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Quantitative Finance · शब्दावली

Merton model क्या है?

Definition 14.2 Rates, Credit, XVA and Risk · अध्याय 14 — Structural Credit Models

The Merton model takes assets following a geometric Brownian motion with volatility $\sigma_V$ and one zero-coupon debt of face $D$ due at $T$ . At $T$ the shareholders receive $\max(V_T-D,0)$ and the creditors $\min(V_T,D)$ . Equity is a European call on the assets struck at the debt, priced by the Black–Scholes formula, and debt is the assets minus the equity:

E_0 = V_0N(d_1)-De^{-rT}N(d_2),\qquad B_0 = V_0-E_0,\qquad d_{1,2} = \frac{\ln(V_0/D)+(r\pm\tfrac12\sigma_V^2)T}{\sigma_V\sqrt T}.

Equity and debt of the chapter’s firm (debt face USD 40 billion due in five years, asset volatility 15.6%) as functions of the asset value. Equity is a call on the assets; debt rises towards the riskless value of the face and falls one-for-one with the assets when they are low. Data: the chapter’s tutorial. — Figure 14.1. Equity and debt of the chapter’s firm (debt face USD 40 billion due in five years, asset volatility 15.6%) as functions of the asset value. Equity is a call on the assets; debt rises towards the riskless value of the face and falls one-for-one with the assets when they are low. Data: the chapter’s tutorial.

उदाहरण

Example 14.3 (A leveraged firm)

An illustrative firm has equity worth USD 10 billion with a volatility of 50%, and debt of face USD 40 billion treated as due in five years; $r = 4\%$ . The inversion gives assets of USD 40.69 billion with a volatility of 15.6%: the equity is more than three times as volatile as the assets because it is levered. The debt is worth USD 30.69 billion, a yield spread of 130 basis points, and the risk-neutral probability that the assets end below the face is 32.7%.

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