Post on 04-Apr-2018
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U.U.D.M. Project Report 2009:4
Examensarbete i matematik, 30 hp
Handledare och examinator: Johan Tysk
Mars 2009
Structured products: Pricing, hedging andapplications for life insurance companies
Mohamed Osman Abdelghafour
Department of MathematicsUppsala University
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Acknowledgement
I would like to express my appreciation to Professor Johan Tysk my supervisor, not
only for his exceptional help on this project, but also for the courses (Financial
Mathematics and Financial Derivatives) that he taught which granted me the
understanding options theory and the necessary mathematical background to come writethis thesis.
I would also like to thank him because he is the one who introduced me to the Financial
Mathematics Master at the initial stage of my studies.
Also thanks to the rest of the professors in the Financial Mathematics and Financial
Economics Programme who provided instruction, encouragement and guidance,
I would like to say Thank you to you all. They did not only teach me how to learn, they
also taught me how to teach, and their excellence has always inspired me.
Finally, I would like to thank my Father, Ramadan for his financial support and
encouragement, my mother, and my wife Nellie who for their patience and continuous
support, when I was studying and writing this thesis.
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Introduction
Chapter 1 Financial derivatives
1.1 What is the structured product?
1.1.1 Equity-linked structured products
1.1.2 Capital-Guaranteed Products
1.2 Financial Derivative topics
1.21 Futures and Forward contracts pricing and hedging
1.2.2 The fundamental exposure types
1.2.3 European type Options
1.2.4 American type options
1.2.5 Bermudian Options
1.2.6 Asian option types
1.2.7 Cliquet options
Chapter 2 interest rate structured products
2.1 Floating Rate Notes (FRNs, Floaters)
2.2 Options on bonds
2.3 Interest Rate Caps and Floors
2.4 Interest rate swap (IRS)
2.5 European payer (receiver) swaption
2.6 Callable/Putable Zero Coupon Bonds2.7 Chapter 3 Structured Swaps
3.1 Variance swaps
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Chapter 1
Introduction
In recent years many investment products have emerged in the financial
markets and one of the most important products are so-called structured products.
Structured products involve a large range of investment products that combine many
types of investments into one product through the process of financial engineering.
Retail and institutional investors nowadays need to understand how to use such
products to manage risks and enhance their returns on their investment.
As structured products investment require some derivatives instruments knowledge.
The author will present some derivative introduction and topics that will be used in themain context of structured products .
Structured investment products are tailored, or packaged, to meet certain financial
objectives of investors. Typically, these products provide investors with capital
protection, income generation and/or the opportunity to generate capital growth.
So the author will present the use of such products and their payoff and analyse the use
of different strategies.
In fact, those products can be considered ready-made investment strategy available for
investors so the investor will save time and effort to establish such complex investmentstrategies.
In the pricing models and hedging, the author will tackle mainly the basic models of
underlying equities and interest rate derivatives and he will give some pricing examples.
Structured products tend to involve periodical interest payments and redemption (which
might not be protected).
A part of the interest payment is used to buy the derivatives part. What sets them apart
from bonds is that both interest payments and redemption amounts depend in a rathercomplicated fashion on the movements of for example basket of assets, basket of
indices exchange rates or future interest rates.
Since structured products are made up of simpler components, I usually break them
down into their integral parts when I need to value them or assess their risk profile and
any hedging strategies.
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This approach should facilitate the analysis and pricing of the individual components.
For many product groups, no uniform naming conventions have evolved yet, and even
where such conventions exist, some issuers will still use alternative names. I use the
market names for products which are common; at the same time, I try to be as accurateas possible. Commonly used alternative names are also indicated in each products
description.
1.1 What are structured products?
Definition: Structured products are investment instruments that combine at least onederivative contract with underlying assets such as equity and fixed-income securities.
The value of the derivative may depend on one or several underlying assets.
Furthermore, unlike a portfolio with the same constituents the structured product is
usually wrapped in a legally compliant, ready-to-invest format and in this sense it is a
packaged portfolio.
Structured investments have been part of diversified portfolios in Europe and Asia for
many years, while the basic concept for these products originated in the United States in
the 1980s.
Structured investments 'compete' with a range of alternative investment vehicles,
such as individual securities, mutual funds, ETFs (exchange traded fund) and
closed-end funds.
The recent growth of these instruments is due to innovative features, better pricing and
improved liquidity.
The idea behind a structured investment is simple: to create an investment product that
combines some of the best features of equity and fixed income namely upside potential
with downside protection.
This is accomplished by creating a "basket" of investments that can include bonds, CDs,
equities, commodities, currencies, real estate investment trusts, and derivative products.
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This mix of investments in the basket determines its potential upside, as well as
downside protection.
The usual components of a structured product are a zero-coupon bond component and
an option component.
The payout from the option can be in the form of a fixed or variable coupon, or can bepaid out during the lifetime of the product or at maturity.
The zero-coupon bond component serves as buffer for yield-enhancement strategies
which profit from actively accepting risk.
Therefore, the investor cannot suffer a loss higher than the note, but may lose significant
part of it.
The zero-coupon bond component is a floor for the capital-protected products.Other products, in particular various dynamic investment strategies, adjust the
proportion of the zero-coupon bond over time depending on a predetermined rule.
1.1.1 Equity-linked structured products
The classification refers to the implicit option components of the product.
In a first step, I distinguish between products with plain vanilla and those with exotic
options components.
While in a second step, exotic products can be uniquely identified and named, a similar
differentiation within the group of plain-vanilla products is not possible.
Their payment profiles can be replicated by one or more plain-vanilla options,
whereby the option types (call or put) and position (long or short) is product-specific.
Therefore, I assign terms to some products that best characterize their payment
profiles.
A classic structured product has the basic characteristics of a bond. As a special-
feature, the issuer has the right to redeem it at maturity either by repayment of its-
nominal value or delivery of a previously fixed number of specified shares.
Most structured products can be divided into two basic types: with and without coupon
payments generally referred to as reverse convertibles and discount certificates.
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In order to value structured products, I decompose them by means of duplication,
i.e., the reconstruction of product payment profiles through several single components.
Thereby, I ignore transactions costs and market frictions, e.g., tax influences.
