# A Novel IoT Sensor Authentication Using HaLo Extraction Method and Memory
Chip Variability

Holden Gordon
Department of Electrical and Computer Engineering
Santa Clara University
Santa Clara, CA 95050
<EMAIL_ADDRESS>
Thomas Lyp
Department of Electrical and Computer Engineering
Santa Clara University
Santa Clara, CA 95050
<EMAIL_ADDRESS>
Calvin Kimbro
Department of Electrical and Computer Engineering
Santa Clara University
Santa Clara, CA 95050
<EMAIL_ADDRESS>
Sara Tehranipoor
Department of Electrical and Computer Engineering
Santa Clara University
Santa Clara, CA 95050
<EMAIL_ADDRESS>

###### Abstract

In this paper, we propose flash-based hardware security primitives as a viable
solution to meet the security challenges of the IoT and specifically
telehealth markets. We have created a novel solution, called the High and Low
(HaLo) method, that generates physical unclonable function (PUF) signatures
based on process variations within flash memory in order to uniquely identify
and authenticate remote sensors. The HaLo method consumes $60\%$ less power
than conventional authentication schemes, has an average latency of only 39ms
for signature generation, and can be readily implemented through firmware on
ONFI 2.2 compliant off-the-shelf NAND flash memory chips. The HaLo method
generates 512 bit signatures with an average error rate of $5.9*10^{-4}$,
while also adapting for flash chip aging. Due to its low latency, low error
rate, and high power efficiency, the HaLo method could help progress the field
of remote patient monitoring by accurately and efficiently authenticating
remote health sensors.

_Keywords_ Physical Unclonable Function (PUF) $\cdot$ Internet of Things (IoT)
$\cdot$ Flash Memory

##  1 Introduction

Internet of Things (IoT) sensors has seen explosive growth over the last
twenty years. The consumer IoT market is estimated to reach 142 billion
dollars by 2026 at a CAGR of $17\%$. Estimates forecast that by 2025, there
will be 152,200 IoT devices connecting to the internet per minute. However,
the expanding IoT landscape has similarly created an increased attack surface.
Malware attacks targeting IoT have increased by $30\%$ with $76\%$ of IoT risk
professionals believing their organization’s IoT security posture leaves them
vulnerable to cyber attacks [1]. This is particularly important for the subset
of IoT devices such as medical sensors that are used in telehealth
applications [2, 3, 4, 5].

Telecommunication in healthcare, or telehealth, provides a means by which
patients can interact with medical professionals and health-related services
virtually. In light of the COVID-19 worldwide pandemic, the need for secure
and readily available remote patient monitoring has never been more important.
Rural and low-income communities, in particular, have been severely impacted
by the lack of accessibility to in-person healthcare. This has created the
need for access to remote patient monitoring and virtual health visits in
order for greater accessibility to premier care. However, the convenience of
connecting medical providers with patients remotely also introduces
significant security and privacy risks. One such risk of using unsecured
medical devices is the potential for a major privacy breach, as they store
sensitive information such as vital signals, diagnosed conditions, therapies,
and a variety of personal data [6]. For example, on the dark web, an
individual’s private health information (PHI) goes for 20 times to 100 times
the value of a social security number or credit card. This creates a strong
financial incentive for malicious actors to illicitly steal healthcare data
from small embedded telehealth sensors which are the most exposed computing
elements due to the tight processing, energy, and latency requirements of IoT
devices [7].

Physical Unclonable Functions (PUFs) can provide a lightweight and tamper-
proof security primitive for IoT devices particularly telehealth sensors [1,
8, 9]. PUFs create cryptographic signatures that can be used for
authentication [10], software attestation, and cryptographic key generation
[11, 12]. These signatures are derived from the sub-micron process variations
present in integrated circuits (ICs) [13]. These signatures are never stored
on the device itself and in many cases are extremely difficult to tamper [14,
15].

In this paper, we propose a novel PUF extraction scheme called the High-Low
method (HaLo), which extracts process variations found in commercially
available NAND flash memory chips. The HaLo method aims at minimizing
authentication latency and maximizing the lifetime of the flash chip by
limiting program/erase cycles. The HaLo method is also proven to work using
commercially available NAND flash chips while being entirely controlled by a
microcontroller that costs under $15, making it one of the most cost effective
published flash-based solutions to our knowledge. Our main contributions are
summarized as follows:

  * •

We have built a novel PUF extraction technique called the HaLo method, which
supports edge deployment on low-cost microcontrollers. This will lower the
cost of entry, and help encourage secure data transmission for remote devices.

  * •

The proposed HaLo method offers a PUF solution for accurate authentication and
has lower latency and lower power consumption than other PUF generation
techniques.

  * •

The HaLo method is compatible with off-the-shelf ONFI 2.2 compliant flash
chips, which could lead to backward compatibility implementation on existing
health sensors.

The organization of the remainder of this paper is as follows. First, Section
II will discuss the preliminary background information required in order to
understand the HaLo method. Next, Section III will give an in-depth look at
related published authentication schemes using process variation, as well as a
look at the advantages of the novel HaLo extraction method. Section IV will
discuss the HaLo extraction method in detail including design constraints and
considerations, and Section V will provide experimental results and validation
of the novel HaLo method. Finally, Section VI will provide details on the
telehealth application proof of concept built with the HaLo extraction method,
and Section VII will give a brief conclusion and summary of the work, along
with details on proposed future work for this project.

##  2 Preliminaries

In this section, we will briefly discuss the general background information
involving our proposed authentication scheme. Specifically, this section
covers an introduction to current wireless authentication protocol solutions,
a brief understanding of process variations found in flash memory chips, and a
general understanding of PUFs [16].

###  2.1 Wireless Authentication Methods

There are a variety of wireless authentication methods that are used by IoT
and Telehealth devices. One of the most common ways to authenticate these
resource constrained devices is by utilizing pre-shared keys [17]. Pre-shared
keys are authentication keys or tokens that are shared with a device prior to
its deployment. These keys are then exchanged with a gateway in order to
authenticate an IoT device. There are several important vulnerabilities within
this model that are solution wishes to address. First, the keys can be
extracted out of firmware by a sophisticated hacker who has access to the
physical device. This has happened in the industry with Link Plugs and other
smaller IoT devices [18]. Secondly, these keys can be cloned through replay
attacks and deauthentication attacks [17, 19]. This was shown to be effective
on all WEP Wifi routers which extracted wifi keys due to the lack of
’freshness’ in the messages sent between endpoints and wifi routers. By using
PUFs and TRNGs, many of these prior security vulnerabilities can be
drastically mitigated by providing random nonces and authentication signatures
[20]. Firstly, the PUFs allow secrets to be embedded in process variations
which are not stored in any nonvolatile memory which prevents hackers from
simply dumping onboard firmware and finding pre-shared keys. Secondly, the
onboard TRNG mitigates replay attacks by preventing attackers from arbitrarily
replaying authentication messages.

###  2.2 MLC NAND Flash Memory Architecture

Figure 1: A) Overview of NAND Flash Cell B) MLC Flash Chip Digitization

Flash NAND memory is a type of non-volatile memory that stores user data in
the physical form of charge on a float gate. Programming these cells requires
electrons to move from the polysilicon channel on/off the floating gate via
electron tunneling, described in figure 1A. It is important to note that this
electron tunneling can damage the tunnel oxide, which means that flash cells
become less reliable after many program/erase cycles (PECs). Most flash memory
is rated anywhere from 1,000 to 10,000 PECs. A flash’s lifetime can increase
drastically by using a memory management controller that distributes PECs
evenly across all cells called wear leveling.

In the case of Multi-Level Cell (MLC) flash chips, 2 bits of data are stored
on each cell, where the current float gate voltage is compared to multiple
threshold voltages in order to determine the cell’s digital value, as shown in
Figure 1B. Flash has high memory density, low cost, and ability to be
programmed and erased electronically without moving parts, flash NAND memory
has exploded in popularity for remote devices and IoT systems.

###  2.3 Process Variations and PUFs for Flash Memory

Like all other silicon-based ICs, flash memory chips are subject to many
different forms of process variations. In general, most process variations are
uncontrollable imperfections caused by limitations in modern lithography
processes. These variations are commonly caused by various disturbs caused by
parasitic capacitance. By performing multiple read or program operations on
sections of the flash chips, disturbs can be induced on these chips causing
random fluctuations and bit flips. These fluctuations depend on each cell’s
relative gate thickness and width. both of which are uncontrollable and
extremely hard to model. Finally, this allows for unique signatures that can
be generated for each page of the flash memory. These signatures are known as
PUFs which can be used for secure key generation and authentication mechanisms
[21]. Each PUF has a challenge and response. The challenge is the input to the
function which then outputs an unclonable response. Key mathematical metrics
are used to describe PUF efficacy which will be touched on in more detail in
Section VI.

###  2.4 Physical Unclonable Functions (PUFs)

A PUF is a function that produces a unique signature based on challenge-
response pairs. These unique signatures often referred to as silicon
fingerprints, are by-products of intrinsic, uncontrollable process variations
found on a silicon-based IC [ProcessVariation]. As mentioned in a previous
section, these process variations are present on every chip, and they can
greatly differ from one another in regards to performance. While understanding
specific instances of process variations can help influence the design for PUF
extraction, the absolute variations do not typically matter.

##  3 Related Works

Flash memory has gained popularity in recent years due to its cheap cost and
high density for storage applications. This is particularly the case for Flash
NAND memory [22]. Rather than including extra CMOS as a PUF, utilizing the
onboard Flash memory can conserve space and energy for an extremely resource-
constrained devices such as those seen in telehealth applications.

When looking at the development of different Flash PUFs, there are several
important trends to recognize. The first Flash PUFs were created from
2012-2017 [23, 24, 25, 26]. These seminal works were predominately focused on
showing how Flash memory was a viable candidate for memory-based PUFs. These
Flash PUFs were important for showing the promise of flash memory, but they
struggled to account for several factors such as aging, temperature, design
constraints. For example, Prabhu et al. [23] required hundreds of thousands of
programs in order for PUF signatures to develop which can take several hours.
Similarly, Kim et al. [27] required very fine-grained flash programming which
required knowledge of sensing voltages that are typically proprietary on
commercial flash chips. Similarly, Wang et al. [24] created both a novel PUF
and TRNG but incurred very high processing overheads that leads to high
latency.

