## Excerpt

## Table of Contents

1 Introduction

2 Complete market

3 The Corridor Option

3.1 The Basic Corridor Option

3.2 Knock-out Option

3.3 Three-Corridor Option

4 Pricing methods

4.1 The Black-Scholes Model

4.2 Monte-Carlo-Simulation

4.2.1 The Model

4.2.2 Empirical Prices

4.3 Analytical Approach

4.3.1 First Model

4.3.2 Analytical Prices

4.3.3 Extension of the Model

5 Conclusion

Bibliography

Appendix: Maple Code

List of Figures

3.1 Corridor

3.2 Payoff Corridor Option

3.3 Knock-out

3.4 Three Corridors

4.1 Simulations

List of Tables

4.1 Prices - Corridor option, Knock-out (K = 10)

4.2 Prices - Corridor option, Knock-out (K2 = 50)

4.3 Prices - Three corridors (K = 10)

4.4 Prices - Three corridors (K2 = 50)

4.5 Three corridors - Percentage of coupon interest (K = 10)

4.6 Comparison: Number of days

4.7 Deviations

ACKNOWLEDGEMENTS

First, I would like to express my sincere gratitude to my supervisor Professor Richard Stockbridge for his excellent guidance and assistance throughout the progress of this thesis. His generous support by explanations and ideas made this thesis possible.

Also, I would like to give special thanks to Professor Eric Key and Professor Chao Zhu for serving on my committee and providing constructive ideas and suggestions.

## Chapter 1 Introduction

The following pages will describe and examine different aspects of the corridor option, a financial instrument that belongs to the class of exotic structures. This option provides different opportunities for an investment regarding risk-attitude and expectation of market development.

Besides the basic structure, two variations of the corridor option will be examined. Both types describe a different way of accruing daily coupons. One ceases to accrue interest when the underlying hits a barrier, while the other type opens new corridors and thus widens the range and the chance to accrue interest after leaving the main corridor. Analyzing these structures will show how the prices of these more or less risky derivatives behave in comparison to the basic corridor option.

After giving an overview of the design and further characteristics of the corridor option and its variations, the thesis will consider two approaches for the pricing of these derivatives. One approach uses Monte-Carlo methods to simulate the present value of an option numerically. The other one uses an analytical approach, which calculates the price of a corridor option considering the generator for the price process and expected occupation measures. To be able to examine complex price structures, the work will be based on the idea of the Black-Scholes model which assumes a complete market. These assumptions will provide a unique measure to determine a fixed price of the corridor option.

An insight will be given into the influence of various parameters on option prices.

By varying the volatility of the underlying as well as the strike price, it can be seen how these two parameters affect the value of an option. Furthermore, different magnitudes of the corridor will be examined. While considering the different prices caused by different parameter values, all other parameters are fixed. Finally, the different variations of the corridor option will be compared.

While the basic corridor option can be described by a complete analytic pricing model, its variations involve more complex methods to describe the price processes with the second approach. The thesis will close with possible ideas to price the two considered variations of the corridor option analytically.

## Chapter 2 Complete market

To be able to examine more complicated price processes all considerations are based on a complete market model. This helps to get a basic understanding of more complex financial instruments, and gives a possibility to build a simplified pricing model that gives us an exact price for each derivative. More specifically, this means essential factors are narrowed against reality to focus on the influences of specific parameters on the option price. Thus the prices of derivatives can be evaluated looking at particular parameters deterministically (like volatility) instead of handling several stochastic factors.

Definition 1 A contingent claim (or derivative security) is a financial contract whose value at expiration date T is determined exactly by the price of the underlying financial assets at time T or within the time interval [0,T].

Definition 2 A market is complete if every contingent claim can be exactly replicated by a unique portfolio.

The prices in a complete market model are unaffected by transaction costs or other market frictions. It is assumed that the interest for lending and borrowing are equal and agents act rationally. It follows from the definition of a complete market that there are no arbitrage possibilities. Pricing by no-arbitrage means that the cost required to set up the replicating portfolio determines the price of the option. Since there is a unique portfolio to replicate each contingent claim in a complete market, every contingent claim has a unique price. This follows from the existence of a unique martingale measure that determines the price of the derivative. The martingale measure is the measure under which the discounted price process is a martingale. Thus the price of the corridor option is given by the expectation of the option under the equivalent martingale measure. Since the corridor option is path- dependent, its price depends on the entire history of the underlying’s value within the life-time of the option. A deterministic drift and volatility of the underlying are assumed in this market model. This aspect will be investigated in the chapter about pricing strategies.

## Chapter 3 The Corridor Option

### 3.1 The Basic Corridor Option

A general call or put option gives someone the right, but not the obligation, to buy or sell an asset at a specific date at a specific price. This implies that only the final value of the underlying at expiry is important to determine the price of the call or put option. However, the payoff of a corridor option depends on the entire path of the underlying asset’s value over the lifetime of the option, which classifies the corridor option as exotic. This financial instrument was first issued by the Swiss association Schweizer Bankverein (SBV) in Germany on April 18th, 1994 [4].

illustration not visible in this excerpt

Figure 3.1: Corridor

A corridor option gives someone the right to receive a daily coupon for each day the underlying stays within a pre-defined range (also called accrual corridor) during the lifetime of the option. This coupon is paid at the end of maturity for a specific strike price. A sample path of the value of an underlying compared to a given lower and upper bound is shown in Figure 3.1 above. Here, the initial value of the underlying is 100 within a corridor of 90 and 110.

