1) Analyse the distribution of the profit&loss (P&L) of delta-hedging strategy for vanilla options in Black-Scholes-Merton (BSM) model and an extension of the Merton jump-diffusion (JDM) model assuming discrete trading and transaction costs 2) Examine the connection between the realized variance and the realized P&L 3) Find approximate solutions for the P&L volatility and the expected total transaction costs 4) Apply the mean-variance analysis to find the trade-off between the costs and P&L variance given hedger's risk tolerance 5) Consider hedging strategies to minimize the jump risk

1. An Approximate Distribution of Delta-Hedging Errors in a
Jump-Diusion Model with Discrete Trading and Transaction
Costs
Artur Sepp
Bank of America Merrill Lynch
Artur.Sepp@baml.com
Financial Engineering Workshop
Cass Business School, London
May 13, 2010
1

2. Motivation
1) Analyse the distribution of the pro

3. tloss (PL) of delta-hedging
strategy for vanilla options in Black-Scholes-Merton (BSM) model
and an extension of the Merton jump-diusion (JDM) model assuming
discrete trading and transaction costs
2) Examine the connection between the realized variance and the
realized PL
3) Find approximate solutions for the PL volatility and the expected
total transaction costs;
Apply the mean-variance analysis to

4. nd the trade-o between the
costs and PL variance given hedger's risk tolerance
4) Consider hedging strategies to minimize the jump risk
2

5. References
Theoretical and practical details for my presentation can be found in:
1) Sepp, A. (2012) An Approximate Distribution of Delta-Hedging
Errors in a Jump-Diusion Model with Discrete Trading and Trans-
action Costs, Quantitative Finance, 12(7), 1119-1141
http://ssrn.com/abstract=1360472
2) Sepp, A. (2013) When You Hedge Discretely: Optimization of
Sharpe Ratio for Delta-Hedging Strategy under Discrete Hedging and
Transaction Costs, Journal of Investment Strategies, 3(1), 19-59
http://ssrn.com/abstract=1865998
3

6. Literature
Discrete 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

7. Plan of the presentation
1) The PL in the BSM model with trading in continuous and dis-
crete time
2) Extended Merton jump-diusion model
3) Analytic approximation for the probability density function (PDF)
of the PL
4) Illustrations
5

8. BSM model
Asset price dynamics under the BSM:
dS(t) = S(t)adW(t); S(t0) = S
Option value U()(t; S), valued with volatility , solves the BSM PDE:
U
()
t +
1
2
2S2U
()
SS = 0; U()(T; S) = u(S)
()
S (t; S):
The delta-hedging portfolio with option delta ()(t; S) U
(h;i)(t; S) = S(t)(h)(t; S) U(i)(t; S)
In general, we have three volatility parameters:
i - for computing option value (implied volatility),
h - for computing option delta (hedging volatility),
a - for the assumed dynamics of the spot S(t) (realized volatility).
Why dierent?:
i - market consensus
h - proprietary view on option delta, risk management limits
a - guess estimated by proprietary and subjective methods
6

9. Illustration using the S500 data
i is illustrated by the VIX - market consensus about the one-month
fair volatility (model-independent)
h is illustrated by the at-the-money one-month implied volatility
a is illustrated by the one-month realized volatility
Time Series from Jan-07 to Mar-10
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
1-day Realized Volatility
1-month Realized Volatility
1-month ATM Volatility
VIX
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
Empirical Frequency
18%
16%
14%
12%
10%
8%
6%
4%
2%
0%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Observed Volatility
Frequency
ATM 1-month volatility
Realized 1-day volatility
In general, the implied volatility is traded at the premium to the real-
ized average volatility
But, over short-term periods, the realized volatility can be jumpy,
which causes signi

10. cant risks for volatility sellers
7

11. Impact on the PL when using dierent volatilities
One-period PL at time t+t and spot S+S with

12. xed (h)(t; S):
(h;i)(t) (h;i)(t+t; S +S) (h;i)(t; S)
= S(h)(t; S)
U(i)(t+t; S +S) U(i)(t; S)
Apply Taylor expansion and use BSM PDE:
(h;i)(t) S
(h)(t; S) (i)(t; S)
+t
2
i
(i)(t; S)
()(t; S) = 1
2S2(t)U
()
SS (t; S) is cash-gamma
= 1
t
S
S
2
is the realized variance
In the limit: ! 2
a as t ! 0
The stochastic term in the PL is caused by mis-match between the
implied and hedging volatilities
The deterministic term in the PL is caused by mis-match between
the implied and actual volatilities
8

