Future value of debt in Merton model

[afterworkguinness]

Hi,
I don't understand the way Malz states the future value of the debt; it seems counter-intuitive and contrary to what I read in de Sevigny and Stulz.

We know the Value of the debt can be modeled as a simultaneous position in a risk-less bond with face value of the risky debt discounted using the risk free rate and a short put on the firm's assets with a strike of the value of the debt. If the asset value is below the value of the debt at maturity, the payoff is max(asset value, 0).

How then do we get D(t) = D - max(D - A(t), 0) ?

I can see D(t) = Min[D, Max(A(t), 0)] but not D(t) = D - max(D - A(t), 0)

 

[David Harper CFA FRM]

Hi @afterworkguinness

It looks equivalent to your expression; if we assume A(t) >0 then yours reduces to D(t) = Min[D, A(t)] and Malz gives such that D(t) = D - max(D - A(t), 0)

• If A(t) > D, then D - max(D - A(t), 0) = D
• If A(t) < D, then D - max(D - A(t), 0) = A(t)

But as Max[D - A(t), 0] is the payoff of a short put option on A(t) with strike of D, D - Max[D - A(t), 0] is the expression of the future payoff at face (D) plus a short put option. I hope that helps,

 

[afterworkguinness]

Thanks for clearing that up.