A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing,...
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A Benchmark Approach to Investing,Pricing and Hedging
Eckhard PlatenUniversity of Technology Sydney
Pl. & Heath (2006, 2010), A Benchmark Approach to Quantitative Finance. Springer
Pl. & Bruti-Liberati (2010), Numerical Solution of Stochastic DifferentialEquations with Jumps in Finance. Springer
Baldeaux &Pl. (2013). Functionals of MultidimensionalDiffusions with Applications to Finance. Springer
1 Best Performing Portfolio as Benchmark
• with focus on long term
• if possible pathwise
• should prefer more for less
• diversified portfolio
• robust, almost model independent construction
• sustainable (strictly positive long only)
• criterion independent from denomination
• criterion independent from sequence of drawdown events
⇒maximize logarithm of portfolio
1
1973 1980 1988 1995 2003 20110
3
6
9
12
15x 10
4
EWI114MCI
EWI141 and MCI
2
1973 1980 1988 1995 2003 20111.5
2
2.5
3
3.5
4
4.5
5
5.5
6
log(EWI114)log(MCI)
logarithms of EWI114 and MCI
3
1976 1984 1993 2002 2011
0
0.02
0.04
0.06
0.08
0.10
0.12
long term growth EWI114long term growth MCI
Long term growth of EWI114 and MCI
4
Key idea:
Make the “best” performing portfolio Sδ∗t numeraire or benchmark
5
Portfolio:
Sδt =
d∑j=0
δjtSjt
Sjt - jth constituent (e.g. cum dividend stocks)
6
Long Term Growth:
limt→∞
1
tlog
(Sδt
Sδ0
)≤ lim
t→∞
1
tlog
(Sδ∗t
Sδ∗0
)a.s. pathwise
Sδ∗ is the numeraire portfolio (NP)Long (1990)
7
Main Assumption:
There exists NP Sδ∗t ,
which means
Sδt
Sδ∗t
≥ Et
(Sδs
Sδ∗s
)(∗)
for all 0 ≤ t ≤ s <∞ and all nonnegative portfolios Sδt ,
Et(·) - real world conditional expectation under information given at time tSδ∗t - numeraire portfolio (NP), see Long (1990)
8
Benchmarking:
make NP numeraire and benchmark
Sδt =
Sδt
Sδ∗t
− benchmarked portfolio
Sjt =
Sjt
Sδ∗t
− jth benchmarked security
9
Existence of NP ⇔ Supermartingale Property
Sδt ≥ Et(S
δs) (∗)
for all 0 ≤ t ≤ s <∞ and all nonnegative Sδt ,
• The key property of financial market!
• model independent
• makes also in the short term “best” performance of Sδ∗ precise
• existence of NP, extremely general, Karatzas and Kardaras (2007)
10
Outperforming Long Term Growth
• long term growth rate
gδ = lim supt→∞
1
tln
(Sδt
Sδ0
)
strictly positive portfolio Sδ
pathwise characteristic
11
Theorem The NP Sδ∗ achieves the maximum long term growth rate.For any strictly positive portfolio Sδ
gδ ≤ gδ∗ .
NP outperforms long term growth pathwise!
• Pathwise outperformance asymptotically over time!
• Ideal long term investment if no constraints!
12
Proof: Consider strictly positive portfolio Sδ , with Sδ0 = Sδ∗
0 = x >
0. By supermartingale property (∗) of Sδ and Doob (1953), for any k ∈1, 2, . . . and ε ∈ (0, 1) one has
expε kP(
supk≤t<∞
Sδt > expε k
)≤ E0
(Sδk
)≤ Sδ
0 = 1.
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For fixed ε ∈ (0, 1)
∞∑k=1
P
(sup
k≤t<∞ln(Sδt
)> εk
)≤
∞∑k=1
exp−ε k <∞.
By Lemma of Borel and Cantelli there exists kε such that for all k ≥ kε
and t ≥ kln(Sδt
)≤ ε k ≤ ε t.
14
Therefore, for all k > kε
supt≥k
1
tln(Sδt
)≤ ε,
which implies
lim supt→∞
1
tln
(Sδt
Sδ0
)≤ lim sup
t→∞
1
tln
(Sδ∗t
Sδ∗0
)+ ε.
Since the inequality holds for all ε ∈ (0, 1), theorem follows.
15
Best Performance of NP also in the Short Term
Outperforming Expected Benchmarked Returns
(∗)⇒ nonnegative Sδ is supermartingale and Sδ∗t = 1
=⇒
• expected benchmarked returns
Et
(Sδt+h − Sδ
t
Sδt
)≤ Et
(Sδ∗t+h − Sδ∗
t
Sδ∗t
)= 0
t > 0, h > 0, Sδ nonnegative
=⇒ No strictly positive expected benchmarked returns possible!
16
Outperforming Expected Growth
• expected growth
gδt,h = Et
(ln
(Sδt+h
Sδt
))for t, h ≥ 0.
17
invest in strictly positive portfolio Sδ and small fraction ε ∈ (0, 112
) insome nonnegative portfolio Sδ ,which yields perturbed portfolio Sδε
• derivative of expected growth in the direction of Sδ:
∂gδε
t,h
∂ε
∣∣∣∣ε=0
= limε→0+
1
ε
(gδε
t,h − gδt,h
)
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Definition Sδ is called growth optimal if
∂gδε
t,h
∂ε
∣∣∣∣ε=0
≤ 0
for all t and h ≥ 0.
19
• classical characterization: maximization of expected logarithmic utilityfrom terminal wealthKelly (1956), Latane (1959), Breiman (1960), Hakansson (1971a), Thorp(1972), Merton (1973a), Roll (1973) and Markowitz (1976)equivalent
20
Theorem
The NP is growth optimalif its expected growth is finite.
NP outperforms expected growth!
21
Proof: Pl. (2011)For t ≥ 0, h > 0, and a nonnegative portfolioSδ we introduce the portfolio
ratio Aδ,ht =
Sδt+h
Sδt
, which is set to 1 for Sδt = 0. For ε ∈ (0, 1
2); and
a nonnegative portfolio Sδ , with Sδt > 0, consider the perturbed portfolio
Sδε with Sδt = Sδ∗
t , yielding portfolio ratio Aδε
t,h = εAδt,h + (1 −
ε)Aδ∗t,h > 0. By the inequality ln(x) ≤ x− 1 for x ≥ 0,
Gδε
t,h =1
εln
(Aδε
t,h
Aδ∗t,h
)≤
1
ε
(Aδε
t,h
Aδ∗t,h
− 1
)=Aδ
t,h
Aδ∗t,h
− 1
and
Gδε
t,h = −1
εln
(Aδ∗
t,h
Aδε
t,h
)≥ −
1
ε
(Aδ∗
t,h
Aδε
t,h
− 1
)=Aδ
t,h −Aδ∗t,h
Aδε
t,h
.
