Tim Xiao
email: tim.xiao@bmo.com

Is the Jump-Diffusion Model a Good Solution for Credit Risk Modeling? The Case of Convertible Bonds

Keywords: jump diffusion, convertible bond, convertible underpricing, convertible arbitrage, default time approach, default probability approach, asset pricing and credit risk modeling

This paper argues that the reduced-form jump diffusion model may not be appropriate for credit risk modeling. To correctly value hybrid defaultable financial instruments, e.g., convertible bonds, we present a new framework that relies on the probability distribution of a default jump rather than the default jump itself, as the default jump is usually inaccessible. As such, the model can back out the market prices of convertible bonds. A prevailing belief in the market is that convertible arbitrage is mainly due to convertible underpricing. Empirically, however, we do not find evidence supporting the underpricing hypothesis. Instead, we find that convertibles have relatively large positive gammas. As a typical convertible arbitrage strategy employs delta-neutral hedging, a large positive gamma can make the portfolio highly profitable, especially for a large movement in the underlying stock price.

Introduction

A company can raise capital in financial markets either by issuing equities, bonds, or hybrids (such as convertible bonds). From an investor's perspective, convertible bonds with embedded optionality offer certain benefits of both equities and bonds -like the former, they have the potential for capital appreciation and like the latter, they offer interest income and safety of principal. The convertible bond market is of primary global importance.

There is a rich literature on the subject of convertible bonds. Arguably, the first widely adopted model among practitioners is the one presented by Goldman [START_REF] Sachs | Valuing convertible bonds as derivatives[END_REF] and then formalized by [START_REF] Tsiveriotis | Valuing convertible bonds with credit risk[END_REF]. The Goldman Sachs' solution is a simple one factor model with an equity binomial tree to value convertible bonds. The model considers the probability of conversion at every node. If the convertible is certain to remain a bond, it is then discounted by a risky discount rate that reflects the credit risk of the issuer. If the convertible is certain to be converted, it is then discounted by the risk-free interest rate that is equivalent to default free. [START_REF] Tsiveriotis | Valuing convertible bonds with credit risk[END_REF] argue that in practice one is usually uncertain as to whether the bond will be converted, and thus propose dividing convertible bonds into two components: a bond part that is subject to credit risk and an equity part that is free of credit risk. A simple description of this model and an easy numerical example in the context of a binomial tree can be found in [START_REF] Hull | Options, Futures and Other Derivatives[END_REF]. [START_REF] Grimwood | The valuation of convertible bonds: a study of alternative pricing models[END_REF] indicate that the Goldman Sachs model is incoherent because it assumes that bonds are susceptible to credit risk but equities are not. [START_REF] Ayache | The valuation of convertible bonds with credit risk[END_REF] conclude that the Tsiveriotis-Fernandes model is inherently unsatisfactory due to its unrealistic assumption of stock prices being unaffected by bankruptcy. To correct this weakness, [START_REF] Davis | Convertible bonds with market risk and credit risk[END_REF], [START_REF] Andersen | Calibration and implementation of convertible bond models[END_REF], [START_REF] Bloomberg | OVCV model description[END_REF], and [START_REF] Carr | A jump to default extended CEV model: an application of Bessel processes[END_REF] etc., propose a jump-diffusion model to explore defaultable stock price dynamics. They all believe that under a risk-neutral measure the expected rate of return on a defaultable stock must be equal to the risk-free interest rate. The jumpdiffusion model characterizes the default time/jump directly.

The jump-diffusion model was first introduced by [START_REF] Merton | Option pricing when underlying stock returns are discontinuous[END_REF] in the market risk context for modeling asset price behavior that incorporates small day-to-day diffusive movements together with larger randomly occurring jumps. Over the last decade, people attempt to propagate the model from the market risk domain to the credit risk arena.

There are two primary types of models that attempt to describe default processes in the literature:

structural models and reduced-form models. The structural models regard default as an endogenous event,

focusing on the capital structure of a firm. The reduced-form models do not explain the event of a default endogenously, but instead characterize it exogenously as a jump process. Many practitioners in the credit trading arena have tended to gravitate toward the reduced-from models given their mathematical tractability and market consistency. [START_REF] Zhou | A jump-diffusion approach to modeling credit risk and valuing defaultable securities[END_REF], [START_REF] Hilberink | Optimal capital structure and endogenous default[END_REF], [START_REF] Chen | Credit spreads, optimal capital structure, and implied volatility with endogenous default and jump risk[END_REF], etc. introduce the jumpdiffusion mechanism into the structural models, while [START_REF] Davis | Convertible bonds with market risk and credit risk[END_REF], [START_REF] Andersen | Calibration and implementation of convertible bond models[END_REF], and [START_REF] Bloomberg | OVCV model description[END_REF], etc. add a default jump to the stock price dynamics. We refer to the formers as the structural jump-diffusion models and the latters as the reduced-form jump-diffusion models.

Although both the structural jump-diffusion model and the reduced-form model contain jumps, these jumps have different meanings: A jump in the structural jump-diffusion model corresponds to a sudden change in the asset value that may or may not cause the firm to default, whereas a jump in the reduced-form model represents the default event itself.

In this paper, we mainly discuss the reduced-form jump-diffusion models. At the heart of the jump-diffusion models lies the assumption that the total expected rate of return to the stockholders is equal to the risk-free interest rate under a risk-neutral measure.

Although we agree that under a risk-neutral measure the market price of risk and risk preferences are irrelevant to asset pricing (see [START_REF] Hull | Options, Futures and Other Derivatives[END_REF]) and thereby the expectation of a risk-free2 asset grows at the risk-free interest rate, we are not convinced that the expected rate of return on a defaultable asset must be also equal to the risk-free rate. We argue that unlike market risk, credit risk actually has a significant impact on asset prices. This is why regulators, such as International Accounting Standards Board (IASB),

Basel Committee on Banking Supervision (BCBS), etc. require financial institutions to report a credit value adjustment (CVA) in addition to the risk-free mark-to-market (MTM) value to reflect credit risk (see [START_REF] Xiao | An accurate solution for credit value adjustment (cva) and wrong way risk[END_REF]. By definition, a CVA is the difference between the risk-free value and the risky value of an asset/portfolio subject to credit risk. CVA implies that the risk-free value should not be equal to the risky value in the presence of default risk. As a matter of fact, we will prove that the expected return of a defaultable asset under a risk-neutral measure actually grows at a risky rate rather than the risk-free rate.

This conclusion is very important for risky valuation.

Because of their hybrid nature, convertible bonds attract different type of investors. Especially, convertible arbitrage hedge funds play a dominant role in primary issues of convertible debt. In fact, it is believed that hedge funds purchase 70% to 80% of the convertible debt offered in primary markets. A prevailing belief in the market is that convertible arbitrage is mainly due to convertible underpricing (i.e., the model prices are on average higher than the observed trading prices) (see [START_REF] Ammann | Are convertible bonds underpriced? An analysis of the French market[END_REF], [START_REF] Choi | Convertible bond arbitrage, liquidity externalities, and stock prices[END_REF][START_REF] Loncarski | The rise and demise of the convertible arbitrage strategy[END_REF], etc.). However, [START_REF] Agarwal | Liquidity provision in the convertible bond market: analysis of convertible arbitrage hedge funds[END_REF] and Batta, et al. (2007) argue that the excess returns from convertible arbitrage strategies are not mainly due to underpricing, but rather partly due to illiquid. [START_REF] Calamos | Convertible arbitrage: Insights and techniques for successful hedging[END_REF] believes that arbitrageurs in general take advantage of volatility. A higher volatility in the underlying equity translates into a higher value of the equity option and a lower conversion premium. Multiple views reveal the complexity of convertible arbitrage, involving taking positions in the convertible bond and the underlying asset that hedges certain risks but leaves managers exposed to other risks for which they reap a reward.

This article makes a theoretical and empirical contribution to the study of convertible bonds. In contrast to the above mentioned literature, we present a model that is based on the probability distribution (or intensity) of a default jump (or a default time) rather than the default jump itself, as the default jump is usually inaccessible (see [START_REF] Duffie | Swap rates and credit quality[END_REF], Jarrow and Protter (2004), etc).

We model both equities and bonds as defaultable in a consistent way. When a firm goes bankrupt, the investors who take the least risk are paid first. Secured creditors have the best chances of seeing the value of their initial investments come back to them. Bondholders have a greater potential for recovering some their losses than stockholders who are last in line to be repaid and usually receive little, if anything.

The default proceedings provide a justification for our modeling assumptions: Different classes of securities issued by the same company have the same default probability but different recovery rates.

Given this model, we are able to back out the market prices.

Valuation under our risky model can be solved by common numerical methods, such as, Monte

Carlo simulation, tree/lattice approaches, or partial differential equation (PDE) solutions. The PDE algorithm is elaborated in this paper, but of course the methodology can be easily extended to tree/lattice or Monte Carlo.

Using the model proposed, we conduct an empirical study of convertible bonds. We obtain a data set from FinPricing (2017). The data set contains 164 convertible bonds and 2 years of daily market prices as well as associated interest rate curves, credit curves, stock prices, implied Black-Scholes volatilities and recovery rates.

