model uncertainty, recalibration, and the emergence of ... · model uncertainty, recalibration, and...

Model Uncertainty, Recalibration, and theEmergence of Delta-Vega Hedging

Sebastian HerrmannETH Zürich

Joint work with Johannes Muhle-Karbe

Financial/Actuarial Mathematics Seminar

University of Michigan, February 12, 2016

Outline

Introduction

Problem Formulation

Results

Illustrations

Summary and Outlook

Introduction

MotivationHedging: Theory vs. Practise

I Hedging problem: Use traded assets to manage the risk associatedwith a short position in a derivative security.

I Classical mathematical finance:I Specify a stochastic model: describes stochastic behaviour of certain

financial variables in terms of deterministic input quantities, the model’sparameters.

I Compute hedging strategy according to some criterion.

I Example: delta hedging in the Black–Scholes model.

I In practise:I Parameters are frequently recalibrated to market prices of traded options.

I Out-of-model hedging: sensitivities with respect to changes in theseparameters are hedged by trading in options.

I Example: Recalibration of the Black–Scholes volatility parameter and vegahedging.

I Logically inconsistent.1 / 21

MotivationOut-of-model hedging

Rebonato ’05:

“Needless to say, out-of-model hedging ison conceptually rather shaky ground: if thevolatility is deterministic and perfectly known,as many models used to arrive at the priceassume it to be, there would be no need toundertake vega hedging. Furthermore, cal-culating the vega statistics means estimatingthe dependence on changes in volatility of aprice that has been arrived at assuming theself-same volatility to be both deterministic andperfectly known. Despite these logical prob-lems, the adoption of out-of-model hedgingin general, and of vega hedging in partic-ular, is universal in the complex-derivativestrading community.”

2 / 21

This paperProblem

I Task: Hedging of a non-traded option by dynamic trading in three liquidassets: stock, bond, and a “call” on the stock.

I Preferences: Moderate risk and uncertainty aversion.I Plausible models: essentially all continuous local martingale dynamics for

stock and call.

I Reference model (“best prediction”): recalibrated Black–Scholes model.

I Uncertainty aversion: Plausible models are weighted according to theirdistance to the reference model.

I Risk aversion: Utility function.

I Goal: Find almost optimal hedging strategies and indifference prices.

3 / 21

This paperResults

I To obtain explicit formulas, study limit for small uncertainty aversion.

I Delta hedging is not asymptotically optimal in general.

I Delta-vega hedging is asymptotically optimal.

I Indifference price corrections:I Determined by disparity between vegas and second-order greeks

(gammas, vannas, volgas) of the non-traded option and the call.

I Averaged over time and states.

I Exposures of delta-vega hedged position with respect to spot volatility andvolatility of implied volatility can be read off.

4 / 21

Problem Formulation

Preferences

Consider an agent with the following beliefs:1. Model uncertainty: True dynamics of stock and the liquidly traded

option are not known precisely.

2. Market efficiency: The market prices liquidly traded options correctly.

3. Moderate risk and uncertainty aversion:I My reference model is my best prediction for the true dynamics (but

could be incorrect).I I take alternative models less seriously the more they deviate from my

reference model.I I am risk-averse.

How to describe the preferences of such an agent?

5 / 21

Variational preferences

I Maccheroni/Marinacci/Rustichini ’06: Decision-theoretic axiomssuggest representation

infP

(EP [U(Y )] + α(P )

)I Utility function U describes risk aversion in a given model.

I Penalty functional α describes uncertainty aversion.

I Classical approach: infinite penalty for all but one model P .

I Worst-case approach: no penalty for a class P of plausible models;infinite penalty otherwise.[Gilboa/Schmeidler ’89; Avellaneda/Levy/Paras ’95; Lyons ’95]

I “Smooth” alternatives?

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Moderate uncertainty aversion?

Multiplier preferences of Hansen/Sargent ’01:I Models penalised by relative entropy with respect to a reference model.

I No strict line between “possible” and “impossible” models.

I Penalty is only finite for absolutely continuous measures. Notapplicable to volatility uncertainty.

This paper:I Choose similar penalty.

