Brownian Bridge™

1. Let $\Omega = \{A, B, C\}$ with $\mathbb{P}(A) = 1/2$ , $\mathbb{P}(B) = 1/4$ , $\mathbb{P}(C) = 1/4$ . Define $X(A) = 4$ , $X(B) = -2$ , $X(C) = 0$ . What is $\mathbb{E}[X]$ ?

3/2

2/3

1

7/4

2. Let $X$ be an integrable random variable and $\varphi(x) = (x - K)^+$ for a fixed strike $K > 0$ . Under the risk-neutral measure $\mathbb{Q}$ with $\mathbb{E}^\mathbb{Q}[S_T] = S_0 e^{rT}$ , Jensen's inequality implies which of the following?

e^{-rT} \mathbb{E}^\mathbb{Q}[(S_T - K)^+] \ge (S_0 - K e^{-rT})^+

e^{-rT} \mathbb{E}^\mathbb{Q}[(S_T - K)^+] \le (S_0 - K e^{-rT})^+

\mathbb{E}^\mathbb{Q}[(S_T - K)^+] = (\mathbb{E}^\mathbb{Q}[S_T] - K)^+

\mathbb{E}^\mathbb{Q}[(S_T - K)^+] \ge 0

only if

S_0 > K

3. The Dominated Convergence Theorem (DCT) justifies computing $\Delta = \partial_S V_0$ by differentiating under the expectation $V_0 = e^{-rT} \mathbb{E}^\mathbb{Q}[(S_T - K)^+]$ . What is the dominating function that makes the DCT applicable here?

g = 1

(the constant function), since

|\partial_S (S_T - K)^+| = \mathbf{1}_{S_T > K} \le 1

g = S_T

, since the payoff is bounded above by the stock price

g = K

, since the strike is fixed and the payoff cannot exceed

K

No dominating function exists; DCT cannot be applied to option payoffs

4. The Monotone Convergence Theorem (MCT) requires the sequence $(f_n)$ to be non-decreasing. Which counterexample demonstrates that the MCT fails without this condition?

f_n = \mathbf{1}_{[n, n+1]}

\mathbb{R}

with Lebesgue measure:

f_n \to 0

but

\int f_n \, d\lambda = 1

f_n = x/n

[0,1]

f_n \to 0

and

\int f_n \, d\lambda \to 0

f_n = \mathbf{1}_{[0, 1/n]}

f_n \to 0

and

\int f_n \, d\lambda \to 0

f_n = n \cdot \mathbf{1}_{[0, 1/n]}

f_n \to 0

everywhere, and

\int f_n \, d\lambda \to 0

5. For the four-outcome uniform space $\Omega = \{1,2,3,4\}$ with $\mathbb{P}(\{k\}) = 1/4$ and $X(k) = k - 5/2$ , compute $\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$ .

5/4

1

3/2

5/2

6. A portfolio's daily P&L $X$ is non-negative with $\mathbb{E}[X] = \$ 100 $. Using Markov's inequality, what is the tightest model-free upper bound on the probability of a daily gain exceeding$ \ $500$ ?

1/5

1/25

1/2500

Markov's inequality cannot bound gains — only losses

7. Which of the following correctly identifies the difference between $X \in L^1$ and $X \in L^2$ on a probability space, and states which condition the Itô integral requires?

L^1

\mathbb{E}[|X|] < \infty

;

L^2

\mathbb{E}[X^2] < \infty

. The Itô integral requires

L^2

so that the Itô isometry holds.

L^1

\mathbb{E}[X] < \infty

;

L^2

\mathbb{E}[X^2] < \infty

. They are equivalent on bounded spaces.

L^1

\mathbb{E}[|X|] < \infty

;

L^2

\mathbb{E}[X^2] < \infty

. The Itô integral requires only

L^1

L^1

and

L^2

coincide on finite probability spaces, so the distinction is purely theoretical.

8. You are pricing an Asian option numerically using Monte Carlo. The payoff is $\Phi = (\bar{S}_T - K)^+$ where $\bar{S}_T$ is the arithmetic average of the stock price at $n$ monitoring dates. The simulation produces a sequence of estimates $\hat{V}_N$ as $N$ (number of paths) increases. Which theorem guarantees $\hat{V}_N \to V_0$ almost surely?

The strong law of large numbers — a direct consequence of the Lebesgue integration theory developed in this moduleThe Monotone Convergence Theorem applied to the increasing sequence of sample averagesThe Dominated Convergence Theorem with dominating function equal to the maximum payoffJensen's inequality with

\varphi(x) = (x - K)^+

Quiz: Lebesgue Integration and Expectation

Quick Quiz