The most important input parameter to be determined is the volatility for valuation. A common approach in the market is to use the at-the-money (ATM) implied Black-Scholes volatility to price convertible bonds. However, most liquid stock options have relatively short maturates (rarely more than 8 years). As a result, some authors, such as [START_REF] Ammann | Are convertible bonds underpriced? An analysis of the French market[END_REF][START_REF] Loncarski | The rise and demise of the convertible arbitrage strategy[END_REF][START_REF] Zabolotnyuk | An empirical comparison of convertible bond valuation models[END_REF], have to make do with historical volatilities. Therefore, we segment the sample into two sets according to the time to maturity: a short-maturity class (0 ~ 8 years) and a long-maturity class (> 8 years).

For the short-maturity class, we use the ATM implied Black-Scholes volatilities for valuation, whereas for the long-maturity class, we calculate the historical volatility as the annualized standard deviation of the daily log returns of the last 2 years and then price the convertible bond based on this real-world volatility.

The empirical results show that the model prices fluctuate randomly around the market prices, indicating the model is quite accurate. Our empirical evidence does not support a systematic underpricing hypothesis. A similar conclusion is reached by [START_REF] Ammann | Simulation-based pricing of convertible bonds[END_REF] who use a Monte-Carlo simulation approach. Moreover, market participants almost always calibrate their models to the observed market prices using implied convertible volatilities. Therefore, underpricing may not be the main driver of profitability in convertible arbitrage.

It is useful to examine the basics of the convertible arbitrage strategy. A typical convertible bond arbitrage employs delta-neutral hedging, in which an arbitrageur buys a convertible bond and sells the underlying equity at the current delta (see [START_REF] Choi | Convertible bond arbitrage, liquidity externalities, and stock prices[END_REF]), Loncarski, et al. (2009), etc.). With delta neutral positions, the sign of Gamma is important. If Gamma is negative, the portfolio profits so long as the underlying equity remains stable. If Gamma is positive, the portfolio will profit from large movements in the stock price in either direction (see [START_REF] Somanath | International financial management[END_REF]).

We study the sensitivities of convertible bonds and find that convertible bonds have relatively large positive gammas, implying that convertible arbitrage can make a profit on a large upside or downside movement in the underlying stock price. Since convertible bonds are issued mainly by start-up or small companies (while more established firms rely on other means of financing), the chance of a large movement in either direction is very likely. Even for very small movements in the underlying stock price, profits can still be generated from the yield of the convertible bond and the interest rebate for the short position.

The rest of this paper is organized as follows: The model is presented in Section 2. Section 3 elaborates the PDE approach; Section 4 discusses the empirical results. The conclusions are provided in Section 5. PDE implementation details, a binomial tree approach and a comparison of models are contained in the appendices.

Model

Convertible bonds can be thought of as normal corporate bonds with embedded options, which enable the holder to exchange the bond asset for the issuer's stock. Despite their popularity and ubiquity, convertible bonds still pose difficult modeling challenges, given their hybrid nature of containing both debt and equity features. Further complications arise due to the frequent presence of complex contractual clauses, such as, put, hard call, soft call, and other path-dependent trigger provisions. Contracts of such complexity can only be solved by numerical methods, such as, Monte Carlo simulation, tree/lattice approaches, or PDE solutions.

From a practitioner's perspective, Monte Carlo is a "last resort" and "least preferred" method, whereas lattice or PDE approaches suffer from the curse of dimensionality: The number of evaluations and computational cost increase exponentially with the dimension of the problem, making it impractical to use in more than two dimensions.

Three sources of randomness exist in a convertible bond: the stock price, the interest rate, and the credit spread. As practitioners tend to eschew models with more than two factors, it is a legitimate question: How can we reduce the number of factors or which factors are most important? [START_REF] Grimwood | The valuation of convertible bonds: a study of alternative pricing models[END_REF] conduct a sensitivity study and find that accurately modeling the equity process appears crucial. This is why all convertible bond models in the market capture, at a minimum, the dynamics of the underlying equity price. Since convertible bonds are issued mainly by start-up or small companies (while more established firms rely on other means of financing), credit risk plays an important role in the valuation. [START_REF] Grimwood | The valuation of convertible bonds: a study of alternative pricing models[END_REF] further note that the interest rate process is of second order importance. Similarly, [START_REF] Brennan | Analyzing convertible bonds[END_REF] conclude that the effect of a stochastic interest rate on convertible bond prices is so small that it can be neglected. Furthermore, [START_REF] Ammann | Simulation-based pricing of convertible bonds[END_REF] notice that the overall pricing benefit of incorporating stochastic interest rates would be very limited and would not justify the additional computational costs. For these reasons, most practical convertible models in the market do not take stochastic interest rate into account.

We consider a filtered probability space (  , F ,   0  t t F , P ) satisfying the usual conditions, where  denotes a sample space, F denotes a  -algebra, P denotes a probability measure, and

  0  t t F denotes a filtration.
The risk-free stock price process can be described as

) ( ) ( ) ( ) ( ) ( t
  dt t S t r t dS E t ) ( ) ( ) (  F (2)
where

  t E F  is the expectation conditional on the t F .
Equation (2) tells us that in a risk-neutral world, the expected return on a risk-free stock is the risk-free interest rate ) (t r , i.e., the discounted stock price under the risk neutral measure is a martingale process.

Next, we turn to a defaultable stock. The defaultable stock process proposed by [START_REF] Davis | Convertible bonds with market risk and credit risk[END_REF], [START_REF] Andersen | Calibration and implementation of convertible bond models[END_REF][START_REF] Andersen | Calibration and implementation of convertible bond models[END_REF][START_REF] Bloomberg | OVCV model description[END_REF], etc., is given by

  ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( t dU t S t dW t S dt t S t h t r t dS         (3) where ) (t U is an independent Poisson process with 1 ) (  t dU
with probability dt t h ) ( and 0 otherwise,

) (t h is the hazard rate or the default intensity, ) (  t S
is the stock price immediately before any jump at time t. The expectation of

) (t dU is dt t h t dU E t ) ( ) ) ( (  F .
The expectation of equation ( 3) is given by

    dt t S t r dt t h t S dt t S t h t r t dS E t ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (     F (4)
It is shown in equation ( 4) that the expected return of a defaultable stock under a jump-diffusion model also grows at the risk-free interest rate. Equation ( 3) is a simpler version of the Merton's Jumpdiffusion model where the number of jumps is 1.

The jump-diffusion model was first proposed in the context of market risk, which naturally exhibits high skewness and leptokurtosis levels and captures the so-called implied volatility smile or skew effects. [START_REF] Ederington | How markets process information: News releases and volatility[END_REF] find that the markets tend to have overreaction and underreaction to the outside news. The jump part of the model can be interpreted as the market response to outside news. If there is not any outside news, the asset price changes according to a geometric Brownian motion. Since the market price of risk and risk preferences are irrelevant to asset pricing within the market risk context, the expected rate of return to the stockholders is equal to the risk-free rate under a risk-neutral measure.

However, we wonder whether it is appropriate to propagate the jump-diffusion model directly from the market risk domain to the credit risk domain, as credit risk actually impacts the valuation of assets. This is why financial institutions are required by regulators to report CVA. In fact, we will show in the following derivation that the expected return of a defaultable asset under a risk-neutral measure is actually equal to a risky rate instead of the risk-free rate. This conclusion is very important for risky valuation.

The world of credit modeling is divided into two main approaches: structural models and reduced-form (or intensity) models. The structural models regard default as an endogenous event, focusing on the capital structure of a firm. The reduced-form models do not explain the event of default endogenously, but instead characterize it exogenously as a jump process. In general, structural models are based on the information set available to the firm's management, such as the firm's assets and liabilities; while reduced-form models are based on the information set available to the market, such as the firm's bond prices or credit default swap (CDS) premia. Many practitioners in the credit trading arena have tended to gravitate toward the reduced-from models given their mathematical tractability. The reducedform models can be made consistent with the risk-neutral probabilities of default backed out from corporate bond prices or CDS spreads/premia.

In the reduced-form models, the stopping (or default) time  of a firm is modeled as a Cox arrival process (also known as a doubly stochastic Poisson process) whose first jump occurs at default and is defined as, It is well-known that the survival probability from time t to s in this framework is defined by

       t
            s t du u h Z t s P s t p ) ( exp ) , | ( : ) , (   (6)
The default probability for the period (t, s) in this framework is defined by

               s t du u h s t p Z t s P s t q ) ( exp 1 ) , ( 1 ) , | ( : ) , (   (7)
We consider a defaultable asset that pays nothing between dates t and T. Let

) (t V and ) (T V
denote its values at t and T, respectively. Risky valuation can be generally classified into two categories: the default time approach (DTA) and the default probability (intensity) approach (DPA).

