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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Importance Sampling for Option Pricing
Steven R. Dunbar
January 25, 2016
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Outline
1 Put Options
2 Monte Carlo Method
3 Importance Sampling
4 Examples
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Put Option
A put option is the right to sell an asset at an establishedprice at a certain time.
The established price is the strike price, K. The certaintime is the exercise time T .
At the exercise time, the value of the put option is apiecewise linear, decreasing function of the asset value.
OptionIntinsicValue
Asset PriceK
VP (S, T ) = max(K − S, 0)
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
What is the price?
For an asset with a random value at exercise time:What is the price to buy a put option before the exercisetime?
Six factors affect the price of a asset option:
the current asset price S;the strike price K;the time to expiration T − t where T is theexpiration time and t is the current time;the volatility of the asset price;the risk-free interest rate; and(the dividends expected during the life of theoption.)
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Geometric Brownian Motion
How do asset prices vary randomly?
Approximate answer is Geometric Brownian Motion:Stock prices can be mathematically modeled with astochastic differential equation
dS(t) = rS dt+σS dW (t), S(0) = S0.
The solution of this stochastic differential equation isGeometric Brownian Motion:
S(t) = S0 exp((r − σ2
2)t+ σW (t)).
Simplest caseS(t) = eW (t).
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Log-Normal Distribution
At time t Geometric Brownian Motion has a lognormalprobability density with parametersm = (ln(S0) + rt− 1
2σ2t) and s = σ
√t.
fX(x;m, s) =1√2πs
exp
(−1
2
[ln(x)−m
s
]2).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4x
lognormal pdf and cdf, m = 1, s = 1.5
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Statistics of Log-Normal Distribution
The mean stock price at any time is
E [S(t)] = S0 exp(rt).
The variance of the stock price at any time is
Var [S(t)] = S20 exp(2rt)[exp(σ2t)− 1].
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Monte Carlo Sample Mean
Assume a security price is modeled by GeometricBrownian Motion, with lognormal pdf.
Draw n (pseudo-)random numbers x1, . . . , xn from thelognormal distribution modeling the stock price S.
Approximate a put option price as the (present-value ofthe) expected value of the functiong(x) = max(K − x, 0), with the sample mean
VP (S, t) = e−r(T−t)E [g(S)] ≈ e−r(T−t)
[1
n
n∑i=1
g(xi)
]= e−r(T−t)gn.
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Central Limit Theorem
The Central Limit Theorem implies that the samplemean gn is approximately normally distributed with meanE [g(S)] and variance Var [g(S)] /
√n,
gn ∼ N(E [g(S)] ,Var [g(S)]).
Recall that for the standard normal distributionP [|Z| < 1.96] ≈ 0.95
A 95% confidence interval for the estimate gn is(E [g(S)]− 1.96
√Var [g(S)]
n,E [g(S)] + 1.96
√Var [g(S)]
n
).
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Estimating the Mean and Variance
A small problem with obtaining the confidence interval:The mean E [g(S)] and the variance Var [g(S)] are bothunknown.
These are respectively estimated with the sample meangn and the sample variance
s2 =1
n− 1
n∑i=1
(g(xi)− gn)2
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Using Student’s t-distribution
The sample quantity
(gn − E [g(X)])
s/√n
has a probability distribution known as the Student’st-distribution, so the 95% confidence interval limits of±1.96 must be modified with the corresponding 95%confidence limits of the appropriate Student-tdistribution.
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Monte Carlo Confidence Interval
The 95% level Monte Carlo confidence interval forE [g(X)](
gn − tn−1,0.975s√n, gn + tn−1,0.975
s√n
).
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Example
Confidence interval estimation to calculate a simplifiedput option price for a simplified security. The simplifiedsecurity has a
risk-free interest rate r = σ2/2,a starting price S = 1,a standard deviation σ = 1.K = 1,time to expiration is T − t = 1.
VP (S, t) = e−r(T−t)∫ ∞0
max(0, K − x)P[eW (T−t) ∈ dx
]13 / 26
ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
R Program for Estimation
#+name Rexample
n <- 10000S <- 1sigma <- 1r <- sigma^2/2K <- 1Tminust <- 1x <- rlnorm(n) #Note use of default meanlog=0, sdlog=1y <- sapply(x, function(z) max(0, K - z ))t.test(exp(-r*Tminust) * y) # all simulation results
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Problems with Monte Carlo
Applying Monte Carlo estimation to a random variablewith a large variance creates a confidence interval that iscorrespondingly large.