From 2018 to 2020, Flash PUF development began marked by the work of [28, 29,
30, 31, 32, 33]. Many of these proposed designs significantly enhanced Flash
PUF architectures that took into account many different factors. These Flash
PUFs incorporate advanced parameters such as aging resistance, temperature
resistance, and unique architectures. For example, work from Clark et al. [29]
designed a Flash PUF that was voltage resistant. Secondly. Poudel et al. [30]
designed a Flash PUF that works on the onboard microcontroller NOR Flash
Memory. Similarly, Larimian et al. [33] verified the machine learning
resilience of their Flash-based PUFs by performing extensive deep learning
tests. Finally, Mahmoodi et al. [31] created one of the most stable and
resilient Flash PUFs by modifying the cells to extract leakage current.

However, several open challenges remain. In order to make PUFs commercially
viable, they must be generated using cheap commercial microcontrollers in
order to keep their price point down. Furthermore, it is helpful that these
PUFs can be deployed on legacy systems by using commercial off-the-shelf flash
memory and these systems should avoid excessively long latency such as those
seen in [23]. Although, even some of the most cutting-edge work do not use
commercial off-the-shelf components. This can reduce the adoption of Flash PUF
technology. Secondly, many of these Flash PUFs require slowly building charge
differences on the floating gate through hundred or thousands of program/erase
cycles. This is effective at generating signatures; however, this drastically
increases the latency required to generate these PUF responses. Finally, many
of the systems that avoid the slow build-up of charge have to use expensive
FPGAs with Gigahertz clocking speeds to generate fine-grained interrupts as
shown in [29]. This is not viable for edge deployments particularly those such
as telehealth-based applications.

Our work seeks to bridge this gap by developing a Flash-based PUF that uses
only ONFI standard commands; works on a 15 dollar microcontroller; uses off
the shelf commodity flash chips [34], generate signatures with low latency
using a novel interrupt technique; and is able to compete with some of the
most resilient PUF structures identified in previous work.

##  4 HaLo Flash PUF Extraction Technique

HaLo extracts process variations found in flash NAND memory cells for
authentication applications. This section will discuss the HaLo experimental
setup, design considerations for PUF extraction, and the HaLo method itself.

###  4.1 Experimental Setup

As mentioned in our summarization of contributions section, it is vital to
keep the experimental setup as minimal as possible in terms of both cost and
complexity. While previous works have mostly required high clock frequencies
and custom chips, our novel design uses a 100 MHz clock and a simple TSOP
Adapter in order to connect to the flash chip. Our simple bill of materials
consists of:

  * •

STM32 based microcontroller with 100 MHz maximum clocking frequency. This will
serve as the memory controller and is used to read/write/erase the flash
memory.

  * •

32 GB MLC Flash NAND memory chip from Micron that does not have an integrated
memory controller. The memory chip can read/store application data and is also
used in the HaLo extraction method

  * •

TSOP Adapter in order to interact with packaged flash NAND chip

The entire experimental setup is purchased for just shy of $20, and all items
were readily available for purchase to be included in a commercial
application. It is worth noting that we used an unmanaged flash chip in order
to give us full control of the device. This extra control allowed us to write
directly to specific memory locations without worrying about wear-leveling
control or error-correcting code (ECC), which are implemented in most managed
flash components. This gave us the highest control on interrupting the
writing, erasing, and reading processes, because we were directly controlling
the chip’s behavior without interference from the memory controller. The HaLo
method only requires the most basic memory access and modification functions
including read page, write page, and erase block.

The experimental setup is shown in Figure 2. The memory chip interface was
Open NAND Flash Interface (ONFI) version 2.2, so the interrupt sequence and
reading of data are ubiquitous across all ONFI 2.2 memory chips. ONFI 2.2
requires 7 control signals and an 8-bit bus for data, as shown with jumper
cables connecting the Flash NAND Chip to the Memory Controller. Additionally,
a USB connection was made between the Memory Controller and a computer. PUF
trial data was transferred via the USB cable so that all statistical testing
could be performed on the computer. A computer is simply an analysis tool, and
no part of the actual PUF extraction method is performed on the computer.

The flash chip tested in this work stores pages consisting of 4096 bytes of
data and 224 extra bytes for ECC. Blocks are organized in chunks of 256 pages,
with each 32 Gb chip having access to 4096 unique blocks [35, 36].

Figure 2: Experimental Setup Diagram

###  4.2 Proposed PUF Extraction: Design Considerations

With the experimental setup described in the previous section, work began on
the PUF extraction method itself. When designing the PUF extraction method,
there are two major metrics that are primarily considered. The first feature
was minimizing the number of program/erase cycles (PECs) required in order to
extract a reliable signature. As mentioned in the Preliminaries section, flash
memory devices have a limited lifetime measured in PECs. As the charge is
tunneled onto and off of the floating gate, the tunnel oxide that separates
the silicon channel and the floating gate begins to deteriorate. As the
deterioration continues, the data held within each cell becomes unreliable due
to the increase of charge leakage on the floating gate. Wear on the cells is
an issue for data retention and general application use, but it also poses an
issue with unreliable signature extraction, because the programming and erase
times for each cell are altered as the cell reaches its end of life. In order
to combat this aging effect, HaLo was designed in order to require as few PECs
as possible.

The second major consideration was lowering the signature extraction latency.
Faster signature extractions allow for faster authentication and lower power
consumption [37]. Telehealth sensors in the scope of this work can be treated
as edge devices that are resource-constrained by both processor speed as well
as limited battery life. In order to prolong the sensor’s battery lifetime as
well as create a reasonably fast authenticated connection, the method needs to
be lightweight and fast.

While additional results and metrics will be discussed in detail in the
Experimental Results and Validation section below, these two design
considerations helped guide the construction of the HaLo method. In the next
section, the HaLo technique will be explained in detail, and additional design
constraints and considerations will be discussed.

###  4.3 PUF Extraction Observations and Techniques

As mentioned in the Related Works section, the PUF extraction must have low
latency and can only use standard edge deployable microcontrollers within the
100MHz frequency range. This introduces several important design challenges in
order to make Flash PUFs achievable on low-cost microcontrollers. First, the
low latency requirement prevents the design from utilizing hundreds of
repeated program and read cycles that slowly increase the charge on the
floating gloat. Therefore, our scheme must use fine-grained interrupts in
order to interrupt the operation of the cells to force the flash into unsteady
positions. However, a single interrupt scheme such as the one seen in Clark et
al. [29] does not have a high enough clock resolution to interrupt the flash
programming fast enough to generate a 50/50 split between ones and zeros. In
fact, signatures that are generated from a single programmed interrupt are
either about $80\%$ 1’s or $80\%$ 0’s. This is highlighted in Figure 3 where
the sysTick clock on the microcontroller is interrupted at different times. As
shown in the figure, the clock does not have the granularity to interrupt the
programming operation on the microcontroller to generate an even distribution
of 1’s and 0’s. This creates two distinct regions detonated by a low
programming interrupt and a high interrupt programming interrupt where the low
interrupt generates signatures with about $80\%$ ones and the high programming
interrupt generates signals with about $80\%$ zeros. The steep drop-off occurs
within a single sysTick clock instruction. This makes interrupting the program
a distinct challenge.

Figure 3: Ratio of 1’s and 0’s for Different Program Interrupt Times

Secondly, the lack of granularity has another limiting effect as well. The
signatures generated from a single interrupt are extremely noisy. This is due
to the interrupting method. This error rate can approach $10\%$ within 80
total P/E cycles. This is due to the lack of granularity in the clock itself.
Due to the max 100 MHz clocking speed, an interrupt at a particular clocking
delay can vary due to the limited frequency. This causes unintended errors for
cells that are sensitive to the interrupt. This necessitates several programs
in the enrollment to ensure that the bytes are stable across a small variation
in the actual interrupt delay.

Although the error increases rapidly and the clocking speed cannot generate
signatures with $50\%$ ones and zeros, several more insights and solutions
were generated to remedy these design challenges. Specifically, a well-defined
enrollment scheme is designed to select only the most stable cells which can
be decoded as one or zero. These can then be used for highly stably
signatures.

###  4.4 Proposed PUF Extraction: Enrollment Scheme

As other work has shown in Poudel et al. [30] unstable cells can be identified
for TRNG bits by applying several reads. These reads apply a smaller voltage
to the floating gate of the flash cells which only slightly disturbs the
cells. This can quickly identify unstable cells that are flagged during
enrollment. Furthermore, approximately $95\%$ of these unstable bits flip
within five reads. Therefore, only applying five reads is sufficient for
identifying stable cells through successive read operations. Figure 4
highlights this observation. This figure graphically shows all 32000 bits on a
single page. All of the yellow lines indicate a flipped bit. Many of these
bits flip within the first five reads which is an effective filtering process
for identifying stable bits within a single read. Combining multiple reads can
help identify unstable bits per page along with multiple programs which helps
to identify unstable bits due to lack of clocking granularity.

Figure 4: Error Increase Over 100 Trials

Secondly, flash cell failure is highly spatially dependent. This is evidenced
by plotting a histogram of the distance between each cell flip or failure.
Failures tend to cluster in groups. This can be modeled as a negative
exponential distribution between errors. Therefore, instead of flagging
individual bits that are stable, bytes are flagged as stable and only if a
byte is completely stable is it passed through the enrollment process.

