An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model with Discrete Trading and Transaction Costs

An Approximate Distribution of Delta-Hedging Errors in aJump-Diffusion Model with Discrete Trading and Transaction

Costs

Artur Sepp

Bank of America Merrill Lynch

[email protected]

Financial Engineering WorkshopCass Business School, London

May 13, 2010

1

Motivation

1) Analyse the distribution of the profit&loss (P&L) of delta-hedgingstrategy for vanilla options in Black-Scholes-Merton (BSM) modeland an extension of the Merton jump-diffusion (JDM) model assumingdiscrete trading and transaction costs

2) Examine the connection between the realized variance and therealized P&L

3) Find approximate solutions for the P&L volatility and the expectedtotal transaction costs;Apply the mean-variance analysis to find the trade-off between thecosts and P&L variance given hedger’s risk tolerance

4) Consider hedging strategies to minimize the jump risk

2

References

Theoretical and practical details for my presentation can be found in:

1) Sepp, A. (2012) An Approximate Distribution of Delta-HedgingErrors in a Jump-Diffusion Model with Discrete Trading and Trans-action Costs, Quantitative Finance, 12(7), 1119-1141http://ssrn.com/abstract=1360472

2) Sepp, A. (2013) When You Hedge Discretely: Optimization ofSharpe Ratio for Delta-Hedging Strategy under Discrete Hedging andTransaction Costs, Journal of Investment Strategies, 3(1), 19-59http://ssrn.com/abstract=1865998

3

http://ssrn.com/abstract=1360472


LiteratureDiscrete trading and transaction costs: Leland (1985), Boyle-Vorst(1992), Avellaneda-Paras (1994), Hoggard-Whalley-Wilmott (1994),Toft (1996), Derman (1999), Bertsimas-Kogan-Lo (2000)

Optimal delta-hedging: Hodges-Neuberger (1989), Edirsinghe-Naik-Uppal (1993), Davis-Panas-Zariphopoulou (1993), Clewlow-Hodges(1997), Zakamouline (2006), Albanese-Tompaidis (2008)

Robustness of delta-hedging: Figlewski (1989) , Crouhy-Galai (1995),Karoui-Jeanblanc-Shreve (1998), Crepey (2004), Carr (2005), Andersen-Andreasen (2000), Kennedy-Forsyth-Vetzal (2009)

Textbooks: Natenberg (1994), Taleb (1997), Lipton (2001), Rebon-ato (2004), Cont-Tankov (2004), Wilmott (2006), Sinclair (2008)

4

Plan of the presentation

1) The P&L in the BSM model with trading in continuous and dis-crete time

2) Extended Merton jump-diffusion model

3) Analytic approximation for the probability density function (PDF)of the P&L

4) Illustrations

5

BSM modelAsset price dynamics under the BSM:

dS(t) = S(t)σadW (t), S(t0) = S

Option value U(σ)(t, S), valued with volatility σ, solves the BSM PDE:

U(σ)t +

1

2σ2S2U

(σ)SS = 0, U(σ)(T, S) = u(S)

The delta-hedging portfolio with option delta ∆(σ)(t, S) ≡ U(σ)S (t, S):

Π(σh,σi)(t, S) = S(t)∆(σh)(t, S)− U(σi)(t, S)

In general, we have three volatility parameters:σi - for computing option value (implied volatility),σh - for computing option delta (hedging volatility),σa - for the ”assumed” dynamics of the spot S(t) (realized volatility).

Why different?:σi - market ”consensus”σh - proprietary view on option delta, risk management limitsσa - ”guess estimated” by proprietary and subjective methods

6

Illustration using the S&500 dataσi is illustrated by the VIX - market ”consensus” about the one-monthfair volatility (”model-independent”)σh is illustrated by the at-the-money one-month implied volatilityσa is illustrated by the one-month realized volatility

Time Series from Jan-07 to Mar-10

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

Jan-07 Mar-07 Jun-07 Sep-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08 Feb-09 May-09 Aug-09 Nov-09 Feb-10

1-day Realized Volatility

1-month Realized Volatility

1-month ATM Volatility

VIX

Empirical Frequency

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%Observed Volatility

Fre

qu

ency

ATM 1-month volatility

Realized 1-day volatility

In general, the implied volatility is traded at the premium to the real-ized average volatilityBut, over short-term periods, the realized volatility can be ”jumpy”,which causes significant risks for volatility sellers

7

Impact on the P&L when using different volatilitiesOne-period P&L at time t+ δt and spot S+ δS with fixed ∆(σh)(t, S):

δΠ(σh,σi)(t) ≡ Π(σh,σi)(t+ δt, S + δS)−Π(σh,σi)(t, S)

= δS∆(σh)(t, S)−(U(σi)(t+ δt, S + δS)− U(σi)(t, S)

)Apply Taylor expansion and use BSM PDE:

δΠ(σh,σi)(t) ≈ δS(∆(σh)(t, S)−∆(σi)(t, S)

)+ δt

(σ2i −Σ

)Γ(σi)(t, S)

