. It is the formula for

computing all earlier option values from the boundary option values

moving backwards down the tree. In the limit of infinitesimally small time

steps and stock moves, the backward equation becomes the Black

Scholes equation. The valuation depends only on the interest rate, the

current stock price, the stock volatility and the stock dividend yield. The

equation is same for all securities whose values are contingent on the

stock price. The only difference is they have different boundary

conditions. If two different portfolios have the same values everywhere

on the boundary, and produce the same cash flows inside this boundary,

the equation tells us that these two portfolios will have the same values

everywhere inside the boundary. This is the principle of static replication.

In summary, we can replicate a target security for all future stock prices

and times with some boundary by constructing a portfolio of securities

with the same cash flows within this boundary and the same values on

the boundary.

3.1.2 Static hedging interpreted by PDE

Continuing with the idea, this method can be employed for any model,

which reduces the problem of pricing a derivative to solving a single PDE

with certain boundary conditions. Considering a model with constant

interest rate and constant dividend yield, the dynamic stock price under

24

equivalent martingale measure is

( ) ( ) ttttt dWStSdtSdrdS ,σ+−= [1]

When ( ) σσ =tx, , it is the Black-Scholes model; when ( ) 1, −∗= γσσ xtx , it

is the constant elasticity of variance (CEV) model as well as the local

volatility model firstly suggested by Dupire (7).

Under the dynamic stock price model of the form [1], the no-arbitrage

PDE to obtain the price of derivative is given by:

( ) ( ) rfS

fSStSfSdr

tf

ttt

tt =

∂∂

+∂∂

−+∂∂

2

222 ,

21σ [2]

Let us define the linear partial differential operator L as follows:

( ) ( ) rS

SStS

Sdrt

Lt

ttt

t −∂∂

+∂∂

−+∂∂

= 2

222 ,

21σ

It is important that, a solution to such a PDE is uniquely determined by a

set of boundary conditions for . In our case, for an up-and-out call

option, the boundary conditions are:

f

( ) ( +−= KSSTf TT, ) for HST <

( ) 00, =tf for Tt ≤

( ) 0, =Htf for Tt ≤

Based on the superposition principle, let L and B be linear partial

differential operators. If satisfy the linear partial differential

equations

kuuu ,,, 21 K

( ) 0=iuL and the boundary conditions ( ) 0=iuB for

and if are any constants then

ki ,,1L=

kcc ,,1 K kkucucucu +++= K2211 satisfies

and ( ) 0=uL ( ) 0=uB .

25

Now suppose that we can evaluate the price of a call option which

satisfies the PDE [2]. That is, we can evaluate the function

at , such that

( )KSTtC t ,,,

( tSt , )

)( )( 0,,, =KSTtCL t for Tt ≤ and 0≥tS

( ) ( +−= KSKSTTC TT ,,, ) for 0>tS

( ) ttS SKSTtCt

=∞→ ,,,lim for Tt ≤

( ) 0,0,, =KTtC for Tt ≤

Given that can be evaluated; this can be used to construct a

portfolio of call options:

( KSTtC t ,,, )

)

)

( ) (∑=

=∏n

iitiiit KStCSt

1,,,, τα such that

( ) ( +−=∏ KSST TT, for HST <

( ) 00, =∏ t for [3] Tt ≤

( ) 0, =∏ Ht j for ( ) 1,,2,1, −=<< njTt jj Lτ

Since ( )( ) 0,,, =itii KStCL τ for ni ,,1K= from the superposition principle,

it is true that . Furthermore, by construction, the boundary

conditions for

( ) 0=∏L

∏ are as given in [3]. Note that, the first two conditions

are exactly the same as those for the up-and-out call option. The last

boundary condition only agrees with the third condition of the up-and-out

call at a finite number of points for jt 1,,2,1 −= nj K . As mentioned

earlier, a solution to a PDE is uniquely determined by a set of boundary

conditions. Therefore, as the number of points increases, jt ( )tSt,∏

26

will tend to be the value of the up-and-out call for all Tt ≤ and .

In other words, matching the value along the boundaries ensures that

the values inside the boundaries are the same. Hence the hedging

portfolio can be taken as the replication for the up-and-out call

option.

