Risk Management > Simulation & Valuation Techniques

Risk Management Monte Carlo Simulation Techniques

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Monte-Carlo Simulation Techniques

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Monte-Carlo Simulation Techniques Pseudo-Random Number Generation ¾ Generating Pseudo-Random Variables ¾ Forecasting Volatilities and Correlations ¾ Monte Carlo Simulation of Diffusions ¾ Monte Carlo Options Pricing ¾ Variation Reduction Techniques ¾

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Monte-Carlo Simulation Techniques

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Modeling Financial Processes ¾

Geometric Brownian Motion dS t ⎛⎡ ⎞ 1 2⎤ = µdt + σdWt St = S 0 exp⎜⎜ ⎢ µ − σ ⎥t + σWt ⎟⎟ St 2 ⎦ ⎝⎣ ⎠

¾

Hull White Stochastic Volatility Model 2 dSt dσ t 1 2 = µdt + σdWt = νdt + ξdWt 2 St σt

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Monte-Carlo Simulation Techniques

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One Factor Interest Rate Models ¾

General Form: dr = m( r) dt + σ( r) dW ƒ Ito Process: • m: drift factor • σ: short rate volatility • dw: ε√t; ε ~ N(0,1)

¾

Model characteristics ƒ All rates move in same direction, but not by same amount ƒ Many different shapes possible (including inverted) ƒ Mean reversion can be built in

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Monte-Carlo Simulation Techniques

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Model Taxonomy Expected Mean Volatility Fits Term Str. change in r Reversion of r Yield Vol m(r) s(r) Vasicek

a[m - r]

yes

CIR Brennan & Schwartz Ho & Lee BDT Hull & White

a[m - r] a[b + L - r] g(t) f(t,r,σ) a(t)[m(t) - r]

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constant

no

no

yes yes

f(r,L)

no no

no no

no limited yes

constant f(time) f(time)

yes yes yes

no yes yes

σ√r

Monte-Carlo Simulation Techniques

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Pseudo-Random Number Generation ¾

Simulation of price and return paths over time ƒ Requires the generation of sequences of random variables. ƒ This is known as Monte Carlo sampling ƒ Generation of “random” (i.e. almost independent) numbers

¾

Pseudo-Random Numbers ƒ Represent as decimal fractions ƒ Interpret as realizations U of the uniform distribution on the unit interval U(0,1)

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Monte-Carlo Simulation Techniques

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Linear Congruential Generator ¾

Most Common Method of Sequential Generation xi +1 = ( axi + c) modulo m

i = 0, 1, 2, ...

ƒ m is a very large number -- the period of the generator ƒ a and c are parameters ƒ

¾

x0 ∈{0, 1,..., m − 1, m}

is the seed provided to start the recursive stream of numbers x0, x1, x2, ... Construct uniform pseudo-random variates ui ~ U(0,1) ui = xi / m i = 0, 1, 2, ... • Sequence is not independent due to the m-long cycle • Okay when the sample number n is small relative to m

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Monte-Carlo Simulation Techniques

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Example For example, if x0 := 35, a := 13, c := 65, and m := 100 the algorithm works as follows: Iteration 0 Iteration 1

Set x0 = 35, a = 13, c = 65, and m = 100. Compute x1 = (a x0 + c) modulo m = [13(35) + 65] modulo 100 = 20 Deliver u1 = x1 / m = 20/100 = 0.2

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Monte-Carlo Simulation Techniques

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Example (continued) Iteration 2

Compute x2 = (a x1 + c) modulo m = [13(20) + 65] modulo 100 = 25 Deliver u2 = x2 / m = 25/100 = 0.25

Iteration 3

Compute x3 = (a x2 + c) modulo m . .

and so on. Copyright © 1997-2006 Investment Analytics

Monte-Carlo Simulation Techniques

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Generating Discrete Random Variables ƒ Conceptually we may generate all n-valued discrete random variables according to the following scheme for S taking 3 values with p.d.f.: Pointer Spin

⎧0.25 s = S1 ⎪0.40 s = S ⎪ 2 f s ( s ) := ⎨ ⎪0.35 s = S3 ⎪⎩0 O.W .