1.1.2 Capital-Guaranteed Products
Capital-guaranteed products have three distinguishing characteristics:
Redemption at a minimum guaranteed percentage of the face value (redemption-
at face value (100%) is frequently guaranteed). No or low nominal interest rates.
Participation in the performance of underlying assets
The products are typically constructed in such a way that the issue price is as close as
possible to the bonds face value (with adjustment by means of the nominal interest
rate).
It is also common that no payments (including coupons) are made until the products
maturity date.
The investors participation in the performance of the underlying asset can take an
extremely wide variety of forms.
In the simplest variant, the redemption amount is determined as the product of the face
value- and the percentage change in the underlying assets price during the term of the
product.
If this value is lower than the guaranteed redemption amount; the instrument is
redeemed at
the guaranteed amount.
This can also be expressed as the following formula:
R=N(1+max(0,ST-S0))
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S0
=N + N . max(0,ST-S0))
S0
where
R: redemption amount
N: face value
S0 : original price of underlying asset
ST : Price of underlying asset at maturity.
Therefore, these products have a number of European call options on the underlying
asset embedded in them.
The number of options is equal to the face value divided by the initial price (cf. the last
term in the formula).
The instrument can thus, be interpreted as a portfolio of zero coupon bonds (redemption
amount and coupons) and European call options.
The possible range of capital-guaranteed products comprises combinations of zero
coupon bonds with all conceivable types of options.
This means that the number of different products is huge.
The most important characteristics for classifying these products are as follows:
(1) Is the bonus return (bonus, interest) proportionate to the performance of
the underlying asset (like call and put options), or does it have a fixed value
once a certain performance level is reached (like binary barrier options)?
(2) Are the strike prices or barriers known on the date of issue?
Are they calculated as in Asian options or in forward start options?
(3) What are the characteristics of the underlying asset? Is it an individual stock,
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an index or a basket?
(4) Is the currency of the structured product different from that of the underlying
asset?
In the sections that follow, a small but useful selection of products is presented.
As there are no uniform names for these products, they are named after the
options embedded in them .
1.2 Derivative introduction and topics
Derivatives are those financial instruments whose values derive from price of theunderlying assets e.g. bonds, stocks, metals and energy.
The derivatives are traded in two main markets: ETM and OTC.
1) The Exchange traded market is a market where individuals trade standardized
derivative contracts.
Investment assets are assets held by significant numbers of people purely for
investment purposes (examples: bonds ,stocks )
2) Over the counter (OTC) is the important alternative to ETM. It is telephone and
computer linked network of dealers ,who do not physically meet.
This market became larger than ETM and structured product are traded in the OTC
market although this market has a huge number of tailored derivative contract.
One of the disadvantages of the OTC markets is that such markets suffer from great
exposure to credit risk.
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1.2.1 Futures and Forward contracts pricing and hedging
Forward contracts are particularly simple derivatives.
It is an agreement to buy or to sell an asset at certain time T for a certain price K.
The pay-off is (ST - K) for long position and (K - ST) for short position .
A future price K is delivery price in a forward contract which is updated daily and F0is
forward price that would apply to the contract today.
The value of a long forward contract, , is =(F0K)erT
Similarly, the value of a short forward contract is (K F0) erT
1 Forward and futures prices are usually assumed the same.
2 When interest rates are uncertain they are, in theory, slightly different:
3 A strong positive correlation between interest rates and the asset price implies the
futures price is slightly higher than the forward price
4 A strong negative correlation implies the reverse
Futures contracts is standardized forward contact and traded in exchange markets for
futures.
Settlement price: the price just before the final bell each day
Open interest: the total number of contracts outstanding Ways Derivatives are used
To hedge risks To speculate (take a view on the future direction of the market) To lock in an arbitrage profit To change the nature of a liability and creating synthetic liability and assets To change the nature of an investment and change the exposure to assets status
without incurring the costs of selling.
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Now I will introduce some important hedging and trading strategies that Structured
product depend on.
Short selling
involves selling securities you do not own. Your broker borrows
the securities from another client and sells them in the market in the usual way, at somestage you must buy the securities back so they can be replaced in the account of the
client. You must pay dividends and other benefits the owner of the securities.by
Other Key Points about Futures
1 They are settled daily
2 Closing out a futures position involves entering into an offsetting trade
3 Most contracts are closed out before maturity
If a contract is not closed out before maturity, it usually settled by delivering the assets
underlying the contract.
$100 received at time T discounts to $100e-RT at time zero when the
continuously compounded discount rate is r
If r is compounded annually
F0=S0 (1 +r )T
(Assuming no storage costs)
If r is compounded continuously instead of annually
F0=S0erT
For any investment asset that provides no income and has no storage costswhen an investment asset provides a known yield q
F0=S0e(rq )T
where q is the average yield during the life of the contract (expressed with Continuous
compounding)
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Valuing a Forward Contract
assume that stock index that pays dividends income on the index the payment is fixed
and known in advance.
1 Can be viewed as an investment asset paying a dividend yield
2 The futures price and spot price relationship is therefore
F0=S0e(rq )T
where q is the dividend yield on the portfolio represented by the index
For the formula to be true it is important that the index represent an investment asset.In other words, changes in the index must correspond to changes in the value of a
tradable portfolio.
Index Arbitrage
When F0>S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures
When F0
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How to hedge using futures
A proportion of the exposure that should optimally be hedged is
h= * (S/ F)
where S is the standard deviation of dS, the change in the spot price during the hedging
period, F is the standard deviation of dF, the change in the futures price during the
hedging period is the coefficient of correlation between dS and dF.
To hedge the risk in a portfolio the number of contracts that should be shorted is where P
is the value of the portfolio, is its beta, and A is the value of the assets.
In practice regression techniques are employed to hedge equity option by using equity
index futures (the author is working in this field).
This technique implemented also in dynamic hedging strategies.
1.2.2 The fundamental exposure types
The fundamental exposure types are the generic option payoffs.
Combining these with a long zero coupon bond gives the primal structured products,
some of which have not failed to go out of fashion.
The following Figure shows clearly the interaction between investment view and payoff .
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1.2.3 European type Options
Let the price process of the underlying asset beS (t),t[0,T].
European optionsgive the holder the right to exercise the option only on the expiration dateT .
