###### Valuation of an Interest Rate Option

Interest rate options are options with an interest rate as the underlying. A... **Read More**

Structural models focus on a firm’s assets and liabilities and define a mechanism for default. The probability of default is endogenous as default normally occurs when the value of the firm’s assets hits a barrier representing default. They are called structural models as default is based on the company’s balance sheet structure.

Structural models allow the interpretation of debt and equity values in terms of options. The probability of default is then modeled using option pricing theory, for example via the Black Scholes-Merton option pricing model.

- The company’s assets trade in frictionless markets that are arbitrage-free.
- The riskless rate of interest, \(r\), is constant over time.
- The value of the firm’s assets at time
*t*follows a lognormal distribution.

Let \(F_t\) be the value of a firm, which is the sum of debt \(B_t\) and equity \(E_t\):

$$ F_t=B_t+E_t $$

Since debt is senior to equity, the value of equity at maturity is given by:

$$ E_T = Max(F_T-K,0) $$

Notice that \({Max}(F_T-K,0)\) is the same as the payoff of a European call on a company’s assets with a strike price of \(K\) and a maturity of \(T\)*.*

According to put-call parity, a firm’s equity can also be interpreted as a long position in the assets, long put option, and short bond:

$$ E_T=F_T-K+Max(K-F_T,0) $$

Owning a firm’s debt is economically equivalent to:

- Owning a riskless bond that pays \(K\) at time \(T\) with certainty.
- Simultaneously selling a European put on the assets of the company with a strike price of \(K\) and expiry at \(T\).

Let \(K\) be the face value of a single risk-free zero-coupon bond that matures at time \(T\).

The value of debt, \(B_T\) is given by:

$$ B_T=F_T-Max(F_T-K,0) $$

According to the put-call parity, debt can be viewed as a long bond and a short put option:

$$ B_T=K- \left\{\begin{matrix} {0\ if\ F_T\geq K} \\ {{K-F}_T\ if\ F_T < K } \end{matrix}\right.=K-Max(K-F_T,0) $$

- The beauty of structural models is that they allow us to use option pricing formulas such as Black Scholes option pricing methods to evaluate default probability and recovery rate.
- Structural models provide both an intuitive economic interpretation and an endogenous explanation of credit defaults.

- The assumption that corporate assets are tradable is unrealistic.
- They assume a simple balance sheet structure. This implies that complex balance sheets cannot be modeled.
- Structural models are not suitable when there is off-balance-sheet financing.
- They ignore the business cycle.
- Since the firm’s assets cannot be observed, credit measures are estimated by implicit estimation procedures prone to intrinsic errors.
- Generally, structural models are analytically complex and computationally intensive.

Contrary to structural models, default is no longer tied to the firm’s assets falling below a threshold level under reduced-form models. Instead, it occurs according to some exogenous hazard rate process.

- They use observed data, both macro and micro, and thus can be ‘fitted’ to data.
- The risk-free interest rate is stochastic.
- The probability of default and recovery rate varies with the business cycle.

- The use of observed data means that historical estimation procedures can be employed to determine credit risk measures.
- Credit risk varies with the business cycle.
- There is no need to specify the firm’s balance sheet.

- Past market conditions may not reflect the future, and estimates derived from observed data may be inappropriate
- Reduced form models do not provide an economic explanation of why default occurs.
- Reduced form models treat default as a random event, which is not the case in reality.

## Question

Which of the following is

most accurateabout reduced-form models?

- Reduced form models provide an economic explanation of the occurrence of default.
- Default occurs according to some endogenous hazard rate process.
- The probability of default and recovery rate vary with the business cycle.

Solution

The correct answer is C.Reduced form models allow the default intensity to change as the firm’s fundamentals and economy changes.

A is incorrect.Reduced form models do not provide an economic rationale behind a default

B is incorrect.The reduced form approach does not consider the endogenous cause of defaults; rather, they depend on exogenous specifications for credit default and debt recovery.Generally speaking, reduced-form models assume an exogenous recovery rate independent of the default probability and the dynamics of a firm’s assets and take as basics the behavior of default-free interest rates, the recovery rate of defaultable bonds default, as well as a stochastic process for default intensity.

Reading 31: Credit Analysis Models

*LOS 31 (d) Explain structural and reduced-form models of corporate credit risk, including assumptions, strengths, and weaknesses.*