7 / 21

General approach

1. Choose a (big) class of plausible models.

2. Choose a reference model (“best prediction”).

3. Choose a penalty functional, e.g., some “distance” to the referencemodel. Describes how seriously you take other models.

4. For each hedging strategy, compute its performance in the above sense(variational preferences).

5. Find an optimal strategy for this criterion.

8 / 21

ModelsI Dynamic trading in stock and a call option: model joint dynamics of

stock S and implied volatility Σ of the call.]Lyons ’97, Schönbucher ’99, . . . ]

I Plausible models: Probability measures P under which (S,Σ) hasdynamics of the form

dSt = StσPt dW 0

t ,

dΣt = νPt dt+ ηPt dW 0t +

√ξPt dW 1

t ,

and call price Ct := C(t, St,Σt) is drift-less.

I Reference model: Recalibrated Black–ScholesI corresponds to νP = ηP = ξP = 0 and σP = Σ.

“The future implied volatility stays at its currently observed level.”

I benchmark case; general approach extends to more realistic referencemodels.

9 / 21

Hedging problem

I Hedge European option with payoff V(ST ).

I Time-t reference value: Vt = V(t, St,Σt).

I P&L with dynamic trading in stock and call:

Y θ,φT = V0 +

∫ T

0

θu dSu +

∫ T

0

φu dCu − V(ST )

I Hedging problem:

v(ψ) = supθ,φ

infP

(EP

[U(Y θ,φT )

]+ αψ(P )

)I Penalty functional αψ(P ) ≥ 0: describes plausibility of a model P , zero

for BS model and positive for alternative models.

10 / 21

Penalty functional

I Penalise “mean-square” deviations from the reference BS model:

αψ(P ) =1

2ψEP

[∫ T

0

U ′(Y θ,φt ){

(νPt )2 + (σPt − Σt)2 + (ηPt )2 + (ξPt )2

}dt

]I Recall:

νP : drift of implied volatilityσP : spot volatilityηP : correlated volatility of implied volatilityξP : uncorrelated squared volatility of implied volatility

I ψ > 0 measures magnitude of uncertainty aversion.

I To obtain explicit results: study limit ψ ↓ 0.

11 / 21

Results

Almost optimality of delta-vega hedging

Under regularity and integrability assumptions. . .I Value has asymptotic expansion of the form

v(ψ) = U(0)− U ′(0)w̃ψ + o(ψ) as ψ ↓ 0.

I Dynamically recalibrated delta-vega hedge is optimal at theleading-order O(ψ):

I # of calls φ? = VΣ/CΣ neutralises vega.I # of shares θ? = VS − φ?CS neutralises delta.

I Correction term w̃ is determined by:

net gamma = φ?CSS − VSSnet vanna = φ?CSΣ − VSΣ

net volga = φ?CΣΣ − VΣΣ

12 / 21

Value expansion

I Recall value expansion: v(ψ) = U(0)− U ′(0)w̃ψ + o(ψ).

I w̃ has probabilistic representation:

w̃ =1

2E

[∫ T

0

g̃(t, St,Σ0) dt

]

g̃(t, S,Σ) = ΣS2 (VSS − φ?CSS) σ̃ –(net gamma)× spot vol deviation+ΣS (VSΣ − φ?CSΣ) η̃ –(net vanna)× correlated vol

+ 12

(VΣΣ − φ?CΣΣ) ξ̃ –(net volga)× uncorrelated vol2.

I σ̃, η̃, ξ̃: deviations of “worst-case model” Pψ from reference BS model:

spot vol correlated vol of IV uncorrelated vol2 of IVσP

ψ ≈ Σ + σ̃ψ ηPψ ≈ η̃ψ ξP

ψ ≈ ξ̃ψ

13 / 21

Exposures

net Greek exposure if net Greek < 0 (“short”)

vega 0 —

gamma φ?CSS − VSS spot volatility above IV

vanna φ?CSΣ − VSΣ positive correlation of IV with stock

volga φ?CΣΣ − VΣΣ positive volatility of IV

14 / 21

Exposures

net Greek exposure if net Greek > 0 (“long”)

vega 0 —

gamma φ?CSS − VSS spot volatility below IV

vanna φ?CSΣ − VSΣ negative correlation of IV with stock

volga φ?CΣΣ − VΣΣ —

14 / 21

Indifference prices

I Indifference ask price for option V :

pa(ψ) = V + w̃ψ + o(ψ)

I w̃ψ is “compensation” for model uncertainty.I Independent of the utility function, like the value V in the complete

reference model.

