Pricing vulnerable options by binomial trees



Pricing Vulnerable Options by Binomial Trees

Shu-Ing Liu

Professor

Department of Finance

Shih Hsin University

Taipei 116, Taiwan, R.O.C.

e-mail:siliu@cc.shu.edu.tw

Tel:02-2236-8225-3433

Fax:02-2236-2265

and

Yu-Chung Liu‡

Graduate Student in Master Degree

‡Department of Mathematics

National Taiwan University

Taipei 106, Taiwan, R.O.C.

e-mail:r93221012@ntu.edu.tw

2006/03/30

Abstract

This paper proposes a simple algorithm extending the discrete CRR model to evaluate vulnerable derivatives. The payoff functions are adopted from the Klein (1996) credit risk framework that includes two stochastic processes, the underlying stock price and the assets value of the option writer. Instead of building a bivariate tree structure for these correlated processes, a univariate binomial tree for the underlying stock price is constructed. The main contribution of this paper is to introduce the concept of expected intrinsic value, which links the relation between the two stochastic processes and helps accomplish the pricing by using the univariate binomial tree model. It is both analytically verified and numerically demonstrated that the proposed binomial tree model contains the Klein (1996) formula as a limiting case. Moreover, comparative static analyses on vulnerable American put options are numerically illustrated.

Keywords: Binomial trees model; Credit risk; Expected intrinsic value; Vulnerable options

JEL Classification: G10, G13, C63

Mathematics Subject Classification (2000): 60F05; 62P05; 91B28

1. Introduction

In recent years, more and more financial institutions trade derivative contracts with their corporate clients and other financial institutions in the over-the-counter (OTC) markets. These off-exchange derivatives have experienced tremendous growth and account for a large amount of derivative contracts. The OTC-issued instruments, however, are usually not guaranteed by a sovereign institution, neither secured with collateral posted. Unlike the major options and futures markets, there is no exchange or clearing house to require OTC-contracts positions to be marked to market on a regular basis or to enforce both sides of an OTC-contract to honor their obligations. The holders of these OTC-contracts are thus subject to credit risk if their counterparty is unable to make the necessary payments at the exercise date. In order to price the products and allocate capital for their derivatives portfolios, the trading property is needed by all participants in the OTC market.

Credit risk represents the possibility that a contractual counterparty does not meet its obligations stated in the contract, thereby causing the creditor a financial loss. It is recognized as a crucial determinant of both prices and promised returns of the financial instruments. A contract involving higher credit risk must promise a higher return to the investor; that is, it should manifest itself in a lower price for otherwise identical non-vulnerable ones. Potential defaults by counterparties in derivatives transactions result in significant credit risk for banks and other financial institutions. Once the possibility of default on outstanding derivative contracts exists, the credit risk should be taken into consideration on pricing the vulnerable derivatives.

The valuation model concerning the issue of counterparty credit risk was first proposed by Johnson and Stulz (1987), assuming that the option itself is the only liability of the option writer, and the default occurs when the option writer's collateral assets can not afford the promised payment in the option contract. They also introduced the term vulnerable option for an option that contains counterparty credit risk. Their approach is an extension of the corporate bond model by Merton (1974). Later, Hull and White (1995) extended the credit risk model of Johnson and Stulz (1987) to bond pricing models related to the first passage time. Considering of the correlation between the underlying asset price and the assets value of the option writer, they assumed that an option writer defaults on its obligations when an exogenously specified boundary is reached by the assets value of the option writer. Klein (1996) further modified Johnson and Stulz (1987) assumption by allowing the option writer to have other liabilities of equal priority payment under the option. Klein (1996) developed close-form solutions for Black-Scholes options subject to credit risk of the option writer's default event. However, most of the literature is focused on pricing of vulnerable European options, while the outstanding options in the real markets are usually of the American style. Due to the increasing popularity in the option market, a theoretical model for pricing vulnerable American options should be of interest and necessary.