22
Because ofAδε
t,h > 0 one obtains forAδt,h −A
δ∗t,h ≥ 0
Gδε
t,h ≥ 0,
and forAδt,h −A
δ∗t,h < 0 because of ε ∈ (0, 1
2) andAδ
t,h ≥ 0
Gδε
t,h ≥ −Aδ∗
t,h
Aδε
t,h
= −1
1− ε+ εAδ
t,h
Aδ∗t,h
≥ −1
1− ε≥ −2.
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Summarizing yields
−2 ≤ Gδε
t,h ≤Aδ
t,h
Aδ∗t,h
− 1,
where by (∗)
Et
(Aδ
t,h
Aδ∗t,h
)≤ 1.
24
By the Dominated Convergence Theorem
∂gδε
t,h
∂ε
∣∣∣∣ε=0
= limε→0+
Et
(Gδε
t,h
)= Et
(lim
ε→0+Gδε
t,h
)
= Et
(∂
∂εln
(Aδε
t,h
Aδ∗t,h
) ∣∣∣∣ε=0
)= Et
(Aδ
t,h
Aδ∗t,h
)− 1.
This proves that NP Sδ∗ is growth optimal.
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Systematic Outperformance
Definition A nonnegative portfolioSδ systematically outperforms a strictlypositive portfolio Sδ if
(i) Sδt0
= Sδt0
;
(ii) P (Sδt ≥ Sδ
t ) = 1 for some t > 0 and
(iii) P(Sδt > Sδ
t
)> 0.
• also called relative arbitrage in Fernholz & Karatzas (2005)
26
Theorem The NP cannot be systematically outperformedby any nonnegative portfolio.
• NP can systematically outperform other portfolios.
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Proof: Consider nonnegative Sδ with Sδt = 1 where Sδ
s ≥ 1 almostsurely at s ∈ [t,∞). By the supermartingale property (∗)
0 ≥ Et
(Sδs − S
δt
)= Et
(Sδs − 1
).
Since Sδs ≥ 1 and Et(S
δs) ≤ 1, it can only follow Sδ
s = 1.=⇒ Sδ
s = Sδ∗s .
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• NP needs shortest expected time to reach a given wealth levelKardaras & Platen (2010)
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Strong Arbitrage
• only market participants can exploit arbitrage with their total wealth
• limited liability
=⇒ nonnegative total wealth of each market participant
Definition A nonnegative portfolio Sδ is a strong arbitrage if Sδ0 = 0
and for some T > 0
P(SδT > 0
)> 0.
• Loewenstein & Willard (2000) - same notion of arbitragethrough economic arguments
• Pl. (2002)-mathematical motivationthrough supermartingale property (∗)
30
Theorem There is no strong arbitrage.
31
Proof:(∗) =⇒
Sδ nonnegative supermartingale
=⇒0 = Sδ
0 ≥ E(SδT
∣∣A0
)= E(Sδ
T )
=⇒P (Sδ
T > 0) = P (SδT > 0) = 0.
32
Classical notion of arbitrage is too restrictive
• Delbaen & Schachermayer (1998)
no free lunches with vanishing risk , existence of risk neutral measure, APT
• Loewenstein & Willard (2000)
free snacks & cheap thrills , some free lunch with vanishing risk
• exploiting weak forms of classical arbitrage requires to allow negativetotal wealth
• pricing via hedging by avoiding “strong arbitrage” makes no sensesince there is no “strong arbitrage” under (∗).
33
How to approximate the NP in the real market?
• diversification leads asymptotically to the NP (Pl. 2005)
• confirm empirically by visualizing diversification effect
• simplest diversification:
equal value weighting = naive diversification
34
Pl. & Rendek (2012)
• investment universe:
market capitalization weighted indices generating the MSCI
• Industry Classification Benchmark (Datastream, Reuters ...):
54 countries with country index
10 industry indices
19 supersector indices
41 sector indices
114 country subsector indices
all market capitalization weighted
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Equi-Weighted Index (EWId)
d constituents
SδEWIdtn
= SδEWIdtn−1
1 +d∑
j=1
ℓd,j∑k=1
πj,kδEWId,tn−1
Sj,ktn − S
j,ktn−1
Sj,ktn−1
,
• Naive diversification over ℓd,j countries
and then over d = 10, 19, 41, 114 world industries
• equal value weighting uncertainties of different economic activities
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1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 20120
50,000
100,000
150,000
MCI
EWI10EWI19
EWI41
EWI114
EWI1
The MCI and five equi-weighted indices: EWI1 (market), EWI10 (industry),EWI19 (supersector), EWI41 (sector), EWI114 (subsector).
37
1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 201210
1
102
103
104
105
106
MCIEWI1
EWI10
EWI19
EWI114
EWI41
The MCI and five equi-weighted indices in log-scale: EWI1 (market), EWI10(industry), EWI19 (supersector), EWI41 (sector), EWI114 (subsector).
38
• all based on the same country industry subsectors of the equity market
• all resulting indices mostly driven by nondiversifiable uncertainty
• Empirically: Better performance through better diversification!
39
Illustrating Diversification Phenomenon:
• return of benchmarked constituent over small period [t, t+ h]
Rj =Sjt+h − S
jt
Sjt
j ∈ 1, 2, . . .
40
Assume for illustration purposes: R1,R2, · · · independent, and
Et(Rj) = 0
Et
((Rj)2
)= σ2h
h > 0, j ∈ 1, 2, . . .
41
• return of portfolio equalssum of weighted returns of constituents
• return of benchmarked EWIℓ with ℓ constituents:
RδEWIℓ =SδEWIℓ
t+h − SδEWIℓt
SδEWIℓt
=1
ℓ
ℓ∑j=1
Rj
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=⇒
Et(RδEWIℓ) = 0
Et
((RδEWIℓ)2
)=
1
ℓ2
ℓ∑j=1
σ2h =1
ℓσ2h
=⇒
limℓ→∞
Et
((RδEWIℓ)2
)= 0
Strong Law of Large Numbers (Kolmogorov)⇒
limℓ→0
RδEWIℓ = 0
almost surely
43
for ℓ >> 1
=⇒ RδEWIℓ ≈ 0
=⇒ SδEWIℓt ≈ 1
=⇒ SδEWIℓt = SδEWIℓ
t Sδ∗t ≈ Sδ∗
t
a.s. for all t
44
Diversification yields approximation of the NP !