Γ(σ)(t, S) = 12S

2(t)U(σ)SS (t, S) is cash-gamma

Σ = 1δt

(δSS

)2is the realized variance

In the limit: Σ→ σ2a as δt→ 0

The stochastic term in the P&L is caused by mis-match between theimplied and hedging volatilitiesThe deterministic term in the P&L is caused by mis-match betweenthe implied and actual volatilities

8

Analysis of the total P&L, P (T ):

P (T ) = limN→∞

N∑n=1

δΠ(σh,σi)(tn)

1) BSM case: σh = σi = σa thus δΠ(σa,σa)(t) = 0, P (T ) = 0No pain - no gain

2) Delta-hedge at implied volatility: σh = σi thus

δΠ(σi,σi)(t) = δt(σ2i − σ

2a

)Γ(σi)(t, S),

P (T ) =(σ2i − σ

2a

) ∫ Tt0

Γ(σi)(t, S)dt

No pain across the path - unpredictable gain at expiry

3) Delta-hedge at the asset volatility: σh = σa, thus

δΠ(σa,σi)(t) = δS(∆(σa)(t, S)−∆(σi)(t, S)

),

P (T ) = U(σi)(t0, S)− U(σa)(t0, S)

Pain across the path - predictable gain at expiry

9

Illustration of a P&L path

P&L path

-0.15%

-0.10%

-0.05%

0.00%

0.05%

0.10%

0.15%

0.20%

0.25%

0.30%

0.00 0.01 0.02 0.02 0.03 0.04 0.05 0.06 0.06 0.07 0.08

t

P&

L, %

P&L, Hedge at Pricing VolatilityP&L, Hedge at Actual Volatility

σi = 23% , σa = 21%, K = S = 1.0, T = 0.08 (one month), N = 100(hourly hedging)

The common practice is to hedge at implied volatility so no proprietaryview on option delta is allowed

In the sequel, we consider the case with hedging at implied volatility

10

The P&L in the discrete timeAssume that the number of trades N is finite and the delta-hedging isrebalanced at times {tn}, n = 0,1, ..., N , with intervals δtn = tn− tn−1Recall that, under the continuous trading, the final P&L is

P (T ) =(σ2i − σ

2a

) ∫ Tt0

Γ(σi)(t, S)dt

Is it correct?:

P (T ; {tn}) =(σ2i − σ

2a

) N∑n=1

Γ(σi)(tn−1, S)δtn

The correct form is:

P (T ; {tn}) =N∑n=1

(σ2i δtn −Σ2

n

)Γ(σi)(tn−1, S)

where Σn =(S(tn)−S(tn−1)

S(tn−1)

)2is the discrete realized variance

Recall that the standard deviation of a constant volatility parameterσ measured discretely is σ/

√2N

11

The Expected P&L

Assume that hedging intervals are uniform with δtn ≡ δt = T/N

The expected P&L:

M1({tn}) =(σ2i − σ

2a

) N∑n=1

δtΓ(σi)(tn−1;S) ≈(σ2i − σ

2a

)TΓ(σi)(t0, S)

In terms of option vega V(t0, S), V(t0, S) = 2σTΓ(σi)(t0, S):

M1({tn}) ≈ (σi − σa)V(t0, S)

The expected P&L equals to the spread between the implied and theactual volatilities multiplied by the option vega

The expected P&L does not depend on the hedging frequency!

12

The P&L volatilityUnder the continuous hedging, the P&L volatility σP&L is affectedonly by the volatility of the asset price:

σP&L ≈ C |σi − σa| V(t0, S),

where C is a small number, C ≡ C(σi, σa, t0, S)

In the ideal BSM model with σi = σa, σP&L = 0

Under the discrete hedging, the P&L volatility σP&L is affected pri-marily by the volatility of the realized variance:

σP&L ≈ σ2a

√1

2N

V(t0, S)

σiFor short-term at-the-money options:

σP&L

U(σi)(t0, S)≈σ2a

σ2i

√1

2N

The P&L volatility is significant for short-term options relative totheir value if N is small

13

Risks of the delta-hedging strategy

1) Mis-specification of volatility parameter (generally, mis-specificationof the model dynamics)

2) Discrete hedging frequency

3) Transaction costs

4) Sudden jumps in the asset price

5) Volatility of the actual volatility σa (volatility of volatility)

For short-term options (with maturity up to one month), 1)-4) aresignificant

For longer-term options, 5) is important while 2) is less significant

In this talk, we concentrate on aspects 1)-4)

14

P&L distribution, Recall that the expected P&L does not dependon the hedging frequency: M1({tn}) ≈ (σi − σa)V(t0, S)

We need to understand how the discrete hedging and jumps affectthe higher moments of the P&L: volatility, skew, kurtosis

For this purpose we study the approximate distribution of the P&Lrather than its higher moments (the latest is the common approachin the existing studies)

P&L distribution of Delta-Hedging Stategy

Fre

qu

ency

ContinuousHedging

DiscreteHedging

P&L volatility (ActualVol, HedgeVol, N)

