U.U.D.M. Project Report 2009:4

Examensarbete i matematik, 30 hpHandledare och examinator: Johan TyskMars 2009

Structured products: Pricing, hedging and applications 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 write this 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 Bonds

2.7 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 the main 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 investment strategies.

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 rather complicated 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 accurate as possible. Commonly used alternative names are also indicated in each product’s description.

1.1 What are structured products?

Definition: Structured products are investment instruments that combine at least one derivative 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 be paid 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 bond’s face value (with adjustment by means of the nominal interest rate). It is also common that no payments (including coupons) are made until the product’s maturity date. The investor’s 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 asset’s 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 the underlying 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 individual’s 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 F0 is forward price that would apply to the contract today. The value of a long forward contract, ƒ, is ƒ = (F0–K)e–rT

Similarly, the value of a short forward contract is (K –F0) e–rT

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 some stage 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 costs when an investment asset provides a known yield q F0= S0e(r–q )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(r–q )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<S0 e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index Index arbitrage involves simultaneous trades in futures and many different stocks Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0and S0 does not holds so F0≤ S0e(r+u )T , where u is the storage cost per unit time as a percent of the so the equality should hold. Otherwise there will be an arbitrage opportunity.

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How to hedge using futures A proportion of the exposure that should optimally be hedged is h= ρ* (σS/ σF) where σS is the standard deviation of dS, the change in the spot price during the hedging period, σF is the standard deviation of dF, the change in the futures price during the hedging period ρ 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, β 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 be S (t),t∈[0,T]. European options give the holder the right to exercise the option only on the expiration date T .

Hence the holder receives the amount (S(T)), where ϕ is a contract function. Moreover, there are two basic types of European options namely 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),(),(21),(),( 2

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sstF

str Fs

stFSs

stFr St

stF

Φ=

=−∂

∂+

∂∂

+∂

∂ σ

in [0 T ]×R+. Here r is the interest rate; σ is the volatility of the underlying assumed fixed parameters. Asset S and Φ(s) = max(s −k ,0) is the contract function. According to the Feynman-Kac theorem PDE solution can represented as an expected value

F(t,s)=e –r(T-t) [ ]),(, TsE st Φ

where the underlying stock S(t ) follows the dynamics

∂ s(u)=r s(u) ∂ u+s(u)σ (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 the interest 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 t ≤ T

before the expiration, date (also called early exercise). The holder of an American option needs to decide whether to exercise immediately or to wait.

If the holder decides to exercise at say t ≤T, then he receives Φ(S(t)) where Φ is the appropriate

contract function. Similarly, this option can also be classified into two basic types: American call options which give the owner the right to buy an underlying asset for a given 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 Asset price 10.00 30.88 51.76 72.65 93.53 114.41 135.29 156.18 177.06 218.82 239.71 260.59 281.47 302.35 323.24 344.12

150.00 50.2736 50.8432 51.4228 52.0323 52.6754 53.3462 54.0368 54.7405 55.4517 56.8819 57.5955 58.3059 59.0117 59.7122 60.4067 61.0947 145.00 45.2736 45.8445 46.4380 47.0762 47.7554 48.4640 49.1912 49.9288 50.6712 52.1546 52.8908 53.6213 54.3452 55.0617 55.7706 56.4715 140.00 40.2736 40.8484 41.4678 42.1490 42.8763 43.6316 44.4018 45.1780 45.9544 47.4945 48.2540 49.0050 49.7469 50.4794 51.2022 51.9154 135.00 35.2736 35.8592 36.5246 37.2678 38.0568 38.8680 39.6871 40.5056 41.3184 42.9168 43.6996 44.4707 45.2300 45.9776 46.7134 47.4379 130.00 30.2737 30.8871 31.6301 32.4580 33.3226 34.1978 35.0704 35.9335 36.7836 38.4395 39.2445 40.0344 40.8097 41.5707 42.3180 43.0522 125.00 25.2742 25.9552 26.8190 27.7556 28.7079 29.6527 30.5807 31.4882 32.3744 34.0837 34.9085 35.7147 36.5033 37.2753 38.0316 38.7729 120.00 20.2792 21.1106 22.1456 23.2106 24.2569 25.2717 26.2528 27.2013 28.1194 29.8737 30.7141 31.5326 32.3307 33.1100 33.8717 34.6170 115.00 15.3132 16.4396 17.6873 18.8874 20.0238 21.1015 22.1277 23.1092 24.0516 25.8369 26.6867 27.5117 28.3141 29.0958 29.8583 30.6032