13. Analysis of the total PL, P(T):
P(T) = lim
N!1
NX
n=1
(h;i)(tn)
1) BSM case: h = i = a thus (a;a)(t) = 0; P(T) = 0
No pain - no gain
2) Delta-hedge at implied volatility: h = i thus
(i;i)(t) = t
2
i 2
a
(i)(t; S);
P(T) =
2
i 2
a
Z T
t0
(i)(t; S)dt
No pain across the path - unpredictable gain at expiry
3) Delta-hedge at the asset volatility: h = a, thus
(a;i)(t) = S
(a)(t; S) (i)(t; S)
;
P(T) = U(i)(t0; S) U(a)(t0; S)
Pain across the path - predictable gain at expiry
9

14. Illustration of a PL path
PL path
0.30%
0.25%
0.20%
0.15%
0.10%
0.05%
0.00%
0.00 0.01 0.02 0.02 0.03 0.04 0.05 0.06 0.06 0.07 0.08
-0.05%
-0.10%
-0.15%
t
PL, %
PL, Hedge at Pricing Volatility
PL, Hedge at Actual Volatility
i = 23% , a = 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 proprietary
view on option delta is allowed
In the sequel, we consider the case with hedging at implied volatility
10

15. The PL in the discrete time
Assume that the number of trades N is

16. nite and the delta-hedging is
rebalanced at times ftng, n = 0; 1; :::;N, with intervals tn = tn tn1
Recall that, under the continuous trading, the

17. nal PL is
P(T) =
2
i 2
a
Z T
t0
(i)(t; S)dt
Is it correct?:
P(T; ftng) =
2
i 2
a
NX
n=1
(i)(tn1; S)tn
The correct form is:
P(T; ftng) =
NX
n=1
2
i tn 2
n
(i)(tn1; S)
where n =
S(tn)S(tn1)
S(tn1)
2
is the discrete realized variance
Recall that the standard deviation p
of a constant volatility parameter
measured discretely is =
2N
11

18. The Expected PL
Assume that hedging intervals are uniform with tn t = T=N
The expected PL:
M1(ftng) =
2
i 2
a
NX
n=1
t(i)(tn1; S)
2
i 2
a
T(i)(t0; S)
In terms of option vega V(t0; S), V(t0; S) = 2e
T(i)(t0; S):
M1(ftng) (i a) V(t0; S)
The expected PL equals to the spread between the implied and the
actual volatilities multiplied by the option vega
The expected PL does not depend on the hedging frequency!
12

19. The PL volatility
Under the continuous hedging, the PL volatility PL is aected
only by the volatility of the asset price:
PL C ji aj V(t0; S);
where C is a small number, C C(i; a; t0; S)
In the ideal BSM model with i = a, PL = 0
Under the discrete hedging, the PL volatility PL is aected pri-
marily by the volatility of the realized variance:
PL 2
a
s
1
2N
V(t0; S)
i
For short-term at-the-money options:
PL
U(i)(t0; S)
2
a
2
i
s
1
2N
The PL volatility is signi

20. cant for short-term options relative to
their value if N is small
13

21. Risks of the delta-hedging strategy
1) Mis-speci

22. cation of volatility parameter (generally, mis-speci

23. cation
of the model dynamics)
2) Discrete hedging frequency
3) Transaction costs
4) Sudden jumps in the asset price
5) Volatility of the actual volatility a (volatility of volatility)
For short-term options (with maturity up to one month), 1)-4) are
signi

24. cant
For longer-term options, 5) is important while 2) is less signi

25. cant
In this talk, we concentrate on aspects 1)-4)
14

26. PL distribution, Recall that the expected PL does not depend
on the hedging frequency: M1(ftng) (i a) V(t0; S)
We need to understand how the discrete hedging and jumps aect
the higher moments of the PL: volatility, skew, kurtosis
For this purpose we study the approximate distribution of the PL
rather than its higher moments (the latest is the common approach
in the existing studies)
PL distribution of Delta-Hedging Stategy
Frequency
ContinuousHedging
DiscreteHedging
PL kurtosis (ActualVol,HedgeVol, N)
PL volatility (ActualVol, HedgeVol, N)
PL skew (ActualVol, HedgeVol, N)
Expected PL =(HedgeVol-ActuaVol)*Vega(HedgeVol)
15

27. Path-dependence of the PL
Under the discrete hedging, the PL is a path-depended function:
P(T; ftng) =
NX
n=1
2
i tn 2
n
(i)(tn1; S)
where n =
S(tn)S(tn1)
S(tn1)
2
is the realized variance
Direct analytical methods are not applicable
What if we use the

28. rst order approximation in (i)(tn1; S)?:
P(T; ftng)
NX
n=1
2
i tn 2
n
G(i)(tn1; S)
where G(i)(tn1; S) = E[(i)(tn1; S)]
The approximate PL is a function of the discrete realized variance
The

