merton-model 0.0.3

Merton model distance to default

The Merton Model

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Introduction

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The Merton model was first developed by economist Robert Merton in 1974.

The model makes a claim about default probabilities based on the capital

structure of the firm. It models the value of equity as a call option on

the total assets of the underlying firm. If assets are below the call

option’s strike price at time T, the firm defaults, the value of equity

goes to zero, and the remaining value of the firm is divided equally

among debt holders.

The model has some limitations. The version of the model we considered

makes the Black-Scholes assumptions, which are generally accepted to be

false. Furthermore, it does not address the possibility of a bankruptcy

before time T.

We used several resources. Of particular utility was a Society of

Actuaries article entitled “Structural Credit Risk Modeling: Merton and

Beyond” by Wang, “Options, Futures, and Other Derivatives” 9th ed. by

Hull, and Journal of Banking & Finance article “Credit spreads with

dynamic debt” by Das and Kim.

Black-Scholes Assumptions and Formula

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In addition to the frame-work of the Merton model, we adopted the

Black-Scholes assumptions but with the underlying asset being the total

assets of the firm. These assumptions are the following. * A constant

continuously compounded risk-free rate r. * The assets of the firm

follow geometric Brownian motion. In particular, if the total assets of

the firm at time :math:t are equal to :math:A_t, then the assets

satisfy the stochastic differential equation

.. math::

dA_t=r A_t dt+ σ_A A_t dW_t

where :math:W_t denotes Brownian motion under the risk-neutral

probability measure :math:Q. * The coefficient in the diffusion term

σ_A is constant. * Portfolios can be instantaneously rebalanced. *

There are no transaction costs or taxes. That includes transaction costs

on short positions. * There are no arbitrage opportunities. That is,

returns in excess of the risk-free rate are proportional to the risk

associated with the position. Under these assumptions, we can use the

popular Black-Scholes call option formula to model the value of the

firm’s equity :math:E_t:

.. math::

E_t= A_t N(d_1)-Ke^{-r(T-t)} N(d_2),

where

.. math::

d_1= \frac{\log(A_t/K)+ (r+ \sigma_A^2/2)(T-t)}{\sigma_A \sqrt{T-t}}\quad\text{and}\quad $d_2=d_1- \sigma_A\sqrt{T-t}.

Parameter Selection

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The formulas in the previous section raise questions regarding parameter

selection. How would one estimate the diffusion coefficient

:math:\sigma_A and strike price :math:K as well as select the length

of the time window? This section aims to address those questions. The

value of :math:\sigma_A is unobservable and we must derive this value

from other traded assets. The diffusion coefficient of equity

:math:\sigma_E is also unobservable, but there is a great deal of

theory behind :math:\sigma_E and an implied value can be extracted

from market data. Then, under the assumption that the value of equity

:math:E_t also follows Brownian motion,

.. math::

\sigma_E E_t- \sigma_A A_t N(d_1) = 0.

Within the Python code, we used fsolve to find :math:\sigma_A. In the

cases were the solver does not converge, we assume

.. math::

A_t =E_t+L_t\quad\text{and}\quad N(d_1) = 0.8.

The value of :math:L_t is the present value of the cash flows from

liabilities. It is calculated using a put option as described in the

next section, but within this put option assume the volatility of the

underlying is :math:\sigma_E instead of :math:\sigma_A; this

overestimates the volatility of the underlying which decreases the value

of :math:L_t. Using all of these assumptions, we find

.. math::

\sigma_A= \frac{E_t}{0.8(E_t+L_t)}\sigma_E.

The value of :math:\sigma_A is most likely smaller than what is

implied from our calculations.

Regarding the time window, we supposed $t = 0 and :math:T = 1. This

choice is somewhat arbitrary. However, modeling default over a shorter

window is more likely to produce reliable results. This is because a

firm’s balance sheet provides less and less information about its future

capital structure the further and further away the present we conduct

our analysis.

Our choice of strike price :math:K was less clear. Denote the current

and noncurrent liabilities of the firm by :math:CL and :math:NCL,

respectively. Then we considered

.. math::

K = \frac{1}{2}CL + \frac{1}{2}NCL,\quad K = CL + \frac{1}{2} NCL,\quad\text{and}\quad K = \frac{3}{2} CL+ \frac{1}{2} NCL.