The DTA involves the default time explicitly. If there has been no default before time T (i.e., T 



), the value of the asset at T is

) (T V . If a default happens before T (i.e., T t  
), a recovery payoff is made at the default time  as a fraction of the market value 3 given by

) (

V

where  is the default recovery rate and ) ( V is the market value at default. Under a risk-neutral measure, the value of this defaultable asset is the discounted expectation of all the payoffs and is given by

    t T T V t D T V T t D E t V F | 1 ) ( ) , ( 1 ) ( ) , ( ) (          (8)
where Y  is an indicator function that is equal to one if Y is true and zero otherwise, and ) , (  t D denotes the stochastic risk-free discount factor at t for the maturity  given by 3 Here we use the recovery of market value (RMV) assumption.

         du u r t D t   ) ( exp ) , ( (9)
Although the DTA is very intuitive, it has the disadvantage that it explicitly involves the default time/jump. We are very unlikely to have complete information about a firm's default point, which is often inaccessible. Usually, valuation under the DTA is performed via Monte Carlo simulation.

The DPA relies on the probability distribution of the default time rather than the default time itself. We divide the time period (t, T) into n very small time intervals ( t  ) and assume that a default may occur only at the end of each very small period. In our derivation, we use the approximation

  y y  1 exp
for very small y. The survival and default probabilities for the period ( t , t t  

) are given by

  t t h t t h t t t p t p          ) ( 1 ) ( exp ) , ( : ) ( ˆ (10)   t t h t t h t t t q t q          ) ( ) ( exp 1 ) , ( : ) ( ˆ (11)
The binomial default rule considers only two possible states: default or survival.



. Under a risk-neutral measure, the value of the asset at t is the expectation of all the payoffs discounted at the risk-free rate and is given by

         t t t t V t t y E t t V t q t t p t t r E t V F F ) ( ) ( exp ) ( ) ( ) ( ) ( ) ( exp ) (             (12) where   ) ( ) ( ) ( 1 ) ( ) ( ) ( t c t r t t h t r t y      
denotes the risky rate and

  ) ( 1 ) ( ) ( t t h t c    is called the (short) credit spread.
Similarly, we have

    t t t t V t t t y E t t V            F ) 2 ( ) ( exp ) ( ( 13
) Note that   t t y   ) ( exp is t t F   -measurable.
                t i t t t t t t V t t i t y E t t V t t t y E t t y E t t V t t y E t V F F F F ) 2 ( ) ) ( exp ) 2 ( ) ( exp ) ( exp ) ( ) ( exp ) ( 1 0                          (14)
By recursively deriving from t forward over T and taking the limit as t  approaches zero, the risky value of the asset can be expressed as

               t T t T V du u y E t V F ) ( ) ( exp ) ( ( 15
)
Using the DPA, we obtain a closed-form solution for pricing an asset subject to credit risk.

Another good example of the DPA is the CDS model proposed by J.P. [START_REF] Morgan | The J. P. Morgan guide to credit derivatives[END_REF].

The derivation of equation ( 15) takes into account all credit characteristics: possibility of a jump to default and recovery rate. It tells us that a defaultable asset under the risk-neutral measure grows at a risky rate. The risky rate is equal to a risk-free interest rate plus a credit spread. If the asset is a bond, the equation is the same as Equation ( 10) in [START_REF] Duffie | Modeling term structure of defaultable bonds[END_REF], which is the market model for pricing risky bonds. The market bond model says that the value of a risky bond is obtained by discounting the promised payoff using the risk-free interest rate plus the credit spread 4 .

Under a risk-neutral measure the market price of risk and risk preferences are irrelevant to asset pricing (see [START_REF] Hull | Options, Futures and Other Derivatives[END_REF]) and thereby the expectation of a risk-free asset grows at the risk-free interest rate. However, credit risk actually has a significant impact on asset prices. This is the reason that regulators, such as IASB and BCBS, require financial institutions to report a CVA in addition to the riskfree MTM value to reflect credit risk. 4 There is a liquidity component in the bond spread. This paper, however, focuses on credit risk only.

In asset pricing theory, the fundamental no-arbitrage theorems do not require expected returns to be equal to the risk free rate, but only that prices are martingales after discounting under the numeraire.

For risk-free valuation, people commonly use a risk-free bond as the numeraire, whereas for risky valuation, they should choose an associated risky numeraire to reflect credit risk. The expected return is that of the numeraire.

If a company files bankruptcy, both bonds and stocks go into a default status. In other words, the default probabilities for both of them are the same (i.e., equal to the firm's probability of default). But the recovery rates are different because the stockholders are the lowest priority in the list of the stakeholders in the company, whereas the bondholders have a higher priority to receive a higher percentage of invested funds. The default proceedings provide a justification for our modeling assumptions: Different classes of securities issued by the same company have the same default probability but different recovery rates.

According to equation ( 15), we propose a risky model that embeds the probability of the default jump rather than the default jump itself into the price dynamics of an asset. The stochastic differential equation (SDE) of a defaultable stock is defined as Equation ( 16) is the direct derivation of equation ( 15). The formula allows different assumptions concerning recovery on default. In particular, 0  s  represents the situation where the stock price jumps to 0, and 1  s  corresponds to the risk-free case. The expectation of equations ( 16) is

  ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) ( 1 )( ( ) ( ) ( t
    dt t S t t h t r t dS E s t ) ( )) ( 1 )( ( ) ( ) (     F (17)
Equation ( 17) says that the expected return of a stock subject to credit risk is equal to a risky rate rather than the risk-free rate. The risky rate reflects the compensation investors receive for bearing credit risk.

PDE Algorithm

The numerical solution of our risky model can be obtained by either PDE methods, tree approaches, or Monte Carlo simulation. In this paper, we introduce the PDE procedure, but of course the methodology can be easily extended to the tree/lattice or Monte Carlo algorithms.

The defaultable stock price process is given by

  ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) ( 1 )( ( ) ( ) ( ) ( t dW t S dt t S t t dW t S dt t S t t h t q t r t dS s            (18)
where )

(t q is the dividend and )) ( 1 )( ( ) ( ) ( ) ( t t h t q t r t s       .
The valuation of a convertible bond normally has a backward nature since there is no way of knowing whether the convertible should be converted without knowledge of the future value. Only on the maturity date, the value of the convertible and the decision strategy are clear. If the convertible is certain to be converted, it behaves like a stock. If the convertible is not converted at an intermediate node, we are usually uncertain whether the continuation value should be treated as a bond or a stock, because in backward induction the current value takes into account the results of all future decisions and some future values may be dominated by the stock or by the bond or by both. Therefore, we arrange the valuation so that the value of the convertible at each node is divided into two components: a component of bond and a component of stock, i.e.

dW S G S dt S G S t G S G S dG                        2 2 2 2 2 1 (19)
Since the Wiener process underlying S and G are the same, we can construct the following portfolio so that the Wiener process can be eliminated.

S G S G X     (20) Therefore, we have dt S G S t G dS S G dG dX                   2 2 2 2 2 1  (21)
In contrast to all previous studies, we believe that the defaultable equity should grow at the risky rate of the equity including dividends, whereas the equity part of the convertible bond should earn the risky rate of the equity excluding dividends, i.e.,

    dt S G S t G dX Sdt S G h q r Gdt h r s s                        2 2 2 2 2 1 ) 1 ( ) 1 (    (22)
So that the PDE of the equity component is given by

    0 ) 1 ( ) 1 ( 2 1 2 2 2 2                G h r S G S h q r S G S t G s s    (23)
Similarly applying Ito Lemma to the bond part of the convertible ) , ( t S B

, we obtain

dW S B S dt S B S t B S B S dB                        2 2 2 2 2 1 (24)
Let us construct a portfolio so that we can eliminate the Wiener process as follows

S B S B Y     (25)
Thus, we have

dt S B S t B dS S B dB dY                   2 2 2 2 2 1  (26)
The defaultable equity should grow at the risky rate of the equity including dividends, while the bond part of the convertible bond grows at the risky rate of the bond. Consequently, we have

    dt S B S t B dY Sdt S B h q r Bdt h r s b                        2 2 2 2 2 1 ) 1 ( ) 1 (    (27)
where b  is the recovery rate of the bond.

The PDE of the bond component is

    0 ) 1 ( ) 1 ( 2 1 2 2 2 2                B h r S B S h q r S B S t B b s    (28)
Equations ( 23) and ( 28) are coupled through appropriate final and boundary conditions reflecting the terms and conditions of each individual convertible and need to be solved simultaneously. Convertible bonds often incorporate various additional features, such as call and put provisions.

The final conditions at maturity T can be generalized as

          otherwise C N P P S if S G p c T T T , 0 , max , min ,   (29)                otherwise C N P P S if C N P P B p c T p c T , 0 , max , min , , max , min  (30)
where N denotes the bond principal, C denotes the coupon, c P denotes the call price, p P denotes the put price and  denotes the conversion ratio. The final conditions tell us that the convertible bond at the maturity is either a debt or an equity. One can use finite difference methods to solve the PDEs ( 23) and ( 28) for the price of a convertible bond.