I Sanity check:I w̃ = 0 if C is a call and V is a (multiple of a) put with matching strikes and

maturities.

I Model-free hedge by put-call parity.

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Illustrations

Log contract

I Payoff: logST ′ .

I Can be replicated statically by a continuum of calls and puts with thesame maturity.

I Simple BS value and Greeks: C(t, S,Σ) = logS − 12Σ2(T ′ − t),

vega gamma vanna volga

CΣ = −Σ(T ′ − t), CSS = −1/S2, CSΣ = 0, CΣΣ = −(T ′ − t).

I Good proxy for a range of traded calls/puts with different strikes, butsame maturity.

We assume now that the liquidly tradedoption is a log contract with maturity T ′ ≥ T .

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Up-and-out call

I Call option (strike K) which becomes worthless if the stock hits abarrier B > K at any time before maturity T .

60 80 100 1200

5

10

15

20

stock price

payoff/v

alu

e

payoff

1 year

1 month

I Very high (absolute) Greeks in the vicinity of the barrier and close tomaturity.

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Hedging an up-and-out call

net cash gamma net cash vanna net volga

60 80 100 120−200

−100

0

100

200

stock price

ne

t ca

sh

ga

mm

a

60 80 100 120−200

−100

0

100

200

stock price

ne

t ca

sh

va

nn

a

60 80 100 120−200

−100

0

100

200

stock price

ne

t vo

lga

60 80 100 1200

0.25

0.5

0.75

1

1.25

1.5

stock price

op

tio

n p

rice

s

BS value

bid/askI Hedge an up-and-out

call with with strike 100 ,barrier 120, andmaturity 1y.

I Maturity of log contract:T ′ =1y

I All plots at t = 0.

18 / 21

Hedging a (plain-vanilla) call


60 80 100 120 140−200

−100

0

100

200

stock price

ne

t ca

sh

ga

mm

a

60 80 100 120 140−200

−100

0

100

200

stock price

ne

t ca

sh

va

nn

a

60 80 100 120 140−200

−100

0

100

200

stock price

ne

t vo

lga

60 80 100 120 1400

0.05

0.1

stock price

bid

−ask s

pre

ad t = 0 I Hedge call with strike

100 and maturity T =1y.

I Maturity of log contract:T ′ =1y

(solid)T ′ =1y 3m (dashed)

19 / 21



60 80 100 120 140−200

−100

0

100

200

stock price

ne

t ca

sh

ga

mm

a

60 80 100 120 140−200

−100

0

100

200

stock price

ne

t ca

sh

va

nn

a

60 80 100 120 140−200

−100

0

100

200

stock price

ne

t vo

lga

60 80 100 120 1400

0.05

0.1

stock price

bid

−ask s

pre

ad t = 0 I Hedge call with strike


I Maturity of log contract:T ′ =1y (solid)T ′ =1y 3m (dashed)

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60 80 100 120 140−200

−100

0

100

200

stock price

ne

t ca

sh

ga

mm

a

60 80 100 120 140−200

−100

0

100

200

stock price

ne

t ca

sh

va

nn

a

60 80 100 120 140−200

−100

0

100

200

stock price

ne

t vo

lga

60 80 100 120 1400

0.05

0.1

stock price

bid

−ask s

pre

ad t = 0.2 I Hedge call with strike



19 / 21



60 80 100 120 140−200

−100

0

100

200

stock price

ne

t ca

sh

ga

mm

a

60 80 100 120 140−200

−100

0

100

200

stock price

ne

t ca

sh

va

nn

a

60 80 100 120 140−200

−100

0

100

200

stock price

ne

t vo

lga

60 80 100 120 1400

0.05

0.1

stock price

bid

−ask s

pre




19 / 21

Summary and Outlook

Summary

I Preferences for moderate uncertainty aversion: models are penalisedaccording to their distance from a reference model.

