The expectation part wasn't bad once I wrote out what each position contributes: a long at time t earns S_T minus S_t, a short earns the negative.
Model the random walk as a sequence of independent steps, define the position at each step, and express the total profit as a sum of products of step signs and final position. Use linearity of expectation and properties of symmetric random walks to compute the mean and variance, possibly leveraging martingale theory or combinatorial identities.
Pro tip: Emphasize that the strategy's profit is a quadratic function of the final position, and use the fact that for a simple symmetric random walk, the expected value of the fourth power of the final position is known. This shows deep understanding and avoids lengthy combinatorial sums.
Let S_t be the position after t steps, with S_0=0 and increments X_i = ±1 with equal probability. At each step i, take a position of +1 if X_i=+1 (long) and -1 if X_i=-1 (short), i.e., position = X_i. Hold until time T and liquidate at S_T.
The profit from the position taken at step i is X_i * (S_T - S_{i-1})? Wait, careful: The position is opened at step i based on X_i, and held until T. The profit is X_i * (S_T - S_i) because you enter at price S_i? Actually, if you open at step i after observing X_i, the entry price is S_i. Then profit = X_i * (S_T - S_i). Sum over i=1 to T. Alternatively, if you open at step i based on X_i, you might enter at S_{i-1}? The problem says 'at each step based on the direction of that step', so likely you observe X_i and then enter at S_i. So profit = sum_{i=1}^T X_i (S_T - S_i).
Note that S_T - S_i = sum_{j=i+1}^T X_j. So total profit = sum_{i=1}^T X_i sum_{j=i+1}^T X_j = sum_{1<=i<j<=T} X_i X_j. This is a sum over all pairs of steps.
By linearity of expectation, E[profit] = sum_{i<j} E[X_i X_j]. Since steps are independent and mean zero, E[X_i X_j] = E[X_i]E[X_j] = 0. Thus expected profit is 0.
Var(profit) = E[profit^2] since mean is 0. profit = sum_{i<j} X_i X_j. Square and take expectation. Use independence: E[X_i X_j X_k X_l] is 1 if each index appears an even number of times, else 0. Count the number of ways to get non-zero contributions. Alternatively, note that profit = (S_T^2 - T)/2? Actually, sum_{i<j} X_i X_j = (S_T^2 - sum X_i^2)/2 = (S_T^2 - T)/2. So profit = (S_T^2 - T)/2. Then E[profit] = (E[S_T^2] - T)/2 = (T - T)/2 = 0. Var(profit) = Var((S_T^2 - T)/2) = (1/4) Var(S_T^2). For simple symmetric random walk, S_T^2 has mean T and variance? Var(S_T^2) = E[S_T^4] - (E[S_T^2])^2. E[S_T^4] = 3T^2 - 2T for T>=2? Actually, for simple random walk, E[S_T^4] = 3T^2 - 2T. So Var(S_T^2) = (3T^2 - 2T) - T^2 = 2T^2 - 2T = 2T(T-1). Thus Var(profit) = (1/4)*2T(T-1) = T(T-1)/2.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the trading strategy and the distribution of one-step returns. Then compute the expectation and variance of a single one-step profit, and finally analyze the cumulative profit after T steps by considering the sum of independent one-step profits.
Pro tip: Emphasize that if one-step profits are independent and identically distributed, the variance of the cumulative profit grows linearly with T, which is crucial for risk assessment. Also, mention that if the expected one-step profit is zero, the cumulative profit is a martingale.
Restate the modified strategy: each position is opened and closed within one step, so profit is realized immediately. Define the one-step profit random variable and its distribution.
Calculate the expected value and variance of the one-step profit using the given distribution or assumptions. If not specified, denote them as μ and σ².
Express the cumulative profit after T steps as the sum of T independent one-step profits. Use properties of expectation and variance for sums of independent random variables.
If T is large, discuss the asymptotic behavior: by the Central Limit Theorem, the cumulative profit is approximately normal with mean Tμ and variance Tσ². Mention implications for risk and scaling.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
The expectation is no longer zero when p is not a half, which makes sense intuitively since the walk drifts.
First, restate the hold-to-end strategy and define the biased random walk with up probability p. Then derive the expected profit and variance using the binomial distribution and properties of random walks, highlighting how they depend on p.
Pro tip: Emphasize the practical implications: when p ≠ 0.5, the expected profit becomes non-zero, and the variance changes, affecting risk management. This shows you understand both theory and application.
Clarify the hold-to-end strategy: you hold the position until the end of N steps, then profit is the final position. Define the biased random walk: each step is +1 with probability p, -1 with probability 1-p.
Let S_N be the final position. E[S_N] = N(2p-1). So expected profit is N(2p-1), which is zero only when p=0.5.
Var(S_N) = N * 4p(1-p). This follows from the variance of a single step: Var(X_i) = 1 - (2p-1)^2 = 4p(1-p).
When p=0.5, expected profit is 0 and variance is N. For p≠0.5, expected profit is non-zero and variance is maximized at p=0.5, decreasing as p moves away from 0.5.
Mention that a biased walk introduces a drift, making the strategy profitable if p>0.5 (or unprofitable if p<0.5). Variance decreases with bias, meaning less uncertainty but also potentially lower risk.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, define the trading rule precisely for the discrete random walk and then adapt it to Brownian motion sampled at intervals Δt. Analyze the expected profit and variance of the strategy as Δt → 0, considering the impact of transaction costs and other frictions. Discuss whether the limit exists and what additional assumptions (e.g., no transaction costs, continuous trading) are needed for the limit to be well-defined.
Pro tip: Emphasize that in the limit, the strategy's profit may diverge or converge to a finite value depending on the rule and costs; highlighting the need for realistic assumptions like transaction costs shows practical insight.
Clearly state the trading rule for the discrete random walk (e.g., buy if the last increment is positive, sell if negative) and how it translates to Brownian motion increments.
For a given sampling interval Δt, calculate the expected profit and variance of the strategy using properties of Brownian motion increments (independent, normal with mean 0 and variance Δt).
Take the limit of the expected profit and variance as Δt approaches zero. Determine whether they converge, diverge, or depend on scaling (e.g., profit per unit time).
Address the role of transaction costs, bid-ask spread, and the feasibility of continuous trading. Explain how these affect the existence of the limit and the strategy's profitability.
Summarize whether the limit exists under idealized assumptions and how realistic frictions alter the conclusion, tying back to practical trading.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.