Combining these observations, the first enrollment strategy was crafted. An
interrupt program is used on the low side and another is applied to the high
side. Then five reads are performed. If bytes remain stable, across the low
interrupt and the high side interrupt they are flagged as stable. After this
technique is applied about 1500 bytes are extracted per page which has a total
of 8000 bytes. Approximately 800 of these bytes are over $98\%$ accurate
during testing. However, approximately 700 bytes are underneath this
threshold. This is due to the fact that many stable bytes are incorrectly
flagged during the enrollment process. However, a slight wrinkle to the first
algorithm is able to generate highly stable PUF responses. This leads to our
new enrollment strategy called the HaLo method.

Figure 5: Histogram of Distance Between Sequential Failures

The HaLo extraction method uses a novel technique to generate highly reliable
signatures that begins with a byte selection process. Reiterating the previous
strategy, rather than aborting the program early and using signatures with
$80\%$ ones, two different types of interrupted programs are performed per
page: the high interrupt and low interrupt. The low program generates a
signature with $80\%$ ones and the high program has signatures with
approximately $20\%$. This is repeated five times on each low programming
interrupt and high programming interrupt. Since errors tend to be close
together, only bytes that are entirely stable are chosen. This creates two
separate sets each of which contains 25 signatures due to the five reads per
five programs. Next, the stable high bytes locations and the stable low byte
locations for a single page are identified. Then bytes that are either highly
resistant to programming or highly susceptible to over-programming are chosen.
This information is captured by selecting bytes that fully resist programming
in the under-programmed section also known as the low side and which bytes are
easily programmed in the over-programmed section also known as the high side.
If a byte group is simultaneously fully programmed in the over-programmed
section and is fully under-programmed in the under-programmed section it is
also removed during the enrollment process. This is because greater than
$97\%$ of errors are from the aforementioned 700 bytes which are stable in
both sections. When referencing Figure 6, the only byte selected from the low
side is Byte 2 and the Byte selected for the high side is Byte 8000. This then
creates a ’map’ for each page of the stable low bytes and stable high bytes.

After enrollment, the telehealth device receives a list of byte locations to
provide a high program and low program along with the page number to apply the
programming. It will do two separate programs: one high and the other low.
Then, the majority bit is selected from each byte and is decoded properly.
Collisions are minimized since ’mapped’ byte values are either highly
resistant to over-partial programming or susceptible to it. However, if it
occurs the byte with the most amount of parity bits when decoded will be
selected. Furthermore, any undefined states (such as equal hamming weights of
four) are decoded as low since more errors come from the high side than the
low side. This process is highlighted in Figure 7. This process reduces the
percent error by several magnitudes of 10 to around $10^{-4}$. A major
downside to this extraction process is that it consumes a tremendous amount of
bits to extract only the most stable bits and performs two programs instead of
one. Specifically, each page consists of 32,000 bits has an output response of
about 500 bits.

Figure 6: This figure highlights the enrollment process where low interrupt
and high interrupt signatures are generated to provide a ’map’ of stable low
byte and high byte values. These values are set along with the page number to
compromise a challenge.  Figure 7: This figure highlights the decoding process
for a device generating a response to a PUF challenge. First, the device
applies the two programming intervals to the page requested in the challenge.
Then, the stable bytes are located and are also sent in the challenge to the
end device. The device then applies the programming and chooses the majority
represented bit in each signature. In the event of a collision the byte that
has the strongest weight is selected. This is why the collision scenario in
the figure decodes to 0.

###  4.5 Proposed PUF Extraction: Challenge and Response

After the HaLo enrollment process is complete a ’map’ of the stable bytes is
stored on the gateway. Each byte in the map is either extremely susceptible to
over-programming or highly resistant to it. With this information, the gateway
will request particular byte locations from the sensor for authentication.
Approximately 256 low byte locations and 256 high byte locations will be sent
along with the specific page number that is used. This compromises
approximately 600 bytes of space which can be sent in the payload of a single
Ethernet frame to the sensor. The microcontroller will then apply two programs
and ten total reads and identifies which bytes are stable and which are not.
If a byte location produces a majority of ones it is subsequently decoded as
one and is assumed to be one of the high side bytes. Inversely, if a byte
produces a majority of zeros than it is assumed to be from the low side and is
decoded as zero. This extraction technique can resist a maximum of three
errors before a byte is incorrectly decoded. Therefore, the gateway will
receive a bit string of ones and zeros to the sensor.

This design allows for two important features for the gateway. Firstly, the
gateway can control how long of an authentication response it needs. This can
allow our application to adapt to various levels of required security. For
example, certain cryptographic applications may only require 100-bit
signatures. The gateway then only has to send 100 bytes to the sensor. On the
other hand, sensitive security tasks that may demand 512-bit signatures can
also be accomplished. Secondly, many other approaches use helper data such as
Hamming Codes, Fire Codes, or more recently Low-Density Parity Codes, to
recover any errors sent back from the device. These codes leak polarity
information about the response values which allows for error correction. Our
scheme leaves out any polarity data and simply sends bytes to read. This makes
our helper data significantly more robust to side-channel or modeling attacks
since the helper data never reveals any polarity information from the bytes it
is requesting. However, it is important to note that this advantage is
realized because the HaLo enrollment algorithm filters bytes from pages very
aggressively. On average, each page returns approximately 700 usable bytes.
Therefore, about $83\%$ bytes are not usable.

Figure 8: Authentication Protocol

The final step in the HaLo extraction method is authenticating a sensor by
issuing a challenge and verifying the response. While most memory based PUF
solutions use a challenge-response scheme that links a challenge value with a
specific page in memory, HaLo extraction includes additional information in
the challenge packet. As mentioned in the PUF Extraction Observations and
Techniques section, the granularity of the interrupted program is not high
enough to generate reliably stable signatures, due to the fact that some cells
appear stable in small sample sizes. In order to combat this, the challenge
issued to the sensor includes the location in memory, as well as a list of 512
stable byte locations on both the high side and low side of the page
programming. It is important to note that these byte indexes do not reveal any
polarity of the bytes themselves – they are simply represent a map of bytes
that were identified as extremely stable in the enrollment process. The basic
authentication procedure is shown in Figure 8.

##  5 Experimental Results and Validations

In this section, we will discuss the relevant PUF metrics of the novel HaLo
extraction process. All of the results generated in this section were gathered
at room temperature using the experimental setup discussed in the Proposed
Flash PUF section. The main characteristics of the HaLo PUF extraction method
are its reliability over multiple challenges, the uniqueness of signatures on
every page, and the minimal time required in order to generate a signature.

###  5.1 PUF Metrics

There are three major metrics for any PUF. They are uniqueness, randomness,
and reliability. Uniqueness defines how different each PUF response is from
the other. This value is best captured through the Inter-Hamming Distance
(Inter-HD). The percentage Inter-HD describes what percentage of bits flip
between two different responses. An ideal value for this is approximately
$50\%$. Our system had an average of $47\%$ which is shown in Figure 10. The
second metric randomness describes how random each signature is. This can be
measured in Shannon entropy per bit. With 256 high bits and 256 low bits, the
average Shannon entropy per bit is approximately 0.999 with an ideal value of
1. Next, reliability is the final measurement. The reliability of this system
is extremely strong. The error does change as the flash cells age; however,
the average percent error is approximately $7.1*10^{-6}$. The max percent
error value is $1.9*10^{-2}$ and the minimum value is $5.9*10^{-7}$. This
reliability is strong enough to possibly support cryptographic key generation
and is able to last through the end of the Flash’s life this is shown in
Figure 11 and Figure 9.

###  5.2 Aging Adaptation

In order to simulate aging, the Flash memory chips were programmed until their
maximum rating which ranges between 3,000 to 4,000 P/E cycles for MLC chips.
The aging causes the oxide to deteriorate from the program voltage stress. The
low side bytes that are susceptible to over programming are barely modified
since these bytes just program and leak charge faster [38]. However, the high
side bytes that are resistant to programming more easily modify the amount of
bit flips in each byte. In general two approaches for this were considered.
First was adapting the polling interval and decrease it since the flash cells
program faster. However, this approach can be difficult to model and control
due to the lack of granularity in the interrupt mechanism. Consequently, the
second approach was taken. In this approach, the number of bytes required for
a decode was changed once the cells reached a particular percentage of life
used. This adapts the error rate and drops it significantly. Instead of taking
the majority (or in our case 5 bits) for a decode. The decode threshold was
varied according to the life of the flash cells. At $50\%$ of the lifetime,
the amount of bits required for a proper decode becomes 6. Then, this changes
to a threshold of 7 at $90\%$ which remains fixed and keeps the error rate
significantly lower. This trend is reflected in Figure 11. All of the
aggregate statistics are reflected in Table LABEL:tabel-stats.

Figure 9: Reliability of 250 Responses at $50\%$ of Life Used Figure 10:
Inter-Hamming Distance Calculations between 250 Responses Figure 11: Error
Rate vs Percentage of Life Used

###  5.3 PUF Latency and Power Consumption

Finally, our PUF will be compared in latency and power consumption to an AES
encryption scheme on one of the most portable AES implementations for
resource-constrained devices, Tiny AES. To just encrypt a preshared key about
22.3 mW of power is used and the encryption process takes around 190 ms.
However, the HaLo extraction technique takes less than 1mW of power and PUF
signatures are generated in 34.8 ms. This relative power difference is shown
in Figure 12. This highlights a significant performance advantage with a
higher security guarantee since the PUF signatures are never stored anywhere
[39]. Furthermore, this low latency and power consumption make the HaLo PUF a
strong candidate for the telehealth application space.

Figure 12: Relative Power Consumption and Latency Comparison between the FPUF and Traditional Tiny AES Encryption Scheme Table 1: PUF Entropy, Reliability, and Randomness Metrics Metric | Minimum | Average | Max
---|---|---|---
Shannon Entropy | 0.99 | 0.99 | 0.99
Reliability | $5.9*10^{-7}$ | $7.1*10^{-6}$ | $1.9*10^{-2}$
Inter-Hamming Distance | $47.2\%$ | $51.0\%$ | $53.1\%$

##  6 Telehealth Application

The first step in developing our authentication protocol was to build our
telehealth application, which involved using a DS18B20 temperature sensor to
collect temperature data and store it in a format that we can then use for
edge deployment. Body temperature collection is just one of the many different
ways remote patient monitoring can be utilized. For our experiment, we used a
Raspberry Pi as a means to collect temperature data in intervals of 10 seconds
and saved the data in a .csv format.