HSt ≤

∏

Based on this method, the static hedging strategy is illustrated in an

alternative way by Nalholm & Poulsen (9). Let ( )KSTtC t ,,, denote the

value of plain vanilla European call option at time , with initial stock

price , maturity T and strike

t

tS K . iα means the weight for

corresponding call option in our hedging portfolio.

Let us consider the following portfolio:

(a) long a plain vanilla European call with maturity and strike T K

(b) 3α of a call with maturity 3τ and strike where 3K

( ) ( 0,,,,,, 33333 ) =+ KHTtCKHtC τα ⇒( )( )333

33 ,,,

,,,KHtCKHTtC

τα −=

If , this portfolio has ( ) HtS =3 valuet −3 0. If we hold this portfolio to

, and the stock price ends below the barrier, its payoff is exactly the

call option.

T

(c) 2α of a call with maturity 2τ and strike where 2K

( ) ( ) ( ) 0,,,,,,,,, 233232222 =++ KHTtCKHtCKHtC τατα

If , this portfolio has ( ) HtS =2 valuet −2 0. If we hold the portfolio to

, its value is also 0 if 3t ( ) HtS =3 , because the 2α maturity 2τ strike

calls are worthless. And if we hold it to , its payoff is the call

option if stock price ends bellow the barrier.

2K T

27

(d) similarly: 1α of a call with maturity 1τ and strike 1K

( ) ( ) ( ) ( ) 0,,,,,,,,,,,, 1331322121111 =+++ KHTtCKHtCKHtCKHtC τατατα

(e) similarly: 0α of a call with maturity 0τ and strike 0K

( ) ( ) ( ) ( ) ( ) 0,,,,,,,,,,,,,,, 03303220211010000 =++++ KHTtCKHtCKHtCKHtCKHtC τατατατα

Note that ( )( )333

33 ,,,

,,,KHtCKHTtC

τα −=

Since the price of a call option decreases as the strike price increases,

we can see that the larger the chosen value for , the larger the

required weight

3K

3α will be. It would be best to choose as smallest

as possible, which must be

3K

H≥ mentioned in the payoff scenario 1 in

the previous section. This is the reason that, the strike price of the

options in hedging portfolio are usually, in theoretical work, is equal to

the barrier level. Therefore, we have HKi = .

In summary, we can rewrite the earlier equations in a matrix form:

( )( )( )( )

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−−

KHTtCKHTtCKHTtCKHTtC

,,,,,,,,,,,,

0

1

2

3

( )( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( ) ⎟⎟

⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

3

2

1

0

330220110000

331221111

332222

333

,,,,,,,,,,,,,,,,,,,,,0,,,,,,00,,,000

αααα

ττττττττττ

KHtCKHtCKHtCKHtCKHtCKHtCKHtCKHtCKHtCKHtC

The weights iα can be solved, since the PDE can give us the option

price accurately and efficiently. Therefore, we can construct ( KSTtC t ,,, )

28

a static hedging portfolio in this way. By increasing the number of

matched zero-value points on the barrier, we can replicate the barrier

option’s payoff for a large number of possible stock price paths; and the

accuracy of our portfolio will be improved.

3.1.3 Solving the PDE by finite difference methods

Since the static hedging problem can be reduced to solve a PDE, it is

very important to find out the efficient and precise option value

from the PDE. In this section, we introduce the backward

time finite difference method to achieve our goal.

( KSTtC t ,,, )

The no-arbitrage PDE is given by

( ) ( ) rfS

fSStSfSdr

tf

ttt

tt =

∂∂

+∂∂

−+∂∂

2

222 ,

21σ

Note that, while ( ) 1*, −= γσσ tt SSt 1=γ is the case of Black-Scholes

model.