The upside constraints at

time ] , 0 [ T t  are                       else B B G G P L

Empirical results

This section presents the empirical results. We use two years of daily data from September 10, 2010 to September 10, 2012, i.e., a total of 522 observation days. This proprietary data are obtained from an investment bank. They consist of convertible bond contracts, market observed convertible prices, interest rate curves, credit curves, stock prices, implied Black-Scholes volatilities, and recovery rates. Convertible bond prices observed in the market will be compared with theoretical prices under different volatility assumptions. The sample is segmented into two sets according to the time to maturity: a short-maturity class (0 ~ 8 years) and a long-maturity class (> 8 years). We first select a convertible bond from each group: a 7-year (or 5-year outstanding) contract and a 20-year (or 17-year outstanding) contract shown in Table 1.

Table 1. Convertible Bonds

We hide the issuer names according to the security policy of the investment bank, but everything else is authentic. In the market, either a conversion price or a conversion ratio is given for a convertible bond,

where conversion ratio = (face value of the convertible bond) / (conversion price). products for certain time horizons. Normally the curve is divided into three parts. The short end of the term structure is determined using the London Interbank Offered Rates (LIBOR). The middle part of the curve is constructed using Eurodollar futures that require convexity adjustments. The far end is derived using mid swap rates. The LIBOR-future-swap curve is presented in Table 2. We bootstrap the curve and get the continuously compounded zero rates. The equity information and recovery rates are provided in Table 3. To determine hazard rates, we need to know the observed market prices of corporate bonds or CDS premia, as the market standard practice is to fit the implied risk-neutral default intensities to these credit sensitive instruments. The corporate bond prices are unfortunately not available for companies X and Y, but their CDS premia are observable as shown in Table 4. Usually the CDS market leads the bond market, in particular during crisis situation. Liquidity in the bond market is typically drying up during a financial crisis. Demand for insurance against default risk, on the other hand, increases if the issuer is experiencing financial stress.

Convertible bond

Consequently, prices and spreads derived from the CDS market tend to be more reliable. Said differently, CDSs on reference entities are often more actively traded than bonds issued by the reference entities.

Unlike other studies that use bond spreads for pricing (see [START_REF] Tsiveriotis | Valuing convertible bonds with credit risk[END_REF], [START_REF] Ammann | Are convertible bonds underpriced? An analysis of the French market[END_REF][START_REF] Zabolotnyuk | An empirical comparison of convertible bond valuation models[END_REF], etc.), we perform risky valuation based on credit information extracted from CDS spreads. Given the recovery rates and the CDS premia, we can compute the hazard rates via a standard calibration process (see J.P. [START_REF] Morgan | The J. P. Morgan guide to credit derivatives[END_REF]). The most important input parameter to be determined is the volatility for valuation. A common approach in the market is to use the at-the-money (ATM) implied Black-Scholes volatility to price convertible bonds. For the 5-year outstanding convertible bond (case 1 in Table 1), we find the ATM implied Black-Scholes volatility is 31.87%, and then price the convertible bond accordingly. The results are shown in Table 5. Our analysis actually indicates an overpricing of 0.42%.

For the 17-year outstanding convertible bond (case 2 in Table 1), however, most liquid stock options have relatively short maturates (rarely more than 8 years). Therefore, some authors, such as [START_REF] Ammann | Are convertible bonds underpriced? An analysis of the French market[END_REF], [START_REF] Loncarski | The rise and demise of the convertible arbitrage strategy[END_REF][START_REF] Zabolotnyuk | An empirical comparison of convertible bond valuation models[END_REF], have to make do with historical volatilities. Similarly, we calculate the historical volatility as the annualized standard deviation of the daily log returns of the last 2 years (from September 10, 2010 to September 10, 2012), and then value the convertible bond based on this real-world volatility. The result shown in Table 5 reports an underpricing of 1.07%. The test results demonstrate that the model prices are very close to the market prices, indicating that the model is quite accurate.

Table 5. Model price vs. market price

This table shows the differences between the model prices and the market prices of the convertible bonds under different volatility assumptions, where Difference = (Model price) / (Market observed price) -1.

The convertible bonds are defined in Table 1.

Case 1 (a 7-year convertible) Case 2 (a 20-year convertible) We repeat this exercise for all contracts on all observation days. For any short-maturity convertible bond, we use the ATM implied Black-Scholes volatility for pricing, whereas for any longmaturity convertible bond, we perform valuation via the historical volatility. The results are presented in Tables 6.

Table 6. Statistics of underpricing for different maturity classes

An observation corresponds to a price snapshot of a convertible bond at a certain valuation date.

Underpricing is referred to as the model price minus the market price. Next, our sample is partitioned into subsamples according to the moneyness of convertibles. The moneyness is measured by the ratio of the conversion value to the equivalent straight bond value or the investment value. The underpricing of each daily observation with respect to the degree of moneyness is shown in Table 7, where moneyness between 0 and 0.9 corresponds to out-of-the-money; moneyness between 0.9 and 1.1 represents around-the-money; and moneyness higher than 1.1 is related to in-themoney.

Maturity

Table 7. Statistics of underpricing for different moneyness classes

The moneyness is measured by dividing the conversion value through the associated straight bond value.

An observation corresponds to a snapshot of the market used to price a convertible bond at a certain valuation date. From Tables 7, it can be seen that the model prices fluctuate randomly around the market prices (sometimes overpriced and sometimes underpriced), indicating the model is quite accurate. Empirically, we do not find support for presence of a systematic underpricing as indicated in previous studies (see [START_REF] Carayannopoulos | Convertible bond prices and inherent biases[END_REF], [START_REF] Ammann | Are convertible bonds underpriced? An analysis of the French market[END_REF], etc.). If there is no underpricing, how has the arbitrage strategy been successful in the past? Maybe convertible arbitrage is not solely based on underpricing In a typical convertible bond arbitrage strategy, the arbitrageur entails purchasing a convertible bond and selling the underlying stock to create a delta neutral position. The number of shares sold short usually reflects a delta-neutral or market neutral ratio. It is well known that delta neutral hedging not only removes small directional risks but also is capable of making a profit on an explosive upside or downside breakout if the position's gamma is kept positive. As such, delta neutral hedging is great for uncertain stocks that are expected to make huge breakouts in either direction. Since convertible bonds are issued mainly by start-up or small companies, the chance of a large movement in either direction is very likely.

Moneyness

Even for very small movements in the underlying stock price, profits can still be generated from the yield of the convertible bond and the interest rebate for the short position.

We calculate the delta and gamma values for the two deals described in table 1. The Greeks vs.

spot equity prices are plotted in Figures 2~ 5. It can be seen that the deltas increase with the underlying stock prices in Figures 2 and4. At low market levels, the convertibles behave like their straight bonds with very small deltas. As the stock price increases, conversion becomes more likely. At certain market levels the convertibles are certain to be converted. In this case, the convertibles are similar to the underlying equities and the deltas are equal to the number of shares (i.e., conversion ratios).

Figure 2. Delta vs. underlying price for a 7-year convertible bond

This graph shows how the delta of the 7-year convertible bond (described in Table 1) changes as the underlying stock price changes. This graph shows how the gamma of the 7-year convertible bond (described in Table 1) changes as the underlying stock price changes.

Delta

Conclusion

This paper aims to value hybrid financial instruments (e.g., convertible bonds) whose values may simultaneously depend on different assets subject to credit risk in a proper and consistent way. The motivation for our model is that if a company goes bankrupt, all the securities (including the equity) of the company default. The recovery is realized in accordance with the priority established by the Bankruptcy Code. In other words, different securities have the same probability of default, but different recovery rates.

Our study shows that risky asset pricing is quite different from risk-free asset pricing. In fact, the expectation of a defaultable asset actually grows at a risky rate rather than the risk-free rate. This conclusion is very important for risky valuation.

We propose a hybrid framework to value risky equities and debts in a unified way. The model relies on the probability distribution of the default jump rather than the default jump itself. The model is quite accurate for pricing convertible bonds.

Empirically, we do not find evidence supporting a systematic underpricing hypothesis. We also find that convertible bonds have relatively large positive gammas, implying that convertible arbitrage can make a profit on a large upside and downside movement in the underlying stock price.

Appendix A. Numeric implementation for PDE

In this section, we describe the numerical method used to solve discrete forms of ( 23) and ( 28). 23) and ( 28) can be rewritten as

  0 ) 1 ( 2 ) 1 ( 2 1 2 2 2 2                         B h r x B h q r x B B b s      (A1)   0 ) 1 ( 2 ) 1 ( 2 1 2 2 2 2                         G h r x G h q r x G G s s      (A2)
The equations (A1) and (A2) can be approximated using Crank-Nicolson rule. We discretize the x to be equally spaced as a grid of nodes 0 ~ M. At the maturity, T G and T B are determined according to (29) and (30). At any time i+1, the boundary conditions are

                          0 ) )) 1 ( ( 5 . 0 1 ( ) ) 1 ( ( 5 . 0 1 ( ) )) 1 ( ( 5 . 0 1 ( ) ) 1 ( ( 5 . 0 1 ( 0 1 0 0 1 0 x when h q r G h q r G h r B h r B s i s i b i b i         (A3)          x when S G B M i M i M  1 1 0 (A4)
Then, we conduct the backward induction. The procedure is as follows.