We then built a TCP/IP dynamic challenge-response authentication scheme using
the PUF. This authentication process starts with the gateway sending a
challenge to the health sensor, denoted Challenge1. This request includes a
challenge location and stable byte locations of the NAND flash memory cell.
The health sensor performs the interrupted program and extracts the stable
byte values. The health sensor then randomly generates a nonce, denoted
Nonce1, and the generated hash value is XORed with Nonce1. The value generated
is sent to the gateway as a response to Challenge1. The gateway knows what
Response1 should be, so it XORs the entire response of the gateway with
Response1 to determine Nonce1. The gateway then randomly generates another
nonce, denoted Nonce2. A second challenge, using a different index of stable
byte locations, is generated by the gateway. Challenge2 is then concatenated
with (Response1 XOR Nonce2), and this message is sent back to the health
sensor. The health sensor is able to separate Challenge2 from (Response1 XOR
Nonce2), and (Response1 XOR Nonce2) is XORed with Response1 to determine
Nonce2. Both the gateway and the health sensor now know the values of
Challenge1, Challenge2, Nonce1, and Nonce2. Using Challenge2, the health
sensor again performs the interrupted program and extracts the stable byte
values. By the end, if the health sensor and the gateway are legitimate
sources, the challenges and responses can be properly decoded and the
transaction is authenticated.

There are many advantages to using this authentication scheme. First, the use
of hashes masks the plaintext values of the data being sent, meaning that
attackers won’t be able to read the data in transit. Second, the use of
randomly generated nonces means that the values being transmitted will always
change with every transaction. Finally, and most importantly, the use of a PUF
provides a unique identifier that the gateway can reliably authenticate, and
attackers won’t be able to model the authentication responses of the PUF.

Here we see the challenge and response pairs that we discussed being sent
between the health sensor and the gateway. Man-in-the-middle and replay
attacks can occur in our any networked environment [40, 41]. However, with
regard to MITM, any modification of the packets being sent would result in a
complete breakdown of the authentication process, ultimately making it obvious
that the connection was tampered with, and would result in the gateway
refusing the connection [42]. With Replay attacks, the use of randomly
generated nonces means the values being sent change with every transaction.
This means that an attacker can’t replay a message with an old pair of nonces,
as the values would be entirely different to the current nonces [43, 44].

##  7 Conclusions and Future Works

As medical care continues to improve in both effectiveness and accessibility
on a global scale, remote patient monitoring is a logical next step for
healthcare investment. We have witnessed a global pandemic that disrupted the
delivery of potentially life-saving medical care for over a year, which may
have been partially mitigated by the adoption of remote patient monitoring
(RPM) technology. While the technology proves useful for many medical
monitoring applications, it is important to carefully consider the security
vulnerabilities and possible exploits of large-scale adoption of RPM. The HaLo
extraction method is a lightweight, fast, and easily implementable security
measure that could help ensure the authenticity of RPM sensor data. The HaLo
method is designed in such a way that it could be implemented on new RPM
sensors with minor software updates, and no additional components. The method
is resource-efficient and offers financial benefits when compared with many
other commercial PUF solutions.

When considering future work for this project, more extensive testing needs to
be done using additional flash chips from different manufacturers.
Additionally, testing must be done in high temperature environments in order
to verify that the reliability remains high regardless of external factors
such as heat. Finally, the HaLo extraction method could be tested on other
flash memory densities (SLC/TLC), or possibly even different flash
architectures (3-D) in order to broaden the possible application space.

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# Solution Representations of Solving Problems for the Black-Scholes equations
and Application to the Pricing Options on Bond with Credit Risk

Hyong-Chol O ∗, Dae-Song Kim, Dae-Song Choe<EMAIL_ADDRESS>Faculty
of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s
Republic of Korea

###### Abstract

In this paper is investigated the pricing problem of options on bonds with
credit risk based on analysis on two kinds of solving problems for the Black-
Scholes equations. First, a solution representation of the Black-Scholes
equation with the maturity payoff function which is the product of the power
function, normal distribution function and characteristic function is
provided. Then a solution representation of a special terminal boundary value
problem of the Black-Sholes equation is provided and its monotonicity is
proved. The simplest case of the structural model of the credit bond is
studied and its pricing formula is provided, and based on the results, the
pricing model of option on corporate bond with credit risk is transformed into
a terminal boundary value problem of the Black-Scholes equation with some
special maturity payoff functions and the solution formula is obtained. Using
it, we provide the pricing formulae of the puttable and callable bonds with
credit risk.

###### keywords:

Black-Scholes equation; option on bond with credit risk; structural model;
puttable bond; callable bond MSC2020:  35C15, 35K15, 35K20, 35Q91

##  1 Introduction

Solving problems for the Black-Scholes equation become pricing models of
various financial derivatives according to the forms of maturity payoff
functions, and it is important to obtain solution representations with
complicated maturity payoff functions. [1] provides solution representations
of the terminal value problems of the Black-Scholes equation with the maturity
payoff functions which are the products of power functions with exponents 0
and 1, cumulative distribution functions of normal distribution and
characteristic functions, and using them provides the pricing formulae of the
exotic options with two expiry dates. [7] provides a solution representation
of the terminal value problem of the Black-Scholes equation with the maturity
payoff function which is the product of the power function with any real
number exponent and cumulative distribution function of normal distribution,
and using it obtains the pricing formulae of some the exotics including
savings plans that provide a choice of indexing.

On the other hand, recent research has been carried out on pricing problems
for complicated composite options such as bond options. [4, 13, 14, 15]
studied the pricing problem of American bond options and [11, 12] studied the
pricing coupon bond options as interest rate-derivatives under the Hull-White
model. These are all the pricing problems of bond options without credit risk.
[9] provides a partial differential equation model of the price of a Discrete
Coupon Bond with Early redemption provision (with credit risk) and an
analytical solution representation, their approach is to analyze together by
adding an early-redemption provision to the pricing model of the discrete-
coupon bonds of [10] rather than from the viewpoint of the bond option.

The purpose of this paper is to investigate the pricing problem of options on
bonds with credit risk based on analysis on two kinds of solving problems for
the Black-Scholes equations. According to experts, the pricing problem of
options on bonds with credit risk is a challenging problem. All corporate
bonds with an early redemption provision may be considered as a bond option
with credit risk.

In this paper, we study a simple structural model of pricing option on bond
with credit risk.

We first provide a solution representation of the Black-Scholes equation with
the maturity payoff function which is the product of the power function,
normal distribution function and characteristic function. This solution
representation includes all the solution representations given in [1, 8, 7] as
special cases. Then we provide a solution representation of a special terminal
boundary value problem of the Black-Sholes equation and prove the monotonicity
of its solution. Unlike the monotonicity results of the solution to the
terminal value problem of the Black-Scholes equation (e.g., [6] et al), the
monotonicity of this solution to the terminal boundary value problem of the
Black-Scholes equation, is not easily derived and is proved based on a
rigorous estimates of the cumulative distribution function of normal
distribution and its derivatives.

Then we provide the simplest case of the structural model of the straight
credit bond (i.e. credit bond without early redemption provision) and its
pricing formula and prove the monotonicity of its pricing function. Then using
it, we transform the pricing model of option on corporate bond with credit
risk to a terminal boundary value problem of the Black-Scholes equation with
some special maturity payoff functions and obtain a solution formula. Using
it, we provide the pricing formulae of the puttable and callable bonds with
credit risk.

The rest of the paper is organized as follows. In Section 2, we provide
solution representations of two kinds of solving problems for the Black-
Scholes equations arising from investigation of pricing options on credit
bonds and study the monotonicity of the solutions. In Section 3 we provide a
pricing formula of the simplest case of the structural model for pricing
credit bond (a terminal boundary value problem of the Black-Scholes equation)
and the pricing formulae of puttable and callable bonds with credit risk.

##  2 Solving problems for the Black-Scholes equations and the monotonicity of
the solutions

###  2.1 A solution representation of a terminal value problem of Black-
Scholes equation with a special maturity payoff functions

The following notation is used:

$\delta\left(\frac{X^{i}}{L},\;\tau_{1},\;\tau_{2},\;\tau_{3}\right)=\left[\ln\frac{X^{i}}{L}+\left(r-q-\frac{\sigma^{2}}{2}\right)\tau_{1}+\sigma^{2}\tau_{2}\right](\sigma\sqrt{\tau_{3}})^{-1}(i\in\mathbb{R}).$

######  Theorem 1.

For the terminal value problem

$\frac{\partial V}{\partial
t}+\frac{1}{2}\sigma^{2}X^{2}\frac{\partial^{2}V}{\partial
X^{2}}+(r-q)X\frac{\partial V}{\partial X}-rV=0,\;\;0<t<T,\;\;X>0$ (1)
$V(X,\;T)=X^{\beta}N[\delta(X^{i}/L,\;\;\tau_{1},\;\;\tau_{2},\;\;\tau_{3})]\cdot
1\\{X>K\\}$ (2)

we have the following solution representation

$V(X,\;\tau;\;\tau_{1},\;\;\tau_{2},\;\;\tau_{3},\;\;i,\;\;L,\;\;K)=X^{\beta}e^{\mu(\beta)\tau}N_{2}(a_{1},\;a_{2};\;\rho).$
(3)