The PDE turns to

( ) rfS

fSSfSdr

tf

tt

tt =

∂∂

+∂∂

−+∂∂

2

222

21 γσ

We transform the PDE into a backward time PDE,

Here, let ( ) ( tt StfSu ,, = )τ , where tT −=τ . So, we have

τ∂∂

−=∂∂ u

tf

, tt S

uSf

∂∂

=∂∂ , 2

2

2

2

tt Su

Sf

∂∂

=∂∂

Therefore, the PDE shifts to

( ) ruS

uSSuSdru

tt

tt =

∂∂

+∂∂

−+∂∂

− 2

222

21 γσ

τ

29

whose boundaries are given by , and minS maxS 0=τ . The area is

presented in the following graph

We subdivide the time axis into pieces, and space axis into

pieces. So

tN xN

tNTk ==∆τ and

xt N

SShS minmax −==∆

Now we can approximate the derivative of time and space as following

( ) ( ) ( )τ

ττττ

τ ∆∆−−

≈∂∂ tt

tSuSu

Su ,,,

( ) ( ) ( )t

ttttt

t SSSuSSu

SSu

∆∆−−∆+

≈∂∂

2,,

,ττ

τ

If mk=τ for some and tNm ,,1L= nhSt = for some then

this can be written as

xNn ,,1L=

( ) ( ) ( )k

nhkmkunhmkunhmku ,,, −−≈

∂∂τ

( ) ( ) ( )h

hnhmkuhnhmkunhmkSu

t 2,,, −−+

≈∂∂

30

If we denote ( )nhmku , as we can rewrite mnu

( )kuu

nhmku mn

mn

1

,−−

≈∂∂τ

( )huu

nhmkSu m

nmn

t 2, 11 −+ −

≈∂∂

Similarly, ( ) 211

2

2 2,

huuu

nhmkS

u mn

mn

mn

t

−+ +−≈

∂∂

Substitute these derivatives into the PDE ( tNm ,,1L= ), we obtain

( )( ) ( ) mn

mn

mn

mn

mn

mn

mn

mn ru

huuu

nhSnhSdrhuu

kuu

=+−

+++−−

+−

− −+−+−

2112

min2

min11

1 221

2γσ

⇓

( )( ) ( ) 12

11

1112

min2

min

11

11

1 221

2+

+−

+++

+−

++

+

=+−

++

⇓

+−−

+−

− mn

mn

mn

mn

mn

mn

mn

mn ru

huuu

nhSnhSdrhuu

kuu γσ

( ) ( ) ( ) ( ) −⎥⎦

⎤⎢⎣

⎡++++⎥

⎦

⎤⎢⎣

⎡+−+−= ++

=γγ σσ 2

min2

212

min2

2

min1

1 21

221 nhS

hkrkunhS

hk

hnhSkdruu m

nmn

mn

( ) ( ) ( ) ⎥⎦

⎤⎢⎣

⎡+++−− +

+γσ 2

min2

2

min1

1 221 nhS

hk

hnhSkdrum

n

Thinking about a fixed , the equation above tells us the value of option

at each share price level , where

m

nh xNn ,,1L= . Get all these equations

together, we are able to gain a matrix form as following.

31

( ) ( ) ( )

( ) ( )( ) ( )( ) ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎥⎦

⎤⎢⎣

⎡−++−+−−

⎥⎦

⎤⎢⎣

⎡+−+−

+

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+

+−

+−

+

+

+

−

−

γ

γ

σ

σ

2min2

2

min1

2min2

2

min1

0

11

12

13

12

11

1

2

3

2

1

122

11

0

00

221

*

hNxShk

hhNxSkdru

hShk

hhSkdru

uu

uuu

F

uu

uuu

mNx

m

mNx

mNx

m

m

m

mNx

mNx

m

m

m

MMM

in a short version, it turns to

11* ++ += mmm BUFU

Where , and mU 1+mU 1+mB are column vectors, and is a matrix. F

( ) 222

2

11 221 TD

hkTD

hkdrAF σ

−−−=

Here, A , , , , are five 1D 2D 1T 2T ( ) ( )11 −×− xx NN matrices.