The purpose of this paper is to develop a univariate binomial tree model to price vulnerable options. Adopting the credit risk framework presented by Klein (1996), a vulnerable binomial tree model, the very extension of the CRR model, is constructed. Just as the Black-Scholes formula is a limiting case of the CRR model, the Klein (1996) formula is served as a limiting case of the proposed binomial model. Besides, the proposed computational algorithms are rather simple and efficient for pricing vulnerable options, especially for vulnerable American options. In fact, the suggested algorithms can be easily extended to evaluate other different vulnerable exotic derivatives.

The rest of this paper is organized as follows. Section 2 describes the credit risk model of Klein (1996) and provides a quick review of the CRR model. Section 3 presents a univariate binomial pricing model that extends the CRR model to pricing vulnerable options under the Klein (1996) credit risk structure. The computational algorithms for both vulnerable European options and vulnerable American options are presented in this section. Numerical evaluations are illustrated in Section 4, as well as a numerical comparison to the Klein (1996) benchmark values. Finally, Section 5 concludes the paper. The appendix gives a proof of the convergence of the suggested binomial model.

2. The valuation model

2.1. Notations and assumptions

The following notations will be employed throughout our financial market models:

[pic]: the time [pic] price of the risky underlying stock.

[pic]: the time [pic] assets value of the option writer.

[pic]: the time [pic] liabilities value of the option writer.

[pic]: the constant riskless interest rate.

[pic]: the strike price of the option contract.

[pic]: the present time point.

[pic]: the maturity date of the option contract.

[pic]: the m-variate normal cumulative distribution function.

[pic]: the m-variate normal distribution, with mean vector[pic]and variance matrix[pic].

A continuous trading economy with trading interval[pic] is considered, where [pic] denotes the present time, and [pic] is the maturity date of the option contract. All random variables introduced are defined on a suitable probability space [pic] with a standard filtration[pic]. The financial market is assumed to be frictionless, arbitrage-free, and complete so that all securities are perfectly divisible; there are no short-sale restrictions, transaction costs, or taxes. The existence and uniqueness of the equivalent martingale measure[pic]is guaranteed by no-arbitrage and completeness assumptions, respectively. Although the completeness of the market may be a rather stronger assumption than other similar models and could be weaken, it gives us at least two advantages: One is that for any contingent claim, even non-tradable, there exists a replicating self-financing strategy for it. The other is that we can simply say "the" equivalent martingale measure instead of "an" equivalent one so that no explicit description of the equivalent martingale measure must be made whenever it is mentioned. Also, it is assumed that the stock upon which the option is written pays no dividends.

2.2. Klein (1996) credit risk pricing model

One of the greatest characters of vulnerable derivatives pricing models is the consideration of the possibility of the option writer that defaults on its obligations from the option contract. The related stochastic processes, the underlying stock price, the assets value of the option writer, and the liabilities value of the option writer, introduce the default event and should be appropriately described.

Klein (1996) proposes a closed-form pricing formula for vulnerable European options under a deterministic interest rate and deterministic liabilities value of the option writer. In the Klein (1996) model, the deflating factor is deterministic and continuously compounded at a riskless interest rate[pic]. The option writer's liabilities value is assumed to be constant so that[pic], for [pic]. Similar to the Black-Scholes framework, the stochastic processes of the underlying stock price,[pic], and option writer's assets value,[pic], are assumed to follow a bivariate geometric Brownian motion satisfying the stochastic difference equations

[pic]

where [pic] and [pic] are respective the instantaneous expected return on the stock and instantaneous volatility of the stock return, and [pic] and [pic] are respective the instantaneous expected return and instantaneous volatility of the assets value of option writer. The[pic]is a bivariate Wiener process under the measure [pic], with [pic]. All the parameters, say [pic], [pic], [pic], [pic], and [pic], are assumed constant.