110.00 10.4857 12.0799 13.5459 14.8645 16.0723 17.1955 18.2514 19.2524 20.2072 22.0037 22.8545 23.6783 24.4780 25.2556 26.0131 26.7521 105.00 6.1194 8.2180 9.8410 11.2294 12.4717 13.6114 14.6736 15.6747 16.6255 18.4066 19.2475 20.0605 20.8486 21.6143 22.3595 23.0861 100.00 2.7763 5.0530 6.6920 8.0687 9.2905 10.4065 11.4440 12.4201 13.3462 15.0793 15.8971 16.6876 17.4538 18.1981 18.9226 19.6289 95.00 0.8696 2.7262 4.1907 5.4529 6.5875 7.6319 8.6080 9.5299 10.4073 12.0549 12.8344 13.5891 14.3215 15.0338 15.7278 16.4051 90.00 0.1638 1.2453 2.3693 3.4191 4.4001 5.3241 6.2009 7.0380 7.8413 9.3633 10.0884 10.7929 11.4787 12.1475 12.8006 13.4392 85.00 0.0159 0.4614 1.1806 1.9564 2.7338 3.4970 4.2412 4.9657 5.6711 7.0287 7.6834 8.3233 8.9495 9.5630 10.1644 10.7546 80.00 0.0007 0.1318 0.5035 1.0009 1.5555 2.1355 2.7253 3.3167 3.9055 5.0661 5.6360 6.1984 6.7530 7.3000 7.8395 8.3716 75.00 0.0000 0.0273 0.1774 0.4466 0.7951 1.1937 1.6237 2.0735 2.5355 3.4779 3.9525 4.4271 4.9005 5.3718 5.8405 6.3061 70.00 0.0000 0.0038 0.0494 0.1684 0.3564 0.5989 0.8823 1.1961 1.5325 2.2510 2.6256 3.0069 3.3929 3.7823 4.1739 4.5667 65.00 0.0000 0.0003 0.0103 0.0516 0.1359 0.2631 0.4283 0.6253 0.8487 1.3560 1.6327 1.9210 2.2189 2.5244 2.8362 3.1530 60.00 0.0000 0.0000 0.0015 0.0122 0.0424 0.0981 0.1807 0.2894 0.4218 0.7477 0.9360 1.1384 1.3529 1.5778 1.8118 2.0535 55.00 0.0000 0.0000 0.0001 0.0021 0.0103 0.0297 0.0640 0.1150 0.1832 0.3692 0.4850 0.6143 0.7560 0.9089 1.0719 1.2439 50.00 0.0000 0.0000 0.0000 0.0002 0.0018 0.0069 0.0182 0.0377 0.0671 0.1586 0.2211 0.2945 0.3785 0.4724 0.5757 0.6878

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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

19

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Calculations of table used to generate 3-D graph

