simple robust hedging with nearby...

Simple Robust Hedging with Nearby Contracts

Liuren Wu and Jingyi Zhu

Baruch College and University of Utah

April 29, 2011Fourth Annual Triple Crown Conference

Liuren Wu (Baruch) Robust Hedging with Nearby Contracts 4/29/2011 1 / 18

Archimedes-style hedging

Give me a lever long enough and a fulcrum on which to place it, and Ishall move the world.— Archimedes, Mathematician and inventor of ancient Greece,287-212BC

In hedging derivatives risk, many think like Archimedes, by making strong,idealistic assumptions on the security dynamics and trading environment.

Black-Merton-Scholes (1973) introduce the dynamic hedging concept,by assuming

The underlying security follows a one-factor diffusion process.One can rebalance the hedge portfolio continuously.

Carr and Wu (2002) propose a static hedge on vanilla options,by assuming

The underlying security follows a one-factor Markovian process.One can deploy an infinite number of short-term options across thewhole continuum of strikes.

Both results are ground breaking, but both rely on strong, Archimedes-styleassumptions.


In reality, the lever is not quite as long

Transaction cost is a fact of life.Both continuous rebalancing and transacting on a continuum of options leadto financial ruin.

Discretization is a must.

One does not know the exact dynamics of the underlying security:(i) how many risk sources and (ii) what the risk exposures are.

Riskfree hedge under model A is not riskfree under model B.Neither hedge is riskfree in reality.

Archimedes (or an Archimedes-style hedger) does not care. All he wants isto guarantee that he’s absolutely right, under his own terms.

Practitioners do care, as they want to be approximately right (so that theydo not lose their shirts), under all conceivable conditions.


Practitioners’ remedy

Use BMS model to perform both delta and vega hedge.

Delta is balanced daily.Vega and stress risk are managed opportunistically.

Acknowledge that this vega is not that vega.

Vega at different strike and maturity ranges are different types of vegas.

Example I: A portfolio with long $10 million vega at 5-yearat-the-money and short $10 million vega at 4-year at-the-money istreated as relatively vega-risk free — vega is netted.

Example II: A portfolio with long $10 million vega at 5-yearat-the-money and short $10 million vega at 1-month 10-delta put istreated as having significant ($20 million) vega exposure — vega isadded.

From academic perspective, stochastic volatility can be generated fromdiffusion risk or jump risk, market risk or credit risk, short-term shockor long-term trend shift...


Our approach: Hedging with nearby contracts

Academics often worry about the exact risk exposure calculation: Should thedelta be calculated under a local vol model or a model with jump?

With the right exposure estimate, one can pick any contracts toneutralize the risk by solving a system of equations.

Traders are less pedantic about the calculation, but intuitively realize that

Achieving vega (duration, etc) neutrality with nearby contracts is saferthan using contracts that are far away (in maturity and/or strike).

We do not attempt to eliminate risks completely under a hypothetical model.

We devise a simple robust hedging strategy that limits loss under all possiblescenarios, regardless of model assumptions.

We do not assume a model, nor do we calculate risk exposures.

We design the hedging portfolio based on affinities in contractcharacteristics (such as strike and maturity), not risk exposures (suchas delta, vega).


Starting example: Hedge with a maturity-strike triangle

We hedge a target vanilla call option at strike K and expiry T , C (K ,T ),with three nearby call option contracts:

Using more than three contracts to hedge is not practical giventransaction cost — Forget about a continuum.

Kd < Kc < Ku, with K ∈ (Kd ,Ku) and ideally Kc = K when available.No theoretical constraints on the maturity choice, but we apply somepractical considerations:

Since often fewer maturities are available than strikes, we focus on twomaturities instead of three, with Kc at one maturity Tc and (Kd ,Ku) atanother maturity To to form a maturity-strike triangle.

It would be nice to sandwich the target maturity T ∈ (Tc ,To), butsince short-term options tend to be more liquid than long-term options,we might need to choose (Tc ,To) < T .

t Tc To T

Kd

K

Ku

Maturity

Stri

ke

t To Tc T

Kd

K

Ku

Maturity

Stri

ke

t Th T

Kd

K

Ku

Maturity

Stri

ke


Assumptions

We assume that there are a finite number of (at least 3) options for us tochoose to form the hedge portfolio.