• diversification holds more generally
• model independent phenomenon
• arises naturally
• can be systematically exploited
45
Naive Diversification Theorem
Pl. & Rendek (2012)
• continuous market, Merton (1972)
• benchmarked constituents
dSjt
Sjt
=∑k
σj,kt dW k
t
W 1,W 2, . . . - Brownian motions
46
• fraction of wealth invested
πjδ,t =
δjt Sjt
Sδt
=δjtS
jt
Sδt
47
• benchmarked portfolio
dSδt
Sδt
=∑j
πjδ,t
dSjt
Sjt
=∑j
πjδ,t
∑k
σj,kt dW k
t
48
• return process of a benchmarked portfolio Sδt
dQδt =
1
Sδt
dSδt
t ≥ 0 with Qδ0 = 0
49
• Quadratic Variation of Return Process
⟨Qδℓ⟩t =∫ t
0
∑k
∑j
πjδℓ,sσj,ks
2
ds
• Tracking rate
Tδℓ(t) =d
dt⟨Qδℓ⟩t =
∑k
∑j
πjδℓ,tσj,kt
2
50
• benchmarked NP equals constant onereturn process of benchmarked NP equals zero
dSδ∗t
Sδ∗t
=∑k
∑j
πjδ∗,tσj,kt dW k
t = 0
=⇒ tracking rate
Tδ∗(t) =d
dt⟨Qδ∗⟩t = 0
51
• ℓth equi-weighted index (EWIℓ)
πjδEWIℓ,t
=
1ℓ
for j ∈ 1, 2, . . . , ℓ0 otherwise.
52
Definition:
A sequence (Sδℓ)ℓ∈1,2,... of strictly positive benchmarked portfolios, withinitial value equal to one, is called a sequence of benchmarked approxi-mate NPs if for each ε > 0 and t ≥ 0 one has
limℓ→∞
P
(Tδℓ(t) =
d
dt
⟨Qδℓ
⟩t> ε
)= 0.
53
• benchmark captures general, systematic or non-diversifiable market un-certainty
• benchmarked primary security accounts capture specific or idiosyncraticuncertainty
• different types of economic activity yield naturally different specific un-certainties
• specific uncertainty can be diversified
54
Particular specific uncertainty drives in reality usually only
returns of restricted number of benchmarked primary security accounts:
55
Definition:
A market is well-securitized if there exists q > 0 and square integrableσ2 = σ2
t , t ≥ 0 with finite mean for all ℓ, k ∈ 1, 2, . . . and t ≥ 0
1
ℓ
∣∣∣∣∣∣ℓ∑
j=1
σj,kt
∣∣∣∣∣∣2
≤1
ℓqσ2
t
P-almost surely.
• k-th uncertainty affects not too many constituents.
56
Naive Diversification Theorem:
In a well-securitized market the sequence of benchmarked equi-weightedindices is a sequence of benchmarked approximate NPs.
Pl. & Rendek (2012)
57
• return process of ℓth benchmarked EWI
QδEWIℓt =
ℓ∑j=1
1
ℓ
∑k
∫ t
0
σj,ks dW k
s
58
• quadratic variation of return process
⟨QδEWIℓ
⟩t=
1
ℓ
∫ t
0
∑k
∣∣∣ 1√ℓ
ℓ∑j=1
σj,ks
∣∣∣2ds• tracking rate
TδEWIℓ(t) =d
dt
⟨QδEWIℓ
⟩t=
1
ℓ
∑k
∣∣∣ 1√ℓ
ℓ∑j=1
σj,kt
∣∣∣2
59
Lemma Assume that for all ε > 0 and t ≥ 0
limℓ→∞
P
1
ℓ
ℓ∑k
∣∣∣ 1√ℓ
ℓ∑j=1
σj,kt
∣∣∣2 > ε
= 0,
then the sequence of equi-weighted indices is a sequence of benchmarkedapproximate NPs.
60
Proof:For ε > 0 and t > 0
0 = limℓ→∞
P
1
ℓ
∑k
∣∣∣ 1√ℓ
ℓ∑j=1
σj,kt
∣∣∣2 > ϵ
= lim
ℓ→∞P
(d
dt
⟨QδEWIℓ
⟩t> ε
)= lim
ℓ→∞P (TδEWIℓ(t) > ε)
61
Proof of Naive Diversification Theorem:
By Markov inequality in a well-securitized market
limℓ→∞
P
1
ℓ
∑k
∣∣∣ 1√ℓ
ℓ∑j=1
σj,kt
∣∣∣2 > ε
= lim
ℓ→∞P
(1
ℓqσ2
t > ε
)≤ lim
ℓ→∞
1
ε
1
ℓqE(σ2
t
)= 0,
NDT follows from previous Lemma.
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1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 201210
1
102
103
104
105
106
EWI114
EWI1145
EWI11480
EWI11440
EWI114200
MCIEWI114240
Logarithms of MCI, EWI114 without transaction cost and EWI114ξ withtransaction costs of 5,40,80,200 and 240 basis points.
63
1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 20120
50,000
100,000
150,000
EWI114
EWI1142
EWI1144
EWI11416
EWI11432
EWI114m reallocated daily and every 2, 4, 8, 16 and 32 days.
