A stochastic programming model for hedging options in a market … · A SP model for hedging options in a market with transaction costs 2 / 20 Outline 1.Introduction and overview

A SP model for hedging options in a market with transaction costs 1 / 20

A stochastic programming model for hedgingoptions in a market with transaction costs

ICSP Bergamo, 2013

Mathias BarkhagenJorgen Blomvall

Linkoping University

2013-07-09

Mathias Barkhagen, Jorgen Blomvall ICSP Bergamo 2013


Outline

1. Introduction and overview of the hedging problem

2. Estimating option implied surfaces

3. PCA of local volatility surfaces

4. Scenario generation

5. SP model

6. Results



How should the option hedging problem be studied?

Important properties that have a large impact on optimal decisions:

Transactions at market prices (bid/ask)

Significant transaction fees

Random variables not normally distributed

Behavior of option market prices

SP provides the tool to be able to solve this type of problem

The optimization model should take these aspects into account



Stylized facts of options markets

Need to model uncertainty in market prices

Stylized facts of equity market returns

Uncertainty in option prices : C(S,K ,T , r , δ, σ)

One volatility per option

Need to have knowledge of the covariance structure of theoption prices

Careful not to introduce arbitrage into the model



Overview of the hedging poblem

Equity market prices

Q

P

Index levelsInterest ratesFutures pricesOption prices

Equity derivativesprices

Interest rates

Estimate models

SP modelOption hedging

Estimate surfaces

Model selection

Monte Carlosimulation

PCA



Plots of option implied surfaces

Surfaces implied by option prices.

600800

10001200

0

0.5

1

0

50

100

150

200

250

300

350

400

Strike

Call price surface

Time−to−maturity

Cal

l pric

e

(a) Call price

600800

10001200

1400

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

Strike

Implied volatility surface


Impl

ied

vola

tility

(b) Implied volatility

600800

10001200

1400

0

0.5

1

0

0.2

0.4

0.6

0.8

1

Strike

Local volatility surface


Loca

l vol

atili

ty

(c) Local volatility

600800

10001200

0

0.5

1

0

0.005

0.01

0.015

0.02

Strike

Implied RND surface


Impl

ied

RN

D

(d) RND



Methods for surface estimation described in theliterature

The standard methods described in the literature assumes someparametric form

Some methods do not guarantee arbitrage-free surfaces

Assuming a parametric form will typically give rise to non-convexoptimization problems with many local optima

⇒ Estimated volatility surfaces can change a lot from one day tothe next

Consequences for the SP model:

The SP model will identify arbitrage and exploit it

The SP model will be confused by all the noise and the largejumps



Estimation of the local volatility surface

Pricing of derivatives via finite difference methods

Extended the methodology in Andersen and Brotherton-Ratcliffe(1997) to a non-uniform grid

The optimization problem is not convex but the optimizationalgorithm does not seem to get stuck in local optima



Discretized optimization problem (local volatilityestimation)

minv ,q,Γ,ze,zb

h(v) + 12zT

e Eeze + 12zT

b Ebzb

s. t. qTi+1 = qT

i Φi(v ,Γ), i = 0, . . . ,M

ge(q) + Feze = xe

xl ≤ gb(q) + Fbzb ≤ xu

q ≥ 0

v ≥ 0,

(1)

where h(v) is the discretized version of

h(v) =12

∫ τnτ

0

∫ xN

x1

(a′′(τ, x)

(∂2v∂τ2

)2

+ b′′(τ, x)

(∂2v∂x2

)2)dτdx . (2)



Empirical demonstration

Estimated surfaces from OMXS30 intraday data from 6 March 2013.

8001000

12001400

16001800

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

Strike

Implied volatility surface


Impl

ied

vola

tility

(a) Implied volatility

8001000

12001400

16001800

0

0.5

1

0

0.2

0.4

0.6

0.8

Strike

Local volatility surface

Time−to−maturityLo

cal v

olat

ility

(b) Local volatility



Demonstration - film of estimated surfaces (2008)



PCA of local volatilities - plots of eigen matrices

Eigen matrices of the first 4 principal components.