29. rst moment of the PL is preserved, how is about its variance?
16

30. MC Simulations of the PL
Apply Monte Carlo to simulate a path of fSng, n = 1; :::N, to compute
PL(T;N) =
NX
n=1
(Sn Sn1)(tn1; Sn1) (U(tn; Sn) U(tn1; Sn1))
RealizedVar(T;N) =
NX
n=1
(Sn Sn1)
Sn1
!2
RealizedGamma(T;N) =
T
N
NX
n=1
(tn1; Sn1)
RealizedVarGamma(T;N) =
NX
n=1
0
@T2
i
N
(Sn Sn1)
Sn1
!2
1
A(tn1; Sn1)
RealizedVarGamma is an approximation to the PL, how is about
RealizedVar?
17

31. MC Simulations of the PL
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 same model parameters are used to compute the delta-hedge
and simulate S(t) using three models:
1) BSM with = 21:02%
2) Merton jump-diusion (MJD) with = 19:30%, = 1:25, =
7:33%, = (1:27%)2
3) Heston SV with V (0) = = (21:02%)2, = 3, = 40%, = 0:7
In MJD: 2
i = 2 +(2 +); in Heston SV: 2
i = V (0)
In MJD and Heston, option value, delta, and gamma are computed
using ecient Fourier inversion methods by Lipton and Lewis
Simulated realizations are normalized to have zero mean and unit
variance (with total of 10,000 paths)
18

32. The PL frequency in the BSM model
N=5
0.10
0.08
0.06
0.04
0.02
0.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Observations
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
N=20
0.10
0.08
0.06
0.04
0.02
0.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Observations
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
N=80
0.10
0.08
0.06
0.04
0.02
0.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Observations
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
N=320
0.10
0.08
0.06
0.04
0.02
0.00
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
-5 -4 -3 -N2orma-1lized O0bserv1ation2s 3 4 5
19

33. The PL frequency in the Merton jump-diusion model
N=5
0.20
0.15
0.10
0.05
0.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Observations
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
N=20
0.20
0.15
0.10
0.05
0.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Observations
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
N=80
0.50
0.40
0.30
0.20
0.10
0.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Observations
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
N=320
0.50
0.40
0.30
0.20
0.10
0.00
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
-5 -4 -3 -N2orma-1lized O0bserv1ation2s 3 4 5
20

34. The PL frequency in the Heston SV model
N=5
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Observations
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
N=20
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Observations
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
N=80
0.10
0.08
0.06
0.04
0.02
0.00
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Observations
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
N=320
0.10
0.08
0.06
0.04
0.02
0.00
Frequency
Realized PL
Realized Variance
Realized Gamma
Realized
VarianceGamma
-5 -4 -3 -N2orma-1lized O0bserv1ation2s 3 4 5
21

35. Observations
The realized variance gamma approximates the realized PL closely
if hedging is daily or more frequent
The frequency of realized variance scales with the frequency of the
PL
This is the key ground for our approximation: use the PDF of the
realized variance to approximate the PDF of the realized PL
The key tool: the Fourier transform of the PDF of the discrete real-
ized variance
On side note: the PL distribution in the BSM and Heston is close
to the normal, while that in the MJD is peaked at zero with heavy
left tail
22

36. Extended Merton Model. The JDM under the historic measure P:
dS(t)=S(t) = (t)dt+(t)dW(t)+
eJ 1
dN(t); S(t0) = S;
N(t) is Poisson process with intensity (t), jumps have the PDF w(J):
w(J) =
XL
l=1
pln(J; l; l);
XL
l=1
pl = 1;
L, L = 1; 2; ::, is the number of mixtures
pl, 0 pl 1, is the probability of the l-th mixture
l and l 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 e
(t),
jump intensity e
(t), risk-neutral drift e
(t):
e(t) = r(t) d(t) e
(t)JQ;
ew
(J) =
XL
l=1
epln(J; el; el);
XL
l=1
epl = 1;
JQ
Z 1
1
eJ 1
ew
(J)dJ =
XL
l=1
epleel+1
2el 1
23

37. Notations
De

38. ne the time-integrated quantities, with 0 t0 t
The variance under P and Q, respectively:
#(t; t0) =
Z t
t0
2(t0)dt0; e #(t; t0) =
Z t
t0
e
2(t0)dt0
The risk-free rate and dividend yield:
r(t; t0) =
Z t
t0
r(t0)dt0; d(t; t0) =
Z t
t0
d(t0)dt0
The drift under P and Q, respectively:
(t; t0) =
Z t
t0
(t0)dt0; e
(t; t0) =
Z t
t0
(t0)dt0
e
The intensity rate under P and Q, respectively:
(t; t0) =
Z t
t0
(t0)dt0; e
(t; t0) =
Z t
t0
(t0)dt0
24