Expected P&L =(HedgeVol-ActuaVol)*Vega(HedgeVol)

P&L skew (ActualVol, HedgeVol, N)

P&L kurtosis (ActualVol,HedgeVol, N)

15

Path-dependence of the P&L

Under the discrete hedging, the P&L is a path-depended function:

P (T ; {tn}) =N∑n=1

(σ2i δtn −Σ2

n

)Γ(σi)(tn−1, S)

where Σn =(S(tn)−S(tn−1)

S(tn−1)

)2is the realized variance

Direct analytical methods are not applicable

What if we use the first order approximation in Γ(σi)(tn−1, S)?:

P (T ; {tn}) ≈N∑n=1

(σ2i δtn −Σ2

n

)G(σi)(tn−1, S)

where G(σi)(tn−1, S) = E[Γ(σi)(tn−1, S)]

The approximate P&L is a function of the discrete realized variance

The first moment of the P&L is preserved, how is about its variance?

16

MC Simulations of the P&L

Apply Monte Carlo to simulate a path of {Sn}, n = 1, ...N , to compute

P&L(T ;N) =N∑n=1

[(Sn − Sn−1)∆(tn−1, Sn−1)− (U(tn, Sn)− U(tn−1, Sn−1))

]

RealizedVar(T ;N) =N∑n=1

((Sn − Sn−1)

Sn−1

)2

RealizedGamma(T ;N) =T

N

N∑n=1

Γ(tn−1, Sn−1)

RealizedVarGamma(T ;N) =N∑n=1

Tσ2i

N−(

(Sn − Sn−1)

Sn−1

)2Γ(tn−1, Sn−1)

”RealizedVarGamma” is an approximation to the P&L, how is about”RealizedVar”?

17

MC Simulations of the P&LCall option with S = K = 1, T = 0.08 and N = 5 (weekly hedging),N = 20 (daily), N = 80 (every 1.5 hour), N = 320 (every 20 minutes)

The same model parameters are used to compute the delta-hedgeand simulate S(t) using three models:

1) BSM with σ = 21.02%

2) Merton jump-diffusion (MJD) with σ = 19.30%, λ = 1.25, ν =−7.33%, υ = (1.27%)2

3) Heston SV with V (0) = θ = (21.02%)2, κ = 3, ε = 40%, ρ = −0.7

In MJD: σ2i = σ2 + λ(ν2 + υ); in Heston SV: σ2

i = V (0)

In MJD and Heston, option value, delta, and gamma are computedusing efficient Fourier inversion methods by Lipton and Lewis

Simulated realizations are normalized to have zero mean and unitvariance (with total of 10,000 paths)

18

The P&L frequency in the BSM model

N=5

0.00

0.02

0.04

0.06

0.08

0.10

-5 -4 -3 -2 -1 0 1 2 3 4 5Normalized Observations

Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma

RealizedVarianceGamma

N=20

0.00

0.02

0.04

0.06

0.08

0.10


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


N=80

0.00

0.02

0.04

0.06

0.08

0.10


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


N=320

0.00

0.02

0.04

0.06

0.08

0.10


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


19

The P&L frequency in the Merton jump-diffusion model

N=5

0.00

0.05

0.10

0.15

0.20


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


N=20

0.00

0.05

0.10

0.15

0.20


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


N=80

0.00

0.10

0.20

0.30

0.40

0.50


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


N=320

0.00

0.10

0.20

0.30

0.40

0.50


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


20

The P&L frequency in the Heston SV model

N=5

0.00

0.02

0.04

0.06

0.08

0.10

0.12


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


N=20

0.00

0.02

0.04

0.06

0.08

0.10

0.12


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


N=80

0.00

0.02

0.04

0.06

0.08

0.10


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


N=320

0.00

0.02

0.04

0.06

0.08

0.10


Fre

qu

ency

Realized P&L

Realized Variance

Realized Gamma


21

Observations

The realized variance gamma approximates the realized P&L closelyif hedging is daily or more frequent

The frequency of realized variance scales with the frequency of theP&L

This is the key ground for our approximation: use the PDF of therealized variance to approximate the PDF of the realized P&L

The key tool: the Fourier transform of the PDF of the discrete real-ized variance

On side note: the P&L distribution in the BSM and Heston is closeto the normal, while that in the MJD is peaked at zero with heavyleft tail

22

Extended Merton Model. The JDM under the historic measure P:

dS(t)/S(t) = µ(t)dt+ σ(t)dW (t) +(eJ − 1

)dN(t), S(t0) = S,

N(t) is Poisson process with intensity λ(t), jumps have the PDF w(J):

w(J) =L∑l=1

pln(J; νl, υl),L∑l=1

pl = 1,

L, L = 1,2, .., is the number of mixturespl, 0 ≤ pl ≤ 1, is the probability of the l-th mixtureνl and υl are the mean and variance of the l-th mixture, l = 1, .., L

Under the pricing measure Q, the JDM for S(t) has volatility σ(t),jump intensity λ(t), risk-neutral drift µ(t):