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

++

++

+

=

rkrk

rkrk

rk

A

10000100

010000100001

LL

MOOOM

LL

( )( ) ⎟

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−+−+

++

+

=

hNShNS

hShS

hS

D

x

x

10000200

03000020000

min

min

min

min

min

1

LL

MOOOM

LL

( )( )

( )

( )( )( )( ) ⎟

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−+−+

++

+

=

γ

γ

γ

γ

γ

2min

2min

2min

2min

2min

2

10000200

03000020000

hNShNS

hShS

hS

D

x

x

LL

MOOOM

LL

32

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

−−

=

01001010

01010001010010

1

LL

MOOOM

LL

T

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

−−

−

=

21001210

01210001210012

2

LL

MOOOM

LL

T

Since we know the boundary values of ( )tSu ,τ , can be solved by

iteration. We still consider an up-and-out call with strike

tNU

K and barrier

level B . In this case, we have

( )( )( )

( )( )( )( ) ⎟

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−+−−+

−+−+−+

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

+

+

+

+

+

−

−

KhNSKhNS

KhSKhSKhS

uu

uuu

U

x

x

Nx

Nx

12

32

min

min

min

min

min

01

02

03

02

01

0

MM and

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

−

−

00

000

1max

2max

3max

2max

1max

maxMM

NtS

NtS

S

S

S

mS

uu

uuu

U

By using ( )111 +−+ −= mmm BUFU , finally we are able to get

( )( )( )

( )( )( )( )⎟

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

−

−

hNTuhNTu

hTuhTuhTu

uu

uuu

U

x

xNNx

NNx

N

N

N

N

t

t

t

t

t

t

1,2,

3,2,,

1

2

3

2

1

MM and ( ) ( )00 ,,0 STuSf =

Therefore, we can obtain the value of ( )KSTtC t ,,, by this method. Our

hedging portfolio can also be solved by the position matrix which is

mentioned in the earlier section.

3.1.4 Implementation and result

So far, we have described the approach of how to static hedge a barrier

33

option with maturity-varying options. Matlab is the main tool to help us

implement these ideas.

As we know, the convergence of the value is the most important issue

when we successfully run a Matlab code, which means the code is

acceptable and the result is stable and convinced.

For instance, we still consider a target up-and-out call with the following

parameters 350 =S , , 30=K 60=H , 2=T , 25.0=σ , and

. Fortunately, the result from Matlab shows us what we expect.

The following figure illustrates the value of our hedging portfolio against

the number of points along the barrier, which are matched between the

payoff of the hedging portfolio and the payoff of the barrier option.

1.0=r

0=d

It is clear that, the hedging portfolio is better replicating the barrier option

while the number of matched points increases. This is also consistent

with the principle discussed in the previous part. On the other hand, the

value of hedging portfolio converges to a constant value; this indicates

34

that the code is convinced and acceptable to make further investigation.

Continue with the example, we aim to static hedge the target up-and-out

call option. After running the code, it shows us the following portfolio

strategy (we take 10-matched-point case as an example).

Quantity Option

Type

Strike Expiration

(years)

Value

1.0000 Call 30 2.0000 11.2905

0.0681 Call 60 0.2000 0.0000

0.0891 Call 60 0.4000 0.0015

0.1199 Call 60 0.6000 0.0187

0.1676 Call 60 0.8000 0.0721

0.2469 Call 60 1.0000 0.1726

0.3915 Call 60 1.2000 0.3212

0.6955 Call 60 1.4000 0.5142

1.4964 Call 60 1.6000 0.7465

4.6893 Call 60 1.8000 1.0126

-9.3145 Call 60 2.0000 1.3074

By this strategy, we plot the value of static hedging portfolio as well as

the exact value of the up-and-out call option.

35

The figure makes us sure, our hedging portfolio can approximately

replicate the target barrier option. The more the matched points we have,

the better the hedging is. And the results also convince us, we can adopt

this method to construct a portfolio with different maturity options, in

order to achieve static hedging a target barrier option.

3.2 Static Hedging with strike-varying options

The method introduced in the previous section is to construct a hedging

portfolio with standard options with varying maturities. In this section, we

talk about another static hedging strategy which is to construct a portfolio

with standard options with varying strikes. This method is initially

suggested by Carr & Chou (10). The idea is to convert the problem of

replicating a barrier option to a problem of replicating a European

security, which turns out to have a non-linear payoff profile.

3.2.1 European security replication

In theory, any European security can be replicated by a combination of

calls, puts, forwards and bonds, which implies we can therefore replicate

the barrier option. Carr & Picron (11) proved this theory. During the

process of prove, two assumptions are mentioned. First, the pay off

function is twice differentiable. Second, there is no arbitrage and

markets are frictionless. The process is as follows, where is the

indicator function.

