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Credit Risk Intro and Merton Default model

In short, credit risk is that borrower is facing some probability of financial distress.

Implicit put option

In reality, banks are lending money to mortgage borrower, or bond holders are lending money to company that issue debt.

For the mortgage borrower or company shareholders, they are actually get a put option.

Why is a put option?

They got money at the first place, it’s like the strike price in the put option. Max(0. K-S)

here the housing price or company asset value is like the stock price, that is the value you are going to lose if you choose to exercise the put option and default.

But this put option does not come with no price, on the company level, that means bankrupt, on the individual level, that person might not going to get a loan for a period of time because their trashed credit.

With this idea of implied put option, bond price can be valued.

bond price = e^(-rT)*[probability of default *At + (1-probability of default)*B]

T = time to maturity

At = asset value at maturity date

B = notional amount of debt

r = corresponding risk free rate

Now, let’s see how the probability of default determined.

Merton default model

Basic idea is to get the distance to default( measured by unit of standard of deviation) and assume the company asset return follow a normal distribution, then normdist that distance and get the probability.

so, assumption:

1. asset value follow fa lognormal distribution and asset return follow a normal distribution

2. asset value is continuous, not jumps

3. no market friction

at time 0, distance to default is ln(S/K)

as time progress, mean return is mu, return volatility is SD.

return exception is (mu -0.5*SD^2)

d2 = distance to default = [ln(S/K) + (mu -0.5*SD^2)*t]/(SD*t^0.5)

true probability of using mu, risk neutral probability if using risk free rate.

N(d2) is the probability of default.

Expected loss(%) = probability of default * loss given default

credit spread = rho – r = (1/T)*ln[1/(1-probability of default*loss given default)

Interest Risk hedging 1

commercial bank usually has a positive duration gap

when interest go up, due to -Δ*ΔR, asset value go down, to hedge we need to short bond future

when interest go down, asset value go up, we might want to long bond future.

insurance company tend to has a negative duration gap

IF interest go up

asset value go up, long bond future

ELSE

asset value go down, short bond future.

But banks are not holding a real bond, it has duration albeit synthetic one

so, basis risk exist.

to minimize the basis risk, we need to measure the sensitivity of change in the underlying bond and change in bank equity exposure.

this sensitivity can be termed as br

, same as Beta in CAPM model( certain portfolio relative to market portfolio) , delta in option Greeks(option price relative stock price) or velocity in physics( distance relative to time), they are just one thing: beta in the OLS regression

as using market portfolio to hedge specific stock

using stock to hedge option

we are using future to hedge bank asset

summarize:

change in equity value = -[(D(A) – D(L)*(L/A)]*A*[ΔR/(1+ R)]

in order to hedge, change in equity must be equal to change in future position:

above = – N*P*Δ(bond)*[ΔR/(1+ R)]

then

Hedging using bond futures

N = [(D(A) – D(L)*(L/A)]*A/[P*Δ(bond)*br]

N: number of future contract needed to hedge bank equity.

P: price of one bond future contract

Hedging using options

N = [(D(A) – D(L)*(L/A)]*A/[P*Δ(bond)*Δ]

 

options stuff #FRM

Strategy

collar  – insurance for long position in stock

straddle – betting for high volatility(with ATM call and put, higher cost)

strangle – betting for higher volatility too but with a lower cost

Binomial Pricing

U = size of going up

D = size of going down

Probability of going up ( risk neutral)

discount back

Option Greeks

delta

to hedge one long call, short delta shares stock

to hedge one long stock, short 1/delta shares call

dela = N(d1)

* probability of default = N(d2)

* expected loss = adjusted exposure * P(d) * loss given degault

Vega

call Vega = put Vega , due to put call parity

Gamma

highest when at the money

Theta

As time more and more approach to maturity, the loss of one day has more and more negative impact on option value

Early exercise of american option on dividend paying stock

American call

benefit: dividend received, PV(dividend)

loss: prepayment of K ,interest loss, K – PV(K)

American put

benefit: interest received, K – PV(K)

loss: dividend loss, PV(dividend)

Black&Scholes

call = S*N(d1) -PV(K)*N(d2)

expected return = mu

mean return = mu – 0.5* SD^2

flaws:

price lognormally distributed, return normally distributed

continuous prices, no jumps

volatility and interest rate is known and stay constant

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miaou – Always Dreaming of Things to Come

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@DC

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