By applying [pic]formula and Girsanov theorem, it can be verified that under the equivalent measure[pic], the martingale measure for the deflating process[pic], the following relationship is established:

[pic]

for [pic], where [pic] is a bivariate Wiener process under the measure [pic]. It can be proved that the covariance of[pic]and[pic]is the same under both measures [pic] and[pic] (see Durret (1996), p93). An immediate result follows: given the current time point [pic], the distribution of [pic] is bivariate normally distributed as[pic]

[pic][pic][pic], (1)

with mean vector and covariance matrix respectively given by

[pic],

and [pic].

The promised amount paid by the option writer depends on the value of the underlying stock price. If the option writer remains solvent throughout the lifetime of the option contract up to time [pic], a full amount of [pic] will be paid out at time [pic]; in case of bankruptcy or default of the option writer, the holder of the option contract may receive only a fraction of [pic], say [pic], instead. The parameter[pic] denotes the value recovered, and is commonly called the recovery rate. In the Klein (1996) model, the recovery rate at the maturity date is assumed to be [pic].The parameter [pic], expressed as a percentage of the option writer's assets value with [pic], represents the deadweight costs associated with the default event. It includes the direct cost of the bankruptcy or of the reorganization process, as well as the indirect effects of distress on business operations of the option writer. The payoff function of a vulnerable call option at the maturity date[pic], with [pic] replaced by[pic], is given by

[pic], (2)

where[pic], and [pic] is the indicator function of an event [pic]. The parameter[pic] is a constant default boundary, allowing the capital forbearance of the counterparty assets. That is, [pic] could be less than [pic] due to the possibility of a option writer continuing in operation even while [pic] is less than[pic].

By the risk-neutral valuation principle, the current time[pic] arbitrage price of the vulnerable European call option is the deflated expected value from time[pic]under the martingale measure[pic]. Therefore, the arbitrage price, with payoff given by equation (2), can be expressed as

[pic], (3)

where[pic]is the expectation operator with respect to the measure[pic], given the filtration [pic].

Under the aforementioned setup, Klein (1996) presented the following pricing formula.

Proposition 1. (Klein Pricing Formula for Vulnerable European Options)

The time[pic] arbitrage price process of a vulnerable European call option, with the payoff function derived from equation (2), is given by

[pic] [pic][pic], (4)

with [pic],[pic],

[pic], [pic], [pic], [pic], [pic], and [pic].

2.3. Brief review of the Cox-Ross-Rubinstein (CRR) model

Most exotic options can be easily and efficiently evaluated by using the binomial tree model, which is a simple and powerful numerical technique for option pricing. In the CRR model, a specified positive increasing sequence [pic] with[pic] is given. The[pic], called the period number, represents the number of periods remaining until expiration. The time interval[pic]is divided into[pic]subintervals of equal length[pic], with[pic][pic]. Trading is supposed to occur at equidistant time points[pic], for[pic].

In the regular CRR model, only the underlying stock price[pic] is assumed random, and all possible outcomes are discretized as a binomial tree. The one-period returns of the stock price are modeled by a family of discrete random variables [pic] defined by [pic], taking values[pic]with probabilities [pic], and [pic], where [pic]. Then the stock price is given by

[pic], for[pic]. (5)

To satisfy an arbitrage-free financial market assumption, the parameters[pic]and[pic] in the nth CRR model are chosen as: [pic],[pic], and [pic], with [pic] satisfies [pic]. By adopting the framework of the CRR model, a suitable binomial tree model with consideration of the credit risk is ready to be developed.

3. Binomial trees for vulnerable options

3.1. An algorithm for pricing vulnerable European options

According to the Klein (1996) model, there are two stochastic processes: the stock price process and the assets value process. While in the CRR model, the binomial tree structure is constructed only for a stochastic process, the stock price. A straight-forward way to handle this problem may be to construct a bivariate binomial tree model for these related processes. However, it is time-consuming to numerically handle a bivariate tree structure. For simplicity, in this paper, the tree structure remains unchanged as given in the CRR model: The jth node at time[pic]is referred to as the [pic] node, and from equation (5), the stock price at the [pic] node is given by

[pic], with[pic], (6)

for [pic].