Time to maturity in days

Asset price

10.00

30.88

51.76

72.65

93.53

114.41

135.29

156.18

177.06

218.82

239.71

260.59

281.47

302.35

323.24

344.12 365.00

150.00 50.1369

50.4222

50.7070

50.9944

51.2892

51.5945

51.9118

52.240

2 52.5795

53.2823

53.6432

54.0076

54.3775

54.7508

55.1223

55.5009

55.8746

145.00 45.1369

45.4222

45.7080

46.0003

46.3059

46.6269

46.9632

47.312

5 47.6731

48.4194

48.8011

49.1892

49.5774

49.9679

50.3604

50.7523

51.1432

140.00 40.1369

40.4223

40.7109

41.0139

41.3380

41.6833

42.0467

42.423

8 42.8114

43.6126

44.0208

44.4286

44.8404

45.2512

45.6630

46.0700

46.4826

135.00 35.1369

35.4228

35.7195

36.0437

36.3985

36.7787

37.1799

37.592

5 38.0153

38.8803

39.3138

39.7493

40.1842

40.6174

41.0462

41.4784

41.9015

130.00 30.1369

30.4252

30.7427

31.1069

31.5107

31.9414

32.3874

32.845

4 33.3088

34.2449

34.7064

35.1730

35.6292

36.0867

36.5394

36.9815

37.4275

125.00 25.1369

25.4361

25.8025

26.2357

26.7093

27.2038

27.7110

28.220

6 28.7273

29.7378

30.2338

30.7226

31.2095

31.6831

32.1547

32.6227

33.0801

120.00 20.1370

20.4775

20.9442

21.4825

22.0492

22.6213

23.1932

23.760

3 24.3155

25.3976

25.9250

26.4351

26.9461

27.4448

27.9303

28.4040

28.8815

115.00 15.1404

15.6142

16.2555

16.9340

17.6075

18.2652

18.9042

19.517

8 20.1226

21.2695

21.8218

22.3553

22.8769

23.3940

23.8969

24.3871

24.8655

110.00 10.1877

10.9996

11.8786

12.7056

13.4827

14.2193

14.9112

15.575

0 16.2035

17.4033

17.9707

18.5184

19.0487

19.5710

20.0822

20.5802

21.0661

105.00 5.55

16 6.90

85 8.00

59 8.95

13 9.80

37 10.5811

11.3015

11.988

5 12.6383

13.8499

14.4199

14.9701

15.5027

16.0197

16.5279

17.0230

17.5060

100.00 2.04

77 3.68

63 4.84

87 5.81

61 6.66

89 7.44

38 8.16

12 8.8337

9.4700

10.6569

11.2155

11.7547

12.2767

12.7835

13.2763

13.7566

14.2254

95.00 0.39

91 1.57

85 2.55

87 3.41

67 4.18

70 4.89

85 5.55

80 6.1817

6.7796

7.8962

8.4223

8.9305

9.4228

9.9010

10.3663

10.8227

11.2710

90.00 0.03

08 0.50

40 1.13

31 1.76

40 2.37

47 2.95

62 3.52

30 4.0567

4.5838

5.5704

6.0468

6.5119

6.9629

7.4012

7.8281

8.2446

8.6580

85.00 0.00

07 0.11

03 0.40

00 0.77

75 1.18

66 1.61

33 2.03

99 2.4694

2.8924

3.7181

4.1253

4.5195

4.9093

5.2991

5.6791

6.0503

6.4132

80.00 0.00

00 0.01

51 0.10

71 0.28

10 0.51

05 0.77

48 1.06

44 1.3671

1.6764

2.3173

2.6393

2.9601

3.2849

3.6015

3.9208

4.2413

4.5550

75.00 0.00

00 0.00

11 0.02

04 0.07

91 0.18

09 0.31

86 0.48

62 0.6751

0.8812

1.3275

1.5648

1.8082

2.0539

2.3044

2.5587

2.8078

3.0680

70.00 0.00

00 0.00

00 0.00

25 0.01

66 0.05

09 0.10

85 0.18

84 0.2895

0.4069

0.6876

0.8464

1.0126

1.1856

1.3687

1.5526

1.7459

1.9406

65.00 0.00

00 0.00

00 0.00

02 0.00

24 0.01

07 0.02

91 0.06

01 0.1045

0.1615

0.3148

0.4074

0.5085

0.6217

0.7413

0.8649

0.9998

1.1342

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 0.51 0.603

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10.00

93.53

177.06260.59

150

145

140

135

130

125

120

115

110

105

100

95908580757065605550

0

10

20

30

40

50

60

Time to maturity

Asset 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.