39. Transition PDF of x(t), x(t) = ln S(t) and X = x(T):
1X
GX(T;X; t; x) =
m1=0
:::
1X
mL=0
P(m1; p1(T; t)):::P(mL; pL(T; t))n(X; ;

40. );
(m1; :::;mL) = x+(T; t)
1
2
#(T; t)+
XL
l=1
mll;

41. (m1; :::;mL) = #(T; t)+
XL
l=1
mll;
where n(x; ;

42. ) is the PDF of a normal with mean and variance

43. P(m; ), P(m; ) = em=m!, is the Poisson probability
Call options are valued by the extension of Merton formula (1976):
U(t; S; T;K) =
1X
m1=0
:::
1X
mL=0
P(m1; p1
e
(T; t)(JQ +1)):::P(mL; pL
e
(T; t)(JQ +1))
C(BSM)(S;K; 1; e

44. (m1; :::;mL); e
(m1; :::;mL); d(T; t));
e
(m1; :::;mL) = r(T; t) e
(T; t)JQ +
XL
l=1
mlel +
1
2
XL
l=1
ml el; e

45. (m1; :::;mL) = e #(T; 25

46. Empirical Estimation
Use closing levels of the SP500 index from January 4, 1999, to
January 9, 2009, with the total of 2520 observations
Fit empirical quantiles of sampled daily log-returns to model quantiles
by minimizing the sum of absolute dierences between the two
Assume time-homogeneous parameters, = 0, and jump variance l
uniform across dierent states
The best

47. t, keeping L as small as possible, is obtained with L = 4
26

48. Empirical Estimation
Model estimates: = 0:1348, = 46:4444
l pl l
p
l pl expected frequency
1 0.0208 -0.0733 0.0127 0.9641 every year
2 0.5800 -0.0122 0.0127 26.9399 every two weeks
3 0.3954 0.0203 0.0127 18.3661 every six weeks
4 0.0038 0.1001 0.0127 0.1743 every

49. ve years
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10
X
Frequency
Empirical PDF
Normal PDF
JDM PDF
0.12
0.08
0.04
0.00
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
-0.04
-0.08
-0.12
JDM quantiles
Data quantiles
27

50. Assumptions for the delta-hedging strategy
1) The synthetic market is based on the continuous-time model of the
asset price dynamics speci

51. ed by either the diusion model (DM) or
the jump-diusion model (JDM) with known model parameters under
P and Q
2) The option value and delta are computed using model parameters
under Q; the PDF of the PL is computed using model parameters
under P
3) The hedging strategy is time based with the set of trading times
ftng, trading intervals (not necessarily uniform) de

52. ned by ftng,
where tn = tn tn1, n = 1; :::;N, and N is the total number of
trades, t0 = 0, tN = T
e
e
Given the discrete time grid ftng, we de

53. ne: n = (tn; tn1), #n =
#(tn; tn1), #n e = #(tn; e tn1), n = (tn; tn1), n = (tn; tn1)
28

54. Discrete model
Discrete version of the DM under P on the grid ftng:
S(tn) = S(tn1) exp
n
1
2
#n +
p
#nn
;
where fng is a collection of independent standard NRVs, n = 1; :::;N
For small n and
p
#n are small, we approximate:
S(tn) S(tn1)
1+n +
p
#nn
Hedging portfolio (tn; S) at rebalancing time tn:
(tn; S) = S(tn)(tn; S) U(tn; S)
The PL at time tn, n:
n
1
2
e #n 2
n
S2(tn1)USS(tn1; S);
2
n
(S(tn) S(tn1))2
S2(tn1)
n +
p
#nn
2
29

55. Transaction costs
Proportional costs is the bid-ask spread:
= 2
Sask Sbid
Sask +Sbid
;
where Sask and Sbid are the quoted ask and bid prices, respectively
Thus, =2 is the average percentage loss per trade amount and:
Sask(t) = (1+=2)S(t); Sbid(t) = (1 =2)S(t)
n is transaction costs due to rebalancing of delta-hedge:
n (=2)S(tn) j(tn; S) (tn1; S)j
30

56. Approximation for the transaction costs
Apply Taylor expansion for US(tn; S):
n (=2)S(tn)USS(tn1; S) jSnj
Apply approximation for Sn, we obtain:
n (=2)S2(tn1)USS(tn1; S)

61. 1
2
+n +
p
#nn
2
1
4

66. De

67. ne the following function:
F(;