At the maturity time T, the intrinsic value of a call option is simply given by [pic] when the stock price [pic], and the arbitrage price of a European call option is evaluated backwards from the intrinsic value at the maturity date. By virtue of our credit risk model, however, not only the stock price contributes to the pricing of a vulnerable call option, but the assets value of the option writer must be also taken into account.

By double expectation property, equation (3) can be re-expressed as [pic]. (7)

Since a binomial tree structure for the stock price is built, the evaluation of the pricing process derived by equation (7) depends mainly upon the conditional expectation

[pic].

The above conditional expectation is referred to as the expected intrinsic value at the maturity date[pic], formally given by

[pic] (8)

[pic]

[pic],

where

[pic], and [pic]. (9)

Let[pic] be the arbitrage value of the vulnerable European call option at the [pic] node. In the case of European options without early exercise, the risk- neutral valuation principle induces

[pic], for[pic], (10)

with initial conditions

[pic], for[pic]. (11)

Beginning with the initial values[pic], and moving backwards throughout every node of the binomial tree, the arbitrage price of a vulnerable European call option at the current time point results. The time[pic] arbitrage price in the suggested nth binomial model is given by[pic]. To help understand how to recursively utilize the proposed binomial tree model, a special case of a two-step tree, with [pic]and[pic], is demonstrated in Figure 1. The backward procedure is summarized in the following four steps:

1. Calculate the expected intrinsic values[pic],[pic], and [pic] from equation (8).

2. Obtain the corresponding initial values[pic],[pic], and [pic] from equation (11).

3. Repeat the backward processes using equation (10) in turn to obtain[pic],[pic],

and finally[pic].

4. The current time[pic] arbitrage price is then given by[pic].

Figure 1. Two-step vulnerable European binomial tree: At each node, [pic] is the arbitrage value of a European call option, and [pic] is the stock price.

By recursively using equation (10), the arbitrage price of the vulnerable European call option at time[pic]can be re-expressed as

[pic], (12)

where [pic].

In order to numerically evaluate equation (12), the conditional expectations at the maturity date given by equation (9) must be calculated in advance. In this paper, a statistical distribution relation between[pic]and [pic]motivated by the Klein (1996) credit risk framework is suggested. Before going on further, a popular property of the bivariate normal distribution is stated without proof. (see Johnson and Wichern (1992)).

Proposition 2. (A Conditional Property of Bivariate Normal Distribution)

Let[pic] be distributed as[pic], with [pic], [pic], and [pic]. Then the conditional distribution of [pic], given [pic], is distributed as [pic], where [pic] [pic], and [pic].

By applying Proposition 2 on the joint distribution of [pic] stated in equation (1), a statistical relation between [pic] and [pic]is assumed as follows.

Assumption 1.

The conditional distribution of [pic], given[pic], under the martingale measure [pic] is assumed to follow a normal distribution:

[pic] [pic] [pic], for[pic], (13)

where [pic], [pic],

[pic], [pic], and [pic].

Under Assumption 1, it can be verified that equation (9) turns out to be

[pic] and [pic], (14)

where [pic] and [pic].

Accordingly, the arbitrage price derived from equation (12) can be re-expressed as [pic] (15)

where[pic], and [pic].

It is well-known that the limiting case of the discrete CRR pricing formula is the Black-Scholes formula. Since the proposed vulnerable binomial tree model is adopted from the Klein (1996) credit risk framework, one would expect that it should converge to Klein (1996) formula as the period number [pic] passes to infinity. The result is stated as follows.

Theorem 1. (Convergence of the Proposed Vulnerable CRR Model to Klein Formula)

Under Assumption 1, the binomial pricing formula of the vulnerable European option given by equation (15) converges to Klein (1996) pricing formula given by equation (4). That is,[pic]

Proof. Cf. Appendix.