Here

$N(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^{2}/2}dt,\mu(\beta)=-r+\beta\left[r-q-\frac{\sigma^{2}}{2}(1-\beta)\right],$
$\tau=T-t,\;\;\rho=i\sqrt{\frac{\tau}{\tau_{3}+i^{2}\tau}},$
$N_{2}(a_{1},\;a_{2};\;\rho)=\frac{1}{2\pi\sqrt{1-\rho^{2}}}\int_{-\infty}^{\;a_{1}}\int_{-\infty}^{\;\;a_{2}}e^{-\frac{y_{1}{}^{2}-2\rho
y_{1}y_{2}+y_{2}{}^{2}}{2(1-\rho^{2})}}dy_{2}dy_{1}$
$a_{2}=\pm\left[\ln\frac{X}{K}+\left(r-q-\frac{\sigma^{2}}{2}\right)\tau+\sigma^{2}\beta\tau\right](\sigma\sqrt{\tau})^{-1}=\pm\delta(X/K,\;\;\tau,\;\;\beta\tau,\;\;\tau)$
$\begin{array}[]{l}{a_{1}=\left[\ln\frac{X^{i}}{L}+\left(r-q-\frac{\sigma^{2}}{2}\right)(\tau_{1}+i\tau)+\sigma^{2}(\tau_{2}+i\beta\tau)\right](\sigma\sqrt{\tau_{3}+i^{2}\tau})^{-1}=}\\\
{\;\;\;\;=\delta(X^{i}/L,\;\;\tau_{1}+i\tau,\;\;\tau_{2}+i\beta\tau,\;\;\tau_{3}+i^{2}\tau)\;.}\end{array}$

###### Proof.

By the formula (3) of [8], the solution of (1) and (2) is written as follows:

$V(X,\;\;\tau)=\frac{e^{-r\tau}}{\sigma\sqrt{2\pi\tau}}\int_{0}^{+\infty}\frac{z^{\beta}}{z}\left[\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\delta(z^{i}/L)}e^{-\frac{y^{2}}{2}}dy\right]\cdot
1\\{z\geq K\\}\cdot
e^{-\frac{1}{2\sigma^{2}\tau}\left[\ln\frac{X}{z}+(r-q-\sigma^{2}/2)\tau\right]^{2}}dz$
$=\frac{e^{-r\tau}}{\sigma\sqrt{2\pi\tau}}\int_{K}^{+\infty}\frac{z^{\beta}}{z}\left[\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\delta(z^{i}/L)}e^{-\frac{y^{2}}{2}}dy\right]\cdot
e^{-\frac{1}{2\sigma^{2}\tau}\left[\ln\frac{X}{z}+(r-q-\sigma^{2}/2)\tau\right]^{2}}dz$
(4)

Here, noting that $\exp(\beta\ln z)=z^{\beta}$ and using the transformation

$y_{2}=\left[\ln\frac{X}{z}+\left(r-q-\frac{\sigma^{2}}{2}+\beta\sigma^{2}\right)\tau\right](\sigma\sqrt{\tau})^{-1}$
$y_{1}=\frac{y\sqrt{\tau_{3}}}{\sqrt{\tau_{3}+i^{2}\tau}}+i\left[\ln\frac{X}{z}+\left(r-q-\frac{\sigma^{2}}{2}+\beta\sigma^{2}\right)\tau\right]\left(\sigma\sqrt{\tau_{3}+i^{2}\tau}\right)^{-1},$

then we have

$z>K\Leftrightarrow
y_{2}<\left[\ln\frac{X}{K}+\left(r-q-\frac{\sigma^{2}}{2}+\beta\sigma^{2}\right)\tau\right]\left(\sigma\sqrt{\tau}\right)^{-1}=a_{2},$

and noting that

$y=y_{1}\sqrt{\frac{\tau_{3}+i^{2}\tau}{\tau_{3}}}-iy_{2}\sqrt{\frac{\tau}{\tau_{3}}},$

then (4) can be written as follows:

$V(X,\;\tau;\;\tau_{1})=\frac{X^{\beta}e^{\mu(\beta)\tau}}{2\pi}\sqrt{\frac{\tau_{3}+i^{2}\tau}{\tau_{3}}}\int_{-\infty}^{a_{1}}\int_{-\infty}^{a_{2}}e^{-\frac{\left(\sqrt{\frac{\tau_{3}+i^{2}\tau}{\tau_{3}}}y_{1}-i\sqrt{\frac{\tau}{\tau_{3}}}y_{2}\right)^{2}+y_{2}{}^{2}}{2}}dy_{2}dy_{1}.$
(5)

Now let $\rho=i\sqrt{\tau/(\tau_{3}+i^{2}\tau)}$, then (5) can be written as
(3) using two dimensionalnormal distribution function with the correlation
matrix $\left(\begin{array}[]{cc}{1}&{\rho}\\\ {\rho}&{1}\end{array}\right)$.
∎

######  Remark 1.

The formula (3) in the case that $\beta=1$or 0,
$i=1,\;\;\tau_{1}=\tau_{3}=T-t,$ $\tau_{2}=\beta(T-t)$ is just the pricing
formula of the second order binary option of [1], the formula (3) in the case
that $\beta$ is any real number and $i=1,\;\;\tau_{1}=\tau_{3}=$ $T-t,$
$\tau_{2}=\beta(T-t)$ is just the pricing formula of the second order power
binary option (the expression (40) at page 21 of [7]) and the formula (3) with
$K=0$ is the just the pricing formula of the power normal distribution
standard option (the expression (12) at page 5 of [7]).

###  2.2 A solution representation of a terminal boundary value problem of the
Black-Scholes equation and the monotonicity of its solution

Consider the following problem:

$\frac{\partial w}{\partial
t}+\frac{1}{2}\sigma^{2}x^{2}\frac{\partial^{2}w}{\partial
x^{2}}+cx\frac{\partial w}{\partial x}=0,\;\;1<x<+\infty,\;\;0<t<T$ (6)
$w(1,\;\;t)=0,\;\;0\leq t<T$ (7) $w(x,\;T)=1,\;\;x>1$ (8)

This is a special terminal boundary value problem of the Black-Scholes
equation and the pricing problem of barrier option with zero interest rate,
dividend rate $-c$ and the maturity payoff 1. Thus it can be solved by the
image solution method [2]. First, appling the formula (3) with
$\beta=0,\;\;i=0,\;\;L=0,\;\;K=1,\;\;r=0,\;\;q=-c$ under consideration
$N_{2}(+\infty,\;a_{2};\;A)=N(a_{2})$, we have the solution of (6) and (8):

$u(x,\;\;t)=N\left(\frac{\ln
x+(c-\sigma^{2}/2)(T-t)}{\sigma\sqrt{T-t}}\right).$

Then using the image solution method, the solution of (6), (7), (8) is
provided by

$w(x,\;\;t)=u(x,\;\;t)-x^{1-2c/\sigma^{2}}u(1/x,\;\;t).$

That is

$w(x,\;\;t)=N\left(\frac{\ln
x+(c-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}\right)-x^{1-\frac{2c}{\sigma^{2}}}N\left(\frac{\ln\frac{1}{x}+(c-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}\right).$
(9)

######  Theorem 2.

The solution of (6), (7), (8) is provided by (9). Furthermore, we have the
following estimates:

$w_{x}(x,\;\;t)>0,~{}0<w(x,\;\;t)<1(x>1,\;\;t<T).$

${Proof.}$ Let $R=c\cdot(T-t),\;\;S=\sigma\sqrt{T-t}(>0)$. Then we have

$w(x,\;\;t)=N\left(\frac{2R-S^{2}}{2S}+\frac{1}{S}\ln
x\right)-x^{1-\frac{2R}{S^{2}}}N\left(\frac{2R-S^{2}}{2S}-\frac{1}{S}\ln
x\right).$

Now, let $\frac{2R-S^{2}}{2S}=\xi$, then $w(x,\;\;t)$ is written as follows

$f(x\;;S,\;\;\xi)=N\left(\xi+\frac{1}{S}\ln
x\right)-x^{-\frac{2\xi}{S}}N\left(\xi-\frac{1}{S}\ln
x\right)\;\;\;(S>0,\;\;\xi\in R\text{: parameters})$

.

Evidently, we have

$f(0+)=f(1)=0,\;\;f(+\infty)=1.$ (10)

And we have

$f^{\prime}(x)=\frac{2}{S}x^{-\frac{2\xi}{S}-1}\left[\frac{1}{\sqrt{2\pi}}\exp\left\\{-\frac{1}{2}\left[\xi-\frac{1}{S}\ln
x\right]^{2}\right\\}+\xi{\kern 1.0pt}N\left(\xi-\frac{1}{S}\ln
x\right)\right]=\frac{2}{S}x^{-\frac{2\xi}{S}-1}\cdot g(x).$ (11)

Thus if $\xi\geq 0$, then $f^{\prime}(x)>0(\forall x>0)$.

Now assume that $\xi<0$. Note that

$g(x)=\frac{1}{\sqrt{2\pi}}\exp\left\\{-\frac{1}{2}\left[\xi-\frac{1}{S}\ln
x\right]^{2}\right\\}+\xi{\kern 1.0pt}N\left(\xi-\frac{1}{S}\ln x\right).$

Then we have

$g(0+)=\xi<0,g(1)=\frac{1}{\sqrt{2\pi}}e^{-\xi^{2}/2}+\xi{\kern
1.0pt}N(\xi),\;\;g(+\infty)=0.$ (12)

The proof of Theorem 2 needs the following Lemmas.

######  Lemma 1.

$f(x)<0,\;\;f^{\prime}(x)<0$ in a right neighborhood of $x=0$. In particular,
if $-S/2<\xi<0$, then $f^{\prime}(0+)=-\infty$; if $\xi=-S/2$, then
$f^{\prime}(0+)=-1<0$ and if $\xi<-S$ $/2$, then $f^{\prime}(0+)=0$.

###### Proof.

It is evident from (11) and the first expression of (12), and the first
expression of (10). ∎

######  Lemma 2.

$g(1)=\frac{1}{\sqrt{2\pi}}e^{-\xi^{2}/2}+\xi{\kern 1.0pt}N(\xi)>0$.