( TSf )

{}⋅I

( ) ( ) { } { }( ) ( ) ( ) { } { }( )KSIKSIkfkfKSIKSISfSf TTTTTT >+≤−+>+≤=

( ) { } ( ) ( )[ ] { } ( ) ( )[ ]kfSfKSISfkfKSIkf TTTT −>+−≤−=

( ) { } ( ) { } ( ) ⎥⎦⎤

⎢⎣⎡>+⎥⎦

⎤⎢⎣⎡≤−= ∫∫

T

T

S

kT

k

ST duufKSIduufKSIkf ''

36

( ) { } ( ) ( ) ( )[ ] { } ( ) ( ) ( )[ ] ⎥⎦⎤

⎢⎣⎡ −+>+⎥⎦

⎤⎢⎣⎡ −+≤−= ∫∫

T

T

S

kT

k

ST dukfufkfKSIdukfufkfKSIkf ''''''

( ) { } ( ) ( ) { } ( ) ( ) ⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡ +>+⎥⎦

⎤⎢⎣⎡

⎥⎦⎤

⎢⎣⎡ −≤−= ∫ ∫∫ ∫

T

T

S

k

u

kT

k

S

k

uT dudvvfkfKSIdudvvfkfKSIkf ''''''

( ) { } ( ) { } ( )∫∫ >+≤−= T

T

S

kT

k

ST dukfKSIdukfKSIkf ''

{ } ( ) { } ( )∫ ∫∫ ∫ >+≤+ T

T

S

k

u

kT

k

S

k

uT dvduvfKSIdvduvfKSI ''''

Since is not dependant on , the second and third terms can be

easily simplified to

( )kf ' u

{ } ( )( ) { } ( )( ) ( )( )KSkfKSkfKSIKSkfKSI TTTTT −=−>+−≤ '''

Next we change the order of integration for the fourth and fifth terms and

integrate with respect to , we get u

( ) ( ) ( )( )kSkfkfSf TT −+= '

{ } ( ) { } ( )∫ ∫∫ ∫ >+≤+ T T

T T

S

k

S

vT

k

S

v

ST dudvvfKSIdudvvfKSI ''''

( ) ( )( )kSkfkf T −+= '

{ } ( )( ) { } ( )( )∫∫ −>+−≤+ T

T

S

k TT

k

S TT dvvSvfKSIdvSvvfKSI ''''

Note that the third term in the last equation is equal to

{ } ( )( ) ( )( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

>

≤−=−≤ ∫∫KSifKSifdvSvvfdvSvvfKSI

T

T

k

S Tk

S TT TT 0

''''

( )( )( )( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

>−

≤−=∫∫

+

+

KSifdvSvvf

KSifdvSvvf

T

k

T

T

k

T

0

''0

''

( )( )∫ +−=k

T dvSvvf0

''

Similarly the forth term can be written as ( )( )∫∞ +−k T dvvSvf ''

37

( )TSf can therefore be simplified as

( ) ( ) ( )( ) ( )( ) ( )( )∫∫∞ ++ −+−+−+=k T

k

TTT dvvSvfdvSvvfkSkfkfSf ''

0

'''

Thus a European security’s payoff can be viewed as the payoff of a static

portfolio with zero-coupon bonds, ( )kf ( )kf ' long forwards and an

infinite number of put and call options

The value of such a portfolio at time is t

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )∫∫∞

++−+=k

k

Tt dvvTtCvfdvvTtPvfTtkBSkfTtBkfV ,,,,,, ''

0

'''

Where

( TtB , )

)

)

is the price of a zero-coupon bond at time with maturity T t

( vTtP ,, is the price of a put at time with maturity T and strike v t

( vTtC ,, is the price of a call at time with maturity T and strike t v

Therefore, the payoff of a European security can be replicated with

positions in zero-coupon bonds, forwards and vanilla European put and

call options. This is the start of this method.