Since we are considering models where :math:T = 1, current

liabilities, i.e. liabilities that are due in one year or less, are

highly relevant. Furthermore, due to the time value of money, the

coefficient in front of CL seems as though it ought to be at least 1. As

a result, within the current iteration of the code, we have

.. math::

K = CL + \frac{1}{2} NCL.

Further Formulas

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Using the Black-Scholes frame-work, the risk neutral probability of

default is

.. math::

P^Q(A_T < K)=N(-d_2).

As stated previously, we could have

.. math::

A_{t_1} < K\quad\text{but} A_T > K

for :math:t\leq t \leq t_1 \leq T. This would trigger a default not

accounted for within our calculations.

Let us derive the value of liabilities :math:L_t. Due to the

assumptions of the model, we suppose :math:L_T = K as long as the firm

does not default. If we purchase the liabilities for price :math:L_t

and a put option with strike price :math:K, then we obtain :math:K

at time :math:T with no risk. If the time :math:t value of the put

option is :math:P_t, then no arbitrage pricing implies

.. math::

L_t + P_t=Ke^{-r(T-t)}.

It follows that

.. math::

L_t = K e^{-r(T-t)}-P_t.

We can obtain a formula for the spread :math:s over the risk-free

rate. Since

.. math::

L_t=K e^{-r(T-t)}-P_t,

and we would discount :math:K at a rate of :math:r + s to obtain

:math:L_t. It follows that

.. math::

K e^{-(r+s)(T-t)}= K e^{-r(T-t)}-P_t.

Hence, the continuously compounded spread is

.. math::

s= -\frac{1}{T-t}\log\left(1 - \frac{P_t}{K}e^{r(T-t)}\right).

Functions

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get_d(V, T, K, r, sigma_V)

V: Value of the firm. T: Time until maturity. K: Strike price; related

to the firm’s debt level. r: Risk-free rate. sigma_V: Firm volatility.

distance(V, T, K, r, sigma_V)

The Merton model distance to default. Corresponds to :math:d_2 in the

Black-Scholes framework.

prob_default(V, T, K, r, sigma_V)

The probability of default using the Merton model. Corresponds to

:math:N(-d_2) in the Black-Scholes framework.

call(V, T, K, r, sigma_V)

Value of a call option under the Black-Scholes framework. Corresponds to

the value of equity under the Merton model. Formula on page 604 of

“Options, Futures, and Other Derivatives” 9th ed. by Hull.

put(V, T, K, r, sigma_V, phi = 1)

Value of a put option under the Black-Scholes framework. Corresponds to

value of insurance required to make debt risk free. Formula on page 604

of “Options, Futures, and Other Derivatives” 9th ed. by Hull.

V: Value of the firm. T: Time until maturity. K: Strike price; related

to the firm’s debt level. r: Risk-free rate. sigma_V: Firm volatility.

phi: Fraction of firm’s value retained in the case of default.

put_ui(V, T, K, H, r, sigma_V, phi = 1)

Value of an up-and-in put option under the Black-Scholes framework.

Formula on page 605 of “Options, Futures, and Other Derivatives” 9th

ed. by Hull.

V: Value of the firm. T: Time until maturity. K: Strike price; related

to the firm’s debt level. H: Value of the firm which activates the

up-and-in option. r: Risk-free rate. sigma_V: Firm volatility. phi:

Fraction of firm’s value retained in the case of default.

put_uo(V, T, K, H, r, sigma_V, phi = 1)

Value of an up-and-out put option under the Black-Scholes framework.

Formula on page 605 of “Options, Futures, and Other Derivatives” 9th

ed. by Hull.

V: Value of the firm. T: Time until maturity. K: Strike price; related

to the firm’s debt level. H: Value of the firm which deactivates the

up-and-out option. r: Risk-free rate. sigma_V: Firm volatility. phi:

Fraction of firm’s value retained in the case of default.

prob_default_mc(V, T, K, r, sigma_V, trails = 10000, steps = floor(12 *

T + 1))

Probability of default if possible for some values of t prior to

maturity. Default occurs if the value of the firm is below K at time Δt,

2Δt, …, or T = steps · Δt. If T is measured in years, steps = floor(12

* T + 1) corresponds to possibility of default at each monthly payment.