22

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 <t < <t =T. Therefore the Bermudan options being a hybrid of European and American options, the value of a Bermudan is greater than or equal to an identical European option but less than or equal to its equivalent American option . I will price some of Bermudan type option like equity Cliquet option .

1.2.6 Asian option types

This type of option depends on the average value of the underlying asset over a time, Therefore, an Asian option is path dependent. Asian options are cheaper relative to their European and American counterparts because of their lower volatility feature

The are broadly three categories:

1) Arithmetic average Asians, 2) Geometric average Asians 3) Combination of 1 and 2 The pay-off can be averaged on a weighted average basis, whereby a given weights is applied to each stock being averaged. This can be useful for attaining an average on a sample with a highly skewed sample population. There are no known closed form analytical solutions arithmetic options, due to the a property of these options under which the lognormal assumptions collapse –so it is not possible to analytically evaluate the sum of the correlated lognormal random variables.

23

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:

Payoff Asian call =max

−

∏=

XS inn

i

/1

1

,0

Payoff Asian 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 =S e(b-r)(T-t)N(d1)-X e-r(T-t)N(d2) and, PG = X e-r(T-t)N(-d2)- S e(b-r)(T-t)N(-d1) where N(x) is the cumulative normal distribution function of:

d1=ln(S/X)+(b+0.52

Aσ )T

Aσ T

d2=ln(S/X)+(b-0.52

Aσ )T

Aσ T

24

The adjusted volatility and dividend yield are given as:

Aσ =σ / 3

b=1/2(r-D-2σ /6)

The payoff of arithmetic Asian options is given as

Payoff Asian call =max(0,(∑=

n

iSi

1/n)-X)

Payoff Asian put= max(0,X-(∑=

n

iSi

1/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 Curran’s model, the value Of an Asian option can be approximated using the following formula:

25

Here is an example of capital guaranteed structured product that has Asian pay off. On the FTSE 100 index using Curran’s model. Average calculated quarterly and the interest rate used are annual compounded and 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 4.00 Number of fixings fixed m 0.00 Risk-free rate ( r ) 4.50% Cost of carry ( b ) 2.00% Volatility (σ ) 26.00% Value 10.7396

26

10.00

114.41

218.82

323.24

200.

00

185.

00

170.

00

155.

00

140.

00

125.

00

110.

00

95.0

0

80.0

0

65.0

0

50.0

0

0.0000

20.0000

40.0000

60.0000

80.0000

100.0000

120.0000

Time to maturity

Asset 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 .

27

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

−−∑

= −

− CFS

SS iCF

n

i i

iii ,,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

28

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∆σ and D = e- t∆σ The adjusted risk-neutral probability for the up state is P = e t∆σ -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

29

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 option prices for various stock prices

30

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 pricePlan vanila CRR

And here is comparison between plan vanilla American call and European Cliquet option prices for various stock prices

0102030405060708090

100110120130140

50.00 70.00 90.00 110.00 130.00 150.00 170.00 190.00 210.00 230.00 250.00

cliquet priceCRR 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 .

31

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.

32

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 options on 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]=σ 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.

33

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)(F0 N(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) -F0 N(-d1)))

Here is pricing example of European bond call option and put option using the Black model and the following parameter .

Bond Data Term Structure Time (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): 100.00Option Life (Years): 3.00Yield Volatility (%): 10.00%

Calculate

PutCall

Quoted Strike

Imply VolatilityBlack - European

Quarterly

34

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

Strike Price

Opt

ion

Pric

e

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

Strike Price

Opt

ion

Pric

e

35

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+1 which is Lδ k Max(Fk -RK ,0)

where δ k = tk+1 - tk.

