Static Hedging and Model Risk for Barrier Options

Static Hedging and Model Risk for
Barrier Options
Morten Nalholm
Department of Applied Mathematics and Statistics
Institute for Mathematical Sciences
University of Copenhagen
www.math.ku.dk/∼nalholm
Static Hedging and Model Risk for Barrier Options – p.1/16

Introduction
Traditional option pricing: Try to replicate whatever with a
dynamically adjusted portfolio in the stock.
Next idea : Replicate exotic options with plain vanilla
options.
Sometimes the hedge may not even require dynamic
trading, ie. it is static.
But the hedges are based on model assumptions so what
about model risk?
Static Hedging and Model Risk for Barrier Options – p.2/16

Barrier Options
A barrier option is a plain vanilla option where something
happens if the stock price hits the barrier B before expiry.
A discontinuous case: The up-and-out call
F ≡ 0
B
0
❆❑
❆
✁ ✁✕ F(B,t) = 0
❈❖
K
❈
F(x,T) ❈
= (x − K) +
Alive region: F solves B/S-PDE
0 T
✛
❝
Static Hedging and Model Risk for Barrier Options – p.3/16

Static Hedging
What Hedging an option by a portfolio bought at initiation,
which has the same payoff as the option and requires
no dynamic trading.
Why Less trading necessary with resulting advantages.
How Two major approaches in the literature
Calendar spreads.
Strike spreads.
Static Hedging and Model Risk for Barrier Options – p.4/16

Literature
Derman, various papers from the mid-90’ies
Carr, numerous papers from the mid-90’ies and on
Andersen, Andreasen & Eliezer (2002)
Nalholm & Poulsen (2005), “Static Hedging and Model Risk
for Barrier Options”
Poulsen (2004), “Exotic Options: Proofs without Formulas”
Nalholm & Poulsen (2005),“Applied Static Hedging of
Barrier Options”
Static Hedging and Model Risk for Barrier Options – p.5/16

Hedging Using Calendar Spreads I
Assume that
Call, Put (spot, time | strike, expiry)
are known functions (enter model risk) and that time points
0 = t 0 < t 1 ... t n−1 < t n = T are given.
Form a PF of the underlying option and α n strike-B expiry-T
calls
α n ∗ Call(B, t n−1 |B, T) + Call(B, t n−1 |K, T) = 0
Value 0 at (B,t n−1 ); same payoff as the barrier option at T
if S(T) < B.
Hedge portfolio by recursion:
α i ∗ Call(B, t i−1 |B; t i ) +
nα j ∗ Call(B, t i−1 |B; t j ) + Call(B, t i−1 |K; T) = 0
j=i+1
Static Hedging and Model Risk for Barrier Options – p.6/16

Hedging Using Calendar Spreads II
(Andersen et al.) This also works for any local volatility
model.
(Andersen et al.) Jumps can be hedged using
hedge-strikes beyond the barrier at each point in time.
Effectively, this sets the hedge value to 0 at possible levels
to which the stock can jump.
(Allen & Padovani and Fink) Stochastic volatility can
operationally be hedged by hedging different volatility levels
at the barrier by use of extra hedge-strikes.
Static Hedging and Model Risk for Barrier Options – p.7/16

Hedging Using Strike Spreads
It can be shown (Carr) that under certain model
assumptions there exists a simple T -claim that is equivalent
to the barrier option along the barrier and at expiry. The
payoff is an adjusted version of the original payoff.
This adjusted payoff can be hedged by varying strike,
expiry-T options.
The method exploits symmetry
(Poulsen) provides a survey.
Static Hedging and Model Risk for Barrier Options – p.8/16

Model risk
How sensitive are the static hedges to model risk?
Guess I: Not very because ...
Static Hedging and Model Risk for Barrier Options – p.9/16

Model risk
How sensitive are the static hedges to model risk?
Guess I: Not very because ...
... natural hedge instruments are used
Static Hedging and Model Risk for Barrier Options – p.9/16

Model risk
How sensitive are the static hedges to model risk?
Guess I: Not very because ...
... natural hedge instruments are used
... Carr says so in Risk 2003: incorporating static or
semi-static positions in options [...] can often reduce
or eliminate model risk
Static Hedging and Model Risk for Barrier Options – p.9/16