We use these options (one is enough) to compute a local volatility,

σ2(K ,To) =2CT (K ,To)

CKK (K ,To).

The concept of local volatility is originally developed by Dupire (94)under a one-factor diffusion setting.

Positive local volatility exists under a much more general setting.

We are not concerned with the dynamics, but rather try to obtain astable estimate of the relation via interpolations and extrapolations:

Details ...

What data are sparse, super-smooth.


Deriving portfolio weights on the maturity-strike triangle

The portfolio weights are obtained via the following steps:

1 Taylor expand both target and hedge options around (K ,To) to first order inT and second-order in K :

C (K ,T ) ≈ C + CT (T − To),C (Kd ,To) ≈ C + CK (Kd − K ) + 1

2CKK (Kd − K )2,C (Ku,To) ≈ C + CK (Ku − K ) + 1

2CKK (Ku − K )2,C (Kc ,Tc) ≈ C + CK (Kc − K ) + CT (K ,To)(Tc − To) + 1

2CKK (Kc − K )2.

2 Replace CKK with CT via the local volatility definition.

3 Choose the three weights (wd ,wc ,wu) to match coefficients on the threeterms: C , CK , and CT .

More instruments can be used to match higher-order expansion terms.Via the local volatility linkage between CT and CKK , we can use 3instruments to match 4 expansion terms, allowing us to go secondorder in strike.

⇒ By matching the coefficients of each term, we do not need to know the valuesor their derivatives (C ,CK ,CT ), which would be model dependent.


The triangle portfolio weights

When the strike spacing are symmetric with Kc = K and Ku −K = K −Kd ,the weights on the isosceles triangle are,

wc =d2 − 1

d2 + α, wd = wu =

1

2(1− wc).

where

d = (|Ko−K |)Kσ(K ,To)

√T−To

is a standardized strike spacing measure.

α = To−Tc

T−Tois a relative maturity spacing measure.

The portfolio weights are approximately static.


From expansion errors to hedging errors

Taylor expansion errors increase with the expansion distance.

Strike distance can be chosen small, but maturity distance is likely large.Hence, we potentially have large expansion errors on maturity.

Hedging errors can be small even if expansion errors are large.

Expansion errors in the target options partially cancel with expansionerrors in the hedge portfolio.

The portfolio weights do not depend on expansion points.

When many strikes are available, one can choose the strike spacingjudiciously to further increase the expansion error cancelation.


Monte Carlo analysis

Simulate four model dynamics.

BS: dSt/St = µdt + σdWt ,MJ: dSt/St = µdt + σdWt +

∫R0(ex − 1) (ν(dx , dt)− λn(x)dxdt) ,

HV: dSt/St = µdt +√

vtdWt ,HW: dSt/St = µdt +

√vtdWt +

∫R0 (ν(dx , dt)− vtλ0n(x)dxdt) ,

dvt = κ (θ − vt) dt − ω√vtdZt , E [dZtdWt ] = ρdt,

Parameters are set to averages of daily calibration results to SPX options.

Maturities and strikes are set similar to that for SPX options.

Maturities: 1,2,3,6,12 months.Strikes are spaced around spot at $1, $1.5, $2, $2.5, and $3, resp.

Perform hedging exercises of different target options with differentmaturity-strike combinations to learn

How does the strike spacing choice affect the hedging performanceunder different model environments?

How does the hedging performance compare with daily delta hedging?


Hedging performance under different strike spacing

BS

0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

1.4

d

RM

SE

(1,2,3)(1,2,6)(1,2,12)

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

dR

MS

E

(1,6,12)(1,3,6)(1,3,12)

0.8 1 1.2 1.4 1.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

d

RM

SE

(2,6,12)(3,6,12)(2,3,6)(2,3,12)

There are 20 more maturity-combinations for each model... and 3 other models...

Choosing the appropriate strike spacing can further reduce hedging error.

The following regression seems to describe the optimal strike choice well:d∗ = a + b α + c(To/T ) + e.

More analytical study can be useful to understand the underlying mechanismof dependence ...


Hedging performance comparison

Th – Average triangle maturityBS MJ

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

T/Th

RM

SE

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

T/Th

RM

SE

HV HW

0 2 4 6 8 10 120

0.2

0.4

0.6

T/Th

RM

SE

0 2 4 6 8 10 120

0.2

0.4

0.6

T/Th

RM

SE

The closer the target is to thetriangle, the better the performance.

