unconditional default probability vs

default intensity/hazard rate

unconditional default is probability of default in a particular period, assuming nothing

default intensity is the chance of default in a period, given survival till the beginning of that period

Prob of defaulting between t and t+ Δt given survival to t λ(t)Δt

Probability of defaulting by time t = Q(t) =

1-e^-λ(t)t

where λ(t) is the average default intensity

λ(t) = s/(1-R)

s = spread between yield and risk free bond

perfect method is to calculate $ value of default and then fully calculate probability for each possible time.

defined as sum of PVs of future exposures to loss

Σu_iv_i

Where u_i = q_i*(1-R)

v_i = value today of intstrument that pays off the exposure to derivative at time t

the value of derviative after allowing for default

f_0^*

f_{0^{* =}} f₀e^-(y*-y)T

with y* being the yield of zero coupon bond of couterparty

and y yield of similar risk free zero coupon bond

so y*-y is the spread

1. Netting - if a company defaults on contract it defaults on all outstanding contracts (in other words, net postive and negative cashflows)
2. Collateralization - periodically revalue contracts and if value is large enough, collateral is required
3. Downgrade Triggers - if credit rating drops below a specific level, owed party has option to close contract.

note this is the probability of the option being exercised

= 1-N(d₂)

d₂ = [ln(S₀/K) + (r-c²/2) * T ]/

(σ*√T)

K is loss to be paid

S₀ = Beginning Assets

r =risk free interest rate

σ_v = volatility of assets

S₀ * N(d₁) - K*e^{^-rt}*N(d₂)

where d₂ =  

d₁ - (σ*√T)

σ_eE_o/(σ_v* V₀) = N(d₁)