I Reference Black–Scholes model is dynamically recalibrated to themarket price of the traded option.

I Impact of uncertainty depends on disparity between the vegas,gammas, vannas, and volgas of the non-traded and traded options.

I Delta-vega hedging is leading-order optimal.

20 / 21

Outlook

I More complex reference models, e.g., stochastic volatility.

I Multiple traded options:I Different maturities: use time-dependent, deterministic volatility−→ “vega-bucketing”.

I Different strikes: BS is over-determined, need, e.g., stochastic volatilityreference model.

I Transaction costs:I Larger spreads for liquid options than for stocks.I Delta-vega hedge should be “buffered” with a suitable no-trade region.I Computation is difficult even without uncertainty (Goodman/Ostrov ’11).

I Hedging instruments change over time:I Always use options close to ATM for hedging.I Tractable model?

21 / 21

Selected references

I M. Avellaneda, C. Friedman, R. Holmes, andD. Samperi.Calibrating volatility surfaces via relative-entropyminimization.Appl. Math. Finance, 4(1):37–64, 1997.

I M. Avellaneda, A. Levy, and A. Paras.Pricing and hedging derivative securities inmarkets with uncertain volatilities.Appl. Math. Finance, 2(2):73–88, 1995.

I M. Avellaneda and A. Paras.Managing the volatility risk of portfolios ofderivative securities: the Lagrangian uncertainvolatility model.Appl. Math. Finance, 3(1):21–52, 1996.

I R. Cont.Model uncertainty and its impact on the pricingof derivative instruments.Math. Finance, 16(3):519–547, 2006.

I J.-P. Fouque and B. Ren.Approximation for option prices under uncertainvolatility.SIAM J. Financ. Math., 5(1):260–383, 2014.

I S. Herrmann and J. Muhle-Karbe.Model uncertainty, dynamic recalibration, andthe emergence of delta-vega hedging.Preprint SSRN:2694718, 2015.

I S. Herrmann, J. Muhle-Karbe, and F. Seifried.Hedging with small uncertainty aversion.Preprint SSRN:2625965, 2015.

I N. El Karoui, M. Jeanblanc, and S. Shreve.Robustness of the Black and Scholes formula.Math. Finance, 8(2):93–126, 1998.

I T. Lyons.Uncertain volatility and the risk-free synthesis ofderivatives.Appl. Math. Finance, 2(2):117–133, 1995.

I F. Maccheroni, M. Marinacci, and A. Rustichini.Ambiguity aversion, robustness, and thevariational representation of preferences.Econometrica, 74(6):1447–1498, 2006.

http://ssrn.com/abstract=2694718

http://ssrn.com/abstract=2625965

Sketch of the proofHeuristics

I Write down the HJBI equation for the stochastic differential game:

vt + sup(θ,φ)

infζ:...

Lψ,(θ,φ),ζv = 0.

ζ = (ν, σ, η, ξ) describes local dynamics of (S,Σ) and must satisfy anonlinear drift condition and ξ ≥ 0.

I Plug in an appropriate ansatz for the asymptotic solution. Expand andsort by powers of the small parameter ψ.

I Min-max problem at the order O(ψ) decouples into a nonlinearlyconstrained quadratic minimisation problem and a quadraticmaximisation problem.

I Maximisation problem: optimised by delta-vega hedge.

I Minimisation problem: nonlinear constraint can be approximated bylinear one; explicit minimiser leads to dynamics for “worst-case model”.

I Insert optimisers back to obtain a PDE for the cash equivalent w̃.22 / 21

Sketch of the proofVerification

I Set up hedging problem on canonical space for continuous paths with alarge class of mutually singular measures as plausible models.

I Analytic part:I Use direct estimates and Lagrange duality to show that candidate solution

and controls solve HJBI equation asymptotically.

I Probabilistic part:I Adapt classical verification arguments for stochastic differential games to

asymptotic setting.I Candidate controls must be modified to satisfy the nonlinear drift condition.

23 / 21

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