B. Binomial tree algorithm

A binomial tree method is equivalent to an explicit difference scheme. Suppose that the stock price S will either move up to the value uS with probability . As the binomial tree is a discrete approximation to the continuous distribution of equation ( 16), the expectation and variance of the discrete distribution should be equal to those of the continuous distribution. This method is commonly referred to as the moment matching technique.

To match the expectation, we have

  ) exp( ) ( ) ( ) 1 ( ) ( ) ( / ) ( 1 t y t S d t S p u t S p t S t S E s i i u i u i i       (B1) or ) exp( ) 1 ( t y d p u p s u u     (B2) where ) 1 ( s s s h q r c q r y         (B3)
where q is the dividend.

To match the variance, we get

        1 ) exp( ) 2 exp( ) ( 2 exp ) 1 ( ) ( ) ( / ) ( 2 2 2 2 2 1           t t
             (B7)   ) exp( 2 ) 2 exp( 4 1 ) exp( ) 2 exp( 1 ) exp( ) 2 exp( 2 2 2 t y t y t t y t t y d s s s s              (B8)
There are many ways to approximate equations (B7) and (B8). The most well-known one is the [START_REF] Cox | Option Pricing: A Simplified Approach[END_REF] type approximation that is up to order t  accuracy and is given by

) exp( t u    (B9) ) exp( t d     (B10)
Equations ( B6), (B9) and (B10) specify the binomial risky tree parameters that are used to map the continuous stock price dynamics into the lattice representation.

Suppose that there is a convertible bond. Let us construct a trading strategy

                (B14)
For a self-financing portfolio, the initial wealth needed to finance this strategy (sometimes called the manufacturing cost of the contingent claim) is

    ) exp( ) ( ) 1 ( ) ( ) exp( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 1 1 1 t y t C p t C p t y t C p t C p t B t S t C t C t C b i d B u i u B u s i d S u i u S u i i i i i S i B i                       (B15)
where u p is defined in (B6).

We split equation (B15) into an equity equation and a bond equation, and get  

) exp( ) ( ) 1 ( ) ( ) ( 1 1 t y t C p t C p t C s i d S u i u S u i S        (B16)   ) exp( ) ( ) 1 ( ) ( ) ( 1 1 t y t C p t C p t C b i d B u i u B u i B        (B17)
Equations ( B16) and (B17) tell us that the fair price of an equity component or a bond component is equal to the expected value of its future payoffs discounted by the associated risky rate. The expected value is calculated using the corresponding values from the latter two nodes (up or down) weighted by the transition probabilities.

5

Unlike the risk-free tree, the risky tree tries to match the discounted value of the replicating portfolio to the discounted value of the convertible bond in order to catch credit risk properly.

C. A comparison of results

Let us now briefly turn to a comparison with previous works. We use a simple example described in Table C1, which is similar to the example used in [START_REF] Tsiveriotis | Valuing convertible bonds with credit risk[END_REF] and [START_REF] Ayache | The valuation of convertible bonds with credit risk[END_REF]. We assume that the interest rate, the bond spread, and the volatility are flat. The hazard rate is [START_REF] Tsiveriotis | Valuing convertible bonds with credit risk[END_REF] model is referred to as the TF model and the [START_REF] Ayache | The valuation of convertible bonds with credit risk[END_REF] model is referred to as the AFV model.

The convertible bond is described in Table C1.

Time steps

This model AFV TF

  continuation value of the equity component. Equation (31) says that the convertible is either in the continuation region or one of the three constraints (called, put or converted).

Figure 1 .

 1 Figure 1. Histogram of convertible bonds by time to maturity

  Figure 3. Gamma vs. underlying price for a 7-year convertible bond

Figure 4 .Figure 5 .

 45 Figure 4. Delta vs. underlying price for a 20-year convertible bond

  For i = penultimateTime to currentTime // determine accrual interest and call/put prices // determine boundary nodes // use the PSOR (Projected Successive Over Relaxation) method to obtain the continuation value of the bond component t B ~ and the continuation value of the equity component t G ~, applying the constraints (31). EndFor The value at node[0][y] is the convertible bond price where the equity price at node[0][y] is equal to the current market stock price.

  By definition, an

		t F  t 	-measurable random
	variable is a random variable whose value is known at time	t t   . Based on the taking out what is
	known and tower properties of conditional expectation, we have

  For most practical problems, zero recovery at default (or jump to zero) is unrealistic. For example, the stock of Lehman Brothers fell 94.3% on September 15, 2008 after the company filed for Chapter 11 bankruptcy. Similarly, the shares of General Motors (GM) plunged 32% on June 1, 2009 after the firm initiated Chapter 11 bankruptcy. A good framework should flexibly allow people to incorporate different recovery assumptions into risky valuation.

	dS	t		r	t		h	t			s	t	S	t	dt		S 	t	dW		t		y	t	S	t	dt		S 	t	dW	(16)
	where s  is the recovery rate of the stock and		(	)		(	)		(	 1 )			s	(	)		is the risky rate.
															y	t			r	t		h	t						t

Case 1 (a 7-year convertible) Case 2 (a 20-year convertible)

	Issuer	X company	Y company
	Notional of bond	100	100
	Annual coupon rate	2.625	5.5
	Payment frequency	Semiannual	Semiannual
	Issuing date	June 9, 2010	June 15, 2009
	Maturity date	June 15, 2017	June 15, 2029
	Conversion price	30.288	13.9387
	Currency	USD	USD
	Day count	30/360	30/360
	Business day convention	Following	Following
	Put price	-	100 at June 20, 2014
	Let valuation date be September 10, 2012. An interest rate curve is the term structure of interest
	rates, derived from observed market instruments that represent the most liquid and dominant interest rate

Table 2 : USD LIBOR-Future-Swap Curve This

 2 table displays the closing prices as of September 10, 2012.

	Instrument Name	Price
	September 19, 2012 LIBOR	0.6049%
	September 2012 Eurodollar 3 month	99.6125
	December 2012 Eurodollar 3 month	99.6500
	March 2013 Eurodollar 3 month	99.6500
	June 2013 Eurodollar 3 month	99.6350
	September 2013 Eurodollar 3 month	99.6200
	December 2013 Eurodollar 3 month	99.5900
	March 2014 Eurodollar 3 month	99.5650
	2 year swap rate	0.3968%
	3 year swap rate	0.4734%
	4 year swap rate	0.6201%
	5 year swap rate	0.8194%
	6 year swap rate	1.0537%
	7 year swap rate	1.2738%
	8 year swap rate	1.4678%
	9 year swap rate	1.6360%
	10 year swap rate	1.7825%
	12 year swap rate	2.0334%
	15 year swap rate	2.2783%
	20 year swap rate	2.4782%
	25 year swap rate	2.5790%
	30 year swap rate	2.6422%

Table 3 . Equity and recovery information

 3

	This table displays the closing stock prices and dividend yields on September 10, 2012, as well as the
	recovery rates
		Company X	Company Y
	Stock price	34.63	23.38
	Dividend yield	2.552%	3.95%
	Bond recovery rate	40%	36.14%
	Equity recovery rate	2%	1%

Table 4 .

 4 CDS premiaThis table displays the closing CDS premia as of September 10, 2012.

	Name	Company X	Company Y
	6 month CDS spread	0.00324	0.01036
	1 year CDS spread	0.00404	0.01168
	2 year CDS spread	0.00612	0.01554
	3 year CDS spread	0.00825	0.01924

  corresponding to either an up movement or a down movement in the stock price. The discounted portfolio should replicate the discounted convertible bond 5 , which yields

	C	d	( t	i	1 	)		C	d B	( t	i	1 	)		C	d E	( t	i	1 	)
												i	1 uS 	( t	i	)	exp(		y	s	t 	)			i	1 	( t B	i	)		C	u B	( t	i	1 	)	exp(		y	b	t 	)		C	u S	( t	i	1 	)	exp(		y	s	t 	)	(B11)
												i	1 	dS	( t	i	)	exp(		y	s	t 	)			i	1 	( t B	i	)		C	d B	( t	i	1 	)	exp(		y	b	t 	)		C	d S	( t	i	1 	)	exp(		y	s	t 	)	(B12)
								Solving for				i	1 1 ,  i  	yields
																									i	1 		C	u S	( t 1 i S ( t 	) ( d t d S  C u  )( )	i	1 	)		C	u B	( t 1 i S ( t 	) ( d t d B  C u  )( )	i	1 	)	exp(	y	s	t 		y	b	t 	)	(B13)
																																													i		i
																													d																		u	d	u
																							uC		( t						)					dC	( t	)	uC	( t	)	dC	( t	)
																													S			i	1									S	i	1	B	i	1	B	i	1
																																																exp(	y	t	)	exp(	y	t	)
															i				1																													s	b
																															B	( t			)(	u			d	)	B	( t	)(	u	d	)
																																					i										i
																																																H		, (  	)	to hold 
	units of the risky stock and 	units of the risky bond. At time	t	the convertible bond value is
																																																i
	C	( t	i	)			C	B	( t	i	)		C	S	( t	i	)		where	C	( B t	i	)	is the bond component and	C	( S t	i	)	is the stock component; the stock
	value is	S	( t	i	)		; and the bond value is	B	( t	i	)	. At time	t	i	1 	, the bond value becomes	( t B	i	)	exp(	y	b	t 	)	where
	y				r		1 ( h				)		is the risky rate of the bond; the stock value becomes either	uS	( t	)	or	dS	( t	)	; and the
	b																			b																										i	i
	convertible											value									has				two	possible	outcomes:	C	u	( t	i	1 	)		C	B u	( t	i	1 	)		C	E u	( t	i	1 	)	or
																																																29

Table C1 . A 5-year convertible bond

 C1 denotes the start time of the accrual period, and e t denotes the end time of the accrual period. The numerical results are shown in TableC2, from which we can see that our model generates lower results than AVF and TF models.