µ(t) = r(t)− d(t)− λ(t)JQ,

w(J) =L∑l=1

pln(J; νl, υl),L∑l=1

pl = 1,

JQ ≡∫ ∞−∞

(eJ − 1

)w(J)dJ =

L∑l=1

pleνl+

12υl − 1

23

NotationsDefine the time-integrated quantities, with 0 ≤ t0 ≤ t

The variance under P and Q, respectively:

ϑ(t; t0) =∫ tt0σ2(t′)dt′, ϑ(t; t0) =

∫ tt0σ2(t′)dt′

The risk-free rate and dividend yield:

r(t; t0) =∫ tt0r(t′)dt′, d(t; t0) =

∫ tt0d(t′)dt′

The drift under P and Q, respectively:

µ(t; t0) =∫ tt0µ(t′)dt′, µ(t; t0) =

∫ tt0µ(t′)dt′

The intensity rate under P and Q, respectively:

λ(t; t0) =∫ tt0λ(t′)dt′, λ(t; t0) =

∫ tt0λ(t′)dt′

24

Transition PDF of x(t), x(t) = lnS(t) and X = x(T ):

GX(T,X; t, x) =∞∑

m1=0

...∞∑

mL=0

P(m1; p1λ(T ; t))...P(mL; pLλ(T ; t))n(X;α, β),

α ≡ α(m1, ...,mL) = x+ µ(T ; t)−1

2ϑ(T ; t) +

L∑l=1

mlνl,

β ≡ β(m1, ...,mL) = ϑ(T ; t) +L∑l=1

mlυl,

where n(x;α, β) is the PDF of a normal with mean α and variance βP(m;λ), P(m;λ) = e−λλm/m!, is the Poisson probabilityCall options are valued by the extension of Merton formula (1976):

U(t, S;T,K) =∞∑

m1=0

...∞∑

mL=0

P(m1; p1λ(T ; t)(JQ + 1))...P(mL; pLλ(T ; t)(JQ + 1))

× C(BSM)(S,K,1; β(m1, ...,mL), α(m1, ...,mL), d(T ; t)),

α(m1, ...,mL) = r(T ; t)− λ(T ; t)JQ +L∑l=1

mlνl +1

2

L∑l=1

mlυl, β(m1, ...,mL) = ϑ(T ; t) +L∑l=1

mlυl

25

Empirical Estimation

Use closing levels of the S&P500 index from January 4, 1999, toJanuary 9, 2009, with the total of 2520 observations

Fit empirical quantiles of sampled daily log-returns to model quantilesby minimizing the sum of absolute differences between the two

Assume time-homogeneous parameters, µ = 0, and jump variance υluniform across different states

The best fit, keeping L as small as possible, is obtained with L = 4

26

Empirical Estimation

Model estimates: σ = 0.1348, λ = 46.4444

l pl νl√υl plλ expected frequency

1 0.0208 -0.0733 0.0127 0.9641 every year2 0.5800 -0.0122 0.0127 26.9399 every two weeks3 0.3954 0.0203 0.0127 18.3661 every six weeks4 0.0038 0.1001 0.0127 0.1743 every five years

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10X

Fre

qu

ency

Empirical PDFNormal PDFJDM PDF

-0.12

-0.08

-0.04

0.00

0.04

0.08

0.12

-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12

JDM quantiles

Dat

a q

uan

tile

s

27

Assumptions for the delta-hedging strategy

1) The synthetic market is based on the continuous-time model of theasset price dynamics specified by either the diffusion model (DM) orthe jump-diffusion model (JDM) with known model parameters underP and Q

2) The option value and delta are computed using model parametersunder Q; the PDF of the P&L is computed using model parametersunder P

3) The hedging strategy is time based with the set of trading times{tn}, trading intervals (not necessarily uniform) defined by {δtn},where δtn = tn − tn−1, n = 1, ..., N , and N is the total number oftrades, t0 = 0, tN = T

Given the discrete time grid {tn}, we define: µn = µ(tn; tn−1), ϑn =

ϑ(tn; tn−1), ϑn = ϑ(tn; tn−1), λn = λ(tn; tn−1), λn = λ(tn; tn−1)

28

Discrete modelDiscrete version of the DM under P on the grid {tn}:

S(tn) = S(tn−1) exp{µn −

1

2ϑn +

√ϑnεn

},

where {εn} is a collection of independent standard NRVs, n = 1, ..., N

For small µn and√ϑn are small, we approximate:

S(tn) ≈ S(tn−1)(1 + µn +

√ϑnεn

)Hedging portfolio Π(tn, S) at rebalancing time tn:

Π(tn, S) = S(tn)∆(tn, S)− U(tn, S)

The P&L at time tn, δΠn:

δΠn ≈1

2

(ϑn −Σ2

n

)S2(tn−1)USS(tn−1, S),

Σ2n ≡

(S(tn)− S(tn−1))2

S2(tn−1)≈(µn +

√ϑnεn

)2

29

Transaction costs

Proportional costs is the bid-ask spread:

κ = 2Sask − Sbid

Sask + Sbid,

where Sask and Sbid are the quoted ask and bid prices, respectively

Thus, κ/2 is the average percentage loss per trade amount and:

