Simulation of Financial Asset Prices and Returns: formula sheet

Key formulas

Simulated price with normal returns (one period)

S(t+1) = S(t) × exp(r), where r = μ + σ × Z

Z is a standard normal draw. Using exp(r) means r is a continuously compounded return and the price cannot go below zero.

Standard normal from a uniform draw

Z = N⁻¹(U), where U is uniform on 0 to 1

N⁻¹ is the inverse standard normal cumulative function. U = 0.5 gives Z = 0.

Mean of simulated outcomes

Mean = (Σ outcomes) ÷ number of trials

Use the same average for any summary statistic, such as the probability of a loss = trials with loss ÷ total trials.

Standard error of the simulated mean

SE = s ÷ √N

s is the standard deviation of the trial results and N is the number of trials. To halve the error you need four times as many trials.

Continuously compounded return

r = ln(S₁ ÷ S₀)

ln is the natural log. Returns over several periods add.

Ending price from a return

S₁ = S₀ × e^r

Use the e^x function on your calculator. On the BA II Plus it is 2nd then LN.

Simulated return from a standard normal draw

r = μ + σ × Z

Z ~ N(0, 1). μ and σ are the mean and standard deviation of the continuously compounded return for that period.

Scaling to a longer horizon

mean = μ × T; variance = σ² × T; st. dev. = σ × √T

Assumes returns in each period are independent with the same distribution.

Expected price under lognormal model

E[S₁] = S₀ × e^(μ + σ²/2)

μ and σ² are the mean and variance of the continuously compounded return over the period. The expected price is above S₀ × e^μ.

GBM-style step with drift m

r = (m − σ²/2) × Δt + σ × √Δt × Z

Here m is the expected rate of price growth. Use this only when the question gives m rather than the mean log return.

Simulated option value (European-style)

Value = e^(−r × T) × (average of simulated payoffs)

Use risk-neutral drift (the risk-free rate) when simulating prices. With discrete compounding, divide by (1 + r)^T instead. For a call, each payoff is max(0, S_T − X).

Simulated VaR

VaR at (1 − α) confidence = loss at the α-th percentile of the simulated profit-and-loss distribution

For 5% VaR with 10,000 trials, the loss at the 500th worst outcome is a rough read-off. VaR is the minimum loss expected in the worst α of cases.

Standard error of a simulation estimate

Standard error ≈ s ÷ √N

s is the standard deviation of the simulated outcomes and N is the number of trials. Quadrupling trials halves the standard error.

Probability from simulation

P(event) ≈ number of trials where event occurs ÷ N

Use this for shortfall or success-rate questions in planning.

Bootstrap resample size

Resample size = original sample size n; draws made with replacement

Each observation can appear more than once, or not at all, in a resample.

Bootstrap standard error of a statistic

SE = standard deviation of the statistic across the B resamples

Compute the statistic on each resample, then take the standard deviation of those B values.

Monte Carlo vs historical vs bootstrap source of random draws

Monte Carlo: assumed distribution | Historical simulation: actual past data | Bootstrap: observed sample, with replacement

Use this to classify any exam description quickly.

Quick revision

• Simulation estimates outcomes by repeating many random trials of a model.
• Monte Carlo draws from an assumed probability distribution.
• Bootstrap draws from observed data with replacement and assumes no distribution.
• Core steps: specify the model, generate random values, compute results, repeat, analyse.
• More trials reduce sampling error in the estimate but do not fix a wrong model.
• A lognormal model keeps simulated prices above zero.
• Continuously compounded returns are often modelled as normal.
• The output is a distribution, so you can read the mean, spread and percentiles.
• Typical uses include valuing complex derivatives and estimating risk measures.
• Results are only as good as the model and inputs: garbage in, garbage out.
• Simulation gives approximate answers, not exact analytical solutions.
• Bootstrap is limited by how well the sample represents the true population.

Common mistakes

• Thinking simulation gives the true answer. Fix: Remember it gives an estimate. Its accuracy depends on the number of trials and on the assumed inputs.
• Believing more trials fix bad assumptions. Fix: More trials reduce sampling error only, in proportion to 1 ÷ √N. Wrong distributions or correlations stay wrong.
• Using S₀ × (1 + r) instead of S₀ × e^r Fix: If the return is continuously compounded, always use the exponential. The two answers will differ in the second decimal place or earlier.
• Scaling volatility by T instead of √T Fix: Variance scales with T. Standard deviation scales with √T. For a quarter, σ_quarter = σ_annual × √0.25 = σ_annual ÷ 2.
• Discounting simulated option payoffs at the expected return of the stock. Fix: For pricing, simulate with the risk-free rate as drift and discount at the risk-free rate.
• Treating simulation output as exact. Fix: Remember it is a statistical estimate with sampling error, and it depends on the assumed inputs.
• Believing more Monte Carlo trials fix a wrong model. Fix: More trials cut sampling error only. Wrong distribution or parameters still produce wrong output.
• Saying bootstrap samples without replacement. Fix: Bootstrap always draws with replacement, and each resample matches the original sample size.

Exam tips

• Know the sequence of steps and their order: specify the model, choose distributions, draw random numbers, compute outcomes, repeat, summarize.
• Remember the standard limitations: it depends on assumptions, gives statistical estimates, and does not give analytical insight.
• Items are standalone with three options, so use the 'exact answer' and 'more trials fix assumptions' traps to eliminate fast.
• Practise one-trial calculations: turn a z-value into a return, then into a price with exp.
• At about 90 seconds per question, do not simulate by hand; only transform one draw.
• Read whether the given mean is for the log return or for the price growth rate. The formulas differ by σ²/2.
• Do the e^x step last and once. Add log returns across periods rather than compounding step by step.
• Use the sign of Z to sense-check the answer: negative Z means a lower price than at the mean, and the price must stay positive.