Hence the holder receives the amount (S(T)), whereis a contract function.
Moreover, there are two basic types ofEuropean optionsnamely European call options
and European put options.
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European Call option: a derivative contract that gives its holder the right to buy the
underlying assets by certain date at certain strike price.
European Put option: a derivative contract that gives its holder the right to sell the underlying assets
by certain date at certain strike price.
Black and Scholes derived a boundary value partial differential equation (PDE) for the value F(t, s) of
an option on a stock.
Pricing of European option
This value F(t , s) solves the Black&Scholes PDE Under risk neutral measure for one underlying asset
only.
)(),(
0),(
),(
2
1),(),( 222
sstF
str Fs
stF
Ss
stF
r St
stF
==
+
+
in [0 T ]R+. Here r is the interest rate; is the volatility of the underlying assumed fixed parameters.
Asset S and(s) =max(sk ,0) is the contract function. According to the Feynman-Kac theorem PDE
solution can represented as an expected value
F(t,s)=er(T-t) [ ]),(, TsE st
where the underlying stock S(t ) follows the dynamics
s(u)=r s(u) u+s(u) (u,s(u)) W(u)
This price process is called geometric Brownian motion. Here W is a Wiener process
where S starts in s at time 0.
For the purpose of option pricing I thus should assume that the underlying stock follows
this dynamics even if in reality we do not expect the value of the stock to grow with theinterest rate r.
The American version of those two options is the same except that it can be exercised
earlier than exercise date.
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1.2.4 An American option
gives the owner the right to exercise the option on or before the Expiration date tT
before the expiration, date (also called early exercise).
The holder of anAmerican optionneeds to decide whether to exercise immediately or to wait.
If the holder decides to exercise at saytT, then he receives (S(t)) where is the appropriate
contract function.
Similarly, this option can also be classified into two basic types:
American call optionswhich give the owner the right to buy an underlying asset for agiven strike price on or before the expiration date, and American put option which gives
the owner the right to sell an underlying asset for a certain strike price on or before the
expiration date.
If the underlying stock pays no dividends, early exercise of an American call option is not
optimal.
On the other hand early exercise of an American put option can be optimal even if the
underlying stock does not pay dividends.
An American option is worth at least as much as an European option. To compare by
examples here are two examples how the two prices compares
For example
Prices of the following options long plain vanilla call option non dividend share for 3
months to expiry date option the two price functions (European and American plain
vanilla option) are plotted here for the same
strikes of 100
current share price 120
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Risk free rate of 10 %
Volatility of 40.
Figure 1.1 is showing the price function of European option using Black and Scholes
formula .
Figure 1.2 is showing the price function of the American option using Bjerksund &
Stensland approximation.for more details about this approximation see the Bjerksund & Stensland
approximation 2002.
The table used to generate the 3 d graph for the American option using Bjerksund approximation
& Stensland approximation.
Time to maturity days
et price 10.00 30.88 51.76 72.65 93.53 114.41 135.29 156.18 177.06
150.00 50.2736 50.8432 51.4228 52.0323 52.6754 53.3462 54.0368 54.7405 55.4517
145.00 45.2736 45.8445 46.4380 47.0762 47.7554 48.4640 49.1912 49.9288 50.6712
140.00 40.2736 40.8484 41.4678 42.1490 42.8763 43.6316 44.4018 45.1780 45.9544
135.00 35.2736 35.8592 36.5246 37.2678 38.0568 38.8680 39.6871 40.5056 41.3184
130.00 30.2737 30.8871 31.6301 32.4580 33.3226 34.1978 35.0704 35.9335 36.7836
125.00 25.2742 25.9552 26.8190 27.7556 28.7079 29.6527 30.5807 31.4882 32.3744
120.00 20.2792 21.1106 22.1456 23.2106 24.2569 25.2717 26.2528 27.2013 28.1194
115.00 15.3132 16.4396 17.6873 18.8874 20.0238 21.1015 22.1277 23.1092 24.0516
110.00 10.4857 12.0799 13.5459 14.8645 16.0723 17.1955 18.2514 19.2524 20.2072
105.00 6.1194 8.2180 9.8410 11.2294 12.4717 13.6114 14.6736 15.6747 16.6255
100.00 2.7763 5.0530 6.6920 8.0687 9.2905 10.4065 11.4440 12.4201 13.3462
95.00 0.8696 2.7262 4.1907 5.4529 6.5875 7.6319 8.6080 9.5299 10.4073
90.00 0.1638 1.2453 2.3693 3.4191 4.4001 5.3241 6.2009 7.0380 7.8413
85.00 0.0159 0.4614 1.1806 1.9564 2.7338 3.4970 4.2412 4.9657 5.6711
80.00 0.0007 0.1318 0.5035 1.0009 1.5555 2.1355 2.7253 3.3167 3.905575.00 0.0000 0.0273 0.1774 0.4466 0.7951 1.1937 1.6237 2.0735 2.5355
70.00 0.0000 0.0038 0.0494 0.1684 0.3564 0.5989 0.8823 1.1961 1.5325
65.00 0.0000 0.0003 0.0103 0.0516 0.1359 0.2631 0.4283 0.6253 0.8487
60.00 0.0000 0.0000 0.0015 0.0122 0.0424 0.0981 0.1807 0.2894 0.4218
55.00 0.0000 0.0000 0.0001 0.0021 0.0103 0.0297 0.0640 0.1150 0.1832
50.00 0.0000 0.0000 0.0000 0.0002 0.0018 0.0069 0.0182 0.0377 0.0671
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Figure 1.1 European call Figure 1.2 American call Bjerksund
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A Trinomial tree has been set up for the American option in case of the American option.
A 500 steps trinomial tree is constructed with matrix of underlying price is as follows.
The following diagram shows how the first node is calculated also I will mention here
how we calculate the relevant probabilities of up and down probabilities and here is part
of algorithm
dt is the time step
n is number of steps
v is the volatility
pu is the up probability
Pd is the down probability
dt =T / n
u =Exp(v * Sqr(2 * dt))
d =1 / u
pu =(Exp(r * dt / 2) - Exp(-v * Sqr(dt / 2))) 2 / (Exp(v * Sqr(dt / 2)) - Exp(-v * Sqr(dt / 2))) 2
pd =(Exp(v * Sqr(dt / 2)) - Exp(r * dt / 2)) 2 / (Exp(v * Sqr(dt / 2)) - Exp(-v * Sqr(dt / 2))) 2
pm =1 - pu pd
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Calculations oftable used togenerate 3-Dgraph
Time tomaturityin days
Assetpr ice
10.0
0
30.8
8
51.7
6
72.6
5
93.5
3
114.