If the LIBOR rate Fk is assumed lognormal distributed with volatility σ k, each caplet can be valued separately using the Black formula. The value of a caplet becomes C=Lδ k P(0, tk+1 ) (Fk N(d1)- RKN(d2))

36

with d1= ln(Fk /RK )+ k

2σ tk/2

tkkσ d2= ln(Fk /RK )- k

2σ tk/2

tkkσ 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

t itC i PN

i∑=

Ftotal =

),(0

t itF i PN

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, in return, 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-rate bond. The forward swap rate Sα,β(t) at time t for the sets of times T and year fractions τ is the rate in the fixed leg of the above IRS that makes the IRS a fair contract at the present time.

37

Sα,β(t) = P(t;Tα )- P(t;T β )

∑+=

β

α

τ1ii 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) swaption is 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 − T2 in our notation) is called the tenor 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 date T1, which is also assumed to be the swaption maturity. Such a value is given by changing sign in formula . Black’s model is used frequently to value European swaption, -

C= r Tx mt

eF

mF −

+

− 1)/1(11

[ ])2()1(* dX NdNF −

P= r Tx mt

eF

mF −

+

− 1)/1(11

[ ])1()2(* dF NdNX −−−

38

d1= ln(F /X )+ 2σ tk/2

Tσ d2 =d1 - Tσ

where F is the strike swap rate and X is the current implied forward swap rate for t1

which 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-08DV01 (Per basis point): -1.25E-09

Gamma01 (Per %): 1.172E-08Vega (per %): 7.45E-08

Swap Option

Black - European

Imply Volatility

Imply Breakeven Rate

Pay Fixed

Rec. Fixed

Calculate

Semi-Annual

39

0

5

10

15

20

25

1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00%

Swap Rate

Opt

ion

Pric

e

40

2.6 Callable/Putable Zero Coupon Bonds Callable (putable) zero coupon bonds differ 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 provision allows 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

41

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 swaps Variance swaps are 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.

42

Market development Variance swap contracts were first mentioned in the 1990’s, 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 direct volatility 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.

43

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 in Europe 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). The short position will profit by the same amount. 1.1: Realised volatility

44

Volatility measures the variability of returns of an underlying asset and in some sense provides a measure of the risk 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 1930’s 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)

45

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 of volatility 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

46

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 σ R(S). σ R(S) is the annualized standard deviation of the Stock’s 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(σ 2

R(S)-Kvar ) Where σ R(S)) is the realized stock volatility (quoted in annual terms) over the life of the contract.

47

(σ 2R(S) =1/T ∫

T

0

σ 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 (σ 2

R(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 [(σ 2

R(S)] Namely, mean value of the underlying variance. Approximation (which is used the second order Taylor expansion for function px) where

E[ σ 2R(S)]≈ )(VE - Var(V)

48

8 E(V)3/2

Where V =σ 2

R(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 1log 2 St

St-1 V=Var(S)= lim Varn (S) n→ ∞

Where we neglected by 1/n ∑=

n

i 1log 2 St

St-1 For simplicity reason only. Inote that iuse Heston (1993) model:

Log St1 = dtrt

t t )2/( 21

11

σ−∫ − + t

t

tt

dwt

∫−

1

1

σ

St1-1

49

E(varn(S))= n )( l o g11

11

2

−=∑

t

tn

t SS

E

(n-1)T snd

E( log 2

11

1

−t

t

SS

)= )(1

11

dtrt

t t∫−

2 _ )(1

11

dtrt

t t∫−

d tEt

t tt

∫−

12

1

)(σ + 41 s dE t

t

t

t

t

221

1

1

1 11

σσ∫∫−−

-E( d tEt

t tt

∫−

12

1

)(σ t

t

tt

dwt

∫−

1

1

σ )+ d tEt

t tt

∫−

12

1

)(σ

50

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 tσt =rt dt+ dwt

st

d tσ 2 =K(θ 2- tσ 2 )dt+γ tσ dwt2

where rt is deterministic interest rate, σ 0 andθ are short and long volatility, k > 0 is a reversion speed, γ > 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 .

51

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´erˆome 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-Johnson Approach, Journal 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 Björk. 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.

52

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-