64
Transaction cost 0 5 40 80 200 240
Reallocation terms 1
Final value 139338.64 130111.93 80543.07 46555.04 8988.23 5194.46
Annualised average return 0.1979 0.1961 0.1834 0.1689 0.1254 0.1109
Annualised volatility 0.1135 0.1135 0.1135 0.1135 0.1134 0.1134
Sharpe ratio 1.4205 1.4046 1.2930 1.1654 0.7822 0.6544
Reallocation terms 2
Final value 124542.04 119369.00 88697.63 63166.73 22808.64 16240.40
Annualised average return 0.1949 0.1938 0.1859 0.1770 0.1500 0.1411
Annualised volatility 0.1134 0.1134 0.1134 0.1134 0.1135 0.1136
Sharpe ratio 1.3955 1.3856 1.3163 1.2369 0.9987 0.9193
Reallocation terms 4
Final value 111899.82 108230.16 85698.25 65628.82 29467.48 22562.42
Annualised average return 0.1921 0.1912 0.1850 0.1780 0.1568 0.1497
Annualised volatility 0.1135 0.1135 0.1134 0.1134 0.1134 0.1135
Sharpe ratio 1.3699 1.3622 1.3080 1.2459 1.0591 0.9967
65
Transaction cost 0 5 40 80 200 240
Reallocation terms 8
Final value 100505.37 97963.66 81881.57 66705.62 36055.10 29367.02
Annualised average return 0.1892 0.1885 0.1837 0.1783 0.1621 0.1566
Annualised volatility 0.1127 0.1127 0.1127 0.1127 0.1128 0.1128
Sharpe ratio 1.3531 1.3471 1.3051 1.2569 1.1119 1.0634
Reallocation terms 16
Final value 98775.24 96892.29 84677.43 72588.14 45711.97 39177.91
Annualised average return 0.1887 0.1882 0.1847 0.1806 0.1684 0.1643
Annualised volatility 0.1130 0.1130 0.1130 0.1130 0.1131 0.1131
Sharpe ratio 1.3463 1.3418 1.3102 1.2740 1.1647 1.1281
Reallocation terms 32
Final value 114592.50 112929.09 101939.59 90678.85 63804.84 56744.28
Annualised average return 0.1927 0.1923 0.1896 0.1865 0.1772 0.1741
Annualised volatility 0.1131 0.1131 0.1131 0.1131 0.1133 0.1133
Sharpe ratio 1.3797 1.3763 1.3522 1.3245 1.2408 1.2127
66
0 50 100 1500
100
200
300
400
500
600
GOP
EWI
MCI
Simulated GOP, EWI and MCI under the Black-Scholes model
67
0 50 100 1500.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
benchmarked MCI
benchmarked EWI
Simulated benchmarked GOP, EWI and MCI under the Black-Scholesmodel
68
Example of a Multi-Asset BS Model
• jth primary security account
dSjt = Sj
t
[(r + σ2
(1 +
1√d
))dt+
σ√d
d∑k=1
dW kt + σ dW j
t
]
t ∈ [0, T ], j ∈ 1, 2, . . . , d, σ > 0
general market risk, specific market risk
69
• NP fractions
πjδ∗,t
=(√
d(1 +√d))−1
j ∈ 1, 2, . . . , d
π0δ∗,t
=(1 +√d)−1
• NP SDE
dSδ∗t = Sδ∗
t
((r + σ2
)dt+
σ√d
d∑k=1
dW kt
)
• simulate over T = 32 years d = 50 risky primary security accounts
σ = 0.15, r = 0.05
70
• Benchmarked primary security accountBS - model
dSjt = d
(Sjt
Sδ∗t
)= Sj
tσdWjt
martingale
71
0
5
10
15
20
25
0 5 10 15 20 25 30 35
time
Simulated primary security accounts.
72
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPB
Simulated GOP and savings account.
73
Equal Weighted Index
SδEWId holds equal fractions
πjδEWId,t
=1
d+ 1
j ∈ 0, 1, . . . , d
• tracking rate for BS example:
T dδEWId
(t) =d∑
k=1
(σ
d+ 1
(1√d− 1
))2
= d
(σ
d+ 1
(1√d− 1
))2
≤σ2
(d+ 1)→ 0
74
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPEWI
Simulated NP and EWI.
75
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPindex
Simulated diversified accumulation index and NP.
76
Diversification in an MMM Setting
Pl. (2001), minimal market model (MMM)
• savings accountS0(d)(t) = expr t
• discounted GOP drift
αδ∗t = α0 expη t
• jth benchmarked primary security account
Sj(d)(t) =
1
Y jt α
δ∗t
77
• square root process
dY jt =
(1− η Y j
t
)dt+
√Y jt dW
jt
• NP
Sδ∗(d)(t) =
S0(d)(t)
S0(d)(t)
• jth primary security account
Sj(d)(t) = Sj
(d)(t)Sδ∗(d)(t)
78
0
50
100
150
200
250
300
350
400
450
0 5 9 14 18 23 27 32
Primary security accounts under the MMM
79
0
10
20
30
40
50
60
0 5 9 14 18 23 27 32
Benchmarked primary security accounts
80
0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
EWI
GOP
NP and EWI
81
0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
Market index
GOP
NP and market index
82
Diversified Portfolios
Fundamental phenomenon of diversification leads naturally to NPalso for more general diversified portfolios than the EWI:
Definition A sequence of strictly positive portfolios (Sδd)d∈1,2,... isa sequence of diversified portfolios if∣∣∣πj
δd,t
∣∣∣ ≤
K2
d12+K1
for j ∈ 0, 1, . . . , d
0 otherwise
a.s. for t ∈ [0, T ].
• still small fractions but more general than EWIs
83
• jth benchmarked primary security account
Sjt =
Sjt
Sδ∗t
dSjt = −Sj
t
∑k
sj,kt dW kt
• sj,kt (j, k)th specific volatilitymeasures specific market risk of Sj
with respect toW k diversifiable uncertainty
• volatility of NP measures general market risk, nondiversifiable uncertainty
84
• kth total specific volatility
skt =∑j
|sj,kt |
Definition A market is called regular if
E((skt)2) ≤ K5
for all t ∈ [0,∞), k ∈ 1, 2, . . . .
85
Tracking Rate
Tδ(t) =d
dt⟨Qδ⟩t =
∑k
d∑j=0
πjδ,t s
j,kt
2
Tδ(t) - quantifies “distance” between Sδ and Sδ∗
Tδ∗(t) = 0
86
Approximate NP
Definition A strictly positive portfolio Sδd is an approximate NP if forall t ∈ [0, T ] and ε > 0
limd→∞
P (Tδd(t) > ε) = 0.
87
Diversification Theorem (Pl. 2005)
In a regular market
any diversified portfolio is an approximate NP.
• model independent
88
Proof:
E (Tδd(t)) ≤d∑
k=1
E
d∑
j=0
|πjδ,t| |s
j,kt |
2
≤(K2)
2
d(1+2K1)
d∑k=1
E((skt)2) ≤ (K2)
2K5
d(1+2K1)d→ 0
Markov inequality =⇒
limd→∞
P (Tδd(t) > ε) ≤ limd→∞
1
εE (Tδd(t)) = 0.
89
Summary on Diversification
• extremely robust approach to asset management
• does not rely on estimating some expected returns
• diversification can be refined
• diversified market indices are in reality similar in their dynamics⇒ proxies for the NPe.g. MSCI, S&P 500, EWI114, EWI142 ...