Sask(t) = (1 + κ/2)S(t), Sbid(t) = (1− κ/2)S(t)

Θn is transaction costs due to rebalancing of delta-hedge:

Θn ≡ (κ/2)S(tn) |∆(tn, S)−∆(tn−1, S)|

30

Approximation for the transaction costsApply Taylor expansion for US(tn, S):

Θn ≈ (κ/2)S(tn)USS(tn−1, S) |δSn|Apply approximation for δSn, we obtain:

Θn ≈ (κ/2)S2(tn−1)USS(tn−1, S)

∣∣∣∣∣(

1

2+ µn +

√ϑnεn

)2−

1

4

∣∣∣∣∣Define the following function:

F (α, β) ≡Eε[∣∣∣∣∣(

1

2+ α+

√βε

)2−

1

4

∣∣∣∣∣]

=

√2β

π

((α+ 1)e

−α2

2β − αe−(1+α)2

2β

)

+ 2(β + α2 + α)

(1

2+N

(α√β

)−N

(1 + α√β

))

Approximate Θn by (the first-order moment matching):

Θn ≈1

2θnS

2(tn−1)USS(tn−1, S),

where θn = κnε2n, κn ≡ κF (µn, ϑn)

31

Approximation for the total P&LThe total P&L, P ({tn}), is obtained by accumulating ∆Πn:

P ({tn}) ≡N∑n=1

(δΠn −Θn)

Apply our approximations:

P ({tn}) ≈N∑n=1

(ϑn −Σ2

n − θn)

Γ(tn−1, S),

where Γ(t, S) is the option cash-gamma: Γ(t, S) = 12S

2(t)USS(t, S)

Recall that Σ2n ≈

(µn +

√ϑnεn

)2

Explicitly, the P&L is a quadratic function of {εn}, n = 1, ..., N :

P ({tn}) =N∑n=1

M(εn;An, Bn, Cn,−Γ(tn−1, S)),

M(ε;A,B,C, o) ≡ o(A+ Cε+Bε2

)An = −ϑn + µ2

n, Cn = 2µn√ϑn, Bn = (ϑn + κn)

32

Expected Cash-GammaFor vanilla calls and puts with strike K:

Γ(t, S) =K

2√

2πϑ(T ; t)exp

− 1

2ϑ(T ; t)

(lnS

K−

1

2ϑ(T ; t)

)2

Under P:

S(tn) = S(t0) exp{µ(tn; t0)−

1

2ϑ(tn; t0) +

√ϑ(tn; t0)εn

}where εn is a standard NRV

The expected cash gamma under P, G(T ; t):

G(tn; t0) ≡ EP [Γ(tn, S)] = Eεn [Γ(tn, S) |S = S(tn)]

=K

2√

2πζ(t0, tn, T )exp

{−

1

2ζ(t0, tn, T )

(lnS

K−

1

2ζ(t0, tn, T )

)2},

where ζ(t0, tn, T ) = ϑ(tn; t0) + ϑ(T ; tn)

If model parameters under Q and P are equal: G(tn; t0) = Γ(t0, S)

33

The Fourier transform of the quadratic function:

Z(A,B,C; o) ≡ Eε[e−M(ε;A,B,C,o)

]=

1√2π

∫ ∞−∞

exp

{−oA− oCx− oBx2 −

x2

2

}dx

=1√

2oB + 1exp

12(oC)2

2oB + 1− oA

Consider the following sum:

I(N) =N∑n=1

M(εn;An, Bn, Cn, on)

where {εn} is a set of independent standard NRVs, An, Bn, Cn are realsand on is complex with <[onBn] > −1/2, n = 1, ..., NCompute the Fourier transform of I(N), GI:

GI ≡ E{εn}[e−I(N)

]=

N∏n=1

Eεn[e−M(εn;An,Bn,Cn,on)

]=

N∏n=1

Z(An, Bn, Cn; on)

= exp

N∑n=1

12(onC)2

n

2onBn + 1− onAn

N∏n=1

1√2onBn + 1

34

The Fourier transform of the P&L distributionDenote the PDF of the P&L P ({tn}) by GP (P ′; {tn})Denote its Fourier transform under P by GP (Ψ; {tn}) with Ψ = ΨR +iΨI, where ΨR and ΨI are reals and i =

√−1:

GP (Ψ; {tn}) ≡ EP[e−ΨP ({tn})

]= EP

exp

−ΨN∑n=1

M(εn;An, Bn, Cn;−1)Γ(tn−1, S)

Approximate GP by:

GP (Ψ; {tn}) ≈ EPexp

−ΨN∑n=1

M(εn;An, Bn, Cn,−1)EP [Γ(tn−1, S)]

= EPexp

−N∑n=1

M(εn;An, Bn, Cn,Ψ∗n)

=N∏n=1

Z(An, Bn, Cn; Ψ∗n),

where Ψ∗n = −ΨG(tn−1; t0)

35

The Fourier transform of the P&L distribution

Simplify:

GP (Ψ; {tn}) ≈ exp

N∑n=1

(−

Ψ∗nµ2n

2Ψ∗n(ϑn + κn) + 1+ ϑnΨ∗n

)×

N∏n=1

1√2Ψ∗n(ϑn + κn) + 1

This expression can be recognized as the characteristic function ofnon-central chi-square distribution with in-homogeneous drift, vari-ance, and shift parameters.

