Models with Deterministic Interest Rates

Picking up on the ideas of Merton (1974), Johnson and Stulz (1987) model the effect of default risk on the value of European options. They assume that the short position in the option is the counterparty’s sole liability and that the counterparty defaults if its asstes are not sufficient to meet the option holder’s claim at maturity. Hence, default may be triggered either by a decline in the counterparty’s assets or by an increase in the option value. In case of default, the option holder receives the entire assets of the counterparty potentially reduced by the cost of default. Johnson and Stulz (1987) also allow for the correlation between the counterparty’s assets and the option’s underlying. However, it is important to note that the Johnson-Stulz model is only suitable if the counterparty’s assets are relatively small compared to the expected option payoff and if the counterparty’s other liabilities are negligible.

Klein (1996), however, considers this assumption to be inappropriate in most situations and thus extends the Johnson-Stulz model by allowing for other liabilities

which rank equally with the option. The counterparty’s total liabilities are assumed to be exogenous and, by construction, must include the short position in the option. Since the structural model of Merton (1974) is used, default may only occur at the option’s maturity. In particular, the counterparty is in default if its assets are less than the total liabilities. In this case, the option holder receives a proportion of his claim which is linked to the value of the counterparty’s assets.

As in the Johnson-Stulz model, Klein (1996) accounts for the correlation between the counterparty’s assets and the option’s underlying. Based on these assumptions, the default risk can only arise from the potential deterioration of the counterparty’s assets, since the total liabilities are fixed.

Klein and Inglis (2001) set up a model which incorporates the features of both Johnson and Stulz (1987) and Klein (1996). In particular, the counterparty’s total liabilities are split into two components: the short position in the option (stochastic) and all other equally ranked liabilities (deterministic). Default occurs if the counterparty’s assets are less than the sum of the option holder’s claim and the market value of the other liabilities at the option’s maturity. The payout ratio in default is linked to the counterparty’s assets and the correlation between the counterparty’s assets and the option’s underlying is retained. In this model, default can be caused either by a decline in the counterparty’s assets or an increase in the option value making the model applicable in various situations.

Liu and Liu (2011) extend the model of Klein (1996) by assuming that the counterparty’s total liabilities are stochastic. Consequently, the counterparty is in default if the assets are not sufficient to meet the total liabilities at the option’s maturity. In case of default, the option holder receives a proportion of his claim which depends on the market value of both the counterparty’s assets and total liabilities.

In this model, the default risk arises either from a decrease in the counterparty’s assets or an increase of the counterparty’s liabilities. Liu and Liu (2011) also account for all possible correlations between the random variables.

In contrast to the previously presented models, Hull and White (1995) use the structural approach of Black and Cox (1976) to account for the default risk.

They assume that all the liabilities of the counterparty are of equal rank. Default

occurs if the counterparty’s assets fall below a determinsitic boundary prior to the option’s maturity. In this case, the option holder receives an exogenously determined proportion of his claim. To keep the model tractable, Hull and White (1995) assume that the counterparty’s default risk and the option’s underlying are independent.

Rich (1996) assumes that the option’s underlying as well as the counterparty’s credit quality (e.g. the counterparty’s assets) and the default boundary (e.g. the counterparty’s liabilities) are characterized by geometric Brownian motions. The correlations between the three stochastic variables are also considered. Since the structural approach of Black and Cox (1976) is applied, the counterparty is in default if the stochastic variable describing the counterparty’s credit quality falls below the default boundary for the first time. Rich (1996) assumes that the payout ratio of the option holder’s claim in case of the counterparty’s default is exogenously given. This assumption is necessary in order to keep the model mathematically tractable.

The model of Hui et al. (2003) extend the models of Hull and White (1995) and Klein (1996). They assume that the counterparty’s total liabilities are time-dependent and are governed by the volatility of the counterparty’s assets. The counterparty is in default if the market value of the assets falls below the market value of the total liabilities at any point in time prior or at the option’s maturity. Furthermore, it is assumed that the option holder receives a exogenously given proportion of his claim if the counterparty defaults.

Hui et al. (2007) can be seen as an extension of Hui et al. (2003), since they assue that the counterparty’s liabilities are governed by its own stochastic process. The counterparty is in default if the market value of the assets falls below the market value of the total liabilities at any point in time prior or at the option’s maturity.

To keep the model mathematically tractable, Hui et al. (2007) assume that payout ratio in case of the counterparty’s default is exogenously specified in order to keep the model mathematically tractable.

Liang and Ren (2007) set up a valuation for vulnerable European options which can be seen as an extension of Johnson and Stulz (1987) and Hull and White (1995). In particular, they assume that the short position is the counterparty’s only liability and that default occurs as soon as the value of the counterparty’s assets falls below

the intrinsic value of the option. Hence, default may occur also prior to the option’s maturity. In contrast to other valuation models based on the Black-Cox approach, Liang and Ren (2007) assume that the payout ratio to the option holder in case of default is endogenously determined.