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Monte-Carlo Simulation Techniques

S1 S2 S3

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Discrete PDF Generation ¾

More generally for x with discrete p.d.f: ƒ We divide the unit interval as f x ( xi ) i = 1, K , N

fx ( x n ) 67 4 4 8

f ( x1 ) 67 4x 4 8

fx ( x N ) 67 4 4 8

0

since

1

N

∑ f (x x

n

n =1

) = 1 n = 1, K , N

and return xn iff un falls in the nth interval

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Monte-Carlo Simulation Techniques

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Example: Bootstrapping Daily Returns ¾

Consider the problem of simulating “Monthly” returns x over N trading days per “Month” given a sequence x1, ..., xN of actual daily returns for one such month

¾

The empirical density function for the return process x assigns probability 1/N to each actual observation -assuming the data independently and identically distributed -- and we think of #{xi < k} as the empirical estimate of P{x < k}

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Monte-Carlo Simulation Techniques

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Resampling/Bootstrapping ¾

Now we use the uniform pseudo-random numbers u0, u1, u2, ... to generate xn n = 1, ..., N iff:

⎡ n −1 n ⎤ ui ∈ ⎢ , ⎥ ⎣ N N⎦ ¾

for as many multiples of N as is required This resampling technique -- known as bootstrapping -may be used to generate as many monthly returns as are required as e.g.: q+ N

∑x

i = q +1

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i

for month q = 1, 2, ...

Monte-Carlo Simulation Techniques

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Generating Continuous Random Variables ¾

Inverse Transform Method ƒ Used for random variables x whose c.d.f. is available in closed form Fx ƒ Based on P{ U ≤ u } = P{ x ≤ Fx− 1 ( u )}

u

-3

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-2

-1

0

F-1(u)

1

Monte-Carlo Simulation Techniques

2

3

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Generating Normal Variates In finance we are most interested in generating standard normal random variables z ~ N(0,1) ¾ Generate x ~ N(µ, σ2) as x=µ +σ z ¾

ƒ Noting that u ~ U(0,1) has mean 1/2 and variance 1/12 the central limit theorem means that n

z = (∑ u i − n / 2) ( n / 12)1/ 2 ≈ N (0,1) i =1

ƒ In practice n := 12 to give

12

z =∑ u i −6 i =1

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Monte-Carlo Simulation Techniques

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Box-Muller Algorithm (1958) ¾

More accurate is the exact Box-Muller (1958) transformation of u1, u2 ~ U(0,1) independent r.v’s as z1=(−2 ln u1 )1/ 2 sin 2π u2 ~ N (0,1) z2 =(−2 ln u1 )1/ 2 cos2π u2 ~ N (0,1)

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Monte-Carlo Simulation Techniques

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Generating Random Vectors ¾

Need to Generate Random Vectors ƒ For Correlated Returns • i.e. Log of price ratios • Assumed multivariate Normal ƒ For Multivariate (Log-Normal) Prices • Used for risk management & stress testing

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Monte-Carlo Simulation Techniques

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Correlated Normal Variates Instantaneous correlation ρ ¾ Generate correlated bivariate Normal variates ¾

ƒ From standard normal variates ε1 and ε2

w1 = ε1 w2 = ρε1 + 1 − ρ ε 2 2

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Monte-Carlo Simulation Techniques

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Procedure for Generating Random Vectors ¾

Given (estimated) covariance matrix Σ ƒ Factor the covariance matrix Σ into Cholesky factors as Σ = A A´ ƒ Generate z ~ N(0,1) ƒ Generate returns x = Az ~ N(0, Σ) ƒ Given a vector f of expected future spot prices generate multivariate lognormal prices p for the next period as xi

pi = f i e

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i = 1, ..., I

Monte-Carlo Simulation Techniques

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Cholesky Factorization ¾

Compute Cholesky factorization Σ = A A´ -where A is lower triangular -- in O(n2) operations as follows

⎡a11 a12 a13⎤ ⎡s11 s12 s13 ⎤ ⎡a11 0 0 ⎤ Σ = ⎢⎢s21 s22 s23⎥⎥ A = ⎢⎢a21 a22 0 ⎥⎥ A' = ⎢⎢0 a22 a23 ⎥⎥ ⎢⎣0 0 a33 ⎥⎦ ⎢⎣s31 s32 s33 ⎥⎦ ⎢⎣a31 a32 a33 ⎥⎦ 2 ⎤ ⎡s11 s12 s13 ⎤ ⎡a11 0 0 ⎤ ⎡a11 a12 a13⎤ ⎡a11 a11a21 a11a31 ⎥ ⎢s s s ⎥ = ⎢a a 0 ⎥ × ⎢0 a a ⎥ = ⎢a a2 + a2 a a + a a ⎢ 11 21 22 21 31 32 22 ⎥ 22 23 ⎥ ⎢ 21 22 23⎥ ⎢ 21 22 ⎥ ⎢ 2 2 ⎥ ⎢⎣s31 s32 s33 ⎥⎦ ⎢⎣a31 a32 a33 ⎥⎦ ⎢⎣0 0 a33 ⎥⎦ ⎢a11a31a21a31 + a32a22 a112 + a22 + a 33 ⎣ ⎦

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Monte-Carlo Simulation Techniques