36

The jump-diffusion model. The Fourier transform of the approxi-mate P&L distribution is (for derivation, see my paper):

GP (Ψ; {tn}) ≡ EP[e−ΨP ({tn})

]≈

N∏n=1

∞∑m1=0

...∞∑

mL=0

P(m1; p1λn)...P(mL; pLλn)Z(An − Σ2n, Bn, Cn;−G(tn−1; t0)Ψ),

An ≡ An(m1, ...,mL) = α2n(m1, ...,mL),

Bn ≡ Bn(m1, ...,mL) = βn(m1, ...,mL),

Cn ≡ Cn(m1, ...,mL) = αn(m1, ...,mL)√βn(m1, ...,mL),

αn(m1, ...,mL) = µn + ν(m1, ...,mL),

βn(m1, ...,mL) = ϑn + υ(m1, ...,mL) + κn(m1, ...,mL),

κn(m1, ...,mL) ≡ κF (µn + ν(m1, ...,mL), ϑn + υ(m1, ...,mL)) ,

ν(m1, ...,mL) ≡L∑l=1

mlνl, υ(m1, ...,mL) ≡L∑l=1

mlυl,

Σ2n = ϑn + λn

L∑l=1

pl

(e2νl+2υl − 2eνl+

12υl + 1

)37

Implications 0

Using the Fourier transform of the approximate distribution of theP&L, we can compute the approximate PDF of P :

GP (P ′; {tn}) =1

π

∫ ∞0<[eΨP ′GP (Ψ)

]dΨI ,

where we apply standard FFT methods to compute the inverse Fouriertransform

Also, we can compute the m-th moment of P by:

EP [(P ({tn}))m]

= (−1)m∂mGP (Ψ; {tn})

∂ΨmR

∣∣∣∣∣ΨR=0,ΨI=0

, m = 1,2, ....

38

Implications I

The expected P&L:

M1({tn}) =N∑n=1

(ϑn − ϑn − κn − µ2

n

)G(tn−1; t0)

If the hedging frequency is uniform with δt = T/N and model param-eters are constant:

M1({tn}) ≈

σ2 − σ2 − κσ√

2N

πT−µ2T

N

TΓ(t0, S)

The break-even volatility σ, so that M1({tn}) = 0:

σ2 = σ2 + κσ

√2N

πT,

which is Leland (1985) volatility adjustment for a short option position

39

Implications IIAssume δt ≡ δtn = T/N and parameters under Q and P are equal with:

µn =∫ tntn−1

µdt ≡ µδt, ϑn = ϑn =∫ tntn−1

σ2dt ≡ σ2δt

The Fourier transform of the P&L distribution

GP (Ψ; {tn}) =1

(2Ψ◦+ 1)N/2exp

(−

ηΨ◦

2Ψ◦+ 1+ sΨ◦

)

Ψ◦ = −cΓ(t0, S)Ψ, c =σ2T

N+ κ

√2

π

σ2T

N, η =

µ2T2

Nc, s =

σ2T

c.

This is the Fourier transform of the non-central chi-square distributionwith N degrees of freedom, non-centrality parameter η, χ(y,N, η):

GP (P ′; {tn}) =1

cχ

(s− P ′

c, N, η

)1{P ′≤s}

where s = σ2TΓ(t0, S), c = cΓ(t0, S)

40

Implications IIIThe P&L M1:

M1 = −Γ(t0, S)

µ2T2

N+ κ

√2σ2TN

π

The P&L volatility M2 is inversely proportional to

√N :

(M2)2 = Γ2(t0, S)(Tσ2)

2Tσ2

N+

4κ√

2Tσ2√πN

+4κ2

π+O

(1

N3/2

)The skewness M3 and the excess kurtosis M4:

M3 = −23/2√N

+O

(1

N

), M4 =

12

N+O

(1

N2/3

)As N →∞, the chi-squared distribution converges to normal:

GP (P ′; {tn}) ≈ n(P ′;M1, (M2)2)

41

Jump-diffusion model IThe expected P&L, M1({tn}):

M1({tn}) =N∑n=1

(ϑn − ϑn + λnEQ[J2]− λnEP[J2]− µ2

n − Kn)G(tn−1; t0)

EQ[J2] ≡∫ ∞−∞

(eJ − 1

)2w(J)dJ

=L∑l=1

pl

(e2νl+2υl − 2eνl+

12υl + 1

)≈

L∑l=1

pl(ν2l + υl

)EP[J2] computed under P by analogy

Kn is the expected proportional transaction costs

Kn ≡κ∞∑

m1=0

...∞∑

mL=0

P(m1; p1λn)...P(mL; pLλn)