41
135.
29
156
.18
177.
06
218.
82
239.
71
260.
59
281.
47
302.
35
323.
24
150.00
50.1369
50.4222
50.7070
50.9944
51.2892
51.5945
51.9118
52.240
252.5795
53.2823
53.6432
54.0076
54.3775
54.7508
55.1223
145.0045.1369
45.4222
45.7080
46.0003
46.3059
46.6269
46.9632
47.
3125
47.6731
48.4194
48.8011
49.1892
49.5774
49.9679
50.3604
140.00
40.1369
40.4223
40.7109
41.0139
41.3380
41.6833
42.0467
42.423
842.8114
43.6126
44.0208
44.4286
44.8404
45.2512
45.6630
135.00
35.1369
35.4228
35.7195
36.0437
36.3985
36.7787
37.1799
37.592
538.0153
38.8803
39.3138
39.7493
40.1842
40.6174
41.0462
130.00
30.1369
30.4252
30.7427
31.1069
31.5107
31.9414
32.3874
32.845
433.3088
34.2449
34.7064
35.1730
35.6292
36.0867
36.5394
125.00
25.1
369
25.4
361
25.8
025
26.2
357
26.7
093
27.2
038
27.7
110
28.220
6
28.7
273
29.7
378
30.2
338
30.7
226
31.2
095
31.6
831
32.1
547
120.00
20.1370
20.4775
20.9442
21.4825
22.0492
22.6213
23.1932
23.760
324.3155
25.3976
25.9250
26.4351
26.9461
27.4448
27.9303
115.00
15.1404
15.6142
16.2555
16.9340
17.6075
18.2652
18.9042
19.517
820.1226
21.2695
21.8218
22.3553
22.8769
23.3940
23.8969
110.00
10.1877
10.9996
11.8786
12.7056
13.4827
14.2193
14.9112
15.575
016.2035
17.4033
17.9707
18.5184
19.0487
19.5710
20.0822
105.00
5.5516
6.9085
8.0059
8.9513
9.8037
10.5811
11.3015
11.988
512.6383
13.8499
14.4199
14.9701
15.5027
16.0197
16.5279
100.00
2.04
77
3.68
63
4.84
87
5.81
61
6.66
89
7.44
38
8.16
12
8.8
337
9.47
00
10.6
569
11.2
155
11.7
547
12.2
767
12.7
835
13.2
763
95.00
0.3991
1.5785
2.5587
3.4167
4.1870
4.8985
5.5580
6.1817
6.7796
7.8962
8.4223
8.9305
9.4228
9.9010
10.3663
90.00
0.0308
0.5040
1.1331
1.7640
2.3747
2.9562
3.5230
4.0567
4.5838
5.5704
6.0468
6.5119
6.9629
7.4012
7.8281
85.00
0.0007
0.1103
0.4000
0.7775
1.1866
1.6133
2.0399
2.4694
2.8924
3.7181
4.1253
4.5195
4.9093
5.2991
5.6791
80.00
0.0000
0.0151
0.1071
0.2810
0.5105
0.7748
1.0644
1.3671
1.6764
2.3173
2.6393
2.9601
3.2849
3.6015
3.9208
75.00
0.0000
0.0011
0.0204
0.0791
0.1809
0.3186
0.4862
0.6751
0.8812
1.3275
1.5648
1.8082
2.0539
2.3044
2.5587
70.00
0.0000
0.0000
0.0025
0.0166
0.0509
0.1085
0.1884
0.2895
0.4069
0.6876
0.8464
1.0126
1.1856
1.3687
1.5526
65.000.00
000.00
000.00
020.00
240.01
070.02
910.06
010.1045
0.1615
0.3148
0.4074
0.5085
0.6217
0.7413
0.8649
0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.0 0.05 0.12 0.17 0.22 0.28 0.35 0.43
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10.00
93.53
177.06
260.59
150
145
140
135
130
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
0
10
20
30
40
50
60
Time to maturity
Ass et price
As we can see here that the trinomial method is value the American option than the
approximation but it will converge as the number of steps increase.
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1.2.5 Bermudan Option
This type of options lies between American and European. They can be exercised at
certain discrete time points for any discrete time t
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A further breakdown of these options concludes that Asians are either based on the average
price of the underlying asset, or alternatively, there is the average strike type.
The payoff of geometric Asian options is given as:
PayoffAsian call =max
=
XS i
nn
i
/1
1
,0
PayoffAsian put=max
=
nn
i
S iX
/1
1
,0
Kemna & Vorst (1990) put forward a closed form pricing solution to geometric averaging
options by altering the volatility, and cost of carry term.
Geometric averaging options can be priced via a closed form analytic solution because of the
reason that the geometric average of the underlying prices follows a lognormal distribution as
well, whereas with arithmetic average rate options, this condition collapses.
The solutions to the geometric averaging Asian call and puts are given as:
CG=Se(b-r)(T-t)N(d1)-X e
-r(T-t)N(d2)
and,
PG=X e-r(T-t)N(-d2)- Se
(b-r)(T-t)N(-d1)
where N(x) is the cumulative normal distribution function of:
d1=ln(S/X)+(b+0.52
A )T
A T
d2=ln(S/X)+(b-0.52
A )T
A T
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The adjusted volatility and dividend yield are given as:
A = / 3
b=1/2(r-D-2
/6)
The payoff of arithmetic Asian options is given as
PayoffAsian call =max(0,(=
n
i
Si1
/n)-X)
PayoffAsian put=max(0,X-(=
n
i
Si1
/n)
Here I will mention one of the approximations to calculate the price of a structured product that
has an Asian structured product .
1) The zero coupon bonds parts are valuated using the relevant spot interest rates.
2)The Asian option for which payments are based on a geometric average are relatively easy
approximations have been developed by Turnbull and Wakeman (1991),Levy (1992) and Curran (1992).