###### Proof.

Since $g^{\prime}(x)=-(S\sqrt{2\pi}\;x)^{-1}\exp\\{-[\xi-S^{-1}\ln
x]^{2}/2\\}\cdot\ln x$, $g(x)$ is strictly increasing for $0<x<1$ and strictly
decreasing for $x>1$, and thus we have $g(1)=$ $\max g(x)$. From the last
expression, we have the conclusion. ∎

Using Lemma 2, Theorem 2 is soon proved. In fact, $g(x)$ is strictly
increasing in the interval $0<x<1$, $g(0+)<0$ from the first expression of
(12) and $g(1)>0$ from Lemma 2, there exists a unique $x_{0}$ in the interval
[0, 1] such that $g(x_{0})=0$ and $g(x)\neq 0$ in the interval
$[1,\;\;+\infty)$. Thus $x_{0}$ is the unique point such that
$f^{\prime}(x_{0})=0$ and therefore $f^{\prime}(x)<0$on $(0,\;\;x_{0})$ and
$f^{\prime}(x)>0$ on $[x_{0},\;\;+\infty)$. So we have
$w_{x}(x,\;\;t)>0$($x>1$) and $0=w(1,\;\;t)\leq
w(x,\;\;t)<w(+\infty,\;\;t)=1$.

##  3 Application to the pricing problem for option on bonds with credit risk.

###  3.1 One factor structural model for straight credit bond

Main Assumptions are as follows:

Assumption 1: Short-term interest rate $r$ is constant.

Assumption 2: The firm value $V$ consists of $m$ shares of traded assets $S$
and $n$ sheets of zero coupon bonds:

$V_{t}=mS_{t}+nB_{t}$

and follows the geometric Brownian motion

$dV=(r-q)Vdt+\sigma VdW\;\text{(under the risk neutral measure)}.$

The firm pays out dividend in the ratio of $q$(a constant) for a unit of firm
value, continuously (volatility $\sigma$: constant)

Assumption 3: Expected default occurs when

$V\leq V_{b}(t)\;\text{(Here}\;V_{b}(t)=be^{-a(T-t)},a,\;b:\text{constants).}$

Assumption 4: Default recovery is $R_{d}=R\cdot e^{-r(T-t)}$ ($0\leq
R<1$:constant).

Assumption 5: Our Corporate bond price is given by a sufficiently smooth
deterministic function $B(V,\;\;t)$. The face value of the bond (the price at
maturity $T$) is 1(unit of currency).

According to the argument in Section 3 of [5], $B=B(V,\;t)$ is the solution to
the following solving problem for the terminal vlue problem of Black-Scholes
equation:

$\frac{\partial B}{\partial
t}+\frac{1}{2}\sigma^{2}V^{2}\frac{\partial^{2}B}{\partial
V^{2}}+(r-q)V\frac{\partial B}{\partial
V}-rB=0,\;\;V_{b}(t)<V<+\infty,\;\;0<t<T$ (13)
$B(V_{b}(t),\;\;t)=e^{-r(T-t)}R,\;\;0<t<T$ (14) $B(V,\;T)=1,\;\;V>V_{b}(T)=b$
(15)

######  Theorem 3.

The solution to the problem (13)$\mathrm{\sim}$(15) is provided as follows:

$B(V,\;\;t)=R\cdot e^{-r(T-t)}+W(V,\;\;t)(1-R)e^{-r(T-t)}$ (16)

Here $W(V,t)$ is the surviving probability of the bond, that is, the
probability of no default at time $t(0\leq t<T)$ and given as follows:

$W(V,\;\;t)=N(d_{1})-\left(\frac{V}{be^{-a(T-t)}}\right)^{1-\frac{2(r-q-a)}{\sigma^{2}}}N(d_{2})$
(17)
$\begin{array}[]{l}{d_{1}=\frac{\ln\frac{V}{be^{-a(T-t)}}+(r-q-a-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}=\frac{\ln\frac{V}{b}+(r-q-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}}\\\
{d_{2}=\frac{\ln\frac{be^{-a(T-t)}}{V}+(r-q-a-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}=\frac{\ln\frac{b}{V}+(r-q-2a-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}}\end{array}$
(18)

Furthermore, we have

$B_{V}(V,\;t)>0,\;\;R\cdot e^{-r(T-t)}<B(V,\;t)<e^{-r(T-t)},\;\;V>V_{b}(t).$

###### Proof.

The problem (13)$\mathrm{\sim}$(15) is a special case of the problem (3.11) of
[5] and thus can be solved by the method of [5]. At time $0\leq t<T$, the bond
holder can receive at least the default recovery $R\cdot e^{-r(T-t)}$
regardless of default and in the case of no default at $t$ he can receive the
remainder $(1-R)e^{-r(T-t)}$. Thus if $W(V,\;\;t)$ is the probability of no
default at time $t$, then our bond price will be represented by (16). Then
$W(V,\;\;t)$ is the solution to the following problem:

$\frac{\partial W}{\partial
t}+\frac{1}{2}\sigma^{2}V^{2}\frac{\partial^{2}W}{\partial
V^{2}}+(r-q)V\frac{\partial W}{\partial V}=0,\;\;V_{b}(t)<V<+\infty,\;\;0<t<T$
$W(V_{b}(t),\;\;t)=0,\;\;0\leq t<T$ $W(V,\;T)=1,\;\;V>b.$

If we use the new variables$V/(be^{-a(T-t)})=x,\;\;W(V,\;\;t)=w(x,\;\;t)$,
then this problem is transformed to the problem (6), (7), (8) with $c=r-q-a$
and thus $w(x,\;\;t)$ is provided by (9). If we return to the original
variables, then we obtain (17), (18).

From Theorem 2, we have $0<W(V,\;\;t)<1$, $W_{V}(V,\;\;t)>0$ and form this
result we have $B_{V}(V,\;t)>0,\;\;R\cdot e^{-r(T-t)}<B(V,\;t)<e^{-r(T-t)}$. ∎

###  3.2 A Pricing Model for Bond Option and Pricing Formula.

Now we assume that the bond issuing company added to the bond defined by the
Assumptions 1$\mathrm{\sim}$5 a new provision that allows the bond holder to
demand early redemption at a predetermined date prior to the maturity.

Assumption 6: At a predetermined date , the bond holder can receive $E$(unit
of currency)(the early redemption premium) and end the bond contract.

By this provision, the bond holder has the credit bond defined by assumptions
1$\mathrm{\sim}$5 together with the right to sell this bond for the exercise
price $E$ at time $T_{1}$. Thus the Assumption 6 provides the bond holder the
bond (put) option with maturity $T_{1}$. This is a bond option on zero coupon
bond with credit risk. If the default event occurs, then the bond holder
receives the default recovery and the bond contract is ended, and thus the
bond option contract is also ended. Therefore this bond option contract is a
kind of barrier option contract and its price is the solution to the following
problem

$\frac{\partial P}{\partial
t}+\frac{1}{2}\sigma^{2}V^{2}\frac{\partial^{2}P}{\partial
V^{2}}+(r-q)V\frac{\partial P}{\partial V}-rP=0,\;\;V>V_{b}(t),\;\;0\leq
t<T_{1}$ (19) $P(V_{b}(t),\;t)=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0\leq
t<T_{1}$ (20)
$P(V,\;T_{1})=(E-B(V,\;T_{1}))^{+},\;\;V>V_{b}(T_{1})=be^{-a(T-T_{1})}.$

Here $B(V,\;T_{1})$ is provided by (16). From Theorem 3, $B(V,\;T_{1})$ is a
strict increasing function on $V$ and we have

$\inf_{V}B(V,\;\;T_{1})=e^{-r(T-T_{1})}R\;,\;\;\sup_{V}B(V,\;\;T_{1})=e^{-r(T-T_{1})}.$

Therefore if $e^{-r(T-T_{1})}R<E<e^{-r(T-T_{1})}$, then there exists a unique
number (called the early redemption boundary) $K\geq V_{b}(T_{1})$
$=be^{-a(T-T_{1})}$ such that

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;B(V,\;T_{1})<E$
if $V<K$; $B(V,\;T_{1})\geq E$ if $V\geq K$.

Thus the terminal value condition is written by

$P(V,\;T_{1})=E\cdot 1\\{V<K\\}-B(V,\;T_{1})\cdot
1\\{V<K\\},\;\;V>V_{b}(T_{1})=be^{-a(T-T_{1})}.$

From (16)$\mathrm{\sim}$(18), the terminal value condition can be written as

$\begin{array}[]{l}{P(V,\;T_{1})=[E-R\cdot e^{-r(T-T_{1})}]\cdot
1\\{V<K\\}-(1-R)e^{-r(T-T_{1})}N(d_{1})\cdot 1\\{V<K\\}}\\\
{\;\;\;\;\;\;\;\;\;\;\;\;\;\;+(1-R)e^{-r(T-T_{1})}\left(\frac{V}{be^{-a(T-T_{1})}}\right)^{1-\frac{2(r-q-a)}{\sigma^{2}}}N(d_{2})\cdot
1\\{V<K\\},\;\;V>be^{-a(T-T_{1})}}\end{array}$ (21)

The problem (19), (20), (21) is a pricing problem for barrier option with
moving barrier. Using new variables , then this problem is changed to the
following problem.