									Bond spread					0.02
									Bond recovery rate		0
									Stock recovery rate		0
	02 . 0	1 /(		) 0		02 . 0	. As the call and put prices are quoted using the clean prices, we need to convert them
	to the dirty prices as
															P	dirty	( t	)		P	clean	( t	)		AI	( t	)	(C1)
	where the accrued interest is give by
															AI	( t	)		C	) ) s s t t , , e t ( ( t  	(C2)
	where C denotes the coupon,		( t	, s s	e	)	denotes the accrual factor or day count fraction for period ( s t , e t )
	where	t	s		t	e t Maturity t  , s					5 years
									Payment frequency		Semiannual
									Coupon payment				4
									Notional					100
									Conversion rule				1 share (ratio) in 0 -5 years
									Clean call price					110 in 2 -5 years
									Clean put price					105 at 3 years
									Spot stock price				100
									Implied volatility of the convertible 0.2
									Interest rate					0.05

Table C2 .

 C2 Model comparison resultsThis table presents the numerical results for model comparison. The

Here, risk-free means free of credit risk, but not necessarily of market risk


Fei Su

Krishnendu Chakrabarty

Vamsee K Pamula

Yield Enhancement of Digital Microfluidics-Based Biochips Using Space Redundancy and Local Reconfiguration*

attributed to the high regularity of memories, PAs and FPGAs, and the ease with which they can be tested and reconfigured to avoid faulty elements. Digital microfluidicsbased biochips are also amenable to redundancy-based yield enhancement. As in the case of memories, they contain regular arrays of small elements, and the elements are simple and identical. Similar to FPGAs, reconfigurability is an inherent property of these devices.

As microfluidics-based biochips become more complex, manufacturing yield will have significant influence on production volume and product cost. We propose an interstitial redundancy approach to enhance the yield of biochips that are based on droplet-based microfluidics. In this design method, spare cells are placed in the interstitial sites within the microfluidic array, and they replace neighboring faulty cells via local reconfiguration. The proposed design method is evaluated using a set of concurrent real-life bioassays.

In this paper, we propose a scheme for incorporating defect tolerance in the design of digital microfluidics-based biochips. While spare rows/columns around a meshconnected array are often used in fault-tolerant processor arrays and FPGAs [6], the property of "fluidic locality" prevents the application of this simple redundancy technique to microfluidic biochips. Due to the absence of programmable interconnects such as switches between microfluidic cells, a droplet is only able to move directly to the adjacent cells. Thus, a faulty cell can only be replaced by its physicallyadjacent cells. Consequently, a complicated "shifted replacement" process is required to utilize the spare cells located in the boundary row/column; this results in an unacceptable increase in the reconfiguration cost.

Introduction

Microfluidics-based biochips promise to revolutionize clinical diagnostics, DNA sequencing, and other procedures involving molecular biology [1]. In contrast to continuousflow microfluidic biochips consisting of permanently etched micropumps, microvalves, and microchannels [1], dropletbased microfluidic biochips relying on the manipulation of individual droplets through electrowetting and referred to as "digital microfluidics" have been demonstrated [2]. Digital microfluidics offer dynamic reconfigurability, whereby groups of cells in a microfluidic array can be reconfigured to change their functionality during the concurrent execution of a set of bioassays. These features make digital microfluidics a promising platform for massively-parallel DNA analysis, automated drug discovery, and real-time biomolecular recognition.

We propose an interstitial redundancy approach to address the above problem. In this approach, spare cells are placed in the interstitial sites within the microfluidic array such that a spare cell can functionally replace any faulty cells that are physically adjacent to it. This defect tolerance method owes its effectiveness to the high utilization of local reconfiguration. We apply this space redundancy technique to a new biochip design with hexagonal electrodes. Microfluidic biochips with different levels of redundancy can be designed to target given yield levels and manufacturing processes. We introduce a metric called "effective yield" to evaluate the yield enhancement provided by these defect-tolerant designs. A set of real-life bioassays, i.e., multiplexed in-vitro diagnostics on human physiological fluids, is used to evaluate the proposed method. Simulation results show that the yield of a digital microfluidics-based biochip can be significantly increased with the addition of interstitial redundancy and the use of local reconfiguration.

Future advances in fabrication technology will allow increased integration of microfluidic components in composite microsystems. The 2003 International Technology Roadmap for Semiconductors (ITRS) anticipates that microfluidic biochips will soon be integrated with electronic components in system-on-chip (SOC) design [3]. It is expected that several bioassays will then be concurrently executed in a single microfluidic array [4]. However, as in the case of integrated circuits, increase in density and area of microfluidics-based biochips will reduce yield, especially for new technology nodes. Low yield is a deterrent to large scale and high-volume production, and it tends to increase production cost. It will take time to ramp up yield learning based on an understanding of defect types in such mixedtechnology SOCs. Therefore, defect-tolerant designs are especially important for the emerging marketplace.

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The organization of the remainder of the paper is as follows. In Section 2, we discuss related prior work. Section 3 presents an overview of digital microfluidics-based biochips. It also introduces a new design based on hexagonal electrodes. Section 4 discusses manufacturing defects and briefly presents a unified test methodology. Reconfiguration techniques for microfluidic biochips are also presented. In Section 5, we introduce various defect-tolerant designs with different levels of redundancy. The defect tolerance of these designs is evaluated in Section 6. In Section 7, multiplexed invitro diagnostics on human physiological fluids is used to _______________________________ * is research was supported by the National Science Foundation under gr t number IIS-0312352 Yield enhancement through space redundancy and reconfiguration has been successfully applied to memories, processor arrays (PAs) and field-programmable gate arrays (FPGAs) [5,[START_REF] Howard | The yield enhancement of Field-Programmable Gate Arrays[END_REF]. The success of these techniques can be Th an evaluate the proposed yield improvement methodology. Finally, conclusions are drawn in Section 8.

Related Prior Work

Defect tolerance techniques have been successfully used for memory chips since the late 1970's [5]. In contrast to memory arrays, few logic circuits have been designed with built-in redundancy. The absence of regularity in these circuits usually leads to high overhead. Regular circuits, such as processor arrays and FPGAs require less redundancy; a number of defect tolerance techniques have been proposed to enhance their yield [START_REF] Howard | The yield enhancement of Field-Programmable Gate Arrays[END_REF][START_REF] Singh | Interstitial redundancy: an area efficient fault tolerance scheme for large area VLSI processor arrays[END_REF]. droplets have been observed with velocities up to 20cm/s [START_REF] Pollack | Electrowetting-based actuation of droplets for integrated microfluidics[END_REF]. Furthermore, based on this principle, microdroplets can be transported freely to any location on a two-dimensional array without the need for pumps and valves. The configurations of the microfluidic array are programmed into a microcontroller that controls the voltages of electrodes in the array.

Microelectromechanical systems (MEMS) is a relatively young field compared to integrated circuits. It employs micromachining techniques, such as surface micromachining and bulk micromachining, in the fabrication process [START_REF] Madou | Fundamentals of Microfabrication[END_REF]. These processes are less mature than standard CMOS manufacturing processes. As a result, the yield for MEMS devices is often less than that for integrated circuits. Attempts have been made in recent years to make MEMS defecttolerant. For example, design-for-manufacturing has been incorporated in the design process for MEMS [START_REF] Dewey | Behavioral modeling of microelectromechinal systems (MEMS) with statistical performance-variability reduction and sensitivity analysis[END_REF].

In the latest generation of microfluidic biochips, hexagonal electrodes are being used to replace the conventional square electrodes design; this close-packed design is expected to increase the effectiveness of droplet transportation in a 2-D array. The top view of a microfluidic array with hexagonal electrodes is shown in Figure 1(b). A droplet can be moved to an adjacent cell in six possible directions. In this paper, we attempt to make this hexagonal array defect-tolerant through space redundancy and local reconfiguration.