In Currans model, the value Of an Asian option can be approximated using the following
formula:
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Here is an example of capital guaranteed structured product that has Asian pay off.
On the FTSE 100 index using Currans model.
Average calculated quarterly and the interest rate used are annual compoundedand volatility is used are annual rate. The main parameters used are as follows
Asset price ( S ) 95.00
Average so far ( SA ) 100.00
Strike price ( X ) 100.00
Time to next average
point (t1) 0.25
Time to maturity ( T ) 5.00
Number of fixings n 4.00Number of fixings fixed
m 0.00
Risk-free rate ( r ) 4.50%
Cost of carry ( b ) 2.00%
Volatility ( ) 26.00%
Value 10.7396
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10.00
114.41
218.82
323.24
200.00
185.00
170.00
155.00
140.00
125.00
110.00
95.00
80.00
65.00
50.00
0.0000
20.0000
40.0000
60.0000
80.0000
100.0000
120.0000
Time to maturity
Ass et price
The frequency with which the value of the underlying asset is sampled varies widely from product to
product.
The averages are usually calculated using daily, weekly or monthly values.
Depending on whether an Asian call or put option is embedded, the redemption amount is
calculated using one of the following formulas:
=Zero coupon bond +Asian option value .
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1.2.7 Cliquet options
Cliquet are option contracts, which provide a guaranteed minimum annual return in
exchange for capping the maximum return earned each year over the life of the contract.
Applications:
Recent turmoil in financial markets has led to a demand for products that reduce risk
while still offering upside potential.
For example, pension plans have been looking at attaching Guarantees to their products
that are linked to equity returns.
Some plans, also in VA life products such as those described.
Pricing Cliquet options
The Pricing framework here will be in the deterministic volatility model .
Cliquet options are essentially a series of forward-starting at-the-money options with a
single premium determined up front, that lock in any gains on specific dates.
The strike price is then reset at the new level of the underlying asset.
I will use the following form, considering a global cap, global floor and local caps at pre-
defined resetting times ti (i =1, . . . , n).
P=exp(-rtn)N.EQ
=
CF
S
SS iCF
n
i i
i
ii,,m i nm a xm a xm i n
1 1
1,
where N is the notional, C is the global cap, F is the global floor, Fi, i =1. . . n the local f
floors, Ci, i =1, . . . , n are the local caps, and S is the asset price following a geometric
Brownian motion, or a jump-diffusion process.
Under geometric Brownian motion with only fixed deterministic annual rate of interest
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I can use the binomial method (CRR) binomial tree to price Cliquet option .
This binomial cliquet option valuation model which maintains the important property of
flexibility, can be used to price European and American cliquets.
The settings for this model are the same as those described in the previous section:
I have the Cox-Ross-Rubinstein (CRR) binomial tree with
U=e t and D =e- t
The adjusted risk-neutral probability for the up state is
P = e t -D
U-D
In addition (1-p) for the downstate probability.
This time, instead of calculating the probability of each payoff, I use the backward valuation approach
described in Hull (2003), Haug (1997)), adjusting it to Cliquet options with no cap or floor applied.
The adjustment is as follows:
For each node that falls under the reset date m, the new strike price is determined.
If the stock price at m is above the original strike, the put will reset its strike price equal to the then-current stock price.
For call options: if the stock price m is below the original strike, the call will reset its strike price equal
to the then-current stock price.
Pricing example
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Current stock price =100
Exercise price =100
Time to maturity =20 year
Time to reset =10 year
Risk-free interest rate =4,5%
Dividend yield =2%
Sigma =20%.
In addition, here is comparison between plan vanilla European call and European Cliquet optionprices for various stock prices
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0
10
20
30
40
50
60
70
80
90
100
110
50.00 70.00 90.00 110.00 130.00 150.00 170.00 190.00 210.00 230.00 250.00
cliquet price
Plan vanila CRR
And here is comparison between plan vanilla American call and European Cliquet option prices
for various stock prices
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
50.00 70.00 90.00 110.00 130.00 150.00 170.00 190.00 210.00 230.00 250.00
cliquet price
CRR vanilla
As you can see from both charts that the price is different only when the stock price is less than 100
strike price for both the American and European option .
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Chapter 2 interest rate structured products
2.1.1 Floating Rate Notes (FRNs, Floaters)
Floating rate notes does not carry a fixed nominal interest rate.The coupon payments are linked to the movement in a reference interest rate (frequently money
market rates, such as the LIBOR) to which they are adjusted at specific intervals, typically on each
coupon date for the next coupon period.
A typical product could have the following features:
The initial coupon payment to become due in six-months time corresponds to the 6-month LIBOR as
at the issue date. After six months the first coupon is paid out and the second coupon payment is
locked in at the then current 6-month LIBOR. This procedure is repeated every six months.
The coupon of an FRN is frequently defined as the sum of the reference interest rate and a spread of
x basis points. As they are regularly adjusted to the prevailing money market rates, the volatility of
floating rate notes is very low.
Replication
Floating rate notes may be viewed as zero coupon bonds with a face value equating the sum of the
forthcoming coupon payment and the principal of the FRN. Because their regular interest rate
adjustments guarantee interest payments in line with market condition.
2.2 Options on bonds
Bond options are an example for derivatives depending indirectly (through price movements of the
underlying bond) on the development of interest rates.
It is common to embed bond options into particular bonds when they are issued to make
them more attractive to potential purchasers.
A callable bond, for example, allows the issuing party to buy back the bond at a
predetermined price in the future.
A putable bond, on the other hand, allows the holder to sell the bond back to the issuer at a certain
future time for a specified price.
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Pricing bond options
The well-known Black-Scholes equation was derived for the pricing of options on stock
prices and it was published in 1973 .
Shortly afterwards, the model has been extended to account for the valuation of optionson commodity contracts such as forward contracts.
In general, this model describes relations for any variable, which is log normally distributed and can
therefore be used for options on interest rates as well.
The main assumption of the Black model for the pricing of options on bonds is that
at time T the value of the underlying asset VT follows a lognormal distribution with the
Standard deviation.
S[ln VT]= T .
Furthermore, the expected value of the underlying at time T must be equal to its forward
price for a contract with maturity T, since otherwise, arbitrage would be possible.
