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Below I am trying to show the relationship between Merton PD and the BSM.

Merton PD = N[ -[ln(V/K)+(μ-0.5σ²)T]/σT ]

The formula inside the bracket (let’s name it D2 since it) is very similar to the formula for d2 in the BSM for pricing call option:

d2 = ln(S/X)+(r-0.5σ²)T]/σT

So we have Merton PD = N(-D2).

David, can you comment if my below interpretation is correct/incorrect?

· We know N(d1) and N(d2) in BSM refer to the call delta and the probability of exercising the call (ITM (i.e., S>X) at expiration) in risk neutral world. What are N(-d1) and N(-d2)? Are they put delta and probability of exercising the put (ITM (i.e., S<X) at expiration?)

· Firm defaults if V (firm value) < D (Debt value). So the Merton formula replaces S with V (it also replaces risk free rate with return on asset). If N(-d2) = probability of S<X, then N(-D2) = probability of V<D, which is the probability of default. Am I correct?

· We can also alter the formula to calculate D1 (merely changing a “-“ sign into a “+” sign). Since N(d1) = delta of call option on the stock = Change in call price on the stock price/Change in stock price (S). N(D1) = delta of call option on the entire firm assets = Change in call option on the entire firm value/Change in firm value. Merton values firm equity (E) as if it were a call option on the firm’s assets.

Thanks.

Sleepybird

Merton PD = N[ -[ln(V/K)+(μ-0.5σ²)T]/σT ]

The formula inside the bracket (let’s name it D2 since it) is very similar to the formula for d2 in the BSM for pricing call option:

d2 = ln(S/X)+(r-0.5σ²)T]/σT

So we have Merton PD = N(-D2).

David, can you comment if my below interpretation is correct/incorrect?

· We know N(d1) and N(d2) in BSM refer to the call delta and the probability of exercising the call (ITM (i.e., S>X) at expiration) in risk neutral world. What are N(-d1) and N(-d2)? Are they put delta and probability of exercising the put (ITM (i.e., S<X) at expiration?)

· Firm defaults if V (firm value) < D (Debt value). So the Merton formula replaces S with V (it also replaces risk free rate with return on asset). If N(-d2) = probability of S<X, then N(-D2) = probability of V<D, which is the probability of default. Am I correct?

· We can also alter the formula to calculate D1 (merely changing a “-“ sign into a “+” sign). Since N(d1) = delta of call option on the stock = Change in call price on the stock price/Change in stock price (S). N(D1) = delta of call option on the entire firm assets = Change in call option on the entire firm value/Change in firm value. Merton values firm equity (E) as if it were a call option on the firm’s assets.

Thanks.

Sleepybird

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