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Cholesky Factorization ¾

¾

Now we use the elements sij of Σ to solve for the elements aij of A -- positive definite For an IxI matrix Σ we use the recursions i −1

aii = [ sii − ∑ aik2 ]1/ 2 k =1 i −1

aij = [ sij − ∑ aik a jk ] / aii

i = 1, ..., I, j = i + 1, i + 2, ..., I

k =1

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Monte-Carlo Simulation Techniques

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Monte Carlo Simulation of Diffusions ¾

Return or price process S given by a stochastic differential equation of the form dSt = µ(St , t)dt +σ (St , t)dWt St ƒ We may apply the theory previously developed in both the univariate and multivariate cases ƒ In the univariate case we consider a real time increment ∆t and simulate a realization of the state St of the process for t=0, 1, 2 ... where t = k∆t

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Monte-Carlo Simulation Techniques

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Simulating a Univariate Diffusion Process ¾

Wiener Process W is given by ∆Wt = ∆tz t t = 0, 1, 2, ...

ƒ Where {zt} are i.i.d. N(0,1) ¾

So

t −1

Wt = ∆t ∑ z t

t = 0, 1, 2, ...

s =0

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Monte-Carlo Simulation Techniques

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Procedure for Univariate Process ¾ ¾ ¾

Generate stream of pseudo-random standard normal variates z0, z1, z2, ... Obtain stream of pseudo-random Wiener process increments ∆W0, ∆W1, ∆W2, ... Produces stream of diffusion process increments ∆S0, ∆S1, ∆S2, ... which are added to give the current state realization as … t −1

S t = ∑ ∆S i + S 0 i =0

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Monte-Carlo Simulation Techniques

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Multiple Stochastic Factors ¾

Example: Spread Option ƒ Difference between two assets

dS 2t dS1t = µ 2dt + σ 2 dW2t = µ 1dt + σ 1dW1t S 2t S1t ¾

Hull White Stochastic Volatility Model

dSt 1 = µdt + σdWt St

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dσ

σ

2 t 2 t

= νdt + ξdW t

Monte-Carlo Simulation Techniques

2

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Multiple Stochastic Factors ¾

Example with mean reverting square root volatility process (Hull White 1988) dS1t = µ 1dt + σ 1dW1t S1t

dS 2t = µ 2dt + σ 2 dW2t S 2t

d σ = α1 (σ − σ )dt + ξ1σ 1t dW 3t 2 1t

2 1t

2 2t

d σ = α 2 (σ − σ )dt + ξ 2σ 2t dW 2 2t

2 2t

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2 2t

Monte-Carlo Simulation Techniques

4t

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Simulating Multivariate Diffusion Processes ¾

Multivariate case of I instruments ƒ St is an I-vector

Covariance Σ and drift µ must be estimated ¾ Volatility σ in the diffusion S.D.E. is its Cholesky factor, i.e. Σ = σσ and ¾

∆S t = µ ( S t , t )∆t + σ ( S t , t )∆Wt St

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t = 0, 1, 2, ...

Monte-Carlo Simulation Techniques

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Monte Carlo Option Pricing ¾

Simulate paths of geometric Brownian motion in terms of ∆S t = rS t ∆t + σS t ∆Wt

t = 0, 1, 2, ...

ƒ This approach must be used for exotic -- path dependent -- options such as lookbacks and Asians ¾

So far there are no reliable methods for simulation of American option prices ƒ Due to the unknown exercise boundary ƒ See Brodie & Glasserman 1996, Rebonato & Cooper 1996

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Monte-Carlo Simulation Techniques

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Monte Carlo Pricing of Vanilla European Options ¾

Valuation under the risk neutral measure given by P ( S , t ):= e − r (T −t ) E[ f (S t , T ) | S t = S ]

ƒ Evaluate the conditional expectation by Monte Carlo methods

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Monte-Carlo Simulation Techniques

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Example: European call ¾

Involves the integral expectation ∞

ln S ( e ∫ − X ) f ln St (s)ds X

• where f ln S is the N{(T-t)r,(T-t)σ2} density ƒ Change variables to invert the normal c.d.f. using table lookups and interpolation ƒ Convert this to a new integral of an integrand g on the unit interval t

1 N −1 I ( g ) := ∫ g (ξ )dξ ≈ ∑ g (un ) N n =0 0 1

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un ∈ [ 0,1]

Monte-Carlo Simulation Techniques

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Variance Reduction Techniques: Antithetic Variates ¾ ¾

These are used to speed convergence of the Monte Carlo approximation and the most popular are the following Antithetic Variates Use both u and 1-u to double sample size cheaply 1 d:=2

0

1

1 Vest = [Vest {u} + Vest {1 − u}] 2 ¾

As long as covariance between V{z} and V{1-z} is negative, the overall variance will be substantially reduced

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Monte-Carlo Simulation Techniques

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Variance Reduction Techniques: Stratification ¾

Random sample ƒ Tends to leave gaps

¾

Stratified sample gives more regular representation ƒ Vi = (i-1) + Ui / 100 • Ui are iid uniform variates • Each Vi is uniformly distributed in the (i-1)th

•

percentile Then Zi = Φ-1(Vi ) falls between (i-1) and ith percentile of Normal distribution

ƒ Downside is loss of independence • Critical for statistical inference Copyright © 1997-2006 Investment Analytics

Monte-Carlo Simulation Techniques

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Variance Reduction Techniques: Control Variates ¾

Control Variates ƒ Correct Monte Carlo estimate of exotic value with vanilla MC error

Vˆ E = V EMC + (V BSMC − V BS ) ¾

Variate – Control Variate correlation ƒ Reduces estimate variance when control and variate are correlated • Based on cancellation of shared estimation errors ƒ No benefit if control is uncorrelated with variate, ƒ If negative correlated, may increase estimate variance!

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Monte-Carlo Simulation Techniques

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Control Variates ¾

Example: Barrier option ƒ Price Vi = discounted payoff for path i

Vi = h( S 0 , St(1i ) ,.........., Stm(i ) ) ƒ MCS estimate V = E[Vi] ¾

1 Vi ∑ n i =1

Assume standard call option control variate ƒ Price known in closed form, or easily evaluated

C = E[C i ] = E[ g ( S 0 , St(1i ) ,......., Stm( i ) ] Copyright © 1997-2006 Investment Analytics

Monte-Carlo Simulation Techniques

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Control Variates ¾

Controlled Estimator 1 n ⎛1 n ⎞ − − V β C C ⎜ ⎟ ∑ ∑ i i n i =1 ⎝ n i =1 ⎠

ƒ Coefficient with smallest variance is:

Cov[Vi , Ci ] β = Var[Ci ] *

ƒ Variance reduction of estimator: 1 – ρ2v,c

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Monte-Carlo Simulation Techniques

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Control Variate Example Correlation = 0.78 ¾ Variance reduction = 61% ¾

100 90 80

Arithmetic Asian

70 60 50 40 30 20 10 0 0

10

20

30

40

50

60

70

80

90

100

Standard Call

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Monte-Carlo Simulation Techniques

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Quasi-Random Numbers Low Discrepancy Sequences ¾ Deterministic sequences generated by number theory ¾

ƒ Halton , Sobel, Faure ƒ Sequences appear random, but not “clumpy” ƒ Behavior is ideal for fast convergence

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Monte-Carlo Simulation Techniques

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Example: Van der Corput Sequence ¾

To obtain nth point in series xn ƒ Restate n in base 2

I

n = ∑ ai 2i i =0

ƒ Transpose digits in ai around “decimal point” I

ai xn = ∑ i +1 i =0 2 ƒ Generates ½, ¼, ¾, 1/8, 5/8, 3/8, 7/8 • Contained in [0,1] • Every consecutive quadruple of points has one point in – (0, ¼), [1/4, ½), ½, ¾), [3/4, 1)

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Monte-Carlo Simulation Techniques

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Other Sequences ¾

Halton ƒ General s-dimensional sequence in [0,1]s hypercube • First dimension is van der Corput base 2 • Second dimension is van der Corput base 3 • S-dimension is van der Corput base sth prime number

¾

Faure ƒ All dimensions use base prime p >= s >= 2 ƒ First dimension sequence is van der Corput base p ƒ Higher dimensions are permutations of 1st dim.

¾

Sobol ƒ All dimensions use 2 as base

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Monte-Carlo Simulation Techniques

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Random vs. Sobel Paskov (1997)

Random points in the unit square

Sobol points in the unit square

Pricing Error for a European Call 0.10

0.05

0.00 0

500

1000

1500

2000

2500

3000

3500

4000

-0.05

Faure

-0.10

Pseudo 1 Pseudo 2

-0.15

-0.20

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Monte-Carlo Simulation Techniques

Slide: 41

Sensitivity Factors with MCS ¾

Approximate using finite difference ratios Delta

∂C C ( P + ∆P) − C ( P − ∆P ) ≈ ∂P 2∆P

¾

Gamma

∂ 2C C ( P + ∆P ) − 2C ( P ) + C ( P − ∆P ) ≈ 2 ∂P ∆P 2

¾

Vega

¾

Theta

∂C C (σ + ∆σ ) − C (σ − ∆σ ) ≈ 2∆σ ∂σ

¾

∂C C (t + ∆t ) − C (t − ∆t ) ≈ ∂t 2∆t Copyright © 1997-2006 Investment Analytics

Monte-Carlo Simulation Techniques

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Summary Monte-Carlo Methods ¾ Random number generation ¾ Simulating diffusion process ¾ Applications to option pricing ¾ Variance reduction techniques ¾

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Monte-Carlo Simulation Techniques

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