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Learning Module 6 of Quantitative Methods 6–9% of the exam

Simulation Methods

3 lessons. About 20 minutes in cram mode, 90 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary

Lessons

1
Why asset prices are lognormal
explain 8 min cram, 32 min full
2
Monte Carlo simulation
describe 6 min cram, 30 min full
3
Bootstrapping: simulating from the data you have
describe 6 min cram, 28 min full

What the exam expects you to do

Explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices when using continuously compounded asset returns.Why asset prices are lognormal
Describe Monte Carlo simulation and explain how it can be used in investment applications.Monte Carlo simulation
Describe the use of bootstrap resampling in conducting a simulation based on observed data in investment applications.Bootstrapping: simulating from the data you have

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

Hide what I know

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

Why asset prices are lognormal

Known

MemorizeNormal versus lognormal

	Normal	Lognormal
Range	−∞ to +∞	0 to +∞
Shape	symmetric	skewed right
Models	continuously compounded returns	prices
Parameters	its own mean and variance	mean and variance of $\ln Y$

Memory hook Read the name as an instruction: log it and it turns normal. Prices have a floor at zero and no ceiling, so the tail runs right.

Know it cold If $\ln Y$ is normal, $Y$ is lognormal. Mean of $Y$ is above its median.

MemorizeSquare-root-of-time rules

Scale by T	Rule
Mean	$\mu T$
Variance	$\sigma^2 T$
Volatility	$\sigma\sqrt{T}$
Daily to annual volatility	× $\sqrt{250}$ ≈ 15.81

Memory hook Variance adds, so the standard deviation (its square root) grows with $\sqrt{T}$ .

Know it cold Never multiply volatility by T. Only continuously compounded returns add; simple returns do not.

Key terms

normal distribution Use it
lognormal distribution Memorize
continuously compounded return Memorize
independently and identically distributed Memorize
stationarity Use it
volatility Memorize

From returns to prices

Price after T periods

P_T = P_0\,e^{r}

Continuously compounded return Memorize normal, can be negative without limit
- Sum of period returns adds across time
- Normal or close to it central limit theorem
Price Memorize lognormal: bounded at 0, skewed right
- Mean above median the long right tail pulls the mean up

What scales with the horizon T (i.i.d. returns)

Continuously compounded return over T periodsneeds independent and stationary returns

Mean Memorize grows with T: $\mu T$
Variance Memorize grows with T: $\sigma^2 T$
Volatility Memorize grows with $\sqrt{T}$ , not T

Formulas

Price from a continuously compounded return Memorize

P_T = P_0\,e^{r_{0,T}}, \qquad r_{0,T} = \ln\left(\frac{P_T}{P_0}\right)

Quick example

A 25.00 share has a continuously compounded return of 8% over the year. $P_T = 25 \times e^{0.08}$ = 27.08.

On your calculator

0.082NDe^x×25= 27.08

2ND second function (the shifted label above a key) · e^x e to the power x

0.08ge^x25× 27.08

g blue shift key · e^x e to the power x

Continuously compounded returns add Memorize

r_{0,T} = r_{0,1} + r_{1,2} + \cdots + r_{T-1,T}

Quick example

A share goes 40.00, 42.40, 41.00, 44.00. Daily returns 5.83% − 3.36% + 7.06% = 9.53%, which equals $\ln(44 \div 40)$ .

On your calculator

44÷40=LN 0.0953, the same total in one step

LN natural logarithm

44ENTER40÷gLN 0.0953

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · LN natural logarithm

Scaling over T periods (i.i.d. returns) Memorize

E(r_{0,T}) = \mu T, \qquad \sigma^{2}(r_{0,T}) = \sigma^{2}T, \qquad \sigma(r_{0,T}) = \sigma\sqrt{T}

Quick example

Daily mean 0.15% and volatility 1.2%. Over 5 days: mean 0.75%, volatility $1.2 \times \sqrt{5}$ = 2.68%.