× F (µn + ν(m1, ...,mL), ϑn + υ(m1, ...,mL))

The hedger should match ϑn and ϑn, EQ[J2] and EP[J2], λn and λn

42

Jump-diffusion model II

Assume constant model parameters equal under Q and P, δt = T/N ,a single jump component, L = 1, with mean and variance ν and υ

Approximate the PDF of the Poisson process by Bernoulli distributiontaking P(0;λn) = 1− λn, P(1;λn) = λn, and P(m;λn) = 0 for m > 1

The expected total transaction costs K(N):

K(N) ≡ Γ(t0, S)N∑n=1

Kn ≈ κΓ(t0, S)

√N√

2σ2T

π+ λT

√2(ν2 + υ)

π

The first term is the expected transaction costs due to the diffusion

The second term represents those due to jumps

43

Jump-diffusion model III

If the hedging frequency is uniform with δt = T/N and model param-eters are constant:

M1({tn}) ≈

σ2 − σ2 + λ(ν2 + υ)− λ(ν2 + υ)− κ

σ√

2N

πT− λ

√2(ν2 + υ)

π

× TΓ(t0, S)

The break-even volatility σ, so that the expected P&L is zero:

σ2 = σ2 + λ(ν2 + υ)− λ(ν2 + υ) + κσ

√2N

πT+ κλ

√2(ν2 + υ)

π,

which is can be interpreted as the Leland volatility adjustment for theMerton JDM

44

Jump-diffusion model IVThe expected P&L without transaction costs:

M1 = −Γ(t0, S)

(µ2T2

N+

2λµνT2

N

)The P&L volatility:

(M2)2 = Γ2(t0, S)(λT

(ν4 + 3υν2 + 3υ2

)+

c

N+O

(1

N2

))where c =

(2σ4 + λσ2(ν2 + υ)− λ2(ν2 + υ)2 + λµ(6νυ + 4ν3)

)T2

When jumps are present the volatility of the P&L cannot be reducedby increasing the hedging frequencyThe constant term is proportional to expected number of jumps λTThe sign of the jump is irrelevant

The skewness M3 and the excess kurtosis M4:

limN→∞

M3 = −O(

1√λT

), limN→∞

M4 = O

(1

λT

)45

Mean-Variance Analysis IThe hedger’s objective is to minimize utility function:

minN

{K(N) +

1

2γV(N)

}where γ is the risk-aversion tolerance, K(N) is the expected totalhedging costs and V(N) is the variance of the P&L

We have shown that for both DM and JDM models:

K(N) = k0 + k1√N, V(N) = v0 +

v1

N

Thus, the optimal solution N∗ is given by:

N∗ =

(v1

γk1

)2/3

For the DM: N∗ = (2π)1/3σ2T(

Γ(t,S)γκ

)2/3,

so that N∗ is proportional to variance σ2T and Γ(t0, S), and inverselyproportional to the risk-aversion tolerance and transaction costs

46

Mean-Variance Analysis II

0

50

100

150

200

250

300

350

400

450

0.01

0.02

0.04

0.08

0.16

0.32

0.64

1.28

2.56

5.12

10.2

4

20.4

8

40.9

6

81.9

2

Risk Aversion

Op

tim

al F

req

uen

cy, N

*

N*, DMN*, JDM

0%

5%

10%

15%

20%

25%

30%

0.01 0.02 0.04 0.08 0.16 0.32 0.64 1.28 2.56 5.12 10.24 20.48 40.96 81.92

Risk Aversion

Co

sts

0%

20%

40%

60%

80%

100%

120%

140%

160%

180%

Vo

lati

lity

Costs K(N*), DMCosts K(N*), JDMVolatility V(N*), DMVolatility V(N*), JDM

Left side: the optimal hedging frequency N∗, Right side: the expectedtransaction costs K(N∗) normalized by the option value as function

of the risk-aversion tolerance and the P&L volatility√V(N∗) under

the DM (DM) and JDM (JDM) using κ = 0.002 for a put option withS(0) = K = 1.0 and maturity T = 0.08

For a small level of the risk tolerance, the expected transaction costsamount to about 20− 30% of the option premium

47

P&L Distribution. Illustrations

Use two models:1) BSM with σ = 21.02%2) Merton jump-diffusion (MJD) with σ = 19.30%, λ = 1.25, ν =−7.33%, υ = (1.27%)2

Apply two cases:1) transaction costs are zero2) transaction costs with rate κ = 0.002

Call option with S = K = 1, T = 0.08 and N = 5 (weekly hedging),N = 20 (daily), N = 80 (every 1.5 hour), N = 320 (every 20 minutes)

The P&L distribution is obtained using the Fourier inversion of the an-alytic approximations (Analytic) and Monte Carlo simulations (Mon-teCarlo) with total of 10,000 paths

48

P&L Distribution in the DM without transaction costs

P&L distribution, N=5

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MonteCarloAnalytic P&L distribution, N=20

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49

P&L Distribution in the JDM without transaction costs


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P&L Distribution in the DM with transaction costs