Increasing the sample size, the reduction is 1√n.
Variance reduction techniques increase the efficiency ofMonte Carlo estimation. Reduce variability with a givennumber of sample points, or for efficiency achieve thesame variability with fewer sample points.
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Importance Sampling
Importance sampling is a variance reduction technique.
Some values in a simulation have more influence on theestimation than others. The probability distribution iscarefully changed to give “important” outcomes moreweight. If “important” values are emphasized by samplingmore frequently, then the estimator variance can bereduced.
The key to importance sampling is to choose a newsampling distribution that “encourages” the importantvalues.
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Choosing a new PDF
Let f(x) be the density of the random variable, so we aretrying to estimate
E [g(x)] =
∫Rg(x)f(x) dx .
We will attempt to estimate E [g(x)] with respect toanother strictly positive density h(x). Then easily
E [g(x)] =
∫Rg(x)
f(x)
h(x)h(x) dx .
or equivalently, we are now trying to estimate
EY
[g(Y )f(Y )
h(Y )
]= EY [g(Y )]
where Y is a new random variable with density h(y).17 / 26
ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Reducing the variance
For variance reduction, determine a new density h(y) soVarY [g(Y )] < VarX [g(X)].
Consider
Var [g(Y )] = E[g(Y )2
]− (E [g(Y )])2
=
∫R
g2(x)f 2(x)
h(x)dx−E [g(X)]2 .
By inspection, we can see that we can makeVar [g(Y )] = 0 by choosing h(x) = g(x)f(x)/E [g(X)].This is the ultimate variance reduction.
Need E [g(X)], what we are trying to estimate!18 / 26
ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Educated Guessing
Importance sampling is equivalent to achange-of-measure from P to Q with
dQdP
=f(x)
h(x)
Choosing a good importance sampling distributionrequires educated guessing. Each instance of importancesampling depends on the function and the distribution.
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Trivial Parameters
Calculate confidence intervals for a Monte Carlo estimateof a European put option price, whereg(x) = max(K − x, 0) and S is distributed as aGeometric Brownian Motion.
To keep parameters simple
risk free interest rate r = σ2/2, thestandard deviation σ = 1,the strike price K = 1 andtime to expiration is 1
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
The quantity to estimate
∫ ∞0
max(0, 1− x)P[eW ∈ dx]
=
∫ ∞0
max(0, 1− x)1√
2πσx√T
exp
(− ln(x)2
2T
)dx .
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
First Change of variable
Want ∫ ∞0
max(0, 1− x)P[eW (1) ∈ dx].
After a first change of variable the integral is
E [g(S)] =
∫ 0
−∞(1− ex)
e−x2/2
√2π
dx
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Another Change of variable
x = −√y for x < 0. Then dx = dy2√yand the
expectation integral becomes∫ ∞0
1− e−√y
√2π√y
e−y/2
2dy .
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Comparative results
n putestimate confintleft confintrightB-S 0.14461MC 100 0.15808 0.12060 0.19555MC 1000 0.15391 0.14252 0.16531MC 10000 0.14519 0.14167 0.14871Norm 100 0.13626 0.099759 0.17276Norm 1000 0.14461 0.133351 0.15587Norm 10000 0.14234 0.138798 0.14588Exp 100 0.14388 0.13614 0.15163Exp 1000 0.14410 0.14189 0.14631Exp 10000 0.14461 0.14392 0.14530
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
A real example
Put option on S & P 500, SPX131019P01575000
S = 1614.96
r = 0.8% (estimated, comparable to 3 year and 5year T-bill rate)σ = 18.27% (implied volatility)T − t = 110/365 (07/01/2013 to 10/19/2013)K = 1575
n = 10,000
Quoted Price: $44.20
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ImportanceSampling for
Option Pricing
Steven R.Dunbar
Put Options
Monte CarloMethod
ImportanceSampling
Examples
Results
Method Value Confidence IntervalBlack-Scholes 44.21273MonteCarlo 45.30217 (43.84440, 46.75994)Importance Sample 44.13482 (43.91919, 44.35045 )
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