A standard corridor option generally involves a payout that is better than the market-LIBOR^{[1]} rates, provided that the underlying of the option stays within the pre-defined band for most of the period. The magnitude of the range can be chosen by the holder according to his expectations of the underlying’s future development. The maximum payoff of the option would be the entire number of days of the option’s lifetime times the daily coupon value. The minimum payoff of zero occurs if the underlying leaves the corridor at the first day and stays outside for the rest of the time. A payoff profile is shown in Figure 3.2. The inital cost is not considered in this figure.

illustration not visible in this excerpt

Figure 3.2: Payoff Corridor Option

The daily coupon is usually based on a common index like the LIBOR. An appropriate index can be chosen depending on the maturity of the option. The following pricing model will consider a corridor option with a maturity of six months, which involves using the six-month LIBOR plus some basis points to provide a coupon that is higher than a market coupon [2]. In general, underlying assets for corridor options are foreign currencies, stock indices, and interest rates.

Since corridor options accrue a coupon interest if the underlying’s value stays within a specific range, investing in these structures are more suitable in stable markets. Thus if a holder expects an asset to remain stable within a specific period of time, he can invest in a corridor option with a self-specified upper and lower bound that represents his expectations. Besides the idea of diversifying the portfolio, this financial instrument has been designed more for trading purposes than for hedging purposes. Compared to a barrier option, the corridor option exists for the whole maturity period, even when the underlying leaves the range, whereas the barrier option is terminated if either of the boundaries is hit at any time. The corridor option is also known as a range accumulation option or warrant, Banking On Overall STability (BOOST), Expected to Accrue Return on Nominal (EARN) Warrant, Range Accrual Option, Hope for A Market STabilization in a givEn Range (HAMSTER)^{[2]} or finally Index Range Note [3].

In the following sections, two variations of the corridor option will be introduced. In later chapters we will compare their prices to the prices of the basic corridor option.

### 3.2 Knock-out Option

A first variation of the corridor option is the knock-out option. In this structure, the holder ceases to accrue coupon interest as soon as the underlying leaves the range for the first time. Even if the underlying subsequently reenters the corridor, coupon interest will no longer accrue. At the end of maturity, the accrued coupon is paid. In Figure 3.3, the time when the underlying leaves the corridor for the first time is marked. Compared to the basic corridor option, this knock-out variation bears a bigger risk, especially if the volatility of the underlying is relatively high. Given the same conditions for the corridor, strike and underlying as the basic corridor option, a lower price for this variation of the corridor option will be expected.

illustration not visible in this excerpt

Figure 3.3: Knock-out

### 3.3 Three-Corridor Option

A second variation of the corridor option is to consider several corridors as shown in Figure 3.4. The idea with this option is to improve the chances of accruing coupon interest by opening a new corridor as soon as the underlying hits one of the boundaries of the first corridor. If the value of the underlying sinks below the lower bound, a second corridor opens with a new range. Similarly, a new upper corridor opens if the underlying hits the upper bound.

We will examine two ways how the coupon interest can be accumulated. In the first case, the holder receives a coupon for each day on which the underlying stays within the first corridor. Once it leaves the first corridor, the coupon of the first corridor is lost and the holder gets a new coupon for each day on which the underlying stays within the second (either upper or lower) corridor. If it leaves the secondary corridor, no coupon will be accrued any longer and the previously accrued amount of coupon interest will be paid at the end of maturity. Another possibility to accrue a daily coupon with this variation of corridor option is the following. Again, the holder receives a coupon for each day on which the underlying stays within the first corridor. This time if it leaves the first corridor, the holder keeps the coupon that he earned up to this point, and has the chance to accrue an additional coupon for each day on which the underlying stays within the new corridor. Again, if it leaves this second corridor, no additional daily coupon can be earned. Both types have knock-out character in the secondary corridor.

The following figure shows a sample path of an underlying that stays within the first corridor ([90,110]) until it hits the upper bound after 30 days. Consequently a new upper corridor opens ([102,118]), in which the asset stays for a further 54 days, after which it leaves the second corridor by hitting the lower bound. In either way of accruing coupon interest, the holder ceases to accrue coupon interest after 84 days. The lower secondary corridor doesn’t come into effect.

illustration not visible in this excerpt

Figure 3.4: Three Corridors

In the following pricing models it will be assumed that the daily coupon, which can be earned in the secondary corridors, is smaller than the coupon of the main corridor.

In the next chapter we examine pricing methods, state the results and point out the respective differences and characteristics for all the above described variations of corridor options.

**[...]**

^{[1]} LIBOR=London Interbank Offered Rate, a common benchmark interest rate index

^{[2]} In German: Hoffnung Auf MarktSTabilitat in Einer Range