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51

P&L Distribution in the JDM with transaction costs


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52

Illustration

P&L volatility

0%

20%

40%

60%

80%

1m 1w 1d 2h 30m 8.5m 2m 0.5mHedging intervals

vola

tilit

y

Volatility, DM no TRC

Volatility, DM with TRC

Volatility, JDM no TRC

Volatility, JDM with TRC

P&L mean return

-200%

-160%

-120%

-80%

-40%

0%1m 1w 1d 2h 30m 8.5m 2m 0.5m

Hedging intervals

retu

rn

Mean, DM with TRC

Mean, JDM with TRC

Left: the P&L volatility under the DM (Volatility, DM no TRC) andJDM (Volatility, JDM no TRC) with no costs, and the P&L volatilityunder the DM (Volatility, DM with TRC) and JDM (Volatility, JDMwith TRC) with proportional costs; Right: the mean of the P&Ldistribution under the DM (Mean, DM with TRC) and JDM (Mean,JDM with TRC) with proportional costs. The x-axes is the hedgingfrequency: 1m - the DHS is applied only t = 0, 1w - weekly, 1d -daily, 2h - every two hours, 30m - every thirty minutes, 8.5m - everyeight minutes, 2m - every two minutes, 0.5m - every thirty seconds.

53

Jump Risk Hedge (Andersen-Andreasen (2000))Define the jump risk U(t, S;T,K) under Q as follows:

U(t, S;T,K) =∫ ∞−∞

U(t, SeJ;T,K)w(J)dJ − U(t, S;T,K)

Use series formula to obtain:

U(t, S;T,K) =∞∑

m1=0

...∞∑

mL=0

P(m1; p1λ(T ; t)(JQ + 1))...P(mL; pLλ(T ; t)(JQ + 1))

×(C(BSM)((JQ + 1)S,K,1; β(m1 + 1, ...,mL + 1), α(m1, ...,mL), d(T ; t))

−C(BSM)(S,K,1; β(m1, ...,mL), α(m1, ...,mL), d(T ; t)))

The hedging portfolio Π(t, S) to hedge U(1)(t, S):

Π(t, S) = S(t)∆(1)(t) + U(2)(t, S)∆(2)(t)− U(1)(t, S),

∆(1)(t) is the number of shares held in the underlying asset∆(2)(t) is the number of units held in the complimentary optionhere, either K(1) 6= K(2) or T (1) 6= T (2)

54

Jump Risk HedgeRequire Π(t, S) to be neutral to both delta and jump risks:

∆(2)(t) =JQS(t)U(1)

S (t, S)− U(1)(t, S)

JQS(t)U(2)S (t, S)− U(2)(t, S)

,

∆(1)(t) = −∆(2)U(2)S (t, S) + U

(1)S (t, S)

To get some intuition, we expand U in Taylor series around J = 0:

∆(2)(t) ≈U

(1)SS (t, S)

U(2)SS (t, S)

The hedging strategy is approximately gamma-neutral

For illustration use the parameters of the JDM with K(1) = K(2) = 1,T (1) = 0.08, K(2) = 0.16 without transaction costs and with trans-action costs κ = 0.002

55

Jump Risk Hedge


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56

Extensions. Variance swap. Consider a portfolio of delta-hedgedpositions, Π(m)(tn, S), with weights w(m), in the same underlying:

Υ(tn, S) =M∑

m=1

w(m)Π(m)(tn, S),

where Π(m)(tn, S) = S(tn)∆(m)(tn, S)− U(m)(tn, S), m = 1, ...,M

The one-period P&L:

δΥ(tn, S) =M∑

m=1

w(m)(ϑ

(m)n −Σ2

n

)Γ(m)(tn−1, S) = A−BΣ2

n,

where A =∑Mm=1w

(m)ϑ(m)n Γ(m)(tn−1, S), B =

∑Mm=1w

(m)Γ(m)(tn−1, S)

For variance swap with unit notional, A ≡ (AF/N2)σ2k , where σ2

k isvariance strike, and B ≡ AF/N (AF is annualization factor, N issampling frequency)

Obtained expressions for the approximate P&L are exact for the P&Ldistribution of the variance swap (within our assumptions)

57

Conclusions

We have presented a quantitative approach to study the P&L distri-bution of the delta-hedging strategy assuming discrete trading andtransaction costs under the diffusion model and jump-diffusion model

The opinions expressed in this presentation are those of the authoralone and do not necessarily reflect the views and policies of Bank ofAmerica Merrill Lynch

58

References

Sepp, A. (2012) An Approximate Distribution of Delta-Hedging Errorsin a Jump-Diffusion Model with Discrete Trading and TransactionCosts, Quantitative Finance, 12(7), 1119-1141http://ssrn.com/abstract=1360472

Sepp, A. (2013) When You Hedge Discretely: Optimization of SharpeRatio for Delta-Hedging Strategy under Discrete Hedging and Trans-action Costs, Journal of Investment Strategies, 3(1), 19-59http://ssrn.com/abstract=1865998

59



Economy & Finance

An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model with Discrete Trading and Transaction Costs