Gradient Estimation and Mountain Range Options

Introduction

Gradient Estimation

Mountain Range Options

Gradient Estimation and Mountain Range Options Andrew O. Hall

Michael Fu

Department of Mathematical Sciences and Network Science Center West Point Robert H. Smith School of Business University of Maryland

SIAM FME 2010

Summary

Introduction

Gradient Estimation

Outline I 1

Introduction

2

Gradient Estimation Indirect Methods Direct Methods Vega for European Call

3

Mountain Range Options Everest Atlas Altiplano/Annapurna

4

Summary

Mountain Range Options

Summary

Introduction

Gradient Estimation

Mountain Range Options

Summary

Monte Carlo Simulation

When hedging, in addition to requiring price information on traded assets, financial engineers or risk managers require estimates of the sensitivity to underliers and model parameters, “the Greeks". In Monte Carlo simulation, the process of finding “the Greeks", calculating calculus derivatives, is referred to as Gradient Estimation.

Introduction

Gradient Estimation

Mountain Range Options

Summary

Gradient Estimation

We begin with J(θ), a performance measure dependent upon θ. Simulation is an effective technique when J(θ) must be estimated, so we calculate a large number of independent trials to find a good estimate J = E[L], where L is the sample performance. Gradient Estimation seeks to estimate the derivative of the sample performance with respect to one of the parameters of the system dJ dE[L] ≈ , dθ dθ in order to estimate the derivative of the performance measure.

Introduction

Gradient Estimation

Mountain Range Options

Gradient Estimation Techniques

Finite Differences Infinitesimal Perturbation Analysis (IPA) or pathwise method Likelihood Ratio or Score Method Weak derivatives

Summary

Introduction

Gradient Estimation

Mountain Range Options

Indirect Methods

Finite Difference Methods

One sided forward difference ˆ + ci ei ) − J(θ) ˆ J(θ ci Two sided symmetric difference ˆ + ci ei ) − J(θ ˆ − ci ei ) J(θ 2ci where ei is the unit vector in the ith direction, and ci is the perturbation in the ith direction.

Summary

Introduction

Gradient Estimation

Mountain Range Options

Indirect Methods

Simultaneous Perturbations

Two similiar estimators are the simultaneous perturbations estimator ˆ + c∆) − J(θ ˆ − c∆) J(θ 2ci ∆i and the random directions gradient estimator ˆ + c∆) − J(θ ˆ − c∆))∆i (J(θ 2ci where ∆ is a d-dimensional vector of perturbations.

Summary

Introduction

Gradient Estimation

Mountain Range Options

Summary

Direct Methods

Beginning with J(θ) = E[L(θ)] = E[L(X1 , X2 , . . . , XT )] we examine the dependance on the parameter θ and categorize as either having either sample or measure dependency.

Introduction

Gradient Estimation

Mountain Range Options

Direct Methods

θ Dependance Where does θ appear: Z E[L(X )] =

Z ydFL (y ) =

L(x)dFX (x)

Does the θ dependence occur in the input random variable distribution (measure) of the input random variable FX Z

1

E[L(X )] =

L(X (θ; u))du Z0 ∞ E[L(X )] = L(x)f (x; θ)dx −∞

Summary

Introduction

Gradient Estimation

Mountain Range Options

Summary

Direct Methods

Pathwise Differentiation

The first of two notions of the derivative which is needed for pathwise gradient estimation is dE[L(X )] = dθ

Z 0

1

dL(X (θ; u)) du dθ

were we take the derivative of a random variable defined by X (θ + ∆θ, ω) − X (θ, ω) dX (θ, ω) = lim dθ ∆θ ∆θ→0 where the family of random variables parameterized by θ are defined on a common probability space such that X (θ) ∼ F (·; θ) s.t. ∀θ ∈ Θ, X (θ) is differentiable w.p.1.

Introduction

Gradient Estimation

Mountain Range Options

Direct Methods

Pathwise Estimator

Assuming the interchange of differentiation and expectation is permissible, dE[L(X )] dθ

Z

dL(X (θ; u)) du dθ

1

dL X (θ) du dX dθ

0

Z = 0

and the estimator is

1

=

dL X (θ) . dX dθ

Summary

Introduction

Gradient Estimation

Mountain Range Options

Direct Methods

Distributional Differentiation

The weak derivative is needed for both LR/SF and weak derivatives gradient estimators Z ∞ dE[L(X )] df (x; θ) = L(x) dx dθ dθ −∞ where

df (x;θ) dθ

is a weak derivative of a measure, df (x; θ) = c(θ)(f (2) (x; θ) − f (1) (x; θ)). dθ

where the two measures are traditionally the Hahn-Jordan decomposition.