Theorem 1 provides a firmly convergence-based version under which the Klein (1996) credit risk pricing formula can be approximated by a suitable binomial tree structure. The suggested binomial pricing model is essentially an extension of the CRR model, with further consideration of credit risk of the option writer. The proposed computational algorithm provides a rather simple and efficient method for pricing vulnerable European options, involving calculations of a univariate normal cumulative distribution function, instead of a bivariate normal cumulative distribution function applied in Klein (1996) formula. The reduction of dimension avoids the computation error caused by a numerical integral of a bivariate normal distribution function.[pic] In addition, the discussed algorithm can be further extended to evaluate other vulnerable options, say, the vulnerable American options, or other exotic options subject to the credit risk. An investigation for pricing vulnerable American options will be carried out in the next section.

3.2. An algorithm for pricing vulnerable American options

In this section, we move to construct a simple binomial tree algorithm for pricing the vulnerable American option. Suppose a binomial tree for the stock price has been constructed as is given in the CRR model. Let[pic]be the arbitrage value of a vulnerable American call option at the [pic] node. To evaluate American options, the early exercise should be taken into consideration. At each node, the arbitrage value of a vulnerable American option must be further compared with the corresponding expected intrinsic value, which is defined as

[pic] (16)

[pic],

for[pic]and[pic]. Assume that there is no default event at the current time[pic], thus[pic].

For vulnerable American options, further assumptions, besides Assumption 1, must be added to evaluate the conditional expectation at each node expressing in equation (16). By extending Assumption 1, the statistical relations between [pic]and[pic], for [pic], are assumed as follows.

Assumption 2.

The conditional distribution of [pic], given[pic], under the martingale measure [pic] is assumed to follow a normal distribution distributed as

[pic] [pic] [pic], for[pic], (17)

where [pic], [pic],

[pic], [pic], and [pic].

Under the conditional distributions stated in Assumption 2, the expected intrinsic value of a vulnerable American call option at the [pic] node can be re-expressed as

[pic] (18)

where

[pic], [pic], (19)

[pic], and [pic].

A special case of a two-step tree, with[pic]and[pic], is demonstrated in Figure 2 to illustrate the relation of a stock price and the corresponding expected intrinsic value. The suggested expected intrinsic value captures not only the effect of early exercise, but also includes the possibilities of the default event at each node. This is achieved by the proposed conditional distribution of the option writer's assets value, given the stock price at each node.

Figure 2. Relation between stock price and expected intrinsic value: At each node, [pic] is the stock price, and [pic] is the corresponding expected intrinsic value of a vulnerable American call option.

Finally, the arbitrage value of the vulnerable American call option at each node is recursively given by

[pic] , (20)

for[pic], with initial conditions

[pic], for [pic]. (21)

Again, beginning with the initial values[pic], and moving backwards throughout the tree by using equation (20), the current arbitrage price of a vulnerable American call option, say[pic], is obtained.

Figure 3 demonstrates a special case of a two-step tree, and helps understand how to utilize the suggested binomial tree in order to recursively evaluate vulnerable American call options. The recursive procedures are summarized in the following four steps:

1. Calculate the expected intrinsic values[pic],[pic], and [pic] from equation (18).

2. Obtain the corresponding initial values[pic],[pic], and [pic] from equation (21).

3. Repeat the backward processes of equation (20) in turn to obtain[pic],[pic] and finally[pic].

4. The current time[pic] arbitrage price is finally given by[pic].

Figure 3. Two-step vulnerable American binomial tree: At each node, [pic] is the arbitrage value of a vulnerable American call option, and [pic] is the expected intrinsic value.

4. Numerical evaluations

In this section, numerical analyses are illustrated by using the aforementioned binomial tree model. A comparison with benchmark values derived from the Klein (1996) formula is provided in Table 1, which evaluates the differences between the Klein (1996) formula and the suggested binomial pricing algorithm. The relative percentage error for pricing of the vulnerable European call option is defined by [pic], where [pic]is the option price calculated from equation (4), and[pic]is obtained from the proposed binomial model. The period number[pic]is specified as [pic], and a set of parameters called the Base Case is given by: [pic], [pic], [pic], [pic], [pic], [pic], [pic], [pic], [pic], [pic], and [pic].

