8/11/2019 Using Equity Prices to Estimate Default Probabilities.pdf

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Using Equity Prices to Estimate DefaultProbabilities

Mauricio Bedoya

javierma36@gmail.com

September 2014

To understand this blog, we must know:

1. Ito Calculus.

The title of this blog is part of chapter 20 (Credit Risk, section 20.6) of Jhon C. Hull books:Option, Future and Other Derivatives; sixth edition.

While you study for the FRM exam, you will probably get stuck with Ito Calculus. Itos Cal-

culus is not testable in FRM curriculum. However, in case you are interested, I will show youhow to achieve equation 20.4 (book equation)1

(E) E(0)=E

V (v) V(0) (1)

First, some conditions:

1. E = F[V]Equity (E) is a function of company assets (V), once debt (D) is pay. Because Equity

cant be negative, we can use Geometric Brownian Motion to characterize its evolution.

dE(t)

E(t)=r dt + (E) dw (2)

2. From the accounting equation, we have:E= V D. In case of no debt, V must alsobe positive. This mean that we can use the Geometric Brownian Motion to characterizeits evolution.

1The following procedure its not implemented in Hulls book.

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dV(t)

V(t)= r dt + (V) dw (3)

One think we have to understand, is that the only source of risk arise from company asset (V)value evolution. This mean that we have only one source of randomness dw.

Next, we will create a portfolio that allow to hedge the risk

(t)=F[V(t)] Q V(t) (4)

with Q equal to the number of assets to buy2 Operating in equation 4, we get

d(t)=F

v

dV +

1

2

Fvv

dV

2

Q

dV

=1

2 Fvv

2(V) V

2 dt + (Fv Q) dV (Risk-less ifFv = Q)

=1

2 Fvv

2(V) V

2 dt

(5)

In equation 5 we eliminate randomness. This mean that the portfolio must earn the risk freerate. Replacing the value of Q and operating, we get

(F[V(t)] Q V(t)) r dt=1

2 Fvv 2(V) V

2 dt

r F[V(t)] =1

2 Fvv

2(V) V

2 + F(v) V(t) r(6)

Next, we can use condition No 1, to get

dE=Fv dV + 12 Fvv dV

2

operating

r E dt + (E) E dw= Fv (r V dt + (V) V dw) +12 Fvv

2(V) V

2 dt

r E dt + (E) E dw= Fv (V) V dw+ (r V Fv+1

2 Fvv

2(V) V

2)

Use equation 6 result

dt

r E dt + (E) E dw= Fv (V) V dw+ r F[V(t)] dt

(E) E dw= Fv (V) V dw

(E) E=Fv (V) V

(7)

Using the result of equation 6 and the condition No1, we get the desired result.

2In this model, we assume that assets are tradable.

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