E[VT]=F0
E[max(V-K),0]=E[V]N(d1)-KN(d2)
E[max(K-V),0]=KN(-d2)-E[V]N(-d1)
where the symbols d1 and d2 are
d1
s
=ln (E[V]/K)+s2/2
d2=d1 =ln (E[V]/K)-s2
/s
2 =d1-s
This is also the main result of Black's model which, for the first time, allowed an
Analytical approach to the pricing of options on any log normally distributed underlying.
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The symbol N(x) denotes the cumulative normal distribution.
For a European call option on a zero-coupon bond this leads to the well-known result for
the value of the option.
The call price is given by
C=P(0,T)(F0N(d1)-KN(d2))
where the value at time T is discounted to time 0 using P(0;T) as a risk free deflator.
The value of the corresponding put option is
P=P(0,T)( KN(-d2) -F0N(-d1)))
Here is pricing example of European bond call option and put option using the Black
model and the following parameter .
Bond Data Term StructureTime (Yrs) Rate (%)
Principal: 100 Coupon Frequency: 0.5 4.500%
Bond Life (Years): 5 1 5.000%
Coupon Rate (%): 6.000% 2 5.500%
Quoted Bond Price (/100): 98.80303 3 5.800%
4 6.100%
Option Data 5 6.300%
Pricing Model:
Strike Price (/100): 100.00
Option Life (Years): 3.00Yield Volatility (%): 10.00%
Calculate
PutCall
Quoted Strike
Imply VolatilityBlack - European
Quarterly
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This is the graph of the call option price against the strike
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
95.00 97.00 99.00 101.00 103.00 105.00
Strik e Price
OptionPrice
This is graph of the put option price against the strike
0
1
2
3
4
5
6
7
95.00 97.00 99.00 101.00 103.00 105.00
Strik e Price
OptionPrice
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2.3 Interest Rate Caps and Floors
Interest rate caps are options designed to provide hedge against the rate of interest on a floating-rate
note rising above a certain level known as cap rate.
A floating rate note is periodically reset to a reference rate, eg. LIBOR.
If this rate exceeds the cap rate, The cap rate applies instead. The tenor denotes the time between
reset dates. The Individual options of a cap are denoted as caplets.
Note that the interest rate is always set at the beginning of the time period, while the payment must
be made at the end of the period.
In addition to caps, floors and collars can be defined analogously to a cap, a floor Provides a payoff if
the LIBOR rate falls below the floor rate, and the components of a floor are denoted as floorlets.
A collar is a combination of a long position in a cap and a short position in a floor. It is used to insure
against the LIBOR rate leaving an interest rate range between two specific levels.
Consider a cap with expiration T, a principal of L, and a cap rate of RK. The reset dates
are t1, t2, ., tn, and tn+1=T.
The LIBOR rate observed at time tk is set for the time Period between tk and tk+1, and the
cap leads to a payoff at time tk+1which is
L kMax(Fk -RK,0)
where k =tk+1- tk.
If the LIBOR rate Fk is assumed lognormal distributed with volatility k, each caplet can be valued
separately using the Black formula. The value of a caplet becomes
C=L k P(0, tk+1) (Fk N(d1)- RKN(d2))
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with
d1=ln(Fk /RK)+ k2 tk/2
tkk
d2=ln(Fk /RK)- k2 tk/2
tkk
For the pricing of the whole cap or floor, the values of each caplet or floorlet have to be
discounted back using discount factor as the numeraire: for N number of floorlet and caplets
Ctotal=
),(0
titC i P
N
i
=
Ftotal =
),(0
titF i P
N
i
=
A Swap is an agreement between two parties to exchange cash flows in the future.
2. Interest rate swap(IRS)A company agrees to pay a fixed interest rate on a specific principal for a number of years and, inreturn, receives a floating interest rate on the same principal (pay fixed receive floating).
The floating interest rate is usually the LIBOR rate.
Such 'plain vanilla' interest rate swaps are often used to transform floating rate to fixed-rate loans or
vice versa.
A swap agreement can be seen as the exchange of a floating-rate (LIBOR) bond with a fixed-ratebond.
The forward swap rate S,(t) at time t for the sets of times T and year fractions is the
rate in the fixed leg of the above IRS that makes the IRS a fair contract at the present time.
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S,(t) = P(t;T)- P(t;T)
+=
1i
i P(t,Ti)
Application
Life insurance companies use the hedge interest rate risk and extend their asset duration in order to
stay matched with their long duration liabilities.
2.5 European payer (receiver) swaptionis an option giving the right (and no obligation)
to enter a payer(receiver) IRS at a given future time, the swaption maturity.
Usually the swaption maturity coincides with the first reset date of the underlying IRS.
The underlying-IRS length (T1 T2in our notation) is called thetenor of the swaption.
Sometimes the set of reset and payment dates is called the tenor structure.
I can write the discounted payoff of a payer swaption by considering the value of the underlying payer
IRS at its first reset dateT1, which is also assumed to be the swaption maturity. Such a value is given
by changing sign in formula .
Blacks model is used frequently to value European swaption,
-
C=r T
x mt
eF
mF
+
1)/1(
11
[ ])2()1(* dX NdNF
P= r Tx mt
eF
mF
+
1)/1(
11
[ ])1()2(* dF NdNX
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d1=ln(F /X)+2 tk/2
T
d2 =d1 - T
where F is the strike swap rate and X is the current implied forward swap rate for t1which is here the maturity of the option element of the swaption and start time of the
swap and time t2 is the time when the swap contract terminate
T=t2- t1
Pricing and applications
Here is example of pricing receiver swaption that life insurer use to hedge their interest rate exposure
in guaranteed annuity option.
