Quiz: Feynman-Kac and the Connection to PDEs

Module 4 of 4 · Hard

Quick Quiz

1. The Feynman-Kac formula states that $u(t,x) = \mathbb{E}\!\left[e^{-\int_t^T r \, ds} \varphi(X_T) \mid X_t = x\right]$ satisfies which PDE?

u_t + \mathcal{L}u + r u = 0

u_t + \mathcal{L}u - r u = 0

-u_t + \mathcal{L}u - r u = 0

u_t + \mathcal{L}u = 0

2. In the Feynman-Kac derivation, we define $Y_s = e^{-\int_t^s r \, du}\, u(s, X_s)$ . For $Y_s$ to be a local martingale, the drift of $Y_s$ must be:

Equal to

r u

Equal to

-r u

ZeroEqual to

\mathcal{L}u

3. The Black-Scholes PDE for a European option on $S_t$ with risk-free rate $r$ and constant volatility $\sigma$ is:

V_t + \mu s V_s + \frac{1}{2}\sigma^2 s^2 V_{ss} - rV = 0

V_t + r s V_s + \frac{1}{2}\sigma^2 s^2 V_{ss} - rV = 0

V_t + r s V_s + \sigma^2 s^2 V_{ss} - rV = 0

V_t + r s V_s + \frac{1}{2}\sigma^2 V_{ss} - rV = 0

4. For a European call with payoff $(S_T - K)^+$ , what is the boundary condition as $s \to 0$ ?

V(t, 0) = K e^{-r(T-t)}

V(t, 0) = 0

V(t, 0) = -K e^{-r(T-t)}

V(t, 0) = K

5. The classical Feynman-Kac theorem applies to pricing under rough volatility models where instantaneous variance is driven by fractional Brownian motion with $H < 1/2$ .

truefalse

6. For American options, in the continuation region (where early exercise is suboptimal), which statement is correct?

The BS PDE holds, and

V > (s-K)^+

The BS PDE holds, and

V = (s-K)^+

The inequality

V_t + rsV_s + \frac{1}{2}\sigma^2 s^2 V_{ss} - rV \geq 0

holdsEarly exercise is always optimal in the continuation region

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