68. ) E

73. 1
2
++
q

74. 2
1
4

79. #
=
s
2

80. (+1)e
2
2

81. e
(1+)2
2

82. !
+2(

83. +2 +)
1
2
+N
p

84. !
N
1+
p

85. !!
Approximate n by (the

86. rst-order moment matching):
n
1
2
nS2(tn1)USS(tn1; S);
where n = n2
n; n F (n; #n)
31

87. Approximation for the total PL
The total PL, P(ftng), is obtained by accumulating n:
P(ftng)
NX
n=1
(n n)
Apply our approximations:
P(ftng)
NX
n=1
e #n 2
n n
(tn1; S);
where (t; S) is the option cash-gamma: (t; S) = 1
2S2(t)USS(t; S)
Recall that 2
n
n +
p
#nn
2
Explicitly, the PL is a quadratic function of fng, n = 1; :::;N:
P(ftng) =
NX
n=1
M(n;An;Bn;Cn;(tn1; S));
M(; A;B;C; o) o
A+C+B2
An = e #n +2
n, Cn = 2n
p
#n, Bn = (#n +n)
32

88. Expected Cash-Gamma
For vanilla calls and puts with strike K:
(t; S) =
K
2
q
2 e #(T; t)
exp
8
:
1
2 e #(T; t)
ln
S
K
1
2
e #(T; t)
2
9=
;
Under P:
S(tn) = S(t0) exp
(tn; t0)
1
2
#(tn; t0)+
q
#(tn; t0)n
where n is a standard NRV
The expected cash gamma under P, G(T; t):
G(tn; t0) EP [(tn; S)] = En [(tn; S) j S = S(tn)]
K
=
q
2(t0; tn; T)
2
exp
(
1
2(t0; tn; T)
ln
S
K
1
2
(t0; tn; T)
2)
;
where (t0; tn; T) = #(tn; t0)+ e #(T; tn)
If model parameters under Q and P are equal: G(tn; t0) = (t0; S)
33

89. The Fourier transform of the quadratic function:
Z(A;B;C; o) E
h
eM(;A;B;C;o)
i
=
1
p
2
Z 1
1
exp
(
oA oCx oBx2
x2
2
)
dx
=
1
p
2oB +1
exp
8
:
1
2(oC)2
2oB +1
oA
9=
;
Consider the following sum:
I(N) =
NX
n=1
M(n;An;Bn;Cn; on)
where fng is a set of independent standard NRVs, An;Bn;Cn are reals
and on is complex with [onBn] 1=2, n = 1; :::;N
Compute the Fourier transform of I(N), GI:
h
i
NY
GI Efng
eI(N)
=
n=1
En
h
eM(n;An;Bn;Cn;on)
i
=
NY
n=1
Z(An;Bn;Cn; on)
= exp
8
:
NX
n=1
0
@
1
2(onC)2
n
2onBn +1
onAn
1
A
9=
;
NY
n=1
1
p
2onBn +1
34

90. The Fourier transform of the PL distribution
Denote the PDF of the PL P(ftng) by GP (P0; ftng)
Denote its Fourier transform under P by bG
P( ; ftng) with = R+
i I, where R and I are reals and i =
p
1:
bG
P( ; ftng) EP
h
e P(ftng)
i
= EP
2
4exp
8
:
NX
n=1
M(n;An;Bn;Cn;1)(tn1; S)
9=
3
5
;
Approximate bG
P by:
P( ; ftng) EP
bG
2
4exp
8
:
NX
n=1
M(n;An;Bn;Cn;1)EP
(tn1; S)
9=
;
3
5
= EP
2
4exp
8
:
NX
n=1
M(n;An;Bn;Cn;
n)
9=
3
5
;
=
NY
n=1
Z(An;Bn;Cn;
n);
where
n = G(tn1; t0)
35

91. The Fourier transform of the PL distribution
Simplify:
bG
P( ; ftng) exp
8 :
NX
n=1
n2
n
2
n(#n +n)+1
+ e #n
n
!9=
;
NY
n=1
1
q
2
n(#n +n)+1
This expression can be recognized as the characteristic function of
non-central chi-square distribution with in-homogeneous drift, vari-
ance, and shift parameters.
36

92. The jump-diusion model. The Fourier transform of the approxi-
mate PL distribution is (for derivation, see my paper):
bG
P( ; ftng) EP
h
e P(ftng)
i
NY
n=1
1X
m1=0
:::
1X
mL=0
P(m1; p1n):::P(mL; pLn)Z(An f
2
n;Bn;Cn;G(tn1; t0) );
An An(m1; :::;mL) = 2
n(m1; :::;mL);
Bn Bn(m1; :::;mL) =

