Deriving the Equity Pricing Equation with Feynman–Kac
Summary
The document sets up a continuous-time model of firm cash flows under a risk-neutral measure and values equity as the discounted cash flows received before default, plus the equity value at the default boundary. Cash flows follow geometric Brownian motion, debt pays a fixed perpetual coupon, and shareholders choose when to default. The question asks how this expectation leads to the stated differential equation for equity value.
It identifies the relevant Itô terms: the drift of equity value depends on its first and second derivatives with respect to cash flows. The pricing equation also includes the current net cash flow, which comes from coupon-adjusted operating cash flows, and the risk-free discount rate. The document poses the derivation problem but does not provide the missing step or solve for equity value. Its setup assumes a fixed default threshold, no taxes, and perpetual fixed-coupon debt, so the equation does not cover more general capital structures or endogenous default choices.
Key ideas
- Equity is represented as discounted cash flows until default plus its value at the default time.
- Geometric Brownian cash flows lead to a second-order differential operator through Itô’s lemma.
- The pricing equation balances net cash flow, risk-free discounting, and the equity value process drift.
- The document asks for the Feynman–Kac derivation but does not supply the full argument or solution.
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Full text
# Model of asset substitution/risk shifting in continuous time
# Model of asset substitution/risk shifting in continuous time
Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free rate, $\beta>0$ is the payout rate, and $\sigma$ is the earnings volatility. Let $X_0 = x$ be the initial value. Now assume that the firm has perpetual debt with coupon $C$ (which is taken to be fixed) and assume there are no taxes. Assume that shareholders can choose a stopping time at which the firm defaults, i.e., $\tau = \inf\{t: X_t \le x_B \}$ (for now assume that $x_B$ is fixed). Let $E_t = E(X_t)$ denote the value of equity at time $t$, then the value of equity at time $t =0$ is given by $$E_0 = E(X_0) = E(x)= \mathbb{E}\left[\int_0^{\tau} e^{-rt}(X_t - C)dt + e^{-r\tau} E(X_{\tau}) \right] \ \ \cdots (1)$$.
Applying the Feyman-Kac formula, we get $$rE(x) = x - C + (r-\beta) xE'(x) + \frac{\sigma^2 x^2}{2}E''(x) \ \ \cdots (2)$$
My question is: how do we get from Equation $(1)$ to $(2)$? What I've done so far is this.
Consider the value of the equity at time $t$, which is given by $$E_t = E(X_t) = \mathbb{E}\left[\int_t^{\tau} e^{-r(s-t)}(X_s - C)ds + e^{-r(\tau-t)} E(X_{\tau}) \right] \ \ \cdots (3)$$ Now use Ito's Lemma on Equation $(3)$ and we get \begin{align} dE_t & = E'(X_t) dX_t + \frac{1}{2}E''(X_t) (dX_t)^2 \\ & = (E'(X_t) X_t (r-\beta) + \frac{1}{2}E''(X_t) \sigma^2 X_t^2) dt+ E'(X_t)\sigma dZ_t \end{align} This is where I am stuck. I can see relevant terms in the above equation that appear in (2) but can't seem to go any further.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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