Microfluidics differs from MEMS in the underlying energy domains and in the working principles. Hence, defect tolerance techniques for MEMS cannot be directly applied to microfluidic biochips. Recent work has focused on a fault classification and a unified test methodology for digital microfluidics-based biochips [START_REF] Su | Testing of droplet-based microelectrofluidic systems[END_REF][START_REF] Su | Concurrent testing of droplet-based microfluidic systems for multiplexed biomedical assays[END_REF]. Faults are classified as either manufacturing or operational, and techniques have been developed to detect these faults by electrostatically controlling and tracking the droplet motion. This cost-effective test methodology facilitates defect tolerance for digital microfluidics-based biochips.

Manufacturing Defects and Reconfiguration Techniques

Digital microfluidics-based biochips are fabricated using standard microfabrication techniques, the details of which are described in [START_REF] Pollack | Electrowetting-Based Microactuation of Droplets for Digital Microfluidics[END_REF]. Microfluidic biochips exhibit behavior resembling that of analog and mixed-signal devices. Therefore, we can classify faults caused by the manufacturing defects as being either catastrophic or parametric, along the lines of fault classification for analog circuits [START_REF] Jee | Carafe: An inductive fault analysis tool for CMOS VLSI circuits[END_REF]. Catastrophic (hard) faults lead to a complete malfunction of the system, while parametric (soft) faults cause a deviation in system performance. A parametric fault is detectable only if this deviation exceeds the tolerance in system performance. However, due to the underlying mixed technology and multiple energy domains, microfluidic biochips exhibit failure mechanisms and defects that are significantly different from failure modes in integrated circuits.

Digital Microfluidics-Based Biochips

Digital microfluidics-based biochips manipulate nanolitervolume droplets using electrowetting. The basic cell in a digital microfluidics-based biochip consists of two parallel glass plates, shown in Figure 1(a). The bottom plate contains a patterned array of individually controllable electrodes, and the top plate contains a ground electrode. The droplets containing biomedical samples are sandwiched between these two plates, and surrounded by a filler medium such as silicone oil. In addition, the droplets are insulated from the electrode array by Parylene C (~800 nm), and a thin layer of hydrophobic Teflon AF 1600 (~50 nm) is coated onto the top and bottom plates to decrease the wettability of the surface.

Catastrophic faults may be caused by the following manufacturing defects:

Dielectric breakdown: The breakdown of the dielectric at high voltage levels creates a short between the droplet and the electrode. When this happens, the droplet undergoes electrolysis preventing further transportation. Electrowetting is the basic principle of microdroplet transportation wherein the interfacial tension of a droplet is modulated with an electric field. A control voltage is applied to an electrode adjacent to the droplet and at the same time the electrode just under the droplet is deactivated. Thus, an accumulation of charge in the droplet/insulator interface over the activated electrode results in a surface tension gradient, which consequently causes the transportation of the droplet. By varying the electrical potential along a linear array of electrodes, nanoliter-volume droplets can be transported along this line of electrodes. The velocity of the droplet can be controlled by adjusting the control voltage (0 ~ 90 V), and Short between adjacent electrodes: If a short occurs between two adjacent electrodes, the two electrodes shorted effectively form one longer electrode. When a droplet resides on this electrode, it cannot overlap its adjacent electrodes. As a result, the actuation of the droplet can no longer be achieved. Open in the metal connection between the electrode and the control source: This defect results in a failure in activating the electrode for transport. Manufacturing defects that cause parametric faults include geometrical parameter deviations. The deviation in insulator thickness, electrode length and height between parallel plates may exceed their tolerance value during fabrication. Reconfiguration can be employed not only after the detection of catastrophic faults, but also after the detection of parametric faults that cause significant performance degradation To test a biochip, stimuli droplets containing the normal conducting fluid (e.g., KCL solution) from the droplet source are transported through the array (traversing the cells) to detect the faulty cells. Reconfiguration techniques can be used to bypass faulty cells and increase yield. These reconfiguration approaches can be divided into two categories. The first category consists of techniques that do not add space redundancy, i.e., spare cells, to the microfluidic array. Instead, they attempt to tolerate the defect by using fault-free unused cells. In order to achieve satisfactory yield using this method, fault tolerance must be considered in the design procedure, e.g., in the placement of microfluidic modules in the array. Consequently, it leads to an increase in design complexity. The second category of reconfiguration techniques is application-independent because it incorporates physical redundancy in the microfluidic array. Built-in spare cells can be utilized to replace a defective cell. Since a faulty cell is replaced by a neighboring spare cell, these techniques are also referred to as local reconfiguration. though it is fault-free; see Figure 2(c).

In order to address the problems resulting from microfluidic locality, a new space redundancy approach, termed interstitial redundancy [START_REF] Singh | Interstitial redundancy: an area efficient fault tolerance scheme for large area VLSI processor arrays[END_REF], is proposed in this paper. In this approach, spare cells are located in the interstitial sites within the microfluidic array such that each spare cell is able to functionally replace any one of the primary cells adjacent to it. In contrast to redundancy based on boundary spare rows/columns, interstitial redundancy offers a simple reconfiguration scheme that effectively utilizes local reconfiguration. We apply interstitial redundancy to a digital microfluidics-based biochip with hexagonal electrodes. Such defect-tolerant microfluidic arrays can incorporate different levels of redundancy depending on the number and location of spare cells. We next introduce some key definitions. Definition 1: A defect-tolerant design for a digital microfluidics-based biochip, denoted DTMB(s, p), has interstitial spare cells such that each non-boundary primary cell can be replaced by any one of s spare cells, and each spare cell can be used to replace any one of p primary cells.

Defect-Tolerant Designs with Different Redundancy Levels

There are several ways to include spare cells in a defecttolerant microfluidic array. The first approach is to include spare rows/columns around the microfluidic array. This is a common redundancy technique for PAs and FPGAs. However, in contrast to these electronic arrays with well-defined roles of logic blocks and interconnect, cells in a microfluidic array can be used for storage, transport or other functional operations on droplets. Due to the absence of separate interconnect entities, droplets can only move to physically-adjacent cells. This property is referred to as microfluidic locality. Consequently, the functionality of a faulty cell can only be assumed by its physically-neighboring cells in the array. Microfluidic locality limits the reconfiguration capabilities of the spare rows/columns if they are not adjacent to the faulty cell. In order to utilize the spare cell in the boundary rows/columns, a series of replacement, referred to as "shifted replacement" is required. In shifted replacement, each faulty cell is replaced by one of its fault-free adjacent cells, which is in turn replaced by one of its adjacent cells, and so on, until a spare cell from the boundary is incorporated in the reconfigured structure. In many cases, this shifted replacement procedure will not only involve the faulty module, but it will also require the reconfiguration of fault-free modules. Therefore, it significantly increases the complexity of the reconfiguration. Figure 2 shows an example of a microfluidic array with a single spare row. If one cell in Module 1 is faulty, Module 1 can be only relocated to bypass the faulty cell, while other modules remain unchanged; see Figure 2(b). However, if there is one faulty cell in Module 3, the shifted replacement of Module 3 causes the reconfiguration of Module 2 even Definition 2: The redundancy ratio (RR) for a defect-tolerant microfluidic array with interstitial redundancy is the ratio of the number of spare cells in the array to the number of primary cells. Clearly, for a DTMB(s, p) array of large size, RR ≈ s/p. A DTMB(1, 6) design is shown in Figure 3(a). A corresponding graph model, derived from the array, is shown in Figure 3(b). Black nodes in the graph represent the primary cells in the microfluidic biochip, while white nodes denote spare cells. An edge between two nodes indicates that the two cells represented by these nodes are physically adjacent in the array. Each primary cell is adjacent to only one spare cell, and every spare cell is adjacent to six primary cells. Therefore, the redundancy ratio for this array approaches 0.1667 as the array size increases.

Other defect-tolerant array designs, e.g., DTMB(2, 6), DTMB(3, 6) and DTMB(4, 4), are shown in Figures 4 + 6. The redundancy ratios of the different designs are listed in Table 1. Since each primary cell is physically adjacent to only one spare cell in DTMB(1, 6), the spare assignment to a faulty cell is straightforward. Thus, its yield can be easily obtained analytically. We can view DTMB(1, 6) as a composition of identical clusters that consist of one spare cell and six primary cells surrounding the spare cell . The yield Y c of any cluster in DTMB(1, 6) is determined by the likelihood of having at most one failed cell among these seven cells, i.e., A biochip with n primary cells can be approximately divided into n/6 clusters. Since the cluster failures are independent, the yield Y for this design is given by ). 1 ( 7 6 7

p p p Y C - + = ( ) . ) 1 ( 6 6 n p - 7 7 6 / n C p p Y + = = Y
Figure 7 shows the yield for DTMB(1, 6) for different values of p and n, and compares it to the yield for a biochip without redundancy. Clearly, interstitial redundancy improves the yield of the microfluidic biochip. For the defect-tolerant design with a higher level of redundancy such as DTMB(2, 6), DTMB(3, 6) and DTMB(4, 4), it is hard to develop an analytical model to determine the yield because their spare assignments are not straightforward. We therefore address this problem using Monte-Carlo simulation. During each run of the simulation, the cells in the microfluidic array, including both primary and spare cells, are randomly chosen to fail with probability p. We then check if these defects can be tolerated via local reconfiguration based on the interstitial spare cells. This checking procedure is based on a graph matching approach as described below. We develop a bipartite graph model to represent the relationship between faulty and spare cells in the microfluidic array. A bipartite graph BG(A, B, E) is a graph whose nodes can be partitioned into two sets, A and B, and each edge in

Estimation of Yield Enhancement

The effectiveness of various defect-tolerant designs can be determined by estimating their enhanced yields. The yield analysis in this paper is based on the following assumption. Assumption: Each single cell in the microfluidic array including each primary and spare cell, has the same defect probability q. Moreover, the failures of the cells are independent. Let p = 1q denote the survival probability.