93. n(m1; :::;mL);
Cn Cn(m1; :::;mL) = n(m1; :::;mL)
q

94. n(m1; :::;mL);
n(m1; :::;mL) = n +(m1; :::;mL);

95. n(m1; :::;mL) = #n +(m1; :::;mL)+n(m1; :::;mL);
n(m1; :::;mL) F (n +(m1; :::;mL); #n +(m1; :::;mL)) ;
XL
(m1; :::;mL)
l=1
mll; (m1; :::;mL)
XL
l=1
mll;
f
2
n = e #n + e
n
XL
l=1
epl
e2el+2el 2eel+1
2el +1
37

96. Implications 0
Using the Fourier transform of the approximate distribution of the
PL, we can compute the approximate PDF of P:
GP (P0; ftng) =
1
Z 1
0
h
e P0 bG
P( )
i
d I;
where we apply standard FFT methods to compute the inverse Fourier
transform
bG
Also, we can compute the m-th moment of P by:
EP
@m (P(ftng))m= (1)m P( ; ftng)
@ mR

101. R=0; I=0
; m = 1; 2; ::::
38

102. Implications I
The expected PL:
M1(ftng) =
NX
n=1
e #n #n n 2
n
G(tn1; t0)
If the hedging frequency is uniform with t = T=N and model param-
eters are constant:
M1(ftng)
0
@e
2 2
s
2N
T
2T
N
1
AT(t0; S)
The break-even volatility e
, so that M1(ftng) = 0:
e
2 = 2 +
s
2N
T
;
which is Leland (1985) volatility adjustment for a short option position
39

103. Implications II
Assume t tn = T=N and parameters under Q and P are equal with:
n =
Z tn
tn1
dt t; #n = e #n =
Z tn
tn1
2dt 2t
The Fourier transform of the PL distribution
bG
P( ; ftng) =
1
(2 +1)N=2
exp
2 +1
+s
!
= c(t0; S) ; c =
2T
N
s
2
+
2T
N
; =
2T2
Nc
; s =
2T
c
:
This is the Fourier transform of the non-central chi-square distribution
with N degrees of freedom, non-centrality parameter , (y;N; ):
GP (P0; ftng) =
1
ec
es P0
ec
;N;
!
1fP0esg
where es = 2T(t0; S), ec = c(t0; S)
40

104. Implications III
The PL M1:
M1 = (t0; S)
0
@2T2
N
s
22TN
+
1
A
The PL volatility M2 is inversely proportional to
p
N:
(M2)2 = 2(t0; S)(T2)
0
@2T2
N
+
p
2T2
p
N
4
+
42
+O
1
N3=2
1
A
The skewness M3 and the excess kurtosis M4:
M3 =
23=2
p
N
+O
1
N
; M4 =
12
N
+O
1
N2=3
As N ! 1, the chi-squared distribution converges to normal:
GP (P0; ftng) n(P0;M1; (M2)2)
41

105. Jump-diusion model I
The expected PL, M1(ftng):
M1(ftng) =
NX
n=1
e #n #n + e
nEQ[J2] nEP[J2] 2
n Kn
G(tn1; t0)
EQ[J2]
Z 1
1
eJ 1
2
ew(J)dJ
=
XL
l=1
epl
e2el+2el 2eel+1
2el +1
XL
l=1
epl
e2
l + el
EP[J2] computed under P by analogy
Kn is the expected proportional transaction costs
Kn
1X
m1=0
:::
1X
mL=0
P(m1; p1n):::P(mL; pLn)
F(n +(m1; :::;mL); #n +(m1; :::;mL))
The hedger should match e #n and #n, EQ[J2] and EP[J2], e
n and n
42

106. Jump-diusion model II
Assume constant model parameters equal under Q and P, t = T=N,
a single jump component, L = 1, with mean and variance and
Approximate the PDF of the Poisson process by Bernoulli distribution
taking P(0; n) = 1 n, P(1; n) = n, and P(m; n) = 0 for m 1
The expected total transaction costs K(N):
K(N) (t0; S)
NX
n=1
Kn (t0; S)
0
@
p
N
s
22T
+T
s
2(2 +)
1
A
The

107. rst term is the expected transaction costs due to the diusion
The second term represents those due to jumps
43

108. Jump-diusion model III
If the hedging frequency is uniform with t = T=N and model param-
eters are constant:
M1(ftng)
0
@e
2 2 + e
(e2 +) (2 + e)
0
@
s
2N
T
s
2(2 +)
1
A
1
A
T(t0; S)
The break-even volatility e
, so that the expected PL is zero:
e
2 = 2 +(2 +) e
(e2 + e)+
s
2N
T
+
s
2(2 +)
;
which is can be interpreted as the Leland volatility adjustment for the
Merton JDM
44