Swap / Cap Data Term Structure
Underlying Type: Time (Yrs) Rate (%)
1 3.961%
Settlement Frequency: 2 3.879%
Principal : 100 3 3.853%
Swap Start (Years): 1.00 4 3.928%Swap End (Years): 30.00 5 3.992%
Swap Rate (%): 1.82% 6 4.118%
7 4.203%
Pricing Model: 8 4.288%
9 4.406%
10 4.618%
Volatility (%): 15.00% 11 4.586%
12 4.482%
13 4.376%
Price: 1.318E-08
DV01 (Per basis point): -1.25E-09
Gamma01 (Per %): 1.172E-08
Vega (per %): 7.45E-08
Swap Option
Black - European
Imply Volatility
Imply Breakeven Rate
Pay Fixed
Rec. Fixed
Calculate
Semi-Annual
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0
5
10
15
20
25
1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.0
Swap Rate
OptionPrice
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2.6 Callable/Putable Zero Coupon Bonds
Callable (putable) zero coupon bondsdiffer from zero coupon bonds in that the Issuer has the right
to buy (the investor has the right to sell) the paper prematurely at a specified price. There are three
types of call/put provisions.
European option:
The bond is callable/putable at a predetermined price on one specified day.
American option:
The bond is callable/putable during a specified period.
Bermuda option:
The bond is callable/putable at specified prices on a number of predetermined occasions.
A call provision allows the issuer to repurchase the bond prematurely at a specified price. In effect,
the issuer of a callable bond retains a call option on the bond. The investor is the option seller.
A put provisionallows the investor to sell the bond prematurely at a specified price.
In other words, the investor has a put option on the bond. Here, the issuer is the option seller.
Call provision
The issuer has a Bermuda call option which may be exercised at an annually changing strike price.
Replication
This instrument breaks into callable zero coupon bonds down into a zero coupon bond and a call
Option.
callable zero coupon bond = zero coupon bond +call option
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where
+long position
- Short position
The decomposed zero coupon bond has the same features as the callable zero coupon bond except for
the call provision. The call option can be a European, American or Bermuda option.
Variance swapsVariance swapsare instruments, which offer investors straightforward and direct exposure to the
volatility of an underlying asset such as a stock or index.
They are swap contracts where the parties agree to exchange a pre-agreed Variance level for the actual
amount of variance realised over a period.
Variance swaps offer investors a means of achieving direct exposure to realised variance without the
path-dependency issues associated with delta-hedged options.
Buying a variance swap is like being long volatility at the strike level; if the market delivers more than
implied by the strike of the option, you are in profit, and if the market delivers less, you are in loss.
Similarly, selling a variance swap is like being short volatility.
However, variance swaps are convex in volatility: a long position profits more from an increase in
volatility than it loses from a corresponding decrease. For this reason variance swaps normally trade
above ATM volatility.
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Market development
Variance swap contracts were first mentioned in the 1990s, but like vanilla options only really took
off following the development of robust pricing models through replication arguments.
The directness of the exposure to volatility and the relative ease of replication through a static portfolio
of options make variance swaps attractive instruments for investors and market-makers alike.
The variance swap market has grown steadily in recent years, driven by investor demand to take directvolatility exposure without the cost and complexity of managing and delta hedging a vanilla options
position.
Although it is possible to achieve variance swap payoffs using a portfolio of options, the variance
swap contract offers a convenient package bundled with the necessary delta-hedging.
This will offer investors a simple and direct exposure to volatility, without any of the path dependency
issues associated with delta hedging an option.
Variance swaps initially developed on index underlings. In Europe, variance swaps on the Euro Stoxx
50 index are by far the most liquid, but DAX and FTSE are also frequently traded.
Variance swaps are also tradable on the more liquid stock underlings especially Euro Stoxx 50
constituents, allowing for the construction of variance dispersion trades.
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Variance swaps are tradable on a range of indices across developed markets and increasingly also on
developing markets.
Bid/offer spreads have come in significantly over recent years and inEurope they are now typically in the region of 0.5 vegas for indices and vegas for single-stocks
although the latter vary according to liquidity factors.
Example 1: Variance swap p/l
An investor want to gain exposure to the volatility of an underlying index (e.g, Dow
Jones FTSE 100 ) over the next year.
The investor buys a 1-year variance swap, and will be delivered the difference between
the realised variance over the next year and the current level of implied variance, multiplied by the
variance notional.
Suppose the trade size is 2,500 variance notional, representing a p/l of 2,500 per point
difference between realised and Implied variance.
If the variance swap strike is 20 (implied variance is 400) and the subsequent variance realised over the
course of the year is(15%)2=0.0225 (quoted as 225),
The investor will make a loss because realised variance is below the level bought.
Overall loss to the long =437,500 =2,500 x (400 225).
Theshort positionwill profit by the same amount.
1.1: Realised volatility
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Volatility measures the variability of returns of an underlying asset and in some sense provides a
measure of therisk of holding that underlying.
In this note I am concerned with the volatility of equities and equity indices, although much of the
discussion could apply to the volatility of other underlying assets such as credit, fixed-income, FX and
commodities.
Figure 3 shows the history of realised volatility on the Dow Jones Industrial Average
over the last 100 years. Periods of higher volatility can be observed, e.g. in the early 1930s as a result
of the Great Depression, and to a lesser extent around 2000 with the build-up and unwind of the dot-
com bubble. Also noticeable is the effect of the 1987 crash, mostly due to an exceptionally large
single day move, as well as numerous smaller volatility spikes
.
Summary of the equity volatility characteristics
The following are some of the commonly observed properties of (equity market) volatility:
Volatility tends to be anti-correlated with the underlying over short time periods
Volatility can increase suddenly in spikes
Volatility can be observed to experience different regimes
Volatility tends to be mean reverting (within regimes)
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This list suggests some of the reasons why investors may wish to trade volatility: as a partial hedge
against the underlying .
Especially for a volatility spike caused by a sudden market sell-off; as a diversifying asset
class; to take a macro view e.g. or a potential change in volatility regime; for to trade a spread ofvolatility between related instruments.
Pricing model and hedging
First let us understand the cash flow structure the following diagram explain the cash flow exchanged
by looking to the following diagram
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Volatility swaps are series of forward contracts on future realized stock volatility, variance.
Swaps are similar contract on variance, the square of the future volatility.
Both these instruments provide an easy way for investors to gain exposure to the future level of
volatility.
A stock's volatility is the simplest measure of its risk less or uncertainty.
Formally, the volatility R(S).
R(S) is the annualized standard deviation of the Stocks returns during the period of
interest , where the subscript R denotes the observed or "realized" volatility for the stock .