90
Continuous Financial Market
• forces us to be more specific
(Ω,A,A, P ) - filtered probability spaceA = (At)t≥0 - filtrationAt - information at time tEt(·) = E(·|At) - conditional expectation
• d sources of continuous traded uncertainty
W 1,W 2, . . . ,W d, d ∈ 1, 2, . . .
independent standard Brownian motions
91
Primary Security Accounts
Sjt - jth primary security account at time t,
j ∈ 0, 1, . . . , d
cum-dividend share value or savings account value,
dividends or interest are reinvested
• vector process of primary security accounts
S = St = (S0t , . . . , S
dt )
⊤, t ∈ [0, T ]
92
• assume unique strong solution of SDE
dSjt = Sj
t
(ajt dt+
d∑k=1
bj,kt dW kt
)
t ∈ [0,∞), Sj0 > 0, j ∈ 1, 2, . . . , d
• predictable, integrable appreciation rate processes aj
• predictable, square integrable volatility processes bj,k
93
• savings account
S0t = exp
∫ t
0
rs ds
predictable, integrable short rate process
r = rt, t ∈ [0,∞)
94
• Market price of risk
θt = (θ1t , θ2t , . . . , θ
dt )
⊤
unique invariant of the market
solution of relation
bt θt = at − rt 1
whereat = (a1
t , a2t , . . . , a
dt )
⊤
1 = (1, 1, . . . , 1)⊤
95
Assumption:
Volatility matrix bt = [bj,kt ]dj,k=1 is invertible
=⇒
• market price of risk
θt = b−1t (at − rt 1)
96
• can rewrite jth primary security account SDE
dSjt = Sj
t
rt dt+
d∑k=1
bj,kt (θkt dt+ dW kt )
t ∈ [0,∞), j ∈ 1, 2, . . . , d
97
• strategy
δ = δt = (δ0t , . . . , δdt )
⊤, t ∈ [0, T ]
predictable and S-integrable,
δjt number of units of jth primary security account
• portfolio
Sδt =
d∑j=0
δjt Sjt
98
• Sδ self-financing⇐⇒
dSδt =
d∑j=0
δjt dSjt
• changes in the portfolio value only due to gains from trading,
• self-financing in each denomination
99
• jth fraction
πjδ,t = δjt
Sjt
Sδt
j ∈ 0, 1, . . . , d
d∑j=0
πjδ,t = 1
100
• portfolio SDE
dSδt = Sδ
t
(rt dt+
d∑k=1
bkδ,t(θkt dt+ dW k
t
))
with kth portfolio volatility
bkδ,t =d∑
j=0
πjδ,t b
j,kt
101
• logarithm of strictly positive portfolio
d ln(Sδt ) = gδt dt+
d∑k=1
bkδ,t dWkt
with growth rate
gδt = rt +
d∑k=1
bkδ,t
(θkt −
1
2bkδ,t
)
102
=⇒ maximize the growth rate
0 =∂gδt∂bkδ,t
= θkt − bkδ∗,t
=⇒ for each k ∈ 1, 2, . . . , d
θkt = bkδ∗,t=
d∑ℓ=1
πℓδ∗,t
bℓ,kt
=⇒θ⊤t = π⊤
δ∗,tbt
103
Since bt invertible =⇒
πδ∗,t = (π1δ∗,t, . . . , πd
δ∗,t)⊤
= (b−1t )⊤ θt
=⇒ Kelly portfolio, growth optimal portfolio, GOP,log-optimal portfolio, NP
dSδ∗t = Sδ∗
t
([rt +
d∑k=1
(θkt )2
]dt+
d∑k=1
θkt dWkt
)
finite
104
=⇒
• GOP
dSδ∗t = Sδ∗
t
(rt dt+
d∑k=1
θkt(θkt dt+ dW k
t
))
• general portfolio
dSδt = Sδ
t
rt dt+ d∑k=1
d∑j=0
πjδ,t b
j,kt
(θkt dt+ dW k
t
)
105
Benchmarked Portfolio
Sδt =
Sδt
Sδ∗t
Ito formula =⇒
dSδt = −Sδ
t
d∑k=1
d∑j=0
πjδ,t
(θkt − b
j,kt
)dW k
t
driftless =⇒ local martingale,“zero expected return locally”Local Efficient Market Hypothesissimilar to Fama (1965)
106
• nonnegative local martingale is supermartingale
=⇒ Kelly portfolio (GOP) is the NP
107
• NP benchmark for investing
• numeraire for pricing
• captures general market risk in risk management
• central in benchmark approach
108
Benchmark Approach
• provides much wider framework for modeling than classical APT
• can capture more phenomena relevant in real world
• contains as special cases classical approachese.g. APT, actuarial approach, CAPM, ...
109
Capital Asset Pricing Model
• risk premium of security V (t)
pV (t) = limh↓0
1
hEt
(V (t+ h)− V (t)
V (t)
)− rt
excess expected return
• NP
dSδ∗t = Sδ∗
t
((rt + pSδ∗ (t)) dt+
d∑k=1
θkt dWkt
)with volatility
|θt| =
√√√√ d∑k=1
(θkt )2 =
√pSδ∗ (t)
110
• underlying security
dSt = St
(rt dt+
d∑k=1
σkt
(θkt dt+ dW k
t
))
risk premium
pS(t) =d∑
k=1
σkt θ
kt
=d
dt
[ln(S), ln(Sδ∗)
]t
= limh↓0
1
hEt
((St+h − St
St
)(Sδ∗t+h − S
δ∗t
Sδ∗t
))
111
• portfolio
dSδt = Sδ
t
(rt dt+ π1
δ,t
d∑k=1
σkt
(θkt dt+ dW k
t
))
risk premium
pSδ(t) = π1δ,t
d∑k=1
σkt θ
kt
= limh↓0
1
hEt
((Sδt+h − Sδ
t
Sδt
)(Sδ∗t+h − S
δ∗t
Sδ∗t
))
=d
dt
[ln(Sδ), ln(Sδ∗)
]t
112
Assumption: Market portfolio SδMP is diversified and approximates NP
SδMPt ≈ Sδ∗
t .