Summary

Introduction

Gradient Estimation

Mountain Range Options

Direct Methods

Likelihood Ratio / Score Function

The first of two methods making use of distributional differentiation is the LR/SF method. Z ∞ dE[L(X )] df (x; θ) = dx L(x) dθ dθ −∞ Z ∞ d ln f (x; θ) = L(x) f (x)dx dθ −∞ and the estimator is L(x)

d ln f (x; θ) . dθ

Summary

Introduction

Gradient Estimation

Mountain Range Options

Direct Methods

Weak Derivatives Alternatively, by representing   df (x; θ) = c(θ) f (2) (x; θ) − f (1) (x; θ) dθ we derive the form of the WD estimator Z ∞ dE[L(X )] df (x; θ) = L(x) dx dθ dθ Z−∞   ∞ = L(x)c(θ) f (2) (x; θ) − f (1) (x; θ) dx −∞ Z ∞ = c(θ) L(x)(f (2) (x; θ)dx −∞  Z ∞ (1) − L(x)f (x; θ))dx . −∞

Summary

Introduction

Gradient Estimation

Mountain Range Options

Summary

Vega for European Call

IPA Example

We calculate vega using the IPA for European Call JT = e−rt (ST − K )+ Assuming lognormal RV ST and Z representing a standard normal RV ST dST dσ

= S0 e(r −δ−σ

2 /2)T +σ

√

TZ

(1)

√

= ST (−σT + T Z )       ST ST 1 = ln − r − δ + σ2 T σ S0 2

(2)

Introduction

Gradient Estimation

Mountain Range Options

Vega for European Call

IPA Example: Cont

Since changes in ST only change JT if ST ≥ K dJT = e−rT 1ST ≥K dST Combining (2) and (3) result in the IPA estimator dJT dσ

dJT dST dST dσ dST 1 = e−rT dσ ST ≥K =

Summary

Introduction

Gradient Estimation

Mountain Range Options

Summary

Vega for European Call

Likelihood Ratio Example

from (1) we have g(x) =

1 1 √ n(d(x)) x ≥ 0, where n(z) = √ e−z/2 xσ T 2π ln x/S0 − (r − δ − σ 2 /2)T √ and d(x) = σ T

so we can write Z E[JT ] = 0

∞

e−rt (x − K )+ g(x)dx.

(3)

Introduction

Gradient Estimation

Mountain Range Options

Vega for European Call

Likelihood Ratio Example: Cont From 3, assuming the order of integration and expectation can be interchanged Z ∞ dJT dg(x) = dx. e−rT (x − K )+ dσ dσ 0 Using the identity dJT = dσ

dg dσ /g

Z

∞

=

d ln g dσ

e−rT (x − K )+

0

d ln g(x) g(x)dx. dσ

so the LR estimator is e−rT (x − K )+ .

d ln g(x) dσ

Summary

Introduction

Gradient Estimation

Mountain Range Options

Mountain Range Options

Exotic Derivatives Traded over the counter Combine characteristics of other options Continuous and discontinuous payoff functions Examples: Everest Atlas Altiplano/Annapurna Himalaya

Summary

Introduction

Gradient Estimation

Mountain Range Options

Summary

Everest

Everest

Mount Everest is the highest point on earth and in the Himalayan Mountain range. Curiously, the Everest option is the pay-out on the worst performer in a basket, normally of 10-25 stocks, with 10-15 year maturity. Given n stocks S1 , S2 , . . . , Sn in a basket, the payoff for an Everest option is ! SiT . JT = min i=1...n Si0

Introduction

Gradient Estimation

Mountain Range Options

Summary

Everest

Everest IPA Estimator

The pathwise gradient estimator is dJT dθ

n X dJT dSiT = dSiT dθ i=1

dJT dSiT

=

1 1 T T Si0 Si ≤Sj ,∀j6=i

(4) (5)

a sufficient condition for the interchange of differentiation and expectation is that the payoff is a continuous function with respect to the parameter.

Introduction

Gradient Estimation

Mountain Range Options

Everest

Everest IPA The IPA for rho and theta: dJT dr

n X 1 dSiT = 1 T ST S 0 dr Si0 ≤ j0 ,∀j6=i i=1 i S i

=

n X i=1

dJT dT

T

S j

SiT 1 T ST . Si0 Si0 ≤ j0 ,∀j6=i S i

S j

n X 1 dSiT = 1 T ST S 0 dT Si0 ≤ j0 ,∀j6=i i=1 i S i

S j

 n  X 1 2 AZ SiT = r − σi + √ 1 . 0 SiT SjT 2 S T ≤ ,∀j6 = i i 0 0 i=1 S i

S j

Summary

Introduction

Gradient Estimation

Mountain Range Options

Everest

Everest LR/SF Estimator

The LR/SF gradient estimator is ! SiT d ln f (S1T , S2T , . . . SnT ; θ) mini=1...n dθ Si0 and in the independence case, ! SiT d ln f (S1T ; θ) d ln f (S2T ; θ) mini=1...n + · · · + . dθ dθ Si0