The period number [pic] ranges from 50 to 2000. In each case of Table 1, the relative percentage error decreases monotonically as the period number increases. It can be also observed that all relative percentage errors are less than [pic] after the period number is larger than 100, less than 0.10% after the period number is over 500, and most of them are less than 0.01% whenever the period number equals 2000.

Table 1. Relative percentage error of vulnerable European call options

|Case |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |

|Base Case |0.3901 |0.1928 |0.0940 |0.0346 |0.0148 |0.0049 |

|[pic] |0.3539 |0.1749 |0.0852 |0.0313 |0.0133 |0.0044 |

|[pic] |0.3734 |0.1854 |0.0912 |0.0346 |0.0158 |0.0063 |

|[pic] |0.4028 |0.2002 |0.0987 |0.0377 |0.0174 |0.0072 |

|[pic] |0.4005 |0.1998 |0.0992 |0.0388 |0.0186 |0.0085 |

|[pic] |0.3845 |0.1905 |0.0933 |0.0350 |0.0155 |0.0058 |

|[pic] |0.4467 |0.2236 |0.1119 |0.0447 |0.0224 |0.0112 |

|[pic] |0.4129 |0.2016 |0.0957 |0.0321 |0.0109 |0.0003 |

|[pic] |0.3756 |0.1862 |0.0913 |0.0343 |0.0153 |0.0058 |

|[pic] |0.6099 |0.1590 |0.1246 |0.0463 |0.0281 |0.0119 |

|[pic] |0.2196 |0.0387 |0.0323 |0.0183 |0.0111 |0.0052 |

|[pic] |0.4448 |0.2226 |0.1113 |0.0444 |0.0221 |0.0110 |

|[pic] |0.4467 |0.2236 |0.1119 |0.0448 |0.0224 |0.0112 |

1. The above calculations are based on the Base Case unless otherwise noted. For instance, the case of [pic] is by replacing the original [pic] in the Base Case and keeps the rest of the parameters unchanged.

2. For each entry, the relative percentage error is computed by:

[(Klein result - the suggested Binomial tree result)/Klein result] x 100 (%)

3. The relative percentage errors are less than 0.25% whenever [pic] is larger than 100, less than 0.05% after [pic] is over 500, and most of them are less than 0.01% when [pic] equals 2000.

4. The relative percentage error is monotonically decreasing as [pic] increases in each case.

The nature of put options between the European-type and the American-type are usually more evident than that of call options.[pic] To visualize the difference between vulnerable European options and vulnerable American options, the following comparative static analyses are focused on put options instead of call options, simply with a fixed period number of 500. The notations "[pic]" denote the price of a non-vulnerable European Black-Scholes put option, "[pic]" and "[pic]" denote approximated prices of the vulnerable put option, while the former for the European-type derived from equation (15), and the latter for the American-type derived from equation (20).

The price of a vulnerable put option is less than that of a non-vulnerable one due to the credit risk consideration, so [pic] always holds. On the other hand, the price of a European option is always less than that of an American one because of the right of early exercise, no matter vulnerable or non-vulnerable. Therefore, the price of a vulnerable European put option is less than that of a vulnerable American one, that is, [pic]. For the vulnerable American option, however, there is a tradeoff between the negative effect of credit risk and the positive effect of the early exercise, so the relation between [pic] and [pic] is not quite clear. When the credit risk effect is weaker than the effect of early exercise, the price of a vulnerable American put option will be higher than that of a non-vulnerable European Black-Scholes put option. On the contrary, the vulnerable American put option costs less than the non-vulnerable European Black-Scholes put option in case that the credit risk effect dominates over the effect of early exercise. These interactive impacts are visualized in the following numerical illustrations.