113
• systematic risk parameter beta
βSδ(t) =ddt
[ln(Sδ), ln(SδMP)
]t
ddt
[ln(SδMP)
]t
=⇒
βSδ(t) ≈ddt
[ln(Sδ), ln(Sδ∗)
]t
ddt
[ln(Sδ∗)
]t
=pSδ(t)
pSδ∗ (t)≈
pSδ(t)
pSδMP (t)
114
=⇒
• fundamental CAPM relation follows directly
pSδ(t) ≈ βSδ(t) pSδMP (t)
115
• CAPM relationship still holds for SδMP with SDE
dSδMPt = SδMP
t
(rt dt+ πt
d∑k=1
θkt(θkt dt+ dW k
t
))
combination of savings account and NP, two fund separation
• βSδ(t) =ddt
[ln(Sδ), ln(SδMP)
]t
ddt
[ln(SδMP)]t
=π1δ,t
∑dk=1 σ
kt θ
kt πt
(πt)2∑d
k=1
(θkt)2
=π1δ,t
∑dk=1 σ
kt θ
kt
πt
∑dk=1
(θkt)2 =
pSδ(t)
pSδMP (t)
116
no utility function involved,no principal agent,no equilibrium assumption
follows simply by Ito calculus and BA
=⇒
• capital asset pricing model (CAPM)
Sharpe (1964), Lintner (1965),
Mossin (1966) and Merton (1973b)
117
Examples for Beta:
• savings accountβS0(t) = 0
• underlying security
βS(t) =
∑dk=1 σ
kt θ
kt
|θt|2
• NPβSδ∗ (t) = 1
118
• portfolio
βSδ(t) =π1δ,t
∑dk=1 σ
kt θ
kt
|θt|2
• If Sδ involves only diversifiable uncertainty, which is not in NP,
then it has zero beta
=⇒
βSδ(t) = 0
119
Portfolio Optimization
• Kelly portfolio (GOP, NP) best long term investment
Kelly (1956), Latane (1959), Breiman (1961),Hakansson (1971b), Thorp (1972)
• optimal portfolio separated into two funds
NP and savings accountTobin (1958), Sharpe (1964)
120
• mean-variance efficient portfolio
Markowitz (1959)
• intertemporal capital asset pricing model
Merton (1973a)
121
Discounted Portfolio
• strictly positive portfolio Sδ ∈ V+
• discounted value
Sδt =
Sδt
S0t
=⇒
• SDE
dSδt =
d∑k=1
ψkδ,t
(θkt dt+ dW k
t
)with kth diffusion coefficient
ψkδ,t =
d∑j=1
δjt Sjt b
j,kt
122
• discounted drift
αδt =
d∑k=1
ψkδ,t θ
kt
link to macro economy
• fluctuations
Mt =
d∑k=1
∫ t
0
ψkδ,s dW
ks
=⇒
Sδt = Sδ
0 +
∫ t
0
αδs ds+ Mt
123
• aggregate diffusion coefficient (deviation)
γδt =
√√√√ d∑k=1
(ψk
δ,t
)2
=⇒ aggregate volatility
bδt =γδt
Sδt
124
Locally Optimal Portfolios
Definition
Sδ ∈ V+ locally optimal, if for all t ∈ [0,∞) and Sδ ∈ V+ with
γδt = γ δ
t
almost surely:
αδt ≤ α
δt .
generalization of mean-variance optimality
Markowitz (1952, 1959)
125
Sharpe Ratio
Sharpe (1964, 1966)
• risk premium
pSδ(t) =αδ
t
Sδt
• aggregate volatility bδt
• Sharpe ratio
sδt =pSδ(t)
bδt=αδ
t
γδt
=mean
deviation
126
• total market price of risk
|θt| =
√√√√ d∑k=1
(θkt)2
Assumption:0 < |θt| <∞
and
π0δ∗,t= 1
almost surely.
127
• Portfolio Selection TheoremFor any Sδ ∈ V+ it follows Sharpe ratio
sδt ≤ |θt|,
where equality when Sδ locally optimal=⇒
dSδt = Sδ
t
bδt|θt|
d∑k=1
θkt(θkt dt+ dW k
t
),
=⇒ fractions
πjδ,t =
bδt|θt|
πjδ∗,t
Markowitz (1959), Sharpe (1964), Merton (1973a)Khanna & Kulldorff (1999), Platen (2002)Sharpe ratio maximization
128
Proof of Portfolio Selection Theorem
Lagrange multiplier λ
consider
L(ψ1δ , . . . , ψ
dδ , λ) =
d∑k=1
ψkδ θ
k + λ
((γ δ)2−
d∑k=1
(ψk
δ
)2)
suppressing time dependence
first-order conditions
∂L(ψ1δ , . . . , ψ
dδ , λ)
∂ψkδ
= θk − 2λψkδ = 0
for all k ∈ 1, 2, . . . , d as well as
129
∂L(ψ1δ , . . . , ψ
dδ , λ)
∂λ=(γ δ)2−
d∑k=1
(ψk
δ
)2= 0
=⇒ locally optimal portfolio S(δ) must satisfy
ψkδ=θk
2λ
for all k ∈ 1, 2, . . . , d.
Thend∑
k=1
(ψk
δ
)2=(γ δ)2
130
=⇒ (γ δ)2
=d∑
k=1
(ψk
δ
)2=
∑dk=1(θ
k)2
4λ2
we have |θ| =√∑d
k=1(θk)2 > 0,
then
ψkδ=γ δ
|θ|θk
for all k ∈ 1, 2, . . . , d
131
=⇒
αδt = γ δ
t
|θt|2
|θt|= γ δ
t |θt|
dSδt = γ δ
t
d∑k=1
θkt|θt|
(θkt dt+ dW kt )
ψkδ,t
=d∑
j=1
δjt Sjt b
j,kt = S
(δ)t
d∑j=1
πj
δ,tbj,kt =
γ δt
|θt|θkt = S
(δ)t bδt
θkt|θt|
invertibility of volatility matrix=⇒
πj
δ,t=
bδt|θt|
d∑k=1
θkt b−1 j,kt
132
Two Fund Separation
• locally optimal portfolio
fraction of wealth in the NP
bδt|θt|
=1− π0
δ,t
1− π0δ∗,t
remainder in savings account
π0δ,t = 1−
bδt|θt|
(1− π0
δ∗,t
)
also known as fractional Kelly strategy:
Kelly (1956), Latane (1959), Thorp (1972), Hakansson & Ziemba (1995)
133
Risk Aversion Coefficient
Jδt =
1− π0δ∗,t
1− π0δ,t
=|θt|bδt
Pratt (1964), Arrow (1965)
• discounted locally optimal portfolio
dSδt = Sδ
t
1
Jδt
|θt| (|θt| dt+ dWt),
where
dWt =
d∑k=1
θkt|θt|
dW kt
1Jδt
fraction in the NP
134
Capital Market Line
• expected rate of return
aδt = rt + pSδ(t)
• for a locally optimal portfolio
aδt = rt + |θt| bδt
• portfolio process at the capital market line
has fraction 1Jδt=
bδt
|θt|in NP
fractional Kelly strategy
135
Markowitz Efficient Frontier
• aggregate volatility
bδt =1− π0
δ,t
1− π0δ∗,t
|θt|
DefinitionMarkowitz efficient portfolio has expected rate of return on efficient frontierif
aδt = rt +
√(bδt )
2 |θt|.