3.2.2 Adjusted payoff function

Before start with adjusted payoff function, it is important to mention the

lemma introduced by Carr & Chou (5); this lemma is also called as

reflection theorem by Poulsen.R (12). The lemma is:

Under the assumptions of the Black-Scholes model, consider a

European security expiring at time with payoff T

( ) ( ) ( )⎭⎬⎫

⎩⎨⎧ ∈

=otherwise

BASSgSf TT

T 0,

1

Further, for , consider another European security with maturity 0>H T

38

and payoff

( )⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧⎟⎟⎠

⎞⎜⎜⎝

⎛∈⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

=otherwise

AH

BHS

SHg

HS

Sf TT

pT

T

0

,222

2

Where ( )2

21σ

qrp −−= and r , and q σ are the constant interest rate,

dividend yield and volatility, respectively.

Then, for any [ Tt ,0 ]∈ , the value of these two securities is equal, when

. Note, HSt = A or B can be or 0 ∞ .

Now we try to use this result to replicate a vanilla barrier option by

European security with specifically chosen payoff function. We still

consider the case of an up-and-out barrier option with payoff function

and the barrier level . ( TSg ) 0SH >

Consider a long position in the following two European securities with

payoff functions at time : T

( ) ( ) ( )⎭⎬⎫

⎩⎨⎧ ∈

=otherwise

HSSgSf TT

T 0,0

1 and

( ) ( )⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧∞∈⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−=

otherwise

HSSHg

HS

Sf TT

pT

T

0

,2

2

In terms of the stock price path, we have two scenarios:

(a)The stock price never cross the barrier over the life of these securities

(b)The stock price does cross the barrier at some time.

Under scenario (a), the payoffs at maturity are ( ) ( TT SgSf =1 ) and

39

( ) 02 =TSf . This gives the combined portfolio’s payoff exactly equal to the

payoff of the target barrier option.

Let’s move on to the scenario (b). Based on the lemma before, when the

stock price hit the barrier, the value of European security with payoff

is exactly the same as the value of a European security with the

payoff function of

( TSf2 )

( ) ( ) ( )⎭⎬⎫

⎩⎨⎧ ∈−

=otherwise

HSSgSf TT

T 0,0

3

Basically, at this time , we can sell the security with payoff

and buy the security with payoff

HSt = ( )TSf2

( )TSf3 with no cost. The new payoff of

the portfolio at maturity is

( ) ( ) ( ) ( ) ( )0

0,00

31 =⎭⎬⎫

⎩⎨⎧ ∈=−

=+otherwise

HSSgSgSfSf TTT

TT

This means that, the payoff of combined portfolio has the same payoff of

the target barrier option under scenario (b). This is a very good news.

Simply, the target up-and-out barrier option can be replicated by one

European security with the payoff function:

( ) ( ) ( )( ) ( )

( )⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∞∈⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−

∈=+= ,

,0: 2

21 HSSHg

HS

HSSgSfSfSf

TT

pT

TT

TTT [4]

The form of [4] is well known as the adjusted payoff function, which is

used to replicate a vanilla barrier option by a European security.

In summary, in order to static hedge a vanilla barrier option, all we need

to do is to statically replicate a European security providing the adjusted

payoff function. Furthermore, known from previous section, we can

statically replicate any European security.

40

3.2.3 Implementation

In this section, we will explain more how this method would be

implemented in practice. We still consider the same example of the

up-and-out call option which is used in the method of static hedging with

maturity-varying options.

The target barrier option is with the following parameters , ,

,

30=K 60=H

2=T 25.0=σ , and 1.0=r 0=d . In order to static hedge this barrier

option, the adjusted payoff function in this case is

( )( ) ( )

( )⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∞∈⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛−

∈−= +

+

,

,02

HSKSH

HS

HSKSSf

TT

pT

TT

T

Plotting the adjusted payoff function would help us better analyze the

case.

Recalling the conclusion in the previous section, a security with such a

payoff function can be made to replicate the up-and-out call. In other

41

words, hedging this payoff function is equivalent to hedging the barrier

option. Therefore, our aim is to hedge this adjusted payoff by using the

plain vanilla options, like calls or puts.

The following figure is helpful to explain how we can achieve our goal.

This payoff function is replicated in two parts.

Firstly, the adjusted payoff below the barrier is exactly replicated by a call

option with strike K and maturity T . Secondly, the upper part of the

function is more complicated, since it shows non-linearity in this region.