A comparative static analysis of vulnerable put options, with the initial stock prices ranging from 25 (deep-in-the-money) to 45 (out-of-money) and other parameters remaining the same as the Base Case, is performed and expressed in Figure 4. The vulnerable European put option always costs less than the non-vulnerable European one, that is, [pic], no matter how the initial stock price changes. In case of out-of-the-money or at-the-money, the early exercise right does not function. Therefore, the vulnerable American put option is less valuable than the non-vulnerable European one. As the initial stock price decreases, the vulnerable American put option gets more and more valuable; eventually, the impact of early exercise eliminated the negative effect of credit risk, and the price of the vulnerable American put option exceeds the price of the non-vulnerable European put option.

Figure 4. Vulnerable put options -- different initial stock price.

1. Parameters are based on the Base Case :[pic],[pic],[pic],[pic], [pic], [pic],[pic],[pic],[pic],[pic], and [pic].

2. The period number [pic] is fixed at 500.

3. When the initial stock price is low, the impact of early exercise dominates the negative effect of credit risk, so the price of the vulnerable American put option exceeds the price of the non-vulnerable European put option. However, as the initial stock price gradually increases, say more that 33, the situation gets reverse.

On the other hand, we vary the initial assets value from 5.0 to 10.0 in Figure 5, with the rest of the parameters kept unchanged in the Base Case. Obviously, the higher the initial assets value goes, the lower the chance of the option writer's default becomes. Thus, the vulnerable European put option will eventually reduce to be the non-vulnerable European Black-Scholes put option as the initial assets value increases. In the meanwhile, the price of the vulnerable American put option is more valuable than that of a non-vulnerable European Black-Scholes one whenever the initial assets value is relatively higher. Nevertheless, as the initial assets value decreases, the effect of default risk gets strong, and thus the non-vulnerable European Black-Scholes put option costs more than the other two kinds of vulnerable put options.

Figure 5. Vulnerable put options -- different initial assets value.

1. Parameters are based on the Base Case :[pic],[pic],[pic],[pic], [pic], [pic],[pic],[pic],[pic],[pic], and [pic].

2. The period number [pic] is fixed at 500.

3. The price of the vulnerable American put option is more valuable than that of a non-vulnerable European Black-Scholes one whenever the initial assets value is relatively high say over 6.5 in this case. However, when the initial assets value is at a low level, the situation gets reverse.

5. Conclusions

In this paper a simple algorithm is developed by utilizing a univariate binomial tree model to price vulnerable options. The payoff functions are adopted from the Klein (1996) credit risk framework that includes two stochastic processes, the underlying stock price and the assets value of the option writer. Instead of building a bivariate tree model for the two correlated processes, a univariate binomial tree model that is an extension of the CRR model, is constructed for the underlying stock price process. It is both analytically verified and numerically demonstrated that the proposed binomial tree model contains the Klein (1996) formula as a limiting case. In addition, the impact of either the initial underlying stock price or the initial assets value of the option writer is numerically illustrated for vulnerable American put options. The proposed binomial tree algorithms are rather simple and efficient for pricing vulnerable options, especially for vulnerable American options, and the algorithms can be easily extended to evaluate other different vulnerable exotic derivatives.

|[pic]Here we simplify the notation[pic] to be[pic]. The filtration [pic] can be simply regarded as the collection of |

|information up to the current time |

|t. For simplicity, the notation [pic] will be omitted throughout this paper. |

|[pic]Approximation techniques for the evaluation of the double integral can be found in Drezner (1978), who provides a way |

|of calculating the bivariate normal integral to an accuracy of four decimal places. Another evaluation of the bivariate |

|normal integral by standard numerical integration is described in Press et al. (2002). |

|[pic]For example, Merton (1976) shows that the price of a non-vulnerable American call option underlying on a stock that |

|does not pay dividends is equal to the price of the non-vulnerable European call option. However, this situation does not |

|happen for put options, no matter vulnerable or non-vulnerable. It is the reason why we choose to provide numerical examples|

|for put options. |

Appendix

Proof of Theorem 1:

First, equation (4) could be decomposed as

[pic],

where [pic], [pic],

[pic], and [pic].

