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:: Firm Value Approach to Pricing Credit Risky Derivatives ::

In a previous article we introduced the Firm value model for pricing credit risk. Re-stated briefly, this model assumes default to be triggered when the couterparty share price hits a prescribed low barrier. The classic Firm model (by Merton) only considers the counterparty share price at the maturity of the derivative (and neglects premature default). If the share price at maturity is below the default barrier the counterparty is deemed to be in a state of default. More recently, this has been extended to encompass the possibility to default at any time up to the maturity of the derivative. These are called structural models. Default is then avoided if the minimum price hit by the counterparty share price is above some pre-specified level. Whichever model is selected, what happens next depends on the specific agreement between the 2 parties.

The recovery rate (if any) must be specified. In the classic firm value model this is usually a percentage of the terminal payoff. In the structural approach the recovery can be a percentage of the intrinsic value at the time of default, in which case the derivative expires prematurely.

Note that it is the counterparty share price that dictates the default,while the derivative is a contract on some other underlying. Hence,the derivative is now a function of 3 variables, the underlying, the counterparty share price and time (t). In this setting the pricing problem is essentially a multi asset pricing problem.

Consider a portfolio long in the derivative and short a quantity   and in the underlying and counterparty share price respectively:  

Over a time step dt the portfolio changes by:

We will assume that both stock price dynamics are driven by the lognormal random walks:

Where W and Z are (possibly) correlated Brownian motions (here we will assume they are uncorrelated for simplicity). Using Ito' s lemma the evolution of the portfolio can be re-written as:

The risk can be eliminated by choosing:

Then, arbitrage arguments dictate that the portfolio should grow at the risk free rate. This leads to the following PDE for the credit risky derivative:

The solution of this PDE would require one time condition and 4 boundary conditions. Although the above holds for any derivative, we will demonstrate the choice of boundary conditions for a risky call option.

Final Condition:

The final condition says that a full pay out is realized if the final counterparty share price is above the default barrier otherwise only a percentage is paid ( = the recovery rate).  

Boundary Condition:

The boundary conditions for the underlying are basically very similar to the ones often used for the standard Black-Scholes PDE. The value of the call is zero when the underlying is 0 (regardless of the state of the counterparty). For high values of the underlying, the value tends asymptotically to the value of the asset.

Boundary condition:

The last 2 boundary conditions are derived to make the derivative insensitive to default risk when the counterparty share price increases (low risk of default). In this case, the price of the derivative should resemble the risk free Black-Scholes price. In an analogy to the first boundary condition above, the price of the derivative at is the minimum possible value, i.e. the defaulted Black-Scholes value.

In the PDE setting structural models can be treated very easily. The treatment is all in the boundary conditions and the situation is analogous to the case for Barrier options. The option pays out as long as the minimum price observed for the counterparty share price is above the barrier. On this barrier, the value of the derivative is the defaulted value. As the counterparty share price increases to infinity, the risk of default diminishes and the value tends to the Black-Scholes price.

At the barrier, the counterparty is deemed to be in a state of default. Hence,the value of the call is a proportion (recovery rate) of the true value, the true value being the value of a risk free call priced with standard Black-Scholes. Similarly, as the counterparty share price increases the risk of default diminishes and the value of the call is just the same as the risk free call.

The model discussed in this article can easily be adapted to the valuation of credit risky bonds. Simply replace V by the value of the bond, S by the value of a risk free bond (under stochastic interest rates) and C is unchanged. As in the derivation of the usual bond pricing formula, the PDE involves a market price of risk since interest rate cannot be hedged in general.

The plots below correspond to the price of a risk free call, Merton risky call and a structural model call under the same parameters. It was assumed that the recovery is zero and there is no correlation between the underlying and the counterparty share prices; this was made for simplicity, although the trends observed in the results are general. The standard Black-Scholes price is invariant with counterparty share price as expected. However, the second graph demonstrates the dependency of the option price on the option writers share price, when that counterparty is deemed to be credit risky. As the counterparty share price falls, the option price also falls to reflect the increasing risk of default. At very high counterparty share prices the risk of default diminishes and the option price reverts to the standard Black-Scholes price. The last graph demonstrates the option price according to the structural model. This is the most realistic model to adopt for pricing since it assumes that default can occur anytime; since this corresponds to a much more risky option the prices are lower than both the first and second graphs (hence giving the investor a higher yield). Once again, for low counterparty prices the value of the option is zero (default with no recovery/rebate). At higher share values the risk of default diminishes and the option price tends to the risk free Black-Scholes values.

Finally, we will summarize some of the pros and cons of the models discussed. The advantages stem from that tractability of the solutions and the ability to easily extend this analysis to more complex cases. In the end the option resembles a multi asset option and these have been around for a long time and have been studies quite extensively. The cons stem from the actual validity of the model. The counterparty share price may not be an indicator of the credit risk. Share prices often neglect the value of the physical assets and goodwill of the issuing company and very often share prices can swing wildly without triggering default. This is clearly observed in market 'mini crashes' where the entire market indices drop as investors leave equity for other 'safer heavens'. The selling of equity can often depress the shares prices temporarily but this not an indication of default.

Written by Samy Mohammed.

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