The easy way to trade volatility is to use volatility swaps, sometimes Called realized volatility forward
contracts, because they provide pure exposure To volatility (and only to volatility). A stock volatility
swap is a forward contract on the annualized volatility.
Its payoff at expiration is equal to
N( 2R(S)-Kvar )
WhereR(S)) is the realized stock volatility (quoted in annual terms) over the life of the contract.
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( 2R(S) =1/T T
0
2(S) ds
Kvar is the delivery price for variance, and N is the notional amount of the swap in dollars per
annualized volatility point squared.
The holder of variance swap at expiration receives N dollars for every point by which the stock's
realized variance has exceeded the variance delivery price Kvar.
Therefore, pricing the variance swap reduces to calculating the realized volatility square.
Valuing a variance forward contract or swap is no different from valuing any other derivative security.
The value of a forward contract P on future realized variance with strike price Kvar is the expected
present value of the Future payoff in the risk-neutral world:
P=E(e-rT ( 2R(S)-Kvar )
where r is the risk-free discount rate corresponding to the expiration date T (Under the
assumption of deterministic risk free rate)and E denotes the expectation.
Thus, for calculating variance swaps we need to know only
E [( 2R(S)]
Namely, mean value of the underlying variance.
Approximation (which is used the second order Taylor expansion for function px)
where
E[ 2R(S)] )(VE - Var(V)
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8 E(V)3/2
Where V = 2R(S)
In addition, Var(V)8 E(V)3/2
this the term of the convexity adjustment.
Thus, to calculate volatility swaps ineed the first and the second term
this variance has unbiased estimator namely:
Varn(S)=n/(n-1)*1/T *=
n
i 1
log2 St
St-1V=Var(S)=lim Varn(S)
n
Where we neglected by 1/n =
n
i1
log2 St
St-1
For simplicity reason only. Inote that iuse Heston (1993) model:
Log St1 =dtr
t
tt )2/(
21
11
+ tt
tt
dw
t
1
1
St1-1
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E(varn(S))= n )( l o g11
1
1
2
=
t
tn
t S
SE
(n-1)T
snd
E( log211
1
t
t
S
S)= )(
1
11
dtr
t
tt
2 _ )(1
11
dtr
t
tt
d tE
t
tt
t
12
1
)( +4
1s dE t
t
t
t
t
221
1
1
1 11
-E( dtEt
tt
t
12
1
)(t
t
tt
dw
t
1
1
)+ dtEt
tt
t
12
1
)(
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Appendix 1
Variance and Volatility Swaps for Heston
Model of Securities Markets
Stochastic Volatility Model.
Let (;F;Ft; P) be probability space with filtration Ft; t [0; T]:
Assume that underlying asset St in the risk-neutral world and variance
follow the following model, Heston (1993) model:
ds tt =rt dt+ dwtst
d t 2 =K(2- t 2 )dt+ t dwt2
where rt is deterministic interest rate, 0 and are short and long volatility,
k >0 is a reversion speed,
>0 is a volatility (of volatility) parameter, w1
and w2 are independent standard Wiener processes.
The Heston asset process has a variance that follows Cox-Ingersoll- Ross (1985) process,
described by the second equation .
If the volatility follows Ornstein-Uhlenbeck process (see, for example, Oksendal (1998)), then Ito's
lemma shows that the variance follows the process described exactly by the second equation .
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References
Leif Andersen, Mark Broadie: A primal-dual simulation algorithm for
Farid AitSahlia, Peter Carr: American Options: A Comparison of Numerical
Methods; Numerical Methods in Finance, Cambridge University Press (1997)
Mark Broadie, J er ome Detemple: American Option Valuation: New
Bounds, Approximations, and a Comparison of Existing Methods, The
Review of Financial Studies, 9, pp. 1211-1250 (1996)
Mark Broadie, Paul Glasserman: Pricing American-style Securities Using
Simulation, Journal of Economic Dynamics and Control, 21, pp. 1323-1352 (1997)
Mark Broadie, Paul Glasserman: A Stochastic Mesh Method for Pricing
High-Dimensional American Options, Working Paper, Columbia University,
New York (1997)
David S. Bunch, Herbert E. Johnson: A Simple and Numerically Efficient
Valuation Method for American Puts Using a Modified Geske-J ohnson
Approach, J ournal of Finance, 47, pp. 809-816 (1992)
Alain Bensoussan, Jaques-Louis Lions: Applicaitons of Variational Inequalities
in Stochastic Control, Studies in Mathematics and its Applications,
12, North-Holland Publishing Co. (1982)
Antonella Basso, Martina Nardon, Paolo Pianca: Optimal exercise of
American options, University of Venice, Italy (2002)
Tomas Bjrk. Arbitrage Theory in Continuous Time, Oxford University Press,
New York 1998
Global Derivatives,http://www.global-derivatives.com/
Avellaneda, M., Levy, A. and Paras, A. (1995): Pricing and hedging derivative
securities in markets with uncertain volatility, Appl. Math. Finance 2, 73-88.
http://www.global-derivatives.com/http://www.global-derivatives.com/http://www.global-derivatives.com/http://www.global-derivatives.com/7/29/2019 Opti Uni Bursa
54/54
Black, F. and Scholes, M. (1973): The pricing of options and corporate
liabilities, J . Political Economy 81, 637-54.
Bollerslev, T. (1986): Generalized autoregressive conditional heteroscedasticity,
J . Economics 31, 307-27.
Brockhaus, O. and Long, D. (2000): "Volatility swaps made simple",RISK, January, 92-96.
Buff, R. (2002): Uncertain volatility model. Theory and Applications.
NY : Springer.
Carr, P. and Madan, D. (1998): Towards a Theory of Volatility Trading.
In the book: Volatility, Risk book publications,
http://www.math.nyu.edu/research/carrp/papers/.
Chesney, M. and Scott, L. (1989): Pricing European Currency Options:
A comparison of modifeied Black-Scholes model and a random variance
model, J . Finan. Quantit. Anal. 24, No3, 267-284.
Cox, J ., Ingersoll, J . and Ross, S. (1985): "A theory of the term
structure of interest rates", Econometrica 53, 385-407.
Demeterfi, K., Derman, E., Kamal, M. and Zou, J . (1999): A guide
to volatility and variance swaps, The Journal of Derivatives, Summer, 9-