$\frac{\partial p}{\partial
t}+\frac{1}{2}\sigma^{2}x^{2}\frac{\partial^{2}p}{\partial
x^{2}}+(r-q-a)x\frac{\partial p}{\partial
x}-rp=0,\;\;b<x<+\infty,\;\;0<t<T_{1}$ (22) $p(b,\;\;t)=0,\;\;0\leq t<T_{1}$
(23) $\begin{array}[]{l}{p(x,\;T_{1})=[E-R\cdot e^{-r(T-T_{1})}]\cdot
1\\{b<x<Ke^{a(T-T_{1})}\\}}\\\
{\;\;\;\;\;\;\;\;\;\;\;\;-(1-R)e^{-r(T-T_{1})}N(d_{1})\cdot
1\\{b<x<Ke^{a(T-T_{1})}\\}}\\\
{\;\;\;\;\;\;\;\;\;\;\;\;+(1-R)e^{-r(T-T_{1})}(x/b)^{1-\frac{2(r-q-a)}{\sigma^{2}}}N(d_{2})\cdot
1\\{b<x<Ke^{a(T-T_{1})}\\}=f_{1}+f_{2}+f_{3}.}\end{array}$ (24)

Here

$d_{1}=\frac{\ln
x/b+(r-q-a-\sigma^{2}/2)(T-T_{1})}{\sigma\sqrt{T-T_{1}}}=\delta(x/b,\;\;T-T_{1},\;\;0,\;\;T-T_{1}),$
$d_{2}=\frac{\ln
b/x+(r-q-a-\sigma^{2}/2)(T-T_{1})}{\sigma\sqrt{T-T_{1}}}=\delta(b/x,\;\;T-T_{1},\;\;0,\;\;T-T_{1}),$
$f_{1}(x)=[E-R\cdot e^{-r(T-T_{1})}]\cdot
1\left\\{b<x<Ke^{a(T-T_{1})}\right\\},\;$
$f_{2}(x)=-(1-R)e^{-r(T-T_{1})}N(d_{1})\cdot
1\left\\{b<x<Ke^{a(T-T_{1})}\right\\},$
$f_{3}(x)=(1-R)e^{-r(T-T_{1})}\left(\frac{x}{b}\right)^{1-\frac{2(r-q-a)}{\sigma^{2}}}N(d_{2})\cdot
1\left\\{b<x<Ke^{a(T-T_{1})}\right\\}.$

The problem (22)$\mathrm{\sim}$(24) is a pricing problem of barrier option
with interest rate $r$, dividend rate $q+a$, constant barrier $b$ and can be
solved by image solution method([2]). That is, if the solution to the problem
(22) and (24) is $u(x,\;\;t)$, then the solution to the problem (22), (23) and
(24) is provided by

$p(x,\;\;t)=u(x,\;\;t)-(x/b)^{1-2(r-q-a)/\sigma^{2}}u(b^{2}/x,\;\;t).$ (25)

Denote the solution to (22) with the terminal value $p(x,\;T_{1})=f_{i}(x)$ by
$u_{i}(x,\;\;t)$.

Since $f_{1}(x)=[E-R\cdot
e^{-r(T-T_{1})}]\cdot(1\\{x>b\\}-1\\{x>Ke^{a(T-T_{1})}\\})$, applying Theorem
1 when $\beta=0,\;\;i=0,\;\;L=0,\;\;r=r,\;\;$$q=q+a$, $K=Ke^{a(T-T_{1})}$ and
when $K=b$, and considering $N_{2}(+\infty,\;a_{2};\;A)=N(a_{2})$, then we
have

$u_{1}(x,\;\;t)=[E-R\cdot e^{-r(T-T_{1})}]\cdot
e^{-r(T_{1}-t)}[N(b_{1})-b_{2})]=[E\cdot e^{-r(T_{1}-t)}-R\cdot
e^{-r(T-t)}][N(b_{1})-N(b_{2})].$

Here

$b_{1}=\delta(\frac{x}{b},\;\;T_{1}-t,\;\;0,\;\;T_{1}-t)=\frac{\ln\frac{x}{b}+(r-q-a-\sigma^{2}/2)(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}.$
$b_{2}=\delta(\frac{x}{Ke^{a(T-T_{1})}},\;\;T_{1}-t,\;\;0,\;\;T_{1}-t)=\frac{\ln\frac{x}{Ke^{a(T-T_{1})}}+(r-q-a-\sigma^{2}/2)(T_{1}-t)}{\sigma\sqrt{T_{1}-t}},$

Since
$f_{2}(x)=-(1-R)e^{-r(T-T_{1})}N(d_{1})\cdot(1\\{x>b\\}-1\\{x>Ke^{a(T-T_{1})}\\})$,
applying Theorem 1 when
$\beta=0,\;\;i=1,\;\;L=b,\;\;$$r=r,\;\;q=q+a$,$\;\;K=Ke^{a(T-T_{1})}$ and when
$K=b$, then we have

$u_{2}(x,\;\;t)=-(1-R)e^{-r(T-t)}[N_{2}(\overline{d}_{1},\;\;b_{1};\rho)-N_{2}(\overline{d}_{1},\;\;b_{2};\rho)].$

Here

$\overline{d}_{1}=\delta(\frac{x}{b},\;\;T-t,\;\;0,\;\;T-t)=\frac{\ln\frac{x}{b}+(r-q-a-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}$
$\rho=\sqrt{(T_{1}-t)/(T-t)}$

Since
$f_{3}(x)=(1-R)e^{-r(T-T_{1})}\left(\frac{x}{b}\right)^{1-\frac{2(r-q-a)}{\sigma^{2}}}N(d_{2})\cdot(1\\{x<Ke^{a(T-T_{1})}\\}-1\\{x<b\\})$,
so applying Theorem 1 when
$\beta=1-2\cdot(r-q-a)/\sigma^{2},$$\;\;i=-1,\;\;L=b^{-1},\;\;r=r,$$\;\;q=q+a$,$\;\;K=Ke^{a(T-T_{1})}$
and when $K=b$, and considering $\mu(\beta)=-r$, then we have

$\begin{array}[]{l}{u_{3}(x,\;\;t)=(1-R)e^{-r(T-T_{1})}e^{\mu(\beta)(T_{1}-t)}(x/b)^{\beta}[N_{2}(\overline{d}_{2},\;\;b_{3};-\rho)-N_{2}(\overline{d}_{2},\;\;b_{4};-\rho)]=}\\\
{\;\;\;\;\;\;\;\;\;\;\;=(1-R)e^{-r(T-t)}(x/b)^{1-2(r-q-a)/\sigma^{2}}[N_{2}(\overline{d}_{2},\;\;b_{3};-\rho)-N_{2}(\overline{d}_{2},\;\;b_{4};-\rho)]}\end{array}.$

Here

$b_{3}=\delta\left(\frac{x}{b},\;T_{1}-t,\;\beta(T_{1}-t),\;T_{1}-t\right)=\frac{\ln\frac{x}{b}-(r-q-a-\frac{\sigma^{2}}{2})(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}$
$b_{4}=\delta\left(\frac{x}{Ke^{a(T-T_{1})}},\;T_{1}-t,\;\beta(T_{1}-t),\;T_{1}-t\right)=\frac{\ln\frac{x}{Ke^{a(T-T_{1})}}-(r-q-a-\frac{\sigma^{2}}{2})(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}$
$\overline{d}_{2}=\delta\left(\frac{b}{x},\;T-2T_{1}+t,\;-\beta(T_{1}-t),\;T-t\right)=\frac{\ln\frac{b}{x}+(r-q-a-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}.$

Therefore the solution to the problem (22) and (24) is given as follows:

$\begin{array}[]{l}{u(x,\;\;t)=[E\cdot e^{-r(T_{1}-t)}-R\cdot
e^{-r(T-t)}][N(b_{1}(x))-N(b_{2}(x))]}\\\
{\;\;\;\;\;\;\;\;\;\;-(1-R)e^{-r(T-t)}[N_{2}(\overline{d}_{1}(x),b_{1}(x);\rho)-N_{2}(\overline{d}_{1}(x),b_{2}(x);\rho)]}\\\
{\;\;\;\;\;\;\;\;\;\;+(1-R)e^{-r(T-t)}(x/b)^{1-2(r-q-a)/\sigma^{2}}[N_{2}(\overline{d}_{2}(x),b_{3}(x);-\rho)-N_{2}(\overline{d}_{2}(x),b_{4}(x);\;-\rho)]}\end{array}.$

And the solution to the problem (22), (23) and (24) is given by (25), so we
have

$\begin{array}[]{l}{p(x,\;\;t)=[Ee^{-r(T_{1}-t)}-R\cdot
e^{-r(T-t)}]\\{N(b_{1}(x))-N(b_{2}(x))}\\\
{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-(x/b)^{1-2(r-q-a)/\sigma^{2}}[N(b_{1}(b^{2}/x))-N(b_{2}(b^{2}/x))\\}}\\\
{\;\;-(1-R)e^{-r(T-t)}\\{N_{2}(\overline{d}_{1}(x),b_{1}(x);\rho)-N_{2}(\overline{d}_{1}(x),b_{2}(x);\rho)-}\\\
{\;\;-(x/b)^{1-2(r-q-a)/\sigma^{2}}[N_{2}(\overline{d}_{1}(b^{2}/x),\;\;b_{1}(b^{2}/x);\;\rho)-N_{2}(\overline{d}_{1}(b^{2}/x),\;\;b_{2}(b^{2}/x);\;\rho)]\\}}\\\
{\;\;+(1-R)e^{-r(T-t)}\\{(x/b)^{1-2(r-q-a)/\sigma^{2}}[N_{2}(\overline{d}_{2}(x),b_{3}(x);-\rho)-N_{2}(\overline{d}_{2}(x),b_{4}(x);\;-\rho)-}\\\
{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-[N_{2}(\overline{d}_{2}(b^{2}/x),\;b_{3}(b^{2}/x)\;;-\rho)-N_{2}(\overline{d}_{2}(b^{2}/x),\;\;b_{4}(b^{2}/x);-\rho)]\\}.}\end{array}$

