Simulation Methods

In a bootstrap valuation of a contingent claim, the present values $C_{i0}$ from 1,000 runs sum to USD 4,250. The bootstrap estimate of the claim's value is closest to:

Consider the following:
I. If one-period continuously compounded returns are i.i.d. with mean $\mu$, then $E(r_{0,T}) = \mu T$.
II. Under the same i.i.d. assumption, $\sigma^2(r_{0,T}) = \sigma^2 T$.
III. If one-period continuously compounded returns are normally distributed, the T-period continuously compounded return is also normally distributed.
How many of the above statements are most accurate according to the CFA Curriculum?

A standard lookback contingent claim has value at maturity equal to the stock's value at maturity minus the minimum value reached during the life of the claim, or USD 0 if that difference is negative. If the minimum value reached is USD 20.11 and the stock ends at USD 23.00, the claim's value is closest to:

Consider the following:
I. Bootstrap treats the observed sample as if it were the population.
II. Each bootstrap resample is the same size as the original sample and is drawn with replacement.
III. Bootstrap requires the analyst to specify probability distributions for key risk factors before drawing observations.
How many of the above statements are most accurate according to the CFA Curriculum?

If a one-year Monte Carlo simulation is split into $K = 4$ equal subperiods, the time increment $\Delta t$ is closest to:

In 654 of 1,000 Monte Carlo trials, the contingent claim payoff was zero. The proportion of trials with a positive payoff is closest to:

A bootstrap resample of size 6 taken from an original sample of six distinct observations contains the following frequencies for the original observations: $2, 2, 1, 1, 0, 0$. The number of original observations omitted from this resample is closest to:

For the contingent claim in the Monte Carlo section, if the final stock price is USD 34 and the average stock price during the claim's life is USD 31, the payoff at maturity is closest to:

Assertion (A): In the six-step Monte Carlo process for valuing the contingent claim, specifying the time grid belongs to the stage of running the simulation rather than specifying it.
Reason (R): The CFA Curriculum separates the process into specification steps 1 through 3 and running steps 4 through 6.

An asset has one-period continuously compounded returns of 1.0%, -2.0%, and 3.0% over three successive periods. The total continuously compounded return over the full horizon is closest to:

If the original observed sample has size 12 and an analyst generates 1,000 bootstrap resamples, each of the same size as the original sample, the total number of draws across all resamples is closest to:

Consider the following:
I. The mean of a lognormal random variable is $\exp(\mu + 0.50\sigma^2)$.
II. The variance of a lognormal random variable is $\exp(2\mu + \sigma^2) \times [\exp(\sigma^2) - 1]$.
III. The expected value of $\exp(X)$, where $X$ is normal with mean $\mu$, is exactly $\exp(\mu)$.
How many of the above statements are most accurate according to the CFA Curriculum?

Assertion (A): Even if one-period continuously compounded returns are not normal, future stock price may still be modeled as lognormal.
Reason (R): Under the central limit theorem, the sum of one-period continuously compounded returns can be approximately normal, and stock price equals current price multiplied by the exponential of that sum.

Assertion (A): Monte Carlo simulation is useful in investment applications partly because the analyst can change assumptions and study model sensitivity.
Reason (R): In Monte Carlo simulation, the analyst controls assumptions by specifying probability distributions for key risk factors that drive the underlying variables.

Assertion (A): Bootstrap resampling mimics sampling from a population by repeatedly drawing samples of the same size, with replacement, from the observed sample.
Reason (R): Bootstrap requires the analyst to specify probability distributions for key risk factors because the underlying population distribution is unknown.

Using the same associated normal distribution with $\mu = 0.07$ and $\sigma^2 = 0.0144$, the variance of the lognormal variable is closest to:

A stock's standard deviation of daily continuously compounded returns is 0.02. Using 250 trading days, the annualized volatility is closest to:

Consider the following:
I. If $\ln Y$ is normally distributed, $Y$ is lognormally distributed.
II. A lognormal distribution is bounded below by 0.
III. A lognormal distribution is symmetric around its mean.
How many of the above statements are most accurate according to the CFA Curriculum?

A continuously compounded return has mean $\mu = 0.07$ and variance $\sigma^2 = 0.0144$. If future price is modeled as a lognormal variable $Y = \exp(X)$, the expected value of $Y$ is closest to:

Assertion (A): Monte Carlo simulation is inferior to analytical methods whenever an analytical pricing formula exists, because Monte Carlo cannot value any security exactly.
Reason (R): The CFA Curriculum describes Monte Carlo simulation as a complement to analytical methods because it provides statistical estimates rather than exact results, whereas analytical methods provide more cause-and-effect insight when available.

Assertion (A): In the contingent-claim example, bootstrap and Monte Carlo share the same first two steps and the same last three steps, differing mainly in the source of the random variable used in the simulation.
Reason (R): Under bootstrap, the simulated random values are drawn from the observed empirical distribution of stock price processes rather than from a specified probability distribution.

A Monte Carlo valuation uses 12 monthly subperiods and 1,000 simulation trials, with one standard normal draw $Z_k$ for each subperiod in each trial. The total number of standard normal draws is closest to:

Consider the following:
I. In the Monte Carlo process, specifying a time grid with K subperiods is part of setting up the simulation.
II. In the Monte Carlo process, drawing K random values for each key risk factor is part of generating the simulation data.
III. In the Monte Carlo process, drawing repeated samples with replacement from the observed sample is the stated method for generating the data.
How many of the above statements are most accurate according to the CFA Curriculum?

Consider the following:
I. Bootstrap can be used to infer population parameter values such as mean, variance, skewness, and kurtosis from repetitive resampling.
II. In simulation using bootstrap, the first two steps and the last three steps are the same as in the Monte Carlo procedure discussed in the CFA Curriculum.
III. Bootstrap simulation, like analytical methods, provides exact results rather than statistical estimates.
How many of the above statements are most accurate according to the CFA Curriculum?

An observed sample contains 8 stock-price changes. Under bootstrap resampling, the size of each resample drawn from this observed sample is closest to:

Assertion (A): If a random variable's natural logarithm is normally distributed, the variable itself is lognormally distributed.
Reason (R): The lognormal distribution is defined in terms of the mean and variance of the lognormal variable itself rather than the associated normal distribution.

Consider the following:
I. Monte Carlo simulation generates a large number of random samples from specified probability distributions.
II. Monte Carlo simulation can be used to test a model’s sensitivity to changes in assumptions about key variables.
III. Monte Carlo simulation provides exact values whenever enough trials are used.
How many of the above statements are most accurate according to the CFA Curriculum?

If a stock's current price is USD 50 and its continuously compounded return to time $T$ is 0.20, the future stock price $P_T$ is closest to: