Methods for Estimating Sensitivities to Parameters in

Methods forEstimating

Sensitivities toParameters in

FinancialModels

Orhan Akal,Dechang

Chen, LukeMohr

Introduction

Finite-differenceapproxima-tions

Pathwisemethod

Likelihoodratio method

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Methods for Estimating Sensitivities toParameters in Financial Models

Orhan Akal, Dechang Chen, Luke Mohr

University of Massachusetts

12/6/2012

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IntroductionA driving problem in Financial Mathematics is the valuation ofderivatives.

DefinitionA derivative is a contract between parties which specifiesconditions under which payments are to be made between thetwo parties. Its value is based on one or more underlying assets.

• Examples of derivatives: options, futures, forwards, swaps• Common underlying assets: stock, commodities,

currencies, bonds, interest ratesThe value of a stock is modeled by a stochastic differentialequation, and the price of an option on the stock is based onthe stock’s value and several parameters.

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IntroductionA driving problem in Financial Mathematics is the valuation ofderivatives.DefinitionA derivative is a contract between parties which specifiesconditions under which payments are to be made between thetwo parties. Its value is based on one or more underlying assets.

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currencies, bonds, interest rates

The value of a stock is modeled by a stochastic differentialequation, and the price of an option on the stock is based onthe stock’s value and several parameters.

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What is an Option?

DefinitionA call option is a financial contract between two parties, thebuyer and the seller of this type of option. The buyer of thecall option has the right, but not the obligation to buy anagreed quantity of an underlying asset from the seller of theoption at a certain time (the expiration date) for a certain price(the strike price). The seller is obligated to sell the commodityor financial instrument should the buyer so decide. The buyerpays a fee (called a premium) for this right.

Notation:• T - expiration date, K - strike price• r - continuously compounded interest rate• S(t) - value of the underlying asset at time t

A standard call option has a discounted payoff ofY = e−rT [S(T )− K ]+, where [x ]+ = max {0, x}.The price (premium) of this option is E[Y ].

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What is an Option?

DefinitionA call option is a financial contract between two parties, thebuyer and the seller of this type of option. The buyer of thecall option has the right, but not the obligation to buy anagreed quantity of an underlying asset from the seller of theoption at a certain time (the expiration date) for a certain price(the strike price). The seller is obligated to sell the commodityor financial instrument should the buyer so decide. The buyerpays a fee (called a premium) for this right.Notation:

• T - expiration date, K - strike price• r - continuously compounded interest rate• S(t) - value of the underlying asset at time t

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What is an Option?

DefinitionA call option is a financial contract between two parties, thebuyer and the seller of this type of option. The buyer of thecall option has the right, but not the obligation to buy anagreed quantity of an underlying asset from the seller of theoption at a certain time (the expiration date) for a certain price(the strike price). The seller is obligated to sell the commodityor financial instrument should the buyer so decide. The buyerpays a fee (called a premium) for this right.Notation:

• T - expiration date, K - strike price• r - continuously compounded interest rate• S(t) - value of the underlying asset at time t

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Black-Scholes Model

dS(t) = rS(t)dt + σS(t)dW (t)

• S(t) is the value of the asset at time t• r is the interest rate• σ is the volatility of the asset• T is the expiration date of the option• K is the strike price of the option• Y = e−rT [S(T )− K ]+ is the discounted payoff of the

option(Note: [x ]+ = max {0, x})

The Black-Scholes model has an explicit solution:

S(T ) = S(0)e(r− 12σ

2)T +σ√

where Z ∼ N (0, 1).

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Black-Scholes Model

S(T ) = S(0)e(r− 12σ

2)T +σ√

where Z ∼ N (0, 1).

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Estimating SensitivitiesWe present methods for estimating the derivatives of derivativeprices, called “Greeks.”These are used in measuring the sensitivity of the derivativeprice to its parameters.Greeks are often used to measure and manage risk.

ExampleDelta,

ddS(0)

is the derivative of the option price with respect to theunderlying asset’s initial value. It is used in some situations tohedge the risk associated with option trading.Three methods to estimate Greeks:

• Finite-difference approximations• Pathwise method• Likelihood ratio method

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Orhan Akal,Dechang

Chen, LukeMohr

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Estimating SensitivitiesWe present methods for estimating the derivatives of derivativeprices, called “Greeks.”These are used in measuring the sensitivity of the derivativeprice to its parameters.Greeks are often used to measure and manage risk.ExampleDelta,

ddS(0)

is the derivative of the option price with respect to theunderlying asset’s initial value. It is used in some situations tohedge the risk associated with option trading.

Three methods to estimate Greeks:• Finite-difference approximations• Pathwise method• Likelihood ratio method

FinancialModels

Orhan Akal,Dechang

Chen, LukeMohr

Introduction

Pathwisemethod

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Estimating SensitivitiesWe present methods for estimating the derivatives of derivativeprices, called “Greeks.”These are used in measuring the sensitivity of the derivativeprice to its parameters.Greeks are often used to measure and manage risk.ExampleDelta,

ddS(0)

is the derivative of the option price with respect to theunderlying asset’s initial value. It is used in some situations tohedge the risk associated with option trading.Three methods to estimate Greeks:

• Finite-difference approximations• Pathwise method• Likelihood ratio method

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Finite Difference Approximation• Random variable Y (θ) discounted payoff of an option,α(θ) is its price.

• α(θ) = E [Y (θ)]

• When θ is the initial price, then α′(θ) is the option’s delta,α′′(θ) is option’s gamma.

• When θ is a volatility parameter α′(θ) is often called”vega”

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Bias• Independent replication of the model at θ; Y1(θ)...Yn(θ)

and n additional replications of the model at parameter(θ + h); Y1(θ + h)...Yn(θ + h) for some h > 0Y n(θ) and Y n(θ + h) are average set of replications

• Forward Difference Estimator∆F ≡ ∆F (n, h) = Y n(θ+h)−Y n(θ)

This estimator has expectationE [∆F ] = h−1[α(θ + h)− α(θ)]

• if α is twice differentiable at θα(θ + h) = α(θ) + α′(θ)h + 1

2α′′(θ)h2 + o(h2)

• The Bias in the forward difference estimatorBias(∆F ) = E [∆F − α′(θ)] = 1

2α′′(θ) + o(h)

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Bias for Central Difference• by simulating at θ − h and θ + h we get central difference

estimator;

∆C ≡ ∆Cn, h = Y (θ+h)−Y (θ−h)2h

• The Bias in central difference estimator;Bias(∆C ) = α(θ+h)−α(θ−h)

2h − α′(θ) = o(h)

which is of smaller order than the Bias for ForwardDifference estimator.

• If α′′(θ) is is itself differentiable at θBias(∆C ) = 1

6α′′′(θ)h2 + o(h2)

• It is obvious that central-difference approximation is muchmore accurate than forward-difference approximation.

• Rounding errors resulting from small values of h limit theaccuracy of finite-difference approximation.

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Variance• The variance of the forward-difference estimator;

Var [∆F (n, h)] = h−2[Y n(θ + h)− Y n(θ)]

• The variance of the central-limit difference estimator;Var [∆C (n, h)] = o(h−2)[Y n(θ + h)− Y n(θ − h)]

• In both cases h−2 might lead severe consequences oftaking h very small.

• The form of the forward- and central-difference estimatorswould lead us to take ever smaller values of h to improveaccuracy(ignore the limits of machine precision). But theeffect of h on bias against its effect on variance shouldoptimized.

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Variance• For simplicity we assume that the pairs (Y (θ),Y (θ + h)

and (Yi (θ),Yi (θ + h), i=1,2,... are i.i.d(independentlyidentically distributed) so thatVar [Y n(θ + h)− Y n(θ)] = 1

n [Y (θ + h)− Y (θ)]

• How the Variance of forward-difference estimator changeswith h is determined by the dependence ofVar [Y (θ + h)− Y (θ)] on h.

• For Var [Y (θ + h)− Y (θ)] three cases ariseO(1),Case(i)

O(h),Case(ii)

O(h2),Case(iii)

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Cases• Case(i) if we simulate Y (θ) and Y (θ + h) independently,

then we getVar [Y (θ + h)− Y (θ)] = Var [Y (θ + h)] + Var [Y (θ)]→2Var [Y (θ)]

assume Var [Y (θ)] is continuous on θ

• Case(ii) simulating Y (θ) and Y (θ + h) using commonrandom numbers. i.e., generating them from the samesequence U1,U2, ... of Unif [0, 1] random variables.

• Case(iii) generally we need not only that Y (θ) andY (θ + h) use the same random numbers, but also that foralmost all values of the random number, the output Y () iscontinuous on θ. We won’t talk on Case(iii) in this section

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Optimal Mean Square Error• Since decreasing h can increase variance while decreasing

bias, minimizing mean square error(MSE), we need tobalance these two conditions.

• Increasing the number of replications n decreases variancewith no effect on bias, whereas h effects both bias andvariance. Need to find optimal relation between two.

• Squaring the Bias(∆F (n, h)) in and adding it to theVar(∆F (n, h)) we denote

MSE (∆F ,i (n, h)) = O(h2) + O(n−1h−2)as a result minimal conditions for convergence are h→ 0and nh2 →∞

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h n optimizationBy considering a a generic estimator ∆ = ∆(n, h) for which

E [∆− α′(θ)] = bhβ + o(hβ) and Var [∆] = σ2

nhη + o(h−η)

Typically β = 1 for Forward-difference estimator and β = 2 forcentral-difference estimator. Taking η = 2 sharpens case(i) andη = 1 sharpens case(ii)Let consider a sequence of estimator ∆(n, hn)

hn = h∗n−γ

then we get MSE (∆) = b2hn2β + σ2

Finally defined

RMSE (∆) = O(n−β

2β+η ) this is reasonable measure of theconvergence rate.

h∗ = ( ησ2

2βb2 )1

2β+η minimizing over h∗ yields that optimal value.

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Table 1.1: The estimators using either (F) orcentral(C)-difference estimator and either independentsampling (i) or common random numbers(ii)

• The table should be understood as follows; if the leadingterms of bias and variance are as indicated in the secondand the third column, then the conclusion in the last threecolumns holds.

• At least asymptotically, ∆C ,ii dominates the other threeestimators because it exhibits the fastest convergence.

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Fig 1.1 RMSE in Forward-difference estimator delta estimatesfor a standard call and digital option as a function of theincrement h, with fixed n=5000

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ExtrapolationFor a smooth function of α(θ), the bias in the finite-differencemethod can be reduced by using extrapolation.This techniqueapplies to all estimators stated before, we are gonna illustratethat on ∆C ,ii Central-difference method using common randomnumbers.

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ExtrapolationA Taylor expansion of α(θ) shows thatα(θ+h)−α(θ−h)

2h = α′(θ) + 16α′′′(θ)h2 + O(h4)

Similarlyα(θ+2h)−α(θ−2h)

4h = α′(θ) + 23α′′′(θ)h2 + O(h4)

The Bias in the combined estimator becomes43 ∆C ,ii (n, h)− 1

3 ∆C ,ii (n, 2h)

is O(h4).

• The RMSE of this estimator is O(n−4/9) if h is taken tobe O(n−1/9). This estimator achieves convergence rate ofnearly n−1/2, best convergence rate so far, with hn nearlyconstant.

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Pathwise Method• Main idea: if Y is discounted payoff of option, θ is a

parameter that it depends on, and α(θ) = E[Y (θ)] is theprice of the option, we estimate α′(θ) by taking thederivative of Y and then expectation, where

Y ′(θ) = limh→0

Y (θ + h)− Y (θ)

• This has expectation E[Y ′(θ)] and is an unbiasedestimator of α′(θ) if

dθY (θ)

ddθE[Y (θ)]

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Pathwise Method• Main idea: if Y is discounted payoff of option, θ is a

parameter that it depends on, and α(θ) = E[Y (θ)] is theprice of the option, we estimate α′(θ) by taking thederivative of Y and then expectation, where

Y ′(θ) = limh→0

Y (θ + h)− Y (θ)

• This has expectation E[Y ′(θ)] and is an unbiasedestimator of α′(θ) if

dθY (θ)

ddθE[Y (θ)]

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Black-Scholes DeltaBlack-Scholes model:

S(T ) = S(0)e(r− 12σ

2)T +σ√

where Z ∼ N (0, 1).

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Black-Scholes DeltaBlack-Scholes model:

S(T ) = S(0)e(r− 12σ

2)T +σ√

where Z ∼ N (0, 1).

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Black-Scholes DeltaThe Black-Scholes delta measures the sensitivity of the optionprice to the initial value of the underlying asset. In order toestimate this with the pathwise method we must find

dYdS(0)

dS(T )

• Y = e−rT [S(T )− K ]+

• dYdS(T )

= e−rT1{S(T )>K}

• dS(T )

dS(0)=

• dYdS(0)

= e−rT S(T )

S(0)1{S(T )>K}

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dYdS(0)

dS(T )

.• Y = e−rT [S(T )− K ]+

• dYdS(T )

= e−rT1{S(T )>K}

• dS(T )

dS(0)=

• dYdS(0)

= e−rT S(T )

S(0)1{S(T )>K}

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dYdS(0)

dS(T )

.• Y = e−rT [S(T )− K ]+

• dYdS(T )

= e−rT1{S(T )>K}

• dS(T )

dS(0)=

• dYdS(0)

= e−rT S(T )

S(0)1{S(T )>K}

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Asian OptionsAn Asian option has discounted payoff

Y = e−rT [S − K ]+, S =1m

m∑i=1

S(ti )

for fixed dates 0 < t1 < · · · < tm ≤ T .

Similar to the Black-Scholes delta,

dYdS(0)

dSdS(0)

= e−rT1{S>K}

dSdS(0)

m∑i=1

dS(ti )

dS(0)=

m∑i=1

S(ti )

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Asian OptionsAn Asian option has discounted payoff

Y = e−rT [S − K ]+, S =1m

m∑i=1

S(ti )

for fixed dates 0 < t1 < · · · < tm ≤ T .Similar to the Black-Scholes delta,

dYdS(0)

dSdS(0)

= e−rT1{S>K}

dSdS(0)

m∑i=1

dS(ti )

dS(0)=

m∑i=1

S(ti )

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Asian OptionsThe pathwise estimator of the option delta is thus

dYdS(0)

= e−rT1{S>K}

This estimator has great practical value because

• This estimator is unbiased.

• S is simulated in estimating the price of the option already,so finding the delta requires little additional effort.

• This method reduces variance and computing timecompared to finite-difference.

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dYdS(0)

= e−rT1{S>K}

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dYdS(0)

= e−rT1{S>K}

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dYdS(0)

= e−rT1{S>K}

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dYdS(0)

= e−rT1{S>K}

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B-S vs. Asian Options

r = .06, σ = 1, T = 1, K = 50, 100000 sample paths, asset price recorded monthly.

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VegaVega measures sensitivity to volatility.

• Black-Scholes vega• S(T ) = S(0)e(r− 1

2σ2)T +σ

• dYdσ =

dYdS(T )

dS(T )

dσ• dS(T )

dσ = (−σT +√

TZ )S(T )

• dYdσ = e−rT (−σT +

√TZ )S(T )1{S(T )>K}

• Asian option vega

• dYdσ = e−rT 1

m∑i=1

dS(ti )

dσ 1{S>K}

• dS(ti )

dσ = (−σ(ti − ti−1) +√

ti − ti−1Zi )S(ti )

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VegaVega measures sensitivity to volatility.

• Black-Scholes vega• S(T ) = S(0)e(r− 1

2σ2)T +σ

• dYdσ =

dYdS(T )

dS(T )

dσ• dS(T )

dσ = (−σT +√

TZ )S(T )

• dYdσ = e−rT (−σT +

√TZ )S(T )1{S(T )>K}

• Asian option vega

• dYdσ = e−rT 1

m∑i=1

dS(ti )

dσ 1{S>K}

• dS(ti )

dσ = (−σ(ti − ti−1) +√

ti − ti−1Zi )S(ti )

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Problems?• Digital options

• Discounted payoff Y = e−rT1{S(T )>K}

• 0 = dYdS(T ) , as Y is a step function in terms of S(T ).

• 0 = E[

dYdS(0)

dS(0) E [Y ]

• Second derivatives

• Note that the digital option is the same as dYdS(T ) for a

standard call option.

• As a result the pathwise derivative is generally inapplicablein estimating second derivatives.

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Problems?• Digital options

• Discounted payoff Y = e−rT1{S(T )>K}

• 0 = dYdS(T ) , as Y is a step function in terms of S(T ).

• 0 = E[

dYdS(0)

dS(0) E [Y ]

• Second derivatives

• Note that the digital option is the same as dYdS(T ) for a

standard call option.

• As a result the pathwise derivative is generally inapplicablein estimating second derivatives.

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Conditions for UnbiasednessLet Y (θ) = f (X1(θ), . . . ,Xm(θ)) for some random variables Xiand integer m. In order for our estimator to be unbiased, i.e,

E[Y ′(θ)] =ddθE [Y ]

we must have

• Y ′(θ) exists with probability 1. It then follows that

Y ′(θ) =m∑

∂f∂xi

(X1(θ), . . . ,Xm(θ))X ′i (θ)

• Y is almost surely Lipschitz in θ, that is

|Y (θ2)− Y (θ1)| ≤ κY |θ2 − θ1|

for a random variable κY with E[κY ] <∞. Thenunbiasedness follows by the dominated convergencetheorem.

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we must have• Y ′(θ) exists with probability 1. It then follows that

Y ′(θ) =m∑

∂f∂xi

(X1(θ), . . . ,Xm(θ))X ′i (θ)

|Y (θ2)− Y (θ1)| ≤ κY |θ2 − θ1|

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we must have• Y ′(θ) exists with probability 1. It then follows that

Y ′(θ) =m∑

∂f∂xi

(X1(θ), . . . ,Xm(θ))X ′i (θ)

|Y (θ2)− Y (θ1)| ≤ κY |θ2 − θ1|

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Discretization for General DiffusionProcesses

dX (t) = a(X (t))dt + b(X (t))dW (t)

with X (0) a fixed inital condition.

For step size h we candiscretize this to

X (i + 1) = X (i) + a(X (i))h + b(X (i))√

Let ∆(i) =dX (i)dX (0)

. Differentiating the discretization yields

∆(i + 1) = ∆(i) + a′(X (i))∆(i)h + b′(X (i))∆(i)√

and ∆(0) = 1.If the discounted payoff of some option is f (X (1), . . . ,X (m))then the pathwise derivative of the option’s delta is

m∑i=1

∂f∂xi

(X (1), . . . ,X (m))∆(i)

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dX (t) = a(X (t))dt + b(X (t))dW (t)

with X (0) a fixed inital condition. For step size h we candiscretize this to

X (i + 1) = X (i) + a(X (i))h + b(X (i))√

∆(i + 1) = ∆(i) + a′(X (i))∆(i)h + b′(X (i))∆(i)√

m∑i=1

∂f∂xi

(X (1), . . . ,X (m))∆(i)

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dX (t) = a(X (t))dt + b(X (t))dW (t)

X (i + 1) = X (i) + a(X (i))h + b(X (i))√

∆(i + 1) = ∆(i) + a′(X (i))∆(i)h + b′(X (i))∆(i)√

and ∆(0) = 1.

If the discounted payoff of some option is f (X (1), . . . ,X (m))then the pathwise derivative of the option’s delta is

m∑i=1

∂f∂xi

(X (1), . . . ,X (m))∆(i)

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dX (t) = a(X (t))dt + b(X (t))dW (t)

X (i + 1) = X (i) + a(X (i))h + b(X (i))√

∆(i + 1) = ∆(i) + a′(X (i))∆(i)h + b′(X (i))∆(i)√

m∑i=1

∂f∂xi

(X (1), . . . ,X (m))∆(i)

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Formulation• Pathwise method: requirement of continuity in the

function of the parameter.• Likelihood Ratio method: Smoothness is not required.• Y = f (X ),X ∈ Rm, θ is a parameter.

• Pathwise X = X (θ)• Likelihood X ∼ gθ(x)

• Expectation:

Eθ(Y ) =

f (x)gθ(x)dx .

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Likelihood Ratio MethodEstimator

• DerivationddθEθ(Y ) =

f (x)ddθgθ(x)dx

f (x)gθ(x)

gθ(x)gθ(x)dx

= Eθ[f (X )

gθ(X )

]• Unbiased LRM estimator

f (X )gθ(X )

gθ(X )= f (X )

d log gθ(X )

d log gθ(X)dθ is called score function.

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Call Option price• Geometric Brownian motion

S(T ) = S(0)e(r− 12σ

2)T +σ√

TZ , Z ∼ N(0, 1)

• Density of S(T )

g(x) =1

xσ√

Tφ(ζ(x)),

φ: standard normal density, ζ(x) =log(x/S(0))−(r− 1

2σ2)T

T .• Discount payoff

Y = e−rT (S(T )− K )+

• Call option price

C = Eg (Y ) = Eg (e−rT (X − K )+)

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Black-Scholes delta• Delta

∆ =∂C∂S(0)

dS(0)Eg (e−rT (X − K )+)

• Estimator of Delta• Score

d log(g(S(T )))

dS(0)=

log(S(T )/S(0))− (r − 12σ

2)TS(0)σ2T .

It can be written as ZS(0)σ

√T since ζ(S(T )) = Z .

• LRM estimator

e−rT (S(T )− K )+ ZS(0)σ

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Estimated vs TrueParameters: r = 5%, σ = 0.3, K = 100, T = 1. Repeat timesN = 100, 000.

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Black-Scholes vega• Vega

ν =∂C∂σ

dσEg (e−rT (X − K )+)

• Estimator of vega• Score

d log(g(S(T )))

dσ = − 1σ− ζ(S(T ))

dζ(S(T ))

dσ =log(S(0)/S(T )) + (r + 1

2σ2)T

σ2√

It can be written as Z 2−1σ − Z

√T .

• LRM estimator

e−rT (S(T )− K )+(Z 2 − 1σ

− Z√

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Black-Scholes gammaLikelihood Ratio Method is applicable to estimate the secondderivatives. The LRM estimator is f (X ) gθ(X)

gθ(X)

• Gamma

Γ =d2C

dS(0)2 =d2

dS(0)2 Eg (e−rT (X − K )+)

• Estimator of gamma• Score

d2g(S(T ))/dS(0)2

g(S(T ))=ζ(S(T ))2 − 1

S(0)2σ2T − ζ(S(T ))

S(0)2σ√

It can be written as Z 2−1S(0)2σ2T −

ZS(0)2σ

√T .

• LRM estimator

e−rT (S(T )− K )+

(Z 2 − 1

S(0)2σ2T −Z

S(0)2σ√

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Asian Call options• Discount payoff

Y = e−rT (S − K )+

• Density of path S(t1), . . . ,S(tm).

g(x1, . . . , xm) = g1(x1|S(0))g2(x2|x1) · · · gm(xm|xm−1),

gj(xj |xj−1) =1

xjσ√

tj − tj − 1φ(ζ(xj |xj−1)),

ζj(xj |xj−1) =log(xj/xj−1)− (r − 1

2σ2)(tj − tj−1)

σ√tj − tj−1

• Asian Call option price

CAsian = Eg (Y ) = Eg (e−rT (X − K )+)

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Asian option delta• Delta

∆ =∂CAsian∂S(0)

dS(0)Eg (e−rT (X − K )+)

• Estimator of delta• Score

∂ log g(S(t1), . . . ,S(tm))

∂S(0)=∂ log g1(S(t1)|S(0))

∂S(0)

=ζ1(S(t1)|S(0))

S(0)σ√

It can be written as Z1S(0)σ

√t1.

• LRM estimator

e−rT (S − K )+ Z1S(0)σ

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Asian option vega• Vega

ν =∂CAsian∂σ

dσEg (e−rT (X − K )+)

• Estimator of vega• Score

∂ log g(S(t1), . . . ,S(tm))

= −m∑

+ ζj(S(tj |S(tj−1))∂ζj∂σ

It can be written as∑m

j −1σ − Zj

√tj − tj−1

• LRM estimator

e−rT (S − K )+m∑

j − 1σ

− Zj√

tj − tj−1

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Limitation of LRM• The need for explicit knowledge of the relevant probability

densities.• The variance is often very large.

Derivative:

ddθEθ(f (X )) ≈ Eθ+h(f (X ))− Eθ(f (X ))

Estimator:

f (X )gθ(X )

gθ(X )≈ f (X )

gθ+h(X )− gθ(X )

hgθ(X ).

The identity

Eθ+h(f (X )) = Eθ(f (X )gθ+h(X )/gθ(X ))

is valid only when gθ is absolutely continuous in θ.

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Variance of Vega estimator

Variance of vega estimators for an Asian option with weekly averaging as a function of number of weeks in

the average.

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Thank you!Reference:P. Glasserman, Monte Carlo Methods in Financial Engineering,Ch. 7.

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