Model risk
How sensitive are the static hedges to model risk?
Guess I: Not very because ...
... natural hedge instruments are used
... Carr says so in Risk 2003: incorporating static or
semi-static positions in options [...] can often reduce
or eliminate model risk
Guess II: Quite a lot because
Static Hedging and Model Risk for Barrier Options – p.9/16

Model risk
How sensitive are the static hedges to model risk?
Guess I: Not very because ...
... natural hedge instruments are used
... Carr says so in Risk 2003: incorporating static or
semi-static positions in options [...] can often reduce
or eliminate model risk
Guess II: Quite a lot because
... OTM options are where model prices differ the
most; these options are used in the hedges
Static Hedging and Model Risk for Barrier Options – p.9/16

Model risk
How sensitive are the static hedges to model risk?
Guess I: Not very because ...
... natural hedge instruments are used
... Carr says so in Risk 2003: incorporating static or
semi-static positions in options [...] can often reduce
or eliminate model risk
Guess II: Quite a lot because
... OTM options are where model prices differ the
most; these options are used in the hedges
... strong assumptions about model, parameter
values and values of the processes are used
Static Hedging and Model Risk for Barrier Options – p.9/16

Models
We investigated different implementations of the strategies
against popular non-Black/Scholes models
Constant elasticity of variance (CEV) where σ = σS α t
Heston’s stochastic volatility model where
σ 2 = a Cox/Ingersoll/Ross square-root model
Merton’s jump diffusion where stock returns are hit by
Poisson arrival of (displaced) lognormal shocks.
Madan, Carr and Chang’s Variance Gamma model
which is an (infinite intensity) pure jump process, that
can be thought of as a time-changed Geometric
Brownian motion.
All calibrated to generate the same 1Y implied skew.
Static Hedging and Model Risk for Barrier Options – p.10/16

Numerical results
Reported number: Standard deviation of hedge error
relative to true option price in percent.
Static hedges use 3 (for D&O-call) and 11 (for U&O-call)
options. Dynamic hedging is done bi-daily.
Mixed ∆: ∆-hedge “barrier - underlying option”.
Smile and uniform scaling: “What would traders do”-type
adjustments to Black/Scholes-based static strike-hedges.
Static Hedging and Model Risk for Barrier Options – p.11/16

DOWN-AND-OUT CALL
Hedge method BS CEV SV JD VΓ
∆; pure 10.1 10.1 15.4 32.6 29.8
∆; mixed 2.0 2.8 4.5 6.1 5.1
STR-static; simple 1.9 14.0 13.3 11.2 11.2
STR-static; unif. scaled 6.2 2.5 2.5 2.5
STR-static; smile-scaled 0.8 1.3 2.3 2.3
CAL-static; simple 1.4 1.8 2.9 2.9 2.8
CAL-static; true CEV 1.5
CAL-static; 1 cond. vol. 3.2
CAL-static; 5 vol. levels 3.1
Static Hedging and Model Risk for Barrier Options – p.12/16

UP-AND-OUT CALL
Hedge method BS CEV SV JD VΓ
∆; pure 95 120 86 133 151
∆; mixed 94 122 93 150 166
STR-static; simple 52 36 38 30 32
STR-static; unif. scaled 55 58 66 70
STR-static; smile-scaled 26 19 31 42
CAL-static; simple 63 75 57 54 57
CAL-static; true CEV 65
CAL-static; 1 cond. vol. 49
CAL-static; 5 vol. levels 32
Static Hedging and Model Risk for Barrier Options – p.13/16

You don’t need that many options to beat ∆-hedging,
misspecification or not.
Static Hedging and Model Risk for Barrier Options – p.14/16

You don’t need that many options to beat ∆-hedging,
misspecification or not.
Discontinuous options are hard to hedge and jumps
and barriers are a nasty combination.
Static Hedging and Model Risk for Barrier Options – p.14/16

You don’t need that many options to beat ∆-hedging,
misspecification or not.
Discontinuous options are hard to hedge and jumps
and barriers are a nasty combination.
∆-hedge-accuracy deteriorates rapidly with
“non-Black/Scholes’ness” ...
Static Hedging and Model Risk for Barrier Options – p.14/16

