Black Scholes Merton Model of Credit Risk Premia Assignment PLEASE follow all instructions and requirements in the word docdata needed is uploaded for (Mat

Black Scholes Merton Model of Credit Risk Premia Assignment PLEASE follow all instructions and requirements in the word docdata needed is uploaded for (Mathematica) Computational Finance Using Excel and Mathematica
© Problem Set 5: Black-Scholes-Merton Model of Credit Risk Premia Using Asset Value and Its
Volatility Estimation
Merton Model of Asset, Debt and Equity Valuation
In this problem, worth 25 points, the Black-Scholes-Merton model of Asset, Debt and Equity
Valuation will be employed to estimate the risk premiums of a firm’s debt using the values below
computed in Problem Set 4:
1. market asset values (as opposed to book asset values),
2. the market implied volatility of asset returns,
3. risk premium on its debt, senior and junior (subordinated debt),
4. solvency ratio,
5. distance to default (as defined by KMV), and
6. probability of insolvency computed assuming V is a lognormal variable.
Data that is needed will come from the firm’s balance sheet and includes:
1.
2.
3.
4.
5.
The book value of assets as of the last annual report to start the estimation,
The sum of debt of the company separated into secured debt and subordinated debt,
The risk free interest rates for 1, 2 and 3 years’
The market capitalization of the firm near the lasts annual report, and
An estimate of the volatility over 1 year of the firm’s stock returns (implied volatility can be
used for a 1 year call option).
Use the firms from Problem Set 4 and compute premiums for 1, 2 and 3 years forward.
Computational Requirements
Use Mathematica to perform your calculations. Two approaches will be used:
1. The Mathematica function of FindRoot or Nsolve using the two equations in the Black-ScholesMerton Model as shown below, and
2. Newtons’s iterative method as developed in Mathematica coding added to BlackBoard.
Write 3-4 pages analysis/summary based on the results and findings
Equations
The underlying assets are assumed to be stochastic and generated by an Ito process in continuous
time. Consistent with this model, equity value of a firm can be considered as a call option on its assets
with a strike price being its total promised debt, B, is:
E = VN (d1 ) ? (B ) exp (? R f T )N (d 2 )
(1)
where
d1
?ln (V B)+ (R
=
f
)
+ 0.5? V2 T
?
?V T
and d 2 = d1 ? ? V T
and
E=
the market value of equity (stock price times number of shares outstanding,
V=
the market value of assets,
B=
the promised value of firm liabilities discounted at the risk-free rate to time T,
Rf =
the risk-free rate with a maturity consistent with the time to asset valuation (bank examination),
?=
the time to expiration of the option and time to maturity of the debt,
?V =
the standard deviation (volatility) of the rate of return on assets,
ln(x) = the natural logarithm of x,
exp(x) = the value e raised to the power of x, and
N(x) = the cumulative standard normal distribution.
Our objective is to estimate two parameters of the contingent claims model of pricing: the market
value of assets, V, and the volatility of asset returns, ?V. To solve for two variables a second equation is
necessary. Ronn and Verma (1986) and Hull (2000, p.630-631) show that by applying Ito’s lemma to
the generating process for the value of assets, the following relationship with observable market value of
equity and its volatility can be used as the second equation in our system:
?EE = N(d1)?VV, and by rearranging, ? V = E? E
VN (d1 )
.
(2)
where ?E is the volatility of the return on equity as computed from the market value of equity and all
other variables are defined as above. Equation (2) shows that asset volatility is derived from leverage
weighted observable volatility of equity. However, it needs to be emphasized that d1 has the asset value
and asset volatility as arguments (see equation (1)).
Nonlinear equations (1) and (2) are used to solve for the implied values of V and ?V via an
iterative process such as that of Newton. An algorithm to solve nonlinear equations based on Newton’s
method with numeric derivatives is presented in Blackboard as a Mathematica Notebook. Simply add
the necessary equations for the risk premiums to this notebook.
This time, the programming of the risk premiums will be up to you. See the Powerpoiunt
side labeled RiskPremia and the word document Hanweck_SpellmanJFSRAug15a.doc on
Blackboard and Appendix A below.
Appendix A: The Yield Spread-Solvency Model
A. The Contingent Claims Model for Bank Subordinated Debt and Equity
In order for subordinated debt yields to be a signal of bank insolvency, the market yields should
monotonically increase as insolvency is approached. This would appear to be the presumed yieldsolvency relationship behind the mandate for subordinated debt. To relate the investors’ subordinated
debt yield relative to solvency we rely on Black and Cox (1976), Smith (1979) and Cox and Rubinstein
(1985) (for references see the Hanweck and Spellman paper on the Blackboard course site). who show
that the market value of subordinated debt, DSub, equals the value of the difference between two
European call options on the value of assets with strike prices of senior debt and total debt and is given
by:1
DSub = c(V, BDep) – c(V, BDep + BSub).