0.5

1

1.5

0

0.2

0.4

0.6

0.8−0.02

0

0.02

0.04

0.06

First eigen matrix, log changes

(a) PC1

0.5

1

1.5

0

0.2

0.4

0.6

0.8−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Second eigen matrix, log changes

(b) PC2

0.5

1

1.5

0

0.2

0.4

0.6

0.8−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Third eigen matrix, log changes

(c) PC3

0.5

1

1.5

0

0.2

0.4

0.6

0.8−0.05

0

0.05

0.1

0.15

Fourth eigen matrix, log changes

(d) PC4



Model for the underlying asset returns

GARCH model without jumps in the index level (GARCH(1,1)) andGARCH-Poisson:

∆Yt+1 = µ∆t + σt√

∆tz1t + ασt

√Nt ∆tz2

t , (3)

σ2t = β0 + β1σ

2t−1 + β2

(∆Yt+1 − γ∆t)2

∆t, (4)

z1t , z

2t ∼ i .i .d . N(0,1),Nt ∼ i .i .d . Po(λ∆t).

Stochastic volatility model without jumps in the index level (log SVLE)and log SVPJLE:

Yt = Yt−1 + µ∆t + (exp(ht−1)∆t)12 ε

yt + Jt , (5)

ht = ht−1 + κ(θ − ht−1)∆t + σh√

∆tεvt , (6)

εyt , ε

vt ∼ i .i .d . N(0,1), E[εyt ε

vt ] = ρ, Jt = qtξt ,

qt ∼ i .i .d . Bernoulli(λy ∆t), ξt ∼ i .i .d . N(µy , σ2y ).



Evaluation of the models

QQ plots for the 4 models.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

Standard Normal Quantiles

Qua

ntile

s of

Inpu

t Sam

ple

QQ plot GARCH(1,1)

(a) GARCH(1,1)

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5−6

−5

−4

−3

−2

−1

0

1

2

3

4

5


Qua

ntile

s of

Inpu

t Sam

ple

QQ plot GARCH−Poisson

(b) GARCH-Poisson

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5−6

−5

−4

−3

−2

−1

0

1

2

3

4

5


Qua

ntile

s of

Inpu

t Sam

ple

QQ plot log SVLE

(c) log SVLE

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5−6

−5

−4

−3

−2

−1

0

1

2

3

4

5


Qua

ntile

s of

Inpu

t Sam

ple

QQ plot log SVPJLE

(d) log SVPJLE



Scenario generation

Mean-reverting AR(1) model for the normalized principalcomponents of the local volatility surface

xk (t) = ln(Ut )V−1/2bk , k = 1, . . . ,10 (7)

xk (t + 1) =e−λk xk (t) + (1− e−λk )xk + εk (t), k = 1, . . . ,10, (8)

whereεk (t) ∼ N(0, σk ). (9)

The model parameters are estimated with regression.

Scenarios for the GARCH Poisson model generated via the inversetransform method.



Hedging model

Two stage SP model without recourse

100 scenarios for each trade date

OMXS30 index options

Hedging of portfolio of sold options with non-standard maturities

Hedging possible with all standardized options and the index

All transactions occur at the bid and ask prices

Transaction fees of 0.5% for options and 0.05% for the index

Option portfolio valued with estimated local volatility surface



SP model

The SP model that is solved for the hedging problem is given by:

minuB ,uS ,x,M

1N − 1

∑Wi<µW

(Wi − µW )2 + λ(τ TBuB + τ T

SuS)

s. t. Wi = Mer∆tk + PTi x + Ii , i = 1, . . . ,N

x = xI + uB − uS,

M = MI − PTAuB + PT

BuS,

µW =1N

N∑k=1

Wk ,

uB,uS ≥ 0.

(10)



Hedging results

Propagation of the wealth of the option portfolio. Hedging performedduring the period 18 June 2009 to 29 December 2009.

0 20 40 60 80 100 120 140−8000

−6000

−4000

−2000

0

2000

4000

(a) Wealth of the portfolio

0 20 40 60 80 100 120 140−40

−30

−20

−10

0

10

20

30

40

50

(b) Delta of the portfolio



Conclusions

Framework that realistically describes the problem of hedginga portfolio of options

Realistic description of the dynamics of market prices

Future improvements:

Increase the quality of the estimated surfaces

Choice of objective function



Thank you for your attention!


Documents

A stochastic programming model for hedging options in a market … · A SP model for hedging options in a market with transaction costs 2 / 20 Outline 1.Introduction and overview