Analyze profits under random walk and Brownian motion

Random-Walk Trading Rules: Expectation and Variance

Setup

• Let (S_t) be a simple random walk for t = 0, 1, ..., T with S_0 = 0.
• Increments X_t = S_t − S_{t−1} are i.i.d. with P(X_t = +1) = P(X_t = −1) = 1/2 unless otherwise stated.
• A long position opened at price S_t and closed at S_T earns S_T − S_t. A short earns −(S_T − S_t).

Tasks

1. Hold-to-T strategy (symmetric walk): For t = 1, ..., T, after observing X_t, open one unit at price S_t: go long if X_t = +1, short if X_t = −1. Hold all positions until time T and liquidate at S_T. Let P_T = total profit = Σ_{t=1}^T [X_t · (S_T − S_t)]. Compute E[P_T] and Var(P_T).
2. One-step strategy (symmetric walk): At each t you take a unit position based on X_t but close immediately at time t+1. Let the one-step profit be Y_t and the cumulative profit be Q_T = Σ_{t=1}^{T−1} Y_t (there are T−1 complete one-step trades). Compute E[Y_t], Var(Y_t), E[Q_T], Var(Q_T).
3. Hold-to-T strategy (biased walk): Now assume P(X_t = +1) = p (≠ 1/2). Compute E[P_T] and Var(P_T).
4. Brownian-price limit: Suppose price follows standard Brownian motion (B_t) with B_0 = 0. Sample at times t_k = kΔ over a fixed horizon H = nΔ, and apply the same hold-to-H trading rule based on observed increments ΔB_k = B_{t_k} − B_{t_{k−1}}. Discuss the limits of E and Var of profit as Δ → 0. State any additional assumptions and whether the limit is well-defined.