You don’t need that many options to beat ∆-hedging,
misspecification or not.
Discontinuous options are hard to hedge and jumps
and barriers are a nasty combination.
∆-hedge-accuracy deteriorates rapidly with
“non-Black/Scholes’ness” ...
... whereas quality of the static hedges are fairly robust
across models.
Static Hedging and Model Risk for Barrier Options – p.14/16

You don’t need that many options to beat ∆-hedging,
misspecification or not.
Discontinuous options are hard to hedge and jumps
and barriers are a nasty combination.
∆-hedge-accuracy deteriorates rapidly with
“non-Black/Scholes’ness” ...
... whereas quality of the static hedges are fairly robust
across models.
You need to take account of the smile/skew in static
hedges ...
Static Hedging and Model Risk for Barrier Options – p.14/16

You don’t need that many options to beat ∆-hedging,
misspecification or not.
Discontinuous options are hard to hedge and jumps
and barriers are a nasty combination.
∆-hedge-accuracy deteriorates rapidly with
“non-Black/Scholes’ness” ...
... whereas quality of the static hedges are fairly robust
across models.
You need to take account of the smile/skew in static
hedges ...
... and not just any ad-hoc adjustment to Black/Scholes
will do.
Static Hedging and Model Risk for Barrier Options – p.14/16

Conclusions and ongoing research
Hedging exotic options with plain vanilla is possible to
understand and do, also in realistic settings.
Static Hedging and Model Risk for Barrier Options – p.15/16

Conclusions and ongoing research
Hedging exotic options with plain vanilla is possible to
understand and do, also in realistic settings.
The reported misspecification analysis was of
“horse-race”.
Static Hedging and Model Risk for Barrier Options – p.15/16

Conclusions and ongoing research
Hedging exotic options with plain vanilla is possible to
understand and do, also in realistic settings.
The reported misspecification analysis was of
“horse-race”.
We are working on a “full Monty” model based
implementation. Many details. Some are not important.
Others are.
Static Hedging and Model Risk for Barrier Options – p.15/16

Conclusions and ongoing research
Hedging exotic options with plain vanilla is possible to
understand and do, also in realistic settings.
The reported misspecification analysis was of
“horse-race”.
We are working on a “full Monty” model based
implementation. Many details. Some are not important.
Others are.
Empirical testing on times-series option data is then a
“no brainer”.
Static Hedging and Model Risk for Barrier Options – p.15/16

Conclusions and ongoing research
Hedging exotic options with plain vanilla is possible to
understand and do, also in realistic settings.
The reported misspecification analysis was of
“horse-race”.
We are working on a “full Monty” model based
implementation. Many details. Some are not important.
Others are.
Empirical testing on times-series option data is then a
“no brainer”.
Single-barrier options aren’t that exotic. OK fine, but
I’ve have never seen anything remotely resembling a
quote. I understand “OTC”, but this is ridiculous.
Static Hedging and Model Risk for Barrier Options – p.15/16

Bibliography
Allen & Padovani (2002). ’Risk management using quasi-static hedging’, Economic
Notes 31(2).
Andersen, Andreasen & Eliezer (2002). ’Static replication of barrier options: some
general results’, Journal of Computational Finance 5.
Carr & Chou (1997). ’Breaking barriers’, Risk Magazine 10.
Derman, Ergener & Kani (1995). ’Static Options Replication’, Journal of Derivatives 2.
Fink (2003). ’An examination of the effectiveness of static hedging in the presence of
stochastic volatility’, Journal of Futures Markets 23.
Heston (1993). ’A closed-form solution for options with stochastic volatility with
applications to bond and currency options’, Review of Financial Studies 6.
Madan, Carr & Chang (1998). ’The variance gamma process and option pricing’,
European Finance Review 2.
Merton (1976). ’Option pricing when the underlying stock returns are discontinuous’,
Journal of Financial Economics 5.
Nalholm & Poulsen (2005). ’Static hedging and model risk for barrier options’, Journal
of Futures Markets forthcoming.
Poulsen (2004). ’Exotic options: proofs without formulas’, working paper.
Static Hedging and Model Risk for Barrier Options – p.16/16