Note that the assumption of equal survival probabilities is reasonable since each cell in the microfluidic array has the same structure. In addition, the assumption of independent failures is valid for random and small spot defects, which result from imperfect materials and from undesirable chemical and airborne particles.

Based on these assumptions, the yield for a defect-tolerant design can be obtained in terms of p. We use both analytical modeling and Monte-Carlo simulation. levels is shown in Figure 10. The number of primary cells is set to 100. As expected, the results show that a microfluidic structure with the higher level of redundancy, such as DTMB(4, 4), is suitable for small values of p. On the other hand, a lower level of redundancy, such as DTMB(1, 6) or DTMB(2, 6), should be used when p is relatively high.

Example: Multiplexed in-vitro Diagnostics

In this section, we evaluate the proposed defect-tolerant design by applying it to a digital microfluidics-based biochip used for multiplexed biomedical assays. The in-vitro measurement of glucose and other metabolites, such as lactate, glutamate and pyruvate, in human physiological fluids plays a critical role in clinical diagnosis of metabolic disorders. For example, the change of regular metabolic parameters in the patient's blood can signal organ damage or dysfunction prior to observable microscopic cellular damages or other symptoms. Recently, the feasibility of performing a colorimetric enzyme-kinetic glucose assay on a digital microfluidic biochip has been successfully demonstrated in experiments [START_REF] Srinivasan | An integrated digital microfluidic lab-on-a-chip for clinical diagnostics on human physiological fluids[END_REF]. E has one node in A and one node in B [START_REF] Asratian | Bipartite Graphs and Their Applications[END_REF]. In our model, nodes in A represent the faulty primary cells in the microfluidic array while nodes in B denote the fault-free spare cells. An edge exists from a node a in A to a node b in B if and only if the faulty primary cell represented by a is physically adjacent to the spare cell represented by b. An example is shown in Figure 8. A maximal matching for this bipartite graph can be obtained using well-known techniques [START_REF] Papadimitriou | Combinatorial Optimization: Algorithms and Complexity[END_REF]. If this maximal matching covers all nodes in A, it implies that all faulty cells can be replaced by their adjacent fault-free spare cells through local reconfiguration. Otherwise, this microfluidic biochip cannot be reconfigured. After 10000 simulation runs, the yield of this microfluidic array is determined from the proportion of successful reconfigurations. The simulation results for DTMB(2, 6), DTMB(3, 6) and DTMB(4, 4) are shown in Figure 9, where n is the number of primary cells.

The glucose assay performed on the biochip is based on Trinder's reaction, a colorimetric enzyme-based method [START_REF] Trinder | Determination of glucose in blood using glucose oxidase with an alternative oxygen acceptor[END_REF]. The enzymatic reactions involved in the assay are:

O 4H ne Quinoneimi TOPS AAP - 4 O 2H O H Acid Gluconic O O H Glucose 2 Peroxidase 2 2 2 2 Oxidase Glucose 2 2 +    →  + + +     →  + +
In the presence of Glucose oxidase, glucose can be enzymatically oxidized to gluconic acid and hydrogen peroxide. Then, in the presence of peroxidase, the hydrogen peroxide reacts with 4-amino antipyrine (4-AAP) and Nethyl-N-sulfopropyl-m-toluidine (TOPS) to form violetcolored quinoneimine, which has an absorbance peak at 545nm. Based on Trinder's reaction, a complete glucose assay can be performed following three steps, namely, transportation, mixing and optical detection. A sample droplet containing glucose and a reagent droplet containing glucose oxidase, peroxidase, 4-AAP and TOPS, are dispensed into the microfluidic array from their respective droplet sources. They are then transported towards a mixer where the sample and the reagent droplets are mixed. The mixed droplet is transported onto a transparent electrode to enable observation of the absorbance of the products of the enzymatic reaction. Absorbance measurements are performed with a green LED and a photodiode. The glucose concentration can be measured from the absorbance, which is related to the concentration of colored quinoneimine in the droplet. Besides glucose assays, other metabolites such as lactate, glutamate, and pyruvate have been detected in a digital microfluidic biochip recently. In addition to current metabolites, a number of reagents can be added onto a chip to enable a multiplexed in-vitro diagnostics platform on different human physiological fluids. Figure 11 shows a recently fabricated microfluidic biochip used for multiplexed biomedical assays [START_REF] Srinivasan | An integrated digital microfluidic lab-on-a-chip for clinical diagnostics on human physiological fluids[END_REF]. In this biochip with square electrodes, SAMPLE1 and SAMPLE2 contain glucose and REAGENT1 and REAGENT2 contain the reagents.

From Figure 9, it is clear that a higher level of redundancy leads to a higher yield. However, adding more redundant cells increases the array area and thereby manufacturing cost. To measure yield enhancement relative to the increased array size, we define the effective yield EY as:

where n is the number of primary cells, and N is the total number of cells in the microfluidic array. The parameter EY represents the tradeoff between yield enhancement and increase in manufacturing cost. The variation of EY with p for different redundancy

), 1 /( ) / ( RR Y N n Y EY + = × =
In this first design and demonstration, only cells used for the bioassays were fabricated; no spare cells were included in be a deterrent to high-volume production and it will increase production cost. The proposed defect-tolerant design approach will therefore be especially useful for the emerging marketplace. the array. Thus, even if one arbitrary cell in this biochip becomes faulty due to a manufacturing defect, this failure cannot be avoided by reconfiguration, and the fabricated biochip has to be discarded. Consequently, the yield for this biochip design is very low. It is only 0.3378 even if the survival probability of a single cell is as high as 0.99. Such low yield makes the first biochip design unsuitable for future mass fabrication and use in clinic diagnostics.

[ In order to improve the yield, we use a defect-tolerant design with interstitial as described in Section 6. To facilitate the comparison, the topological structure of primary cells in the first design is directly mapped to a DTMB(2, 6) design. The new defect-tolerant design has the same number of primary cells used for multiplexed biomedical assays as the original design; see Figure 12(a). There are 252 primary cells (108 of them used in assays) and 91 spare cells in this defecttolerant biochip.

Figure 1 :

 1 Figure 1: (a) Basic cell used in a digital microfluidics-based biochip; (b) Digital microfluidics-based biochip with hexagonal electrodes.

Figure 2 :

 2 Figure 2: Example of a microfluidic array with one single spare row.

Figure 3 :

 3 Figure 3: Top view and graph model of DTMB(1, 6).

  (a) A DTMB(2, 6) design. (b) An alternative DTMB(2, 6) design. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

Figure 7 :

 7 Figure 7: Yield for DTMB(1,6).

Figure 4 :

 4 Figure 4: DTMB(2, 6) designs. ... ... ... ... ... ... ... ...

Figure 5 :

 5 An DTMB(3, 6) design.

  ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

Figure 6 :

 6 Figure 6: An FTMB(4, 4) design.

Figure 8 :

 8 Figure 8: Example of maximal bipartite matching model.

Figure 9 :

 9 Figure 9: Yield estimation for DTMB(2,6), DTMB(3,6) and DTMB(4,4).

Figure 10 :

 10 Figure 10: Effective yield for different levels of redundancy.

Figure 11 :Figure 13 :

 1113 Figure 11: Fabricated biochip used for multiplexed bioassays.

  and N. F. De Rooij, "Microfluidics meets MEMS", Proceedings of the IEEE, vol. 91, pp. 930-953, 2003.

Figure 12 :

 12 (a) A defect-tolerant design based on DTMB(2, 6). (b) An example of local reconfiguration. [2] M. Pollack et al., "Electrowetting-based actuation of liquid droplets for microfluidic applications", Applied Physics Letters, vol. 77, pp. 1725-1726, 2000.

Table 1 :

 1 Redundancy ratios for the different defect-tolerant architecture.

	Design	DTMB(1, 6)	DTMB(2, 6)	DTMB(3, 6)	DTMB(4, 4)
	RR	0.1667	0.3333	0.5000	1.0000

  3] International Technology Roadmap for Semiconductor (ITRS), http://public.itrs.net/Files/2003ITRS/Home2003.htm. [4] V. Srinivasan et al., "Clinical diagnostics on human whole blood, plasma, serum, urine, saliva, sweat, and tears on a digital microfluidic platform", Proc. µTAS, pp. 1287-1290, 2003. [5] I. Koren and Z. Koren, "Defect tolerance in VLSI circuits: techniques and yield analysis", Proceedings of the IEEE, vol. 86, pp. 1819-1836, 1998.

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