109. Jump-diusion model IV
The expected PL without transaction costs:
M1 = (t0; S)
2T2
N
+
2T2
N
!
The PL volatility:
(M2)2 = 2(t0; S)
T
4 +32 +32
+
^c
N
+O
1
N2
where ^c =
24 +2(2 +) 2(2 +)2 +(6 +43)
T2
When jumps are present the volatility of the PL cannot be reduced
by increasing the hedging frequency
The constant term is proportional to expected number of jumps T
The sign of the jump is irrelevant
The skewness M3 and the excess kurtosis M4:
lim
N!1
M3 = O
1
p
T
!
; lim
N!1
M4 = O
1
T
45

110. Mean-Variance Analysis I
The hedger's objective is to minimize utility function:
min
N
(
K(N)+
1
2
V(N)
)
where
is the risk-aversion tolerance, K(N) is the expected total
hedging costs and V(N) is the variance of the PL
We have shown that for both DM and JDM models:
K(N) = k0 +k1
p
N; V(N) = v0 +
v1
N
Thus, the optimal solution N is given by:
N =
v1
k1
!2=3
For the DM: N = (2)1=32T
(t;S)
2=3
,
so that N is proportional to variance 2T and (t0; S), and inversely
proportional to the risk-aversion tolerance and transaction costs
46

111. Mean-Variance Analysis II
450
400
350
300
250
200
150
100
50
0
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
Optimal Frequency, N*
N*, DM
N*, JDM
30%
25%
20%
15%
10%
5%
0%
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
Costs
180%
160%
140%
120%
100%
80%
60%
40%
20%
0%
Volatility
Costs K(N*), DM
Costs K(N*), JDM
Volatility V(N*), DM
Volatility V(N*), JDM
Left side: the optimal hedging frequency N, Right side: the expected
transaction costs K(N) normalized by the option value as function
of the risk-aversion tolerance and the PL volatility
q
V(N) under
the DM (DM) and JDM (JDM) using = 0:002 for a put option with
S(0) = K = 1:0 and maturity T = 0:08
For a small level of the risk tolerance, the expected transaction costs
amount to about 20 30% of the option premium
47

112. PL Distribution. Illustrations
Use two models:
1) BSM with = 21:02%
2) Merton jump-diusion (MJD) with = 19:30%, = 1:25, =
7:33%, = (1:27%)2
Apply two cases:
1) transaction costs are zero
2) transaction costs with rate = 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 PL 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

113. PL Distribution in the DM without transaction costs
PL distribution, N=5
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
P'
-140% -100% -60% -20% 20% 60% 100%
Frequency
0.12
MonteCarlo
Analytic PL distribution, N=20
0.10
0.08
0.06
0.04
0.02
0.00
P'
-140% -100% -60% -20% 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=80
0.20
0.16
0.12
0.08
0.04
0.00
-140% -100% -60% -20% P' 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=320
0.40
0.30
0.20
0.10
0.00
-140% -100% -60% -20% P' 20% 60% 100%
Frequency
MonteCarlo
Analytic
49

114. PL Distribution in the JDM without transaction costs
PL distribution, N=5
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-180% -140% -100% -60% -20% P' 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=20
0.12
0.10
0.08
0.06
0.04
0.02
0.00
P'
-180% -140% -100% -60% -20% 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=80
0.20
0.16
0.12
0.08
0.04
0.00
P'
-180% -140% -100% -60% -20% 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=320
0.40
0.30
0.20
0.10
0.00
P'
-180% -140% -100% -60% -20% 20% 60% 100%
Frequency
MonteCarlo
Analytic
50

115. PL Distribution in the DM with transaction costs
PL distribution, N=5
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-180% -140% -100% -60% -20% P' 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=20
0.12
0.10
0.08
0.06
0.04
0.02
0.00
P'
-180% -140% -100% -60% -20% 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=80
0.16
0.12
0.08
0.04
0.00
P'
-180% -140% -100% -60% -20% 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=320
0.30
0.20
0.10
0.00
P'
-180% -140% -100% -60% -20% 20% 60% 100%
Frequency
MonteCarlo
Analytic
51

116. PL Distribution in the JDM with transaction costs
PL distribution, N=5
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-180% -140% -100% -60% -20% P' 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=20
0.12
0.10
0.08
0.06
0.04
0.02
0.00
P'
-180% -140% -100% -60% -20% 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=80
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
P'
-180% -140% -100% -60% -20% 20% 60% 100%
Frequency
MonteCarlo
Analytic
PL distribution, N=320
0.30
0.25
0.20
0.15
0.10
0.05
0.00
P'
-180% -140% -100% -60% -20% 20% 60% 100%
Frequency
MonteCarlo
Analytic
52