On the other hand, equation (15) could be also divided into four terms as

[pic],

where [pic],

[pic],

[pic],

and [pic],

with [pic]. It suffices to show that [pic], for[pic]

Let[pic]and[pic]be random variables with[pic] and [pic]. Hence, the stock price at the maturity date could be re-written as[pic]. After some algebra, we have

[pic][pic],

[pic],

[pic], [pic],

where [pic], [pic],

[pic], and [pic].

Both[pic] and [pic] can be derived from the CRR model (see Bingham and Kiesel (2004)). To complete the proof of Theorem 1, it is sufficient to show the following results.

Proposition 3. [pic], [pic], [pic], and

[pic].

To prove Proposition 3, some notations and preliminary results are given as follows.

Notation 1. (little oh)

Given two sequences[pic]and[pic]such that[pic], we say[pic], which is read "[pic] is little oh of [pic]", if[pic].

Notation 2. (standardized[pic] and[pic])

[pic],[pic], [pic],

[pic], [pic], and [pic].

By the central limit theorem, both [pic] and [pic] hold.

Notation 3.

[pic], [pic], [pic], and [pic], where [pic], [pic], [pic], [pic], [pic], and [pic].

Lemma 1. [pic] , [pic], [pic], and [pic].

Proof of Lemma 1:

By definition,[pic], and [pic][pic], thus [pic] [pic]. Further, [pic][pic], and [pic] [pic]; therefore, [pic], and [pic] [pic]. □

Lemma 2. [pic], and[pic].

Proof of Lemma 2: By definition,[pic]

[pic]. From results of Lemma 1, it follows that

[pic], and

[pic]. □

Lemma 3. [pic], [pic], [pic], and [pic].

Proof of Lemma 3: By Lemma 1, [pic], [pic], and Lemma 3 follows. □

Lemma 4. [pic]. □

Proof of Proposition 3: By definition of[pic], [pic]

[pic]

[pic][pic].

Similarly,

[pic], [pic],

and [pic]. By the central limit theorem, it follows that

[pic]

[pic] [pic][pic],

[pic]

[pic]

[pic] [pic].

This completes the proof. □

References

Ammann, M., Credit Risk Valuation: methods, models, and applications, 2nd ed., 2001 (Springer-Verlag: New York).

Bingham, N. H. and Kiesel, R., Risk-Neutral Valuation, pricing and hedging of financial derivatives,

2nd ed., 2004 (Springer-Verlag: New York).

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Durret, R., Stochastic Calculus, 1996 (CRC Press: New York).

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Jarrow, R. A. and Turnbull, S. M., Pricing Derivatives on Financial Securities Subject to Credit Risk. J. Finance, 1995, 50, 53-85.

Johnson, H. and Stulz, R., The pricing of Options with Default Risk. J. Finance, 1987, 42, 267-280.

Johnson R. A., and Wichern, D. W., Applied Multivariate Statistical Analysis, 3rd ed., 1992 (Prentice-Hall: New Jersey).

Klein, P., Pricing Black-Scholes options with correlated credit risk. J. Banking Finance, 1996, 20, 1211-1129.

Klein, P. and Inglis, M., Pricing vulnerable European options when the option's payoff can increase the risk of financial distress. J. Banking Finance, 2001, 25, 993-1012.

Lando, D., Credit Risk Modeling: theory and applications, 2004 (Princeton University Press: New Jersey).

Merton, R. C., On the Pricing of Corporate Debt: The Risk Structure of Interest Rates. J. Finance, 1974, 2, 449-470.

Merton, R. C., Option Pricing when Underlying Stock Returns Are Discontinuous. J. Finan. Econ., 1976, 3, 125-144.

Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical Recipes in C++: The Art of Scientific Computing, 2nd ed., 2002 (Cambridge University Press: Cambridge, UK).

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