136
Theorem Any locally optimal portfolio Sδ is Markowitz efficient port-folio.
pSδ(t) =1− π0
δ,t
1− π0δ∗,t
|θt|2 = bδt |θt|
Sharpe ratio maximizationrisk attitude expressed via Jδ
t = |θt|bδt
short term view
137
-0.04 0 0.04 0.08 0.12
0.05
0.09
-0.04 0 0.04 0.08 0.12
0
0.05
0.09
Efficient frontier
138
Efficient Growth Rates
• efficient growth rate frontier
of locally optimal portfolio
gδt = rt +
√|bδt |2 |θt| −
1
2|bδt |
2 = rt +|θt|2
Jδt
(1−
1
2 Jδt
)
• maximum
gδ∗t = rt +
1
2|θt|2
• long term view
139
0 0.04 0.08 0.12 0.16
0.05
0 0.04 0.08 0.12 0.16
0
0.05
Efficient growth rates
140
Market Portfolio
SδMPt =
n∑ℓ=1
Sδℓt
portfolio of tradable wealth of ℓth investor Sδℓ
Remark:
If each investor forms a nonnegative, locally optimal portfolio with her or his
total tradable wealth, then MP is locally optimal .
141
dSδMPt =
n∑ℓ=1
dSδℓt
=n∑
ℓ=1
(Sδℓt − δ0ℓ
)(1− π0
δ∗,t
) d∑k=1
θkt (θkt dt+ dW kt )
= SδMPt
(1− π0
δMP,t
)(1− π0
δ∗,t
) d∑k=1
θkt (θkt dt+ dW kt )
• CAPM relationship still holds when MP locally optimal
142
Practical Feasibility ofSample Based Markowitz Mean-Variance Approach:
• NP theoretically known for given model
• Markowitz (1952)
• Best & Grauer (1991)
• DeMiguel, Garlappi & Uppal (2009)
practical applicationrequires for 50 assets observation window of about 500 years
• not realistic to be applied
• theoretically can be reconciled under strong assumptions(MP locally optimal)
143
Maximum Drawdown Constrained Portfolios
X - set of nonnegative continuous discounted portfolios
• running maximumforX = Xt, t ≥ 0 ∈ X
X∗t = sup
u∈[0,t]
Xu
• drawdownX∗
t −Xt
144
• relative drawdownXt
X∗t
• maximum relative drawdown
• express attitude towards risk by restricting toX ∈ αX , where
Xt
X∗t
≥ α ,α ∈ [0, 1)
pathwise criterion
145
• drawdownforX ∈ αXXt ≥ αX∗
t
=⇒X∗
t −Xt ≤ (1− α)X∗t
146
• maximum drawdown constrained portfolioα ∈ [0, 1),X ∈ X
αXt = α(X∗t )
1−α + (1− α)Xt(X∗t )
−α
= 1 +
∫ t
0
(1− α)(X∗s )
−αdXs ∈ [αX∗t , (X
∗t )
1−α]
Grossman & Zhou (1993), Cvitanic & Karatzas (1994),Kardaras, Obloj & Pl. (2012)
147
=⇒ SDEdαXt
αXt
= απt
dXt
Xt
fraction
απt = 1−(1− α)Xt
X∗t
α+ (1− α)Xt
X∗t
model independentdepends only on Xt
X∗t
and ααSδ∗
t - locally optimal portfolio
148
1871 1899 1927 1955 1983 2011−1
0
1
2
3
4
5
6
ln(X)
ln(X*)
logarithm of discounted S&P500 and its running maximum
149
1871 1899 1927 1955 1983 20110
0.2
0.4
0.6
0.8
1
1.2
relative drawdown
Relative drawdown of discounted S&P500
150
1871 1899 1927 1955 1983 2011−0.5
0
0.5
1
1.5
2
2.5
ln(αX)
ln(αX*)
Logarithm of maximum drawdown constrained portfolio, α = 0.6
151
1871 1899 1927 1955 1983 20110
0.2
0.4
0.6
0.8
1
1.2
relative drawdown
Relative drawdown for drawdown constrained portfolio, α = 0.6
152
1871 1899 1927 1955 1983 20110
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
fraction in X
Fraction of wealth in S&P500 for α = 0.6
153
1871 1899 1927 1955 1983 20110
0.002
0.004
0.006
0.008
0.01
0.012
0.014
relative hedge error
Relative hedge error, α = 0.6, monthly hedging
154
• Long term growth rateFor α ∈ [0, 1),X ∈ αX
limt→∞
(log(Xt)
Gt
)≤ 1− α = lim
t→∞
(log(αSδ∗
t )
Gt
)
log(Sδ∗t ) = Gt +
∫ t
0
|θs|dWs
Kardaras, Obloj & Pl. (2012)
155
• asymptotic maximum long term growth rate
limt→∞
1
tlog(αSδ∗
t ) = (1− α) limt→∞
1
tlog(Sδ∗
t )
αg = (1− α)g
restricted drawdown =⇒ reduced maximum growth ratelong term view with short term attitude towards riskrealistic alternative to Markowitz mean-variance approachand utility maximization
156
1871 1900 1928 1956 1984 2012−1
0
1
2
3
4
5
6
Logarithm of drawdown constrained portfolios
157
1879 1906 1932 1959 1985 2012−0.01
0
0.01
0.02
0.03
0.04
0.05
Long term growth of drawdown constrained portfolios
158
Expected Utility Maximization
• utility function U(·), U ′(·) > 0 U ′′(·) < 0
• examples
power utility
U(x) =1
γxγ
for γ = 0 and γ < 1
log-utilityU(x) = ln(x)
159
5 10 15 20x
-15
-10
-5
5
U
Examples for power utility (upper graph) and log-utility (lower graph)
160
• assume S0 is scalar Markov process
• maximize expected utility
vδ = maxSδ∈V+
S0
E(U(SδT
) ∣∣∣A0
)
V+S0
set of strictly positive, discounted, fair portfolios
Sδ0 = S0 > 0
161
Theorem
Benchmarked, expected utility maximizing portfolio:
Sδt = u(t, S0
t ) = E(U ′−1
(λ S0
T
)S0T
∣∣∣At
),
λ s.t. S0 = Sδ0 S
δ∗0 = u(0, S0)S
δ∗0
two fund separation with risk aversion coefficient
J δt =
1
1− S0t
u(t,S0t )
∂u(t,S0t )
∂S0t
.