On your calculator

5√x×1.2= 2.6833

√x square root

5g√x1.2× 2.6833

g blue shift key · √x square root

Annualized volatility Memorize

\sigma_{annual} = \sigma_{period}\sqrt{\text{periods per year}}, \quad \text{daily: } \sqrt{250}

Quick example

Weekly volatility 0.9%. Annual: $0.9 \times \sqrt{52}$ = 6.49%. Multiplying by 52 (46.80%) is the trap.

On your calculator

52√x×0.9= 6.4900

√x square root

52g√x0.9× 6.49

g blue shift key · √x square root

Mean of a lognormal variable Use it

E(Y) = e^{\mu + \sigma^{2}/2}

Quick example

Annual return mean 5%, standard deviation 20%, price 100. Mean price $100 \times e^{0.07}$ = 107.25, above the median 105.13.

On your calculator

0.2x²÷2+0.05=2NDe^x×100= 107.25

x² square · 2ND second function (the shifted label above a key) · e^x e to the power x

0.2ENTER×2÷0.05+ge^x100× 107.25

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Monte Carlo simulation

Known

MemorizeThe six steps, in order

#	Step	Part
1	Specify the quantity of interest and start values	Specify
2	Specify the time grid (K sub-periods)	Specify
3	Specify the model and distributions of the risk factors	Specify
4	Draw random numbers, convert to prices or paths	Run
5	Compute the outcome (and discount it)	Run
6	Repeat for N trials, then summarize	Run

Memory hook Three to set up, three to run. Nothing is drawn until the model and distributions are chosen.

Know it cold Result = average of the discounted trial outcomes. More trials cut the error, they do not fix wrong assumptions.

MemorizeUses and limits

Uses	Limits
Complex or path-dependent claims	Only as good as the assumed distributions
Portfolio risk and VaR	Statistical estimate, not exact
Retirement and liability planning	Shows outcomes, not causes
Sensitivity of a model to inputs	Misses events outside the assumed distribution

Memory hook "Garbage in, garbage out." The simulation draws only from what you tell it.

Know it cold Use a formula where one exists. Simulate where it does not.

Key terms

Monte Carlo simulation Memorize
risk factor Use it
random number generation Know the idea
simulation trial Use it
contingent claim Use it
sensitivity analysis Use it

Six steps of a Monte Carlo simulation

Monte Carlo simulationspecify it, then run it

Specify the simulation Memorize steps 1 to 3
- 1. Quantity of interest what to value or measure, and the starting values
- 2. Time grid K sub-periods of length $\Delta t$
- 3. Model and distributions a distribution for each risk factor, with parameters
Run the simulation Memorize steps 4 to 6
- 4. Draw and convert random numbers into prices or paths
- 5. Compute the outcome payoff or balance, discounted if valuing
- 6. Repeat, then summarize N trials; average or read the distribution

What Monte Carlo is used for, and where it falls short

Monte Carlo simulationa complement to analytical methods

Uses Memorize
- Valuing complex claims path-dependent, embedded options, no formula
- Risk and return distribution of portfolio gains and losses, VaR
- Planning chance a retirement pot or liability plan fails
- Sensitivity analysis change an assumption, rerun, compare
Limits Memorize
- Depends on assumptions wrong distributions, wrong answers
- Statistical, not exact an estimate with an error, not a proof
- No cause and effect shows what happens, not why
- Heavy to run needs computing power

Formulas

Simulated price step Use it

\Delta S = \mu\, S_{prev}\,\Delta t + \sigma\, S_{prev}\, Z

Quick example

Price 80.00, drift 6% a year, step of a quarter, step volatility 4%, draw $Z = -0.5$ . $\Delta S = 1.20 - 1.60$ = −0.40, so the next price is 79.60.

On your calculator

0.06×80×0.25+(0.04×80×0.5+|-)= −0.4

+|- change sign

0.06ENTER80×0.25×0.04ENTER80×0.5CHS×+ −0.4

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Payoff of an average-price claim, one trial Use it

C_T = \max\left(S_T - \bar{S},\, 0\right), \qquad C_0 = \frac{C_T}{(1+r)^{t}}

Quick example

Final price 41.00, average over the life 38.50, risk-free rate 2%, one year. Payoff 2.50, present value $2.50 \div 1.02$ = 2.45.