(1)
where DSub is the market value of subordinated debt, V is the unobserved market value of assets, BDep is
the present value of the promised value of senior debt (deposits for most banks) discounted at the riskfree rate to period T and BSub is the promised value of subordinated debt discounted at the risk free rate.
Applying continuous time approximations to these relationships and the assumptions of the
Black-Scholes-Merton options-pricing model gives the market value of the subordinated debt as:
DSub = V N (d1 ) ? BDep exp (? R f ? )N (d 2 ) ? VN (g1 ) + (BSub + BDep )exp (? R f ? )N (g 2 )
(2)
This can be simplified to the relationship as presented in Gorton and Santomero (1990):
DSub, = V ?N (d1 ) ? N (g1 )? ? BDep exp (? R f ? )N (d 2 ) + (BSub + BDep )exp (? R f ? )N (g 2 )
where,
? ?V
?
?
2
?ln ? BDep ? + R f + 0.5? V ? ?
?
?
d1 = ? ?
(
)
?V ?
? ?V
?
?
2
?ln ? (BDep + BSub )? + R f + 0.5? V ? ?
?
?
g1 = ? ?
(
and d 2 = d1 ? ? V ? ,
)
?V ?
and g 2 = g1 ? ? V ?
V=
the market value of assets,
B=
the promised value of bank liabilities discounted at the risk-free rate to time T,
Rf =
the risk-free rate with a maturity consistent with the time to asset valuation (bank examination),
?=
the market-perceived time until receivership,
1
The valuation equation for subordinated debt is derived in Black and Cox (1976) and Smith (1979). Cox and Rubinstein
(1985) show that the subordinated debt value equals the difference between two European call options:
Dj = c (V, Bs) – c (V, B).
?V =
the standard deviation (volatility) of the rate of return on assets,
ln(x) = the natural logarithm of x,
exp(x) = the value e raised to the power of x, and
N(x) = the cumulative standard normal distribution.
Consistent with the above model of subordinated debt, the equity value of a firm can be
considered as a call option on its assets with a strike price being its total promised debt, B, is:
E = VN (g1 ) ? (B ) exp (? R f ? )N ( g 2 )
(3)
where E is the market value of equity and g1, g2 and all other variables are defined as above.
Our objective is to estimate three parameters of the contingent claims model of pricing: the
market value of assets, V, the volatility of asset returns, ?V, and the investors’ expected time to
resolution, ?. To solve for three variables a third equation is necessary. Ronn and Verma (1996) and
Hull (2000, p.630-631) show that by applying Ito’s lemma to the generating process for the value of
assets, the following relationship with observable market value of equity and its volatility can be used as
the third equation in our system:
?EE = N(g1)?VV
(4)
where ?E is the volatility of equity of the bank and all other variables are as defined above. Equation (4)
shows that the volatility of assets, ?V, can be considered a leverage adjusted value of the volatility of
equity:
?V = ?E(E/(VN(g1))
(5)
The market value of assets can be approximated by the market value of equity and the market value of
debt. The value of N(g1) is the likelihood of the normalized value of asset return being less than g1 or the
value of the recovery of assets upon default (see Hull, 2000, p. 631).
The relationship in equation (1) can be stated in terms of an interest rate risk premium, or spread,
defined as the difference between the yield to maturity on the risky debt, RSub, and a default risk-free
security such as a U.S. Treasury security of the same remaining maturity, Rf. This is done by
recognizing that the market value of the subordinated debt is the continuous discounted value of the
promised amount at the market rate of interest on the debt and treating it as a zero coupon debt
instrument (all values are evaluated at time t before expiration):
DSub = BSub exp (? RSub? )
(6)
Substituting from equation (2) above, the default risk premium, RSub – Rf, is:
RSub ? R f = ? ln ?V BSub exp (R f ? )?N (d1 ) ? N ( g1 )? ? (BDep BSub )N (d 2 ) +
(B
Sub
+ BDep )
BSub
N (g 2 )
?
? (7)
Equation (7) is used to simulate the theoretical risk premium, assuming values for all the parameters. In
addition, equation (7) is used as one of the three equations to estimate the unobserved parameters of
interest: V, ?V, and ?.2
2
Subordinated debt spreads are used in the study instead of debt prices because they have been more prominent in previous
studies. Debt prices could also be used in the estimation with equal efficiency.

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