Structural Model - Merton Model

Prerequisite:

• Some knowledge of probability
• Some knowledge of statistics
• Some knowledge of stochastic process

In previous blogs, we show that credit risk is undoubtedly essential in finance. Understanding credit risk can not only help us manage risk but also price credit derivatives like CDS. Sadly, if you misuse the model, the consequences can be disastrous. (See AIG)

This post aims to explain the Merton Model. Though it has many flaws (all models have flaws), the Merton Model is the ancestor of the structural model family and is still widely used to understand credit risk. Further, we can use empirical data to answer questions like “What factors lead to the default?”

What does Structural mean?

In his paper, Bob Merton initially aims to quantify corporate liabilities.

Here, Merton imposes the simplest form of liability structure on a firm, that is, assets = equity + debts(liabilities)

Merton’s Assumptions

N.b. we use different notations here. See more in section 3 of the original paper

1. The market is frictionless: no transaction cost, no tax (open the door, IRS!), etc.
2. The market is competitive: Everyone is a price-taker. No one can manipulate the price by massive buys or sells
3. One can borrow or lend at the same interest rate
4. One can short-sell
5. Trading takes place in continuous time: you can trade at anytime
6. By the Modigliani-Miller theorem, asset value is invariant to its capital structure
7. The interest rate evolution is known with certainty. Simply: $B(0, T)=e^{-rT}$ , where $B(0, T)$ is the present value of a zero-coupon bond that pays 1 dollar at time T
8. The asset value process is a geometric Brownian motion, i.e. $dA = (\alpha A - c)dt + \sigma Adz$ , where
   1. $A$ is the asset price (which is a stochastic process)
   2. $c$ is the dividend/interest payment
   3. $\alpha$ is the expected rate of return
   4. $\sigma^2$ is the variance of the rate of return
   5. $dz$ is a standard Weiner process (Brownian motion)

Similarity to Black-Scholes

You may find a lot of the same or similar assumptions in the Black-Scholes model. Especially that, asset value evolves as a geometric Brownian motion. This is a very bussing assumption, but also very sus.

Valuation of risky bond

Remember that on the firm’s balance sheet, there are Equity, Asset, and Liability.

Players in a firm:

Now, let’s say there are only two major players in the firm:

1. Bondholders, who hoard homogeneous debts (homogeneous means they mature at the same time)
2. Shareholders, who hoard equities (and of course get dividends)

Rules of the Jungles:

1. The firm is obligated to pay $B$ dollars to bondholders at the time T.
2. If the firm defaults (doesn’t have to be at time T), bondholders take over the firm and shareholders get nothing.
3. No dividends to shareholders, or new shares, or share repurchase prior to the bond maturity

All holders are equal, but some holders are more equal upon default. — Benton Li

At times t we use $E_t, A_t, L_t$ to denote the price of the equity, asset, and bond respectively. N.b. at time T, the bondholder expects to receive $B$ dollars.

What happens at time T?

If you are a shareholder, you might strategically decide to go default or not.

If $A_T > B$, shareholders are better off liquidating the assets and paying the bondholders. Otherwise, per rule 2, shareholders will receive nothing if they go default. In shareholders’ best interests, they rather receive $A_T - B > 0$, which is better than nothing.

Otherwise, $A_T \leq B$. In this case, shareholders might rather go default than pay bondholders using shareholders‘s money (assume they are rational asf in terms of money and have absolutely no emotional attachment to their company). Per rule 2, bondholders take over the firm. Intuitively, bondholders will liquidate all the assets, and get back the loss given default (LGD).

Payoff Valuation

Therefore, we can say that at time T:

• Bondholders’ payoff $L_T = min(A_T, B) = B - min(0,B-A_T)$
• Shareholders’ payoff $E_T =max(0, A_T-B)$

What does it bondholders’ payoff look like?

• A covered European put option on a firm asset $A_t$ with a strike price $B$

What does it shareholders’ payoff look like?

• A European call option on a firm asset $A_t$ with a strike price $B$

Now let’s plug in our favourite Black-Scholes model here:

where $d_1=\dfrac{ln(\dfrac{A_t}{B}) + (r + \dfrac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}} , \\d_2=\dfrac{ln(\dfrac{A_t}{B}) + (r - \dfrac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}} = d_1 - \sigma\sqrt{T-t}$

Interpreting the model and beyond

$L_t= Be^{-r(T-t)}N(d_2) + A_tN(-d_1) \\ L_t= Be^{-r(T-t)} - ((1-N(d_2))Be^{-r(T-t)} - A_tN(-d_1))$

We can view $1-N(d_2)$ as the probability of default (PD) But can we use $1-N(d_2)$ as PD and compute EAD?

Probably not. There are too many unrealistic assumptions. For example, what kind of asset has a lognormal return? Can you really liquidate your asset at the mark-to-market price? Also, is it really okay to assume the volatility of the asset is constant? In the real world, shareholders would love to strategically screw the bondholders. For example, upon default, equity might be more privileged than CoCo bonds.

Even more importantly: How do I know the fair market price of the assets?

These questions give us a hint of what to improve. Nevertheless, Merton's model is a great founding stone for structural models. There are many models that extend the Merton Model:

• Anderson, Sundaresan, and Tychon (1996): Liquidation is perhaps the last resort. Shareholders and bondholders might be civil. They could negotiate and propose a new contract after default.
• Huang and Huang (2003): Empirically data shows that equity risk premium tends to be negatively correlated to the return of the assets. HH model imposes a new structure on asset price evolution
• The infamous MKMV: See more at The Moody's KMV EDF RiskCalc v3.1 Model (moodys.com)

Advanced reading

Hypothetical security

Now suppose there exists a magic security $Y$, which can be seen as a derivative whose underlying asset value $A$. For simplicity, we can say $Y$ is a mixture portfolio of debt and equity

$Y$’s value $F(A,t)$ is a function of asset value $A$ and time

by Ito Lemma, we have that

N.b. subscript means partial derivative.

E.g. $F_A$ means the partial derivative of $F$ with respect to $A$.

E.g. $F_{AA}$ means the partial derivative of $F_A$ with respect to $A$

Further, to satisfy no-arbitrage and no risk

See full proof in Merton’s paper

Nota Bene:

Merton's model was first introduced in Bob Merton’s PhD dissertation.

I took Fixed Income and Asset Pricing Theory with Prof. Robert Jarrow, who was once a student of Bob Merton’s.

Reference:

Merton, Robert C. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” The Journal of Finance, vol. 29, no. 2, 1974, pp. 449–70. JSTOR, https://doi.org/10.2307/2978814. Accessed 24 Nov. 2023.

Robert Jarrow (Cornell University, USA) and Arkadev Chatterjea (Indiana University Bloomington, USA). An Introduction to Derivative Securities, Financial Markets, and Risk Management

Ronald W. Anderson, Suresh Sundaresan, Pierre Tychon. Strategic analysis of contingent claims, European Economic Review, 1996, vol. 40, issue 3-5, 871-881 Huang, Jing-Zhi Jay and Huang, Ming, How Much of Corporate-Treasury Yield Spread is Due to Credit Risk? (September 12, 2012). Forthcoming in the Review of Asset Pricing Studies, 14th Annual Conference on Financial Economics and Accounting (FEA); Texas Finance Festival; 2003 Western Finance Association Meetings, Available at SSRN: https://ssrn.com/abstract=307360 or http://dx.doi.org/10.2139/ssrn.307360