117. Illustration
PL volatility
80%
60%
40%
20%
0%
1m 1w 1d 2h 30m 8.5m 2m 0.5m
Hedging intervals
volatility
Volatility, DM no TRC
Volatility, DM with TRC
Volatility, JDM no TRC
Volatility, JDM with TRC
PL mean return
0%
-40%
-80%
-120%
-160%
-200%
1m 1w 1d 2h 30m 8.5m 2m 0.5m
Hedging intervals
return
Mean, DM with TRC
Mean, JDM with TRC
Left: the PL volatility under the DM (Volatility, DM no TRC) and
JDM (Volatility, JDM no TRC) with no costs, and the PL volatility
under the DM (Volatility, DM with TRC) and JDM (Volatility, JDM
with TRC) with proportional costs; Right: the mean of the PL
distribution under the DM (Mean, DM with TRC) and JDM (Mean,
JDM with TRC) with proportional costs. The x-axes is the hedging
frequency: 1m - the DHS is applied only t = 0, 1w - weekly, 1d -
daily, 2h - every two hours, 30m - every thirty minutes, 8.5m - every
eight minutes, 2m - every two minutes, 0.5m - every thirty seconds.
53

118. Jump Risk Hedge (Andersen-Andreasen (2000))
De

119. ne the jump risk U(t; S; T;K) under Q as follows:
U(t; S; T;K) =
Z 1
1
U(t; SeJ; T;K)ew
(J)dJ U(t; S; T;K)
Use series formula to obtain:
1X
U(t; S; T;K) =
m1=0
:::
1X
mL=0
P(m1; p1
e
(T; t)(JQ +1)):::P(mL; pL
e
(T; t)(JQ +1))
C(BSM)((JQ +1)S;K; 1; e

120. (m1 +1; :::;mL +1); e
(m1; :::;mL); d(T; t))
C(BSM)(S;K; 1; e

121. (m1; :::;mL); e
(m1; :::;mL); d(T; t))
The hedging portfolio (t; S) to hedge U(1)(t; S):
(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 option
here, either K(1)6= K(2) or T(1)6= T(2)
54

122. Jump Risk Hedge
Require (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)
(1)
SS (t; S)
U
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 = 0:002
55

123. Jump Risk Hedge
PL distribution, N=5
0.30
0.25
0.20
0.15
0.10
0.05
0.00
P'
-80% -40% 0% 40%
Frequency
TRC
NOC
PL distribution, N=20
0.30
0.25
0.20
0.15
0.10
0.05
0.00
P'
-80% -40% 0% 40%
Frequency
TRC
NOC
PL distribution, N=80
0.30
0.25
0.20
0.15
0.10
0.05
0.00
P'
-80% -40% 0% 40%
Frequency
TRC
NOC
PL distribution, N=320
0.30
0.25
0.20
0.15
0.10
0.05
0.00
P'
-80% -40% 0% 40%
Frequency
TRC
NOC
56

124. Extensions. Variance swap. Consider a portfolio of delta-hedged
positions, (m)(tn; S), with weights w(m), in the same underlying:
(tn; S) =
MX
m=1
w(m)(m)(tn; S);
where (m)(tn; S) = S(tn)(m)(tn; S) U(m)(tn; S), m = 1; :::;M
The one-period PL:
(tn; S) =
MX
m=1
w(m)
e #
(m)
n 2
n
(m)(tn1; S) = A B2
n;
where A =
PMm
(m)
n (m)(tn1; S), B =
=1 w(m) e #
PMm
=1 w(m)(m)(tn1; S)
For variance swap with unit notional, A (AF=N2)2
k , where 2
k is
variance strike, and B AF=N (AF is annualization factor, N is
sampling frequency)
Obtained expressions for the approximate PL are exact for the PL
distribution of the variance swap (within our assumptions)
57

125. Conclusions
We have presented a quantitative approach to study the PL distri-
bution of the delta-hedging strategy assuming discrete trading and
transaction costs under the diusion model and jump-diusion model
The opinions expressed in this presentation are those of the author
alone and do not necessarily re
ect the views and policies of Bank of
America Merrill Lynch
58

126. References
Sepp, A. (2012) An Approximate Distribution of Delta-Hedging Errors
in a Jump-Diusion Model with Discrete Trading and Transaction
Costs, Quantitative Finance, 12(7), 1119-1141
http://ssrn.com/abstract=1360472
Sepp, A. (2013) When You Hedge Discretely: Optimization of Sharpe
Ratio for Delta-Hedging Strategy under Discrete Hedging and Trans-
action Costs, Journal of Investment Strategies, 3(1), 19-59
http://ssrn.com/abstract=1865998
59