In general, we can replicate this non-linear part by a hedging portfolio

that includes a finite number of vanilla call or put options. No doubt, this

approximation becomes better and better if we increase the number of

points at which the hedging portfolio payoff is matched to the adjusted

payoff.

Let us consider the following portfolio:

(a) long a plain vanilla European call with strike K and maturity T .

42

(b) 4α of a call with maturity and strike where T 4K

( ) ( ) ( )44444 xfKxKx =−+− ++α

⇒ ( ) ( ) ( )++ −−=− KxxfKx 44444α

This is to match the point of ( )4xf between hedging portfolio payoff

and the adjusted payoff.

(c) 3α of a call with maturity and strike where T 3K

( ) ( ) ( ) ( )33333434 xfKxKxKx =−+−+− +++ αα

⇒ ( ) ( ) ( ) ( )+++ −−=−+− KxxfKxKx 33333434 αα

This is to match the point of ( )3xf .

(d) 2α of a call with maturity and strike where T 2K

( ) ( ) ( ) ( ) ( )22222323424 xfKxKxKxKx =−+−+−+− ++++ ααα

⇒ ( ) ( ) ( ) ( ) ( )++++ −−=−+−+− KxxfKxKxKx 22222323424 ααα

This is to match the point of ( )2xf .

(e) 1α of a call with maturity and strike where T 1K

( ) ( ) ( ) ( ) ( ) ( 11111212313414 xfKxKxKxKxKx =−+−+−+−+− +++++ αααα )

⇒ ( ) ( ) ( ) ( ) ( ) ( +++++ −−=−+−+−+− KxxfKxKxKxKx 11111212313414 αααα )

This is to match the point of ( )1xf .

In summary, we can get the matrix form as following

( ) ( )( ) ( )( ) ( )( ) ( )

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−−−−−−

+

+

+

+

KxxfKxxfKxxfKxxf

11

22

33

44

43

( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ⎟⎟

⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−−−−−

−−−

++++

+++

++

+

1

2

3

4

11213141

223242

3343

44

000000

αααα

KxKxKxKxKxKxKx

KxKxKx

Therefore, we can easily obtain the positions required for the calls so as

to match the adjusted payoff for final stock price of , , and . 4x 3x 2x 1x

Here, we have four matched points, if we use more points between

hedging portfolio payoff and adjusted payoff, better approximation will be

gained.

44

Chapter 4 Summary

In this project, we discuss some hedging strategies which are widely

used in the real financial world. We start from Delta Hedging that is the

most basic tool to hedge risks. However, our aim in this project is to

hedge more complicated options, for instance, barrier options. Since the

delta of a barrier option is extremely large around the barrier region,

Delta Hedging does not work in this case. Therefore we investigate

another hedging technique called Static Hedging. We describe how to

statically hedge a barrier option both in discrete and in continuous time.

In this paper, two different static hedging strategies are considered. One

is static hedging with maturity varying options; the other is static hedging

with strike varying options. We also show how to implement these

hedging techniques using computers. Furthermore, static hedging is

proved to be a good choice to hedge complicated barrier options.

45

References

1. John C. Hull, 2003. "Options, Futures, and Other Derivatives" 5th

editon, Prentice Hall

2. Matlab

3. F.Black and M.Scholes, 1973. "The Valuation of Options and

Corporate Liabilities"

4. Tomas Bjork, 2004. "Arbitrage Theory in Continuous Time" 2nd edition,

Oxford

5. Carr.P. & Chou.A., 1997b. "Hedging complex barrier options"

6. Derman.E. Ergener.D. & Kani.I., 1995. "Static options replication"

7. Dupire.B., 1994. "Pricing with a smile"

8. Lotter.G., 2004. "Computational differential equations"

9. Nalholm.M. & Poulsen.R., 2005. "Static replication and model risk:

Razor's edge or trader's hedge?"

10. Carr.P. & Chou.A., 1997a. "Breaking barriers"

11. Carr.P. & Picron.J., 1999. "Static hedging of timing risk"

12. Poulsen.R., 2006. "Barrier options and their static hedges: simple

46

derivations and extensions"

47