Here if we consider the relationships

$x=Ve^{a(T-t)},\;\;\frac{x}{b}=\frac{V}{be^{-a(T-t)}},\;\;\frac{x}{Ke^{a(T-T_{1})}}=\frac{Ve^{a(T_{1}-t)}}{K},$

then we have

$b_{1}(x)=\frac{\ln\frac{V}{be^{-a(T-t)}}+(r-q-a-\sigma^{2}/2)(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}:=b_{1}$
$b_{2}(x)=\frac{\ln\frac{V}{K}+(r-q-\frac{\sigma^{2}}{2})(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}:=b_{2}$
$b_{1}(b^{2}/x)=\frac{\ln\frac{be^{-a(T-t)}}{V}+(r-q-a-\sigma^{2}/2)(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}:=\tilde{b}_{1}$
$b_{2}(b^{2}/x)=\frac{\ln\frac{be^{-a(T-T_{1})}}{K}\cdot\frac{be^{-a(T-t)}}{V}+(r-q-\sigma^{2}/2)(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}:=\tilde{b}_{2}$
$\overline{d}_{1}(x)=\frac{\ln\frac{V}{be^{-a(T-t)}}+(r-q-a-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}(=d_{1}\;\text{in}\;\eqref{eq18}):=d$
$\overline{d}_{1}(b^{2}/x)=\frac{\ln\frac{be^{-a(T-t)}}{V}+(r-q-a-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}(=d_{2}\;\text{in}\;\eqref{eq18}):=\tilde{d}$
$\overline{d}_{2}(x)=\frac{\ln\frac{be^{-a(T-t)}}{V}+(r-q-a-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}(=d_{2}\;\text{in}\;\eqref{eq18}=\tilde{d})$
$b_{3}(x)=\frac{\ln\frac{V}{be^{-a(T-t)}}-(r-q-a-\frac{\sigma^{2}}{2})(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}(=-\tilde{b}_{1})$
$b_{4}(x)=\frac{\ln\frac{Ve^{a(T-t)}}{Ke^{a(T-T_{1})}}-(r-q-a-\frac{\sigma^{2}}{2})(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}:=-\tilde{b}_{5}$
$b_{3}(b^{2}/x)=\frac{\ln\frac{be^{-a(T-t)}}{V}-(r-q-a-\frac{\sigma^{2}}{2})(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}=-b_{1}$
$b_{4}(b^{2}/x)=\frac{\ln\frac{be^{-a(T-T_{1})}}{K}\cdot\frac{be^{-a(T-t)}}{V}-(r-q-a-\frac{\sigma^{2}}{2})(T_{1}-t)}{\sigma\sqrt{T_{1}-t}}:=-b_{5}$
$\overline{d}_{2}(b^{2}/x)=\frac{\ln\frac{V}{be^{-a(T-t)}}+(r-q-a-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}=d_{1}\;\text{in}\;\eqref{eq18}=d$

and

$\displaystyle P(V,\;t)$ $\displaystyle=p(Ve^{a(T-t)},\;\;t)$
$\displaystyle=[E\cdot e^{-r(T_{1}-t)}-R\cdot
e^{-r(T-t)}]\\{N(b_{1})-N(b_{2})-\left(\frac{V}{be^{-a(T-t)}}\right)^{1-2(r-q-a)/\sigma^{2}}[N(\tilde{b}_{1})-N(\tilde{b}_{2})]\\}$
$\displaystyle-(1-R)e^{-r(T-t)}\Big{\\{}N_{2}(d,\;b_{1};\;\rho)-N_{2}(d,\;b_{2};\;\rho)-\left(\frac{V}{be^{-a(T-t)}}\right)^{1-2(r-q-a)/\sigma^{2}}\cdot$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\cdot[N_{2}(\tilde{d},\;\tilde{b}_{1};\;\rho)-N_{2}(\tilde{d},\;\tilde{b}_{2};\;\rho)]\;+[N_{2}(d,\;-{b}_{1}\;;\;-\rho)-N_{2}(d,\;\;-{b}_{5}\;;\;-\rho)]$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\left(\frac{V}{be^{-a(T-t)}}\right)^{1-2(r-q-a)/\sigma^{2}}[N_{2}(\tilde{d},\;-\tilde{b}_{1};\;-\rho)-N_{2}(\tilde{d},\;-\tilde{b}_{5};\;-\rho)]\Big{\\}}.$
(26)

Thus we have proved the following theorem.

######  Theorem 4.

Assume that $e^{-r(T-T_{1})}R<E<e^{-r(T-T_{1})}$. The price for bond put
option (the solution to the problem (19), (20), (21)) is given by (26).

######  Remark 2.

Assumption 6 gives the holder of the bond with expiry date $T$ the right to
receive early repayment of debts at the date $T_{1}$ prior to the expiry date.
The repayment $E$ must satisfy $e^{-r(T-T_{1})}R<E<e^{-r(T-T_{1})}$. (26) is
just the early redemption premium which the holder should pay more due to this
early redemption provision. Thus the price in the time interval
$[0,\;\;T_{1}]$ of the puttable bond (the bond with early redemption
provision) is the sum of $B$(given by (16)) and $P$(given by (26)), and the
price in the time interval $(T_{1},\;\;T]$ is equal to $B$.

Next, we assume that the bond issuing company added to the bond defined by the
Assumptions 1$\mathrm{\sim}$5 a new provision that allows the company to pay
back the debt early at a predetermined date prior to the maturity.

Assumption 7: At a predetermined date , the bond issuing company can give
$E$(unit of currency) and call the bond.

By this provision, the bond issuing company has the right to purchase this
bond defined by the Assumptions 1$\mathrm{\sim}$5 for the exercise price $E$
at time $T_{1}$. Thus the Assumption 7 provides the company the bond (call)
option with maturity $T_{1}$. This is a bond option on zero coupon bond with
credit risk. If the default event occurs, then the bond holder receives the
default recovery and the bond contract is ended, and thus the bond option
contract is ended, too. Therefore this bond option contract is a kind of
barrier option contract and its price $P=C(V,\;\;t)(0\leq t\leq T_{1})$
satisfies (19) and (20), and we have the following terminal value condition

$P_{T_{1}}=C(V,\;T_{1})=(B(V,\;T_{1})-E)^{+},\;\;V>V_{b}(T_{1})=be^{-a(T-T_{1})}.$

Here $B(V,\;T_{1})$is given by (16) and thus there exists a unique number
$K\geq V_{b}(T_{1})=be^{-a(T-T_{1})}$such that

$B(V,\;T_{1})<E\;\;\text{if}\;V<K,\;\;\;B(V,\;T_{1})\geq E\;\;\text{if}\;V\geq
K.$

And the terminal value condition of the option is written by

$C(V,\;T_{1})=B(V,\;T_{1})\cdot 1\\{V\geq K\\}-E\cdot 1\\{V\geq K\\}.$

Thus from (16)$\mathrm{\sim}$(18), we have

$\begin{array}[]{l}{C(V,\;T_{1})=[R\cdot e^{-r(T-T_{1})}-E]\cdot 1\\{V\geq
K\\}+(1-R)e^{-r(T-T_{1})}N(d_{1})\cdot 1\\{V\geq K\\}}\\\
{\;\;\;\;\;\;\;\;\;\;\;\;\;\;-(1-R)e^{-r(T-T_{1})}\left(\frac{V}{be^{-a(T-T_{1})}}\right)^{1-\frac{2(r-b-a)}{\sigma^{2}}}N(d_{2})\cdot
1\\{V\geq K\\}}\end{array}$ (27)

Using the same method of Theorem 4, we can obtain the following theorem.

######  Theorem 5.

If $e^{-r(T-T_{1})}R<E<e^{-r(T-T_{1})}$, then the price of the bond (call)
option (the solution to the problem (19), (20) and (27)) is represented as
follows:

$\displaystyle P(V,\;t)$ $\displaystyle=p(Ve^{a(T-t)},\;\;t)$
$\displaystyle=[R\cdot e^{-r(T-t)}-E\cdot
e^{-r(T_{1}-t)}]\left[N(b_{2})-\left(\frac{V}{be^{-a(T-t)}}\right)^{1-2(r-q-a)/\sigma^{2}}N(\tilde{b}_{2})\right]+$
$\displaystyle+(1-R)e^{-r(T-t)}\Big{[}N_{2}(d,\;\;b_{2};\;\rho)-\left(\frac{V}{be^{-a(T-t)}}\right)^{1-2(r-q-a)/\sigma^{2}}N_{2}(\tilde{d},\;\;\tilde{b}_{2};\;\rho)$
$\displaystyle+N_{2}(d,\;\;-{b}_{5};\;-\rho)-\left(\frac{V}{be^{-a(T-t)}}\right)^{1-2(r-q-a)/\sigma^{2}}N_{2}(\tilde{d},\;-\tilde{b}_{5};\;-\rho)\Big{]}.$
(28)

######  Remark 3.

Assumption 7 gives the bond issuing company(debtor) the right to call the bond
from the bond holder at the date $T_{1}$ prior to expiry date with the early
repayment $E$ and obliged the bond holder. (28) is just the premium that the
company should pay to the holder due to this early repayment provision. So the
price in the time interval $[0,\;\;T_{1}]$ of this callable bond is equal to
the difference of $P$(given by (26)) from $B$(given by (16)), and the price in
the time interval $(T_{1},\;\;T]$ is equal to $B$.

Numerical Experiments

When
$r=0.04,\;\;q=0,\;\;\sigma=0.5\;;\;\;b=100,\;\;a=0;\;\;R=0.7,\;\;T=2,\;\;T_{1}=1,\;\;E=0.9$,
Figure 1 is the plot (t, B) of the price given by (16) of ordinary credit bond
without early redemption provision. Figure 2 shows the plot (V, B) for firm
value V at time T1 and 0 and the early redemption boundary. Figure 3 is the
time plot (t, P) of the bond option price (early redemption premium) given by
(26), and Figure 4 is the graph (V, P) for firm value V at some fixed time.
Figure 5 is the time plot of the price of the puttable bond (with early
redemption provision).

Figure 1. Plot (t, B) in [0, T] when V=139, 199, 300;

Figure 2. Plot (V, B) when t =T1${}_{\ }$, early redemption boundary K=199.109

Figure 3. Plot (t, P) in [0, T1] when V=139, 199, 300

Figure 4. Plot (V, P) when t = 0, T1/2

Figure 5. Time Plot of the price of the puttable bond, (t, P+B) in [0, T1],
(t, B) in [T1, T] when V=139, 199, 300

The results of the experiment show that the theoretical treatment in the above
is correct. Figure 1 and Fig. 2 show that increasing property for firm value
variable of the price function of ordinary credit bond without early
redemption provision and uniqueness of early redemption boundary. Fig. 3 and
Fig. 4 show the early redemption premium which the holder of puttable bond
should pay more due to this early redemption provision. Fig. 5 shows that the
price of our puttable bond increases with respect to the firm value variable.

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