Post on 17-Apr-2020
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
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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.
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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
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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
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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.
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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
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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.
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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