Daily delta hedging withunderlying futures

T BS MJ HV HW

2 0.21 0.85 0.88 1.083 0.17 0.76 0.95 1.076 0.12 0.60 0.89 1.02

12 0.08 0.43 0.68 0.86

BS: Triangles withT/Th < 3 perform better.

MJ,HV,HW: All 30triangles perform betterthan delta hedging.

Not always nearby: Thetarget-hedge maturity ratiocan be 12 times.


Hedging error sample paths

Compare the best delta (top panels) with the worst triangle (bottom panels):BS MJ HV HW

0 5 10 15 20 25 30−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Days

He

dg

ing

Err

or

0 5 10 15 20 25 30−6

−5

−4

−3

−2

−1

0

1

Days

He

dg

ing

Err

or

0 5 10 15 20 25 30−5

−4

−3

−2

−1

0

1

2

Days Forward

He

dg

ing

Err

or

0 5 10 15 20 25 30−10

−8

−6

−4

−2

0

2

Days Forward

He

dg

ing

Err

or

0 5 10 15 20 25 30−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Days Forward

He

dg

ing

Err

or

0 5 10 15 20 25 30−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Days Forward

He

dg

ing

Err

or

0 5 10 15 20 25 30−2

−1.5

−1

−0.5

0

0.5

1

1.5

Days Forward

He

dg

ing

Err

or

0 5 10 15 20 25 30−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Days Forward

He

dg

ing

Err

or

Delta hedge: Negative gamma → Large move leads to large loss.

Triangle: Reasonably symmetric.


Root mean squared hedging error

Compare the best delta (dashed line) with the worst triangle (solid line):BS MJ HV HW

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Days

RM

SE

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Days

RM

SE

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Days

RM

SE

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Days

RM

SE

Delta hedge effectiveness depends crucially on dynamics.The performance is very good under BS, but deteriorates drastically in thepresence of jumps/stochastic volatility.RMSE under HW is 9 times RMSE under BS.

Triangle: Performance is stable across all model environments.RMSE are between 0.2-0.4 under all models.


A historical exercise on SPX options

Daily data on SPX options from January 1996 to March 2009.

Choose 158 starting dates with a set of options expiring in exactly 30 days.

At each starting date, group options into 4 maturity groups: (i) 1 month(30) days, (ii) 2 months (59 or 66 days), (iii) 3-5 months (87-157 days), (iv)one year (276-402 days).

Based on the 4 maturity groups, form 14 target-hedge portfolio maturitycombinations: 4 with Tc < To < T , 4 with To < Tc < T , and 6 withTc = To < T .

Choose the target option strike K closest to the spot level.

Choose optimal strike spacing based on the regression results fromsimulation. Map the optimal strike spacing to the cloest available strikes.

Construct the local volatility surface from the observed option impliedvolatilities. Compute weights for each portfolio.

Track the hedging error over 30 days as in the simulation.

Perform delta hedging for comparison.


Hedging performance comparison on SPX options

Triangle performance:

0 2 4 6 8 100.2

0.4

0.6

0.8

T/Th

RM

SE

The hedging errors are slightlylarger than the HV&HW casedue to constraints in strikeavailability and/or morecomplicated dynamics.

Daily delta-hedge performance:

RMSE on 2, 4, and12-month options is0.63, 0.63, 0.66.

11 of the 14 trianglesperform better, even whentarget maturity is 6 timesof hedge maturity.


Concluding remarks

Existing hedging practices are mostly based on risk-exposures.

The issue: It is hard to know exactly what the risk exposures are.

Different model assumptions can all match current market prices, butcan imply quite different hedging ratios.

We propose a simple, robust hedging strategy that does not depend on riskexposures, but only on affinities of contracts characteristics.

We explore the idea using the example of hedging one option with threeoptions that form a maturity-strike triangle. The results are promising.

Many triangles can be constructed flexibly to balance contractavailability, transaction cost, and hedging efficiency.

Both simulation and a historical run on SPX options show that mosttriangles outperform delta hedge under realistic environments.

A lot more can be explored on this new idea: (i) other markets, (ii) largeportfolio, (iii) optimal contract placement, (iv) hedging error behavior anddependence on contract placements ...


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