Probability obtained using Monte Carlo simulation.

V: Value of the firm. T: Time until maturity. K: Strike price; related

to the firm’s debt level. r: Risk-free rate. sigma_V: Firm volatility.

trails: Monte Carlo simulations used to obtain approximation; antithetic

values also considered. steps: Number of times the firm can default

within each simulation.

spread(V, T, K, r, sigma_V, phi = 1)

CDS spread given default only possible at T. Uses the Black-Scholes

framework. Value tends to be below market CDS spreads.

V: Value of the firm. T: Time until maturity. K: Strike price; related

to the firm’s debt level. r: Risk-free rate. sigma_V: Firm volatility.

phi: Fraction of firm’s value retained in the case of default.

spread_mc(V, T, K, r, sigma_V, phi = 1, trails = 10000, steps = floor(12

* T + 1))

CDS spread if default possible for some values of t prior to maturity.

Default occurs if the value of the firm is below K at time Δt, 2Δt, …,

or T = steps · Δt. If T is measured in years, steps floor(12 * T + 1)

corresponds to monthly payments. In the case of default, partial payment

assumed to be made at time T. Probability obtained using Monte Carlo

simulation.

V: Value of the firm. T: Time until maturity. K: Strike price; related

to the firm’s debt level. r: Risk-free rate. sigma_V: Firm volatility.

phi: Fraction of firm’s value retained in the case of default. trails:

Monte Carlo simulations used to obtain approximation; antithetic values

also considered. steps: Number of times the firm can default within each

simulation.

spread_das(V, T, K_1, K_2, H, r, sigma_V, phi = 1)

CDS spread if default possible only at time T, but firm increases debt

if its value goes above barrior. Inspired by the Journal of Banking &

Finance article “Credit spreads with dynamic debt” by Das and Kim.

V: Value of the firm. T: Time until maturity. K_1: Strike price if firm

does not increase its debt level. K_2: Strike price if firm increases

its debt level. H: Value of the firm which triggers it to increase its

debt level from K_1 to K_2. r: Risk-free rate. sigma_V: Firm volatility.

phi: Fraction of firm’s value retained in the case of default.

obj(sigma_V, sigma_E, V, T, K, r)

Objective function to be minimized. Used to obtain the volatility of the

firm’s value.

sigma_V: Firm volatility. sigma_E: Equity volatility. V: Value of the

firm. T: Time until maturity. K: Strike price; related to the firm’s

debt level. r: Risk-free rate.

get_sigma_V(sigma_E, V, T, K, r)

Obtain the volatility of the firm’s value. Uses the Black-Scholes

framework and fsolve.

sigma_E: Equity volatility. V: Value of the firm. T: Time until

maturity. K: Strike price; related to the firm’s debt level. r:

Risk-free rate.

get_K(CL, NCL, T = 1, r = 0)

If the value of the firm drops below K at maturity, then the firm is

considered to be in default under the Merton model. Formula overweights

current liabilities because they must be paid within the year or one

business cycle.

CL: Current liabilities. NCL: Noncurrent liabilities T: Time until

maturity. r: Risk-free rate.

References

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Wang, Yu. “Structural Credit Risk Modeling: Merton and Beyond.” Society

of Actuaries, June 2009, Structural Credit Risk Modeling: Merton and

Beyond (soa.org). Accessed June 2021. “Default Probability by Using the

Merton Model for Structural Credit Risk.” MathWorks, Default Probability

by Using the Merton Model for Structural Credit Risk - MATLAB & Simulink

(mathworks.com). Accessed June 2021. Sundaresan, Suresh. “A Review of

Merton’s Model of the Firm’s Capital Structure with its Wide

Applications.” Columbia Business School, Merton_review_cap_structure.pdf

(columbia.edu). Accessed June 2021. Hull, John. “Options, Futures, and

Other Derivatives” 9th ed.  Das, S. and Kim. Journal of Banking &

Finance article “Credit spreads with dynamic debt”