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# Understanding the relationship between Merton Probability of Default (PD) and the Black-Scholes Mode

#### sleepybird

##### Active Member
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

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi sleepybird,

My post here explains this in detail: http://www.bionicturtle.com/forum/threads/merton-model-a-summary-of-the-issues.5646/
including that as N(d2) is risk neutral probability of option expiring ITM, N(-d2) = N(-distance to default) = probability of default (analogous to option expiring OTM, as equity is a call option on firm assets), except riskfree rate in BSM is replaced by actual asset drift in Merton. Feel free to follow-up on that post, if i can provide further clarification, but i think it covers all of your angles, thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thank you Shakti!

I do hope that helps show how Merton models default as an OTM call option (strike = future firm asset value), such that N(d2) = ITM becomes N(-d2). And N(-d2) becomes N(-DD) with replacement of the riskfree rate by actual drift; a seemingly trivial change but wonderfully illustrative of Jorion's Chapter 1 distinction between risk-neutral pricing (valuation) versus risk measurement (i.e., precise center of distribution discounted to present versus future distribution of actual tail)

#### sleepybird

##### Active Member
Thanks for the video. That's exactly the explanation I'm looking for.

#### sleepybird

##### Active Member
David,
To solve for firm value and firm volatility, we simultaneously solve the BSM formula and the Ito's Lemma formula E*SigE=V*SigV*N(d1) through some kind of iterative process using computer. But can you please explain how the Ito's formula was derived? The book gave no such explanation. I know this is probably not testable (am I correct to say that?) but I'm very interested in knowing how it is derived. I searched the forum, there seems no mention of it.

I attempted to solve below but got stuck:

Firm equity is a function of it's assets (V):
E=f(V)

Taylor series:
dE=f'(V) dV + 0.5*f''(V)*dV^2
f'(V)=first derivative=delta
dE=delta*dV + 0.5*f''(V)*dV^2
f''(V)=second derivative=? (I know for bond it is the convexity)

GBM:
dV=mu*dt*V + SigV*dW*V
dE=mu*dt*E + SigE*dW*E

Plug into earlier formula:
mu*dt*E + SigE*dW*E=delta*(mu*dt*V + SigV*dW*V) + 0.5*f''(V)*(mu*dt*V + SigV*dW*V)^2

I don't know what to do next. How to get rid of dW and f''(V)?

Your help would be extremely appreciated. Thanks.

#### ShaktiRathore

##### Well-Known Member
Subscriber
dE=[f'(V)*mu*V + 0.5*f''(V)*sigma^2*V^2]dt+SigV*dW*V*f'(V).........u mistook this formula
dE=f'(V)*mu*V*dt+SigV*dW*V*f'(V)+0.5*f''(V)*sigma^2*V^2*dt
dE=dV*f'(V)+0.5*f''(V)*sigma^2*V^2*dt
dE=dV*f'(V)+(dE-delta*dV)*sigma^2*dt
dE(1-sigma^2*dt)=dV*delta*(1-sigma^2*dt)
dE=dV*delta
delta=dE/dV

#### sleepybird

##### Active Member
shaki
Thanks. You showed how to derive delta, can you how to derive the Ito formula? Also how did you get rid of the term dW and f''(x)? Sorry I don't have strong background in math.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi sleepybird,

Hull Chapter 13 Appendex derives Ito's Lemma, in detail, over two pages (also the assigned McDonald's text, Derivatives, walks though it a more carefully).

Ito process and derivation of Ito's Lemma are not testable (ie, clearly out of scope). Of course you are correct, Ito is a necessary ingredient, however in my opinion, an understanding of it is not essential to the intuition behind the first issue (the first step is Merton really, which occurs BEFORE any of the steps I articulated above!) that you identify: Merton needs both equity value and equity volatility (two unknowns) and our BSM Merton gives use one equation, as you say, the second "E*SigE=V*SigV*N(d1)," which (as Hull show's) is simply: volatility (equity)*Equity = dEquity/dVolatilty * volatility (asset)*Asset. Is Ito critical to believing this? I don't think so. Rather, i think the key mathematical issue created by this first step in Merton is that we have two unknowns in two non-linear equations.

On the other hand, Ito is the mathematical justification for the lognormal property (the stochastic process GBM which is assumed in BSM and, importantly, the evolution of firm's asset value in Merton), so in practice, it's how we would rigorously understand why the "- variance(asset)/2" term counter-intuitively introduces into the equation (or in BSM d2 for that matter). Thanks,

#### Mauricio Bedoya (Pass all Levels of CFA)

##### New Member
Hello Sleepybird.

Demonstration below. If you have any doubt, just reply. Firs, some notation:
eq 1: E = F (F is only a function os company assets)
eq 2: W = F - D*V; F (firm value), D(Number of assets to buy), F(firm asset function value)
eq 3: dV=r*dt*V + SigV*dW*V (risk neutral world)
eq 4: dE=r*dt*E + SigE*dW*E (risk neutral world)

eq 3 and eq 4, share same source of risk
Fv : first derivative; Fvv: second derivative

Step 1 "Apply Itô to eq 2"
dw = Fv*dv + 0.5 *Fvv * dv^2 - D * Fvv
replace dv and agrupate terms
dw = [r*V *Fv + 0.5*(SigV*V)^2 *Fvv - r*V*D]*dt + SigV*V*dW*[Fv - D]; now make riskless Fv = D
this makes the portfolio to return the risk free rate
[F - Fv*V]*r]dt = [r*V *Fv + 0.5*(SigV*V)^2 *Fvv - r*V*Fv]*dt; operate to get

eq 5: rf = r*V *Fv + 0.5*(SigV*V)^2 *Fvv

Step 2 "Apply Itô to eq 1"
dE = Fv*dv + 0.5*(SigV*V)^2 *Fvv; now replace dE, dV and dv^2 and agrupate

r*E*dt + SigE*E*dW = [r*V*Fv + 0.5*(SigV*V)^2 *Fvv]*dt + Fv*SigV*V*dW

in brakets we have eq5 *dt. Cause E = F, we get

SigE*E*dW = Fv*SigV*V*dW; now cancel dW

FINAL RESULT: SigE*E = Fv*SigV*V (book result)

If you are interested in pde (partial derivatives) expressions in Black Scholes world, save this procedure. You will see it, thousands of times