On your calculator

41−38.5=÷1.02= 2.451

41ENTER38.5−1.02÷ 2.451

ENTER enter (on the HP, puts the number on the stack)

Monte Carlo estimate of value Use it

\hat{V} = \frac{1}{N}\sum_{i=1}^{N} C_{i,0}

Quick example

Five discounted payoffs 1.79, 0, 0.84, 0 and 2.40. Mean = 5.03 ÷ 5 = 1.01. A real run uses thousands of trials.

On your calculator

1.79+0.84+2.4=÷5= 1.006

1.79ENTER0.84+2.4+5÷ 1.006

ENTER enter (on the HP, puts the number on the stack)

Standard error of the estimate Use it

SE = \frac{s}{\sqrt{N}}

Quick example

Discounted payoffs have a standard deviation of 3.20. With 10,000 trials, SE = 3.20 ÷ 100 = 0.032. Four times the trials halves it.

On your calculator

3.2÷10000√x= 0.032

√x square root

3.2ENTER10000g√x÷ 0.032

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Bootstrapping: simulating from the data you have

Known

MemorizeMonte Carlo versus bootstrap

	Monte Carlo	Bootstrap
Source of draws	a distribution you specify	the observed data
Distribution assumed	yes	none
Real fat tails and skew	only if modeled	kept, as in the sample
Unseen extremes	possible	not possible
Main risk	wrong distribution	sample not representative

Memory hook Bootstrap = pull yourself up by your own data. Nothing new goes in, so nothing new comes out.

Know it cold Draw at random from the observed sample, with replacement. To estimate a statistic, each resample has the same size as the sample; to simulate a path, draw one observation per step.

Key terms

bootstrap Memorize
resampling Use it
sampling with replacement Memorize
empirical distribution Use it
historical simulation Know the idea

Bootstrap simulation, step by step

Bootstrap simulationresample the data you have

Set up
- Collect the sample Memorize observed returns, for example 240 months
- Match the time grid steps as long as the data periods
Draw
- Pick one observation at random Memorize record it
- Put it back Memorize with replacement, so it can be drawn again
- Repeat to build a path for example 12 draws for a year
Run
- Compute the outcome of the path
- Repeat thousands of times, then summarize same as Monte Carlo

Where the random draws come from

Simulationsame steps, different source of randomness

Monte Carlo Memorize draws from a distribution you specify
- Can produce unseen extremes
- Risk: the wrong distribution
Bootstrap Memorize draws from the observed data, with replacement
- Keeps the real skew and fat tails
- Cannot go beyond the sample range
- Risk: the past is not the future

Formulas

Return of a simulated path Use it

R_{path} = (1 + r_1)(1 + r_2)\cdots(1 + r_k) - 1

Quick example

A bootstrap draws monthly returns of 2%, −1% and 3%. $1.02 \times 0.99 \times 1.03 - 1$ = 4.01%.

On your calculator

1.02×0.99×1.03−1= 0.0401

1.02ENTER0.99×1.03×1− 0.0401

ENTER enter (on the HP, puts the number on the stack)

Most extreme bootstrap path Use it

R_{worst} = (1 + r_{min})^{k} - 1, \qquad R_{best} = (1 + r_{max})^{k} - 1

Quick example

Sample returns range from −8% to +15%. Five draws can lose at most $0.92^5 - 1$ = −34.09% and gain at most $1.15^5 - 1$ = 101.14%.

On your calculator

0.92y^x5=−1= −0.3409

y^x power

0.92ENTER5y^x1− −0.3409

ENTER enter (on the HP, puts the number on the stack) · y^x power

Cheat sheet · PDF · 3 A4 pages

Simulation Methods on 3 pages

What to memorize, the concept maps, every formula with a worked example and keystrokes for the TI BA II Plus and HP 12C, how each idea is asked, and the traps. Print it, or keep it on your phone.

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