Pricing of Embedded Options: Implementing Stochastic Interest Rates & Stochastic Spread
(2022) In Master's Theses in Mathematical Sciences MASM02 20221Mathematical Statistics
- Abstract
- Given the current market climate, in an era of negative interest-rates, the Hull-White model has regained popularity in the eyes of investors. This thesis aims to extend this model to incorporate credit risk, to allow the modelling of credit derivatives such as diff swaps, defaultable corporate bonds and credit default swaps. This process can be achieved in a number of ways, by utilising either two one-factor models or one two-factor model. Additionally, notable generalisation procedures are outlined to extend the modelling framework to incorporate popular one-factor short rate alternatives such as Black-Karasinski. Finally, calibration methods of the input variables are discussed as well as determining sensitivity metrics directly from... (More)
- Given the current market climate, in an era of negative interest-rates, the Hull-White model has regained popularity in the eyes of investors. This thesis aims to extend this model to incorporate credit risk, to allow the modelling of credit derivatives such as diff swaps, defaultable corporate bonds and credit default swaps. This process can be achieved in a number of ways, by utilising either two one-factor models or one two-factor model. Additionally, notable generalisation procedures are outlined to extend the modelling framework to incorporate popular one-factor short rate alternatives such as Black-Karasinski. Finally, calibration methods of the input variables are discussed as well as determining sensitivity metrics directly from the lattice trees themselves, used in hedging these derivatives which is of particular interest in practical applications. (Less)
- Popular Abstract
- All people, at some point in their life will likely seek a loan or mortgage from a bank, friend or private institution. Interest rates govern the cost of these luxuries, and are influenced by a range of factors including political influence, inflation and government intervention. This process is difficult enough to model itself, but it is not the only factor. A financial institution will also assess the inherent cost of risk (among other things), the ability of the borrower to repay their loan.
Corporations face the same dilemma. In an effort to raise capital, corporations often issue stocks or bonds rather than borrowing from banks or other financial institutions. Bonds are of particular interest, as they are cheaper and do not afford... (More) - All people, at some point in their life will likely seek a loan or mortgage from a bank, friend or private institution. Interest rates govern the cost of these luxuries, and are influenced by a range of factors including political influence, inflation and government intervention. This process is difficult enough to model itself, but it is not the only factor. A financial institution will also assess the inherent cost of risk (among other things), the ability of the borrower to repay their loan.
Corporations face the same dilemma. In an effort to raise capital, corporations often issue stocks or bonds rather than borrowing from banks or other financial institutions. Bonds are of particular interest, as they are cheaper and do not afford the owner of the bond any control of the company. Additionally, the interest paid from the issuer to the owner of the bond is usually less than that offered by the bank. This allows the corporation in invest in their growth and other projects.
That being said there is at least one distinguishing feature between a company and a government, a risk of default. Most governments are assumed to be risk-free and this is reflected in the cost and yield of their bonds. Corporate bonds are an attractive investing opportunity to a lender, where a greater risk is likelier to return a greater reward. Similar to stocks, bonds also have option features, where it is possible for the issuer to call their bond at a pre-determined price at pre-specified dates. There is one exception, the price of this exercisibility is embedded in the option price. It is not a stand alone product as in the case of the stock market.
Calculating the fair price of the bond, and in turn the cost of this embedded option is far from trivial. Apart from financial statements, there are additional factors such as market risk, operational and systematic risk as well as increased regulations which could influence the fair price of the bond. Thus, modelling this inherent default risk into a suitable yield curve is an art form.
Once all the relevant data has been obtained, one must still contend with a modelling framework to capture all the underlying information. A family of models, denoted affine term-structure models, are popular in practice. These models are particularly useful at relating the prices of bonds to market data, more specifically the yield curve. One of it's most popular children, the Hull-White model is making a revival in recent years due to the nature of the market, with many government curves yielding close to zero or in fact, negative interest rates.
This thesis will attempt to model the aforementioned underlying risks in the popular one-factor Hull-White interest rate model. Notably, this model will be extended to incorporate the default risk of corporations and other financial institutions. In doing so, the fair price of a corporate defaultable bond will be modelable. Additionally, the framework will provide a template to model a range of other credit derivatives such as credit default swaps. Finally, a routine will be established to allow this framework to be further extended under the class of affine term-structure models. This will outline the flexibility of the modelling framework from an investors' perspective, allowing the capture of the inherent risks under different modelling assumptions. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9080695
- author
- Müller, Jan
- supervisor
- organization
- course
- MASM02 20221
- year
- 2022
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- Option pricing, Callable bonds, Affine term structure models, Hull-White one-factor, Hull White two-factor, Trinomial trees, Short rate, Default intensity, Swaption volatilities, Black-76, Credit derivatives, Calibration, Optimisation.
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUNFMS-3108-2022
- ISSN
- 1404-6342
- other publication id
- 2022:E19
- language
- English
- id
- 9080695
- date added to LUP
- 2022-05-24 16:37:10
- date last changed
- 2022-06-10 15:49:12
@misc{9080695,
abstract = {{Given the current market climate, in an era of negative interest-rates, the Hull-White model has regained popularity in the eyes of investors. This thesis aims to extend this model to incorporate credit risk, to allow the modelling of credit derivatives such as diff swaps, defaultable corporate bonds and credit default swaps. This process can be achieved in a number of ways, by utilising either two one-factor models or one two-factor model. Additionally, notable generalisation procedures are outlined to extend the modelling framework to incorporate popular one-factor short rate alternatives such as Black-Karasinski. Finally, calibration methods of the input variables are discussed as well as determining sensitivity metrics directly from the lattice trees themselves, used in hedging these derivatives which is of particular interest in practical applications.}},
author = {{Müller, Jan}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{Pricing of Embedded Options: Implementing Stochastic Interest Rates & Stochastic Spread}},
year = {{2022}},
}