Summary

Introduction

Gradient Estimation

Mountain Range Options

Everest

Everest LR/SF The score function is d ln f (S1T , S2T , . . . SnT ; θ) (St − µ(θ))T Σ−1 dµ(θ) √ = . dθ dθ T For vega, rho and Theta, with µi = log Si0 + (r − 21 kΣi k2 )T , we obtain dµi dσ dµi dr dµi dT

= −kΣi kT , = T = r − kΣi k2

Summary

Introduction

Gradient Estimation

Mountain Range Options

Summary

Everest

Illustrative Results: Comparison of Everest Gradient Estimators

IPA has smaller standard errors, but requires continuity of performance measures

LR/SF has higher standard errors, but is applicable for a larger class of performance measures

Introduction

Gradient Estimation

Mountain Range Options

Everest

Illustrative Results: Step Sizes for Indirect Estimators Estimates calculated for Si = 0.001

Indirect estimates for varying Si

Summary

Introduction

Gradient Estimation

Mountain Range Options

Atlas

Atlas

The Atlas option removes a fixed number of stocks from the basket with n1 and n2 detailing the number of stocks to be removed from the minimum and maximum of the ordering. Given two numbers n1 , n2 where n1 + n2 < n, and n stocks S1 , S2 , . . . , Sn in a basket with strike K , the payoff for the Atlas option is +  n−n T X2 R(j) − K . JT =  n − (n1 + n2 ) j=1+n1

Summary

Introduction

Gradient Estimation

Mountain Range Options

Summary

Atlas

Atlas IPA

IPA estimator for rho and theta T dS(i)

n

X dJT 1 = 1 0 Pn−n2 dr (n − (n1 + n2 ))S(i) i=1

dJT = dT

n X i=1

RT (j) j=1+n1 (n−(n1 +n2 )) >K

1 1 0 Pn−n2 (n − (n1 + n2 ))S(i) j=1+n

dr T dS(i)

RT (j) >K (n−(n 1 1 +n2 ))

dT

11+n1 ≤i≤n−n2 . 11+n1 ≤i≤n−n2 .

Introduction

Gradient Estimation

Mountain Range Options

Summary

Altiplano/Annapurna

The Altiplano/Annapurna pays a coupon if none of the stocks in the basket hits a certain level before expiration. In the event that one or more stocks hits the critical level, the Altiplano/Annapurna is a call option on the basket of stocks. The limit could be either a floor or a ceiling for the stocks in the basket. Given n stocks S1 , S2 , . . . , Sn in a basket, a coupon amount, C, a limit L, and strike K, with barrier period beginning at t1 and ending at t2 , the P&L for an Altiplano option is Sit ≤ L ∀i, t ∈ {t1 , t2 } Si0  + n X SjT  − K  otherwise. 0 S j j=1

C if max

Introduction

Gradient Estimation

Mountain Range Options

Altiplano/Annapurna

Altiplano LR/SF Estimator

The LR/SF gradient estimator is  C1 

n X SjT



Sj0

j=1

max

St i ≤L S0 i

+ ∀i,t∈{t1 ,t2 }

+



− K 1

max

×

St i >L S0 i

∀i,t∈{t1 ,t2 }



d ln f (S1T , S2T , . . . SnT ; θ) dθ

Summary

Introduction

Gradient Estimation

Mountain Range Options

Future Work

Implement additional multi-stock option and multi-index options Implement weak derivative gradient estimator Implement additional models of stock price evolution and correlation Examine hedging strategies

Summary

Introduction

Gradient Estimation

Mountain Range Options

Summary

References

Fu and Hu, Sensitivity Analysis for Monte Carlo Simulation of Options Prices, (Probability in the Engineering and Informational Sciences 1995) Broadie and Glasserman, Estimating Security Price Derivatives Using Simulation (Management Science 1996) Fu, Chapter 19: Gradient Estimation, Handbooks in OR & MS, Vol. 13: Simulation Fu, What you should know about Simulation and Derivatives (Naval Research Logistics 2008) Glasserman, Monte Carlo Methods in Financial Engineering (2004)

Introduction

Gradient Estimation

Mountain Range Options

Additional Literature

Chen and Fu, Efficient Sensitivity Analysis for Mortgage-Backed Securities, 2001 Chen and Fu, Hedging Beyond Duration and Convexity, WinterSim 2002 M.B. Giles, Monte Carlo evaluation of sensitivities in computation finance, 2007 Giles and Glasserman, Smoking Adjoints: Fast Monte Carlo Greeks, Risk 2006 Glasserman and Liu, Estimating Greeks in Simulating Lévy-Driven Models, 2008 Chen and Glasserman, Sensitivity estimates for portfolio credit derivatives using Monte Carlo

Summary