=⇒ locally optimal portfolio
162
Proof:
vδ = E(U(Sδ
T ))− λ
(E(Sδ
T )− Sδ0
)= E
(F (Sδ
T ))
F (SδT ) = U(Sδ
T )− λ(SδT
Sδ∗T
− Sδ0
)
F ′(SδT ) = U ′(Sδ
T )−λ
Sδ∗T
= 0
163
=⇒
U ′(SδT ) =
λ
Sδ∗T
SδT = U
′−1
(λ
Sδ∗T
)= U
′−1(λS0T )
SδT = U
′−1(λS0T )S
0T
164
• log-utility function U(x) = ln(x)
U ′(x) =1
x
U ′−1(y) =1
y
U ′′(x) = −1
x2
utility concave
u(t, S0t ) = E
(U ′−1
(λ S0
T
)S0T
∣∣∣At
)= E
(1
λ S0T
S0T
∣∣∣At
)=
1
λ
Lagrange multiplier
165
λ =Sδ∗0
S0
risk aversion coefficientJ δt = 1
• expected log-utility
vδ = E(ln(Sδ∗T
) ∣∣∣A0
)= ln(λ) + ln(S0) +
1
2
∫ T
0
E((θ(s, Sδ∗
s ))2 ∣∣∣A0
)ds
if local martingale part forms a martingale
166
• power utility
U(x) =1
γxγ
for γ < 1, γ = 0
U ′(x) = xγ−1
U ′−1(y) = y1
γ−1
U ′′(x) = (γ − 1)xγ−2
concave
u(t, S0t ) = E
( λ
Sδ∗T
) 1γ−1
1
Sδ∗T
∣∣∣∣At
= λ1
γ−1 E
((Sδ∗T
) γ1−γ
∣∣∣At
)
167
Sδ∗ geometric Brownian motion
u(t, S0t ) = λ
1γ−1
(Sδ∗t
) γ1−γ
×E(exp
γ
1− γ
(θ2
2(T − t) + θ (WT −Wt)
) ∣∣∣∣At
)
= λ1
γ−1(Sδ∗t
) γ1−γ exp
θ2
2
γ
(1− γ)2(T − t)
168
Lagrange multiplier
λ = Sγ−10 Sδ∗
0 exp
θ2
2
γ
1− γT
S0t = (Sδ∗
t )−1
∂u(t, S0t )
∂S0=u(t, S0
t )
S0t
γ
γ − 1
risk aversion coefficientJ δt = 1− γ
expected utility
vδ = E
(1
γ
(SδT
)γ ∣∣∣∣A0
)= exp
−θ2
2
γ
1− γT
(S0)
γ
169
Various Approaches to Asset Pricing
• Actuarial Pricing Approach (APA)
• Capital Asset Pricing Model (CAPM)
• Arbitrage Pricing Theory (APT)
• Utility Indifference Pricing
• Benchmark Approach (BA)
170
Classical
Law of One Price
“All replicating portfolios of a payoff have the same price!”
Debreu (1959), Sharpe (1964), Lintner (1965),
Merton (1973a, 1973b), Ross (1976), Harrison & Kreps (1979),
Cochrane (2001), . . .
will be, in general, violated under the benchmark approach.
171
-8
-7
-6
-5
-4
-3
-2
-1
0
1
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(savings bond)ln(fair zero coupon bond)
ln(savings account)
Logarithms of savings bond, fair zero coupon bond and savings account
172
• benchmarked value
Sδt =
Sδt
Sδ∗t
Supermartingale Property
For nonnegative Sδ
Sδt ≥ Et
(Sδs
)0 ≤ t ≤ s <∞
Sδ supermartingale
173
• One needs consistent pricing concept
• All benchmarked nonnegative securities are supermartingales
(no upward trend)
174
Definition Price is fair if, when benchmarked, forms martingale
(no trend)
Sδt = Et
(Sδs
)0 ≤ t ≤ s <∞.
175
Lemma The minimal nonnegative supermartingale that reaches a givenbenchmarked contingent claim is a martingale.
see Revuz & Yor (1999)
176
0
0.02
0.04
0.06
0.08
0.1
0.12
1930 1940 1950 1960 1970 1980 1990 2000
time
benchmarked savings bondbenchmarked fair zero coupon bond
Benchmarked savings bond and benchmarked fair zero coupon bond
177
Benchmark Approach:
Law of the Minimal Price
Pl. (2008)
Theorem If a fair portfolio replicates a nonnegative payoff, then this rep-resents the minimal replicating portfolio.
• least expensive
• minimal possible hedge
• economically correct price in a competitive market
178
• contingent claim
HT
E0
(HT
Sδ∗T
)<∞
• SδHt fair if
SδHt =
SδHt
Sδ∗t
= Et
(HT
Sδ∗T
)
• real world expectationbest performing portfolio as numeraire
=⇒ Direct link with real world and economy !
179
Law of the Minimal Price =⇒
Corollary
Minimal price for replicableHT is fair and given by
real world pricing formula
SδHt = Sδ∗
t Et
(HT
Sδ∗T
).
• Du & Pl. (2013) benchmarked risk minimizationfor nonreplicable claims
180
• most pricing concepts become unified and generalized by real worldpricing
=⇒
actuarial pricingrisk neutral pricingpricing with stochastic discount factorpricing with numeraire changepricing with deflatorpricing with state pricing densitypricing with pricing kernelutility indifference pricingpricing with numeraire portfolioclassical risk minimizationbenchmarked risk minimization
181
• Equivalent to risk neutral pricing are pricing via:
stochastic discount factor, Cochrane (2001);
deflator, Duffie (2001);
pricing kernel, Constantinides (1992);
state price density, Ingersoll (1987);
numeraire portfolio as in Long (1990)
182
Actuarial Pricing
WhenHT independent of Sδ∗T
=⇒ rigorous derivation of
• actuarial pricing formula
SδHt = P (t, T )Et(HT )
with zero coupon bond as discount factor
P (t, T ) = Sδ∗t Et
(1
Sδ∗T
)
183
Risk Neutral Pricing
• real world pricing formula =⇒
SδH0 = E0
(ΛT
S00
S0T
HT
)Radon-Nikodym derivative (density)
Λt =S0t
S00
supermartingale
1 = Λ0 ≥ E0(ΛT )
=⇒
SδH0 ≤
E0
(ΛT
S00
S0THT
)E0(ΛT )
184
similar for any numeraire (here S0 savings account)
185
0
10
20
30
40
50
60
70
80
90
100
1930 1940 1950 1960 1970 1980 1990 2000
time
Discounted S&P500 total return index
186
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1930 1940 1950 1960 1970 1980 1990 2000
time
Radon-Nikodym derivativeTotal RN-Mass
Radon-Nikodym derivative and total mass of putative risk neutral measure
187
• special case when savings account is fair:
=⇒ ΛT = dQdP
forms martingale; E0(ΛT ) = 1;
equivalent risk neutral probability measureQ exists;
Bayes’ formula =⇒risk neutral pricing formula
SδH0 = EQ
0
(S00
S0T
HT
)
Harrison & Kreps (1979), Ingersoll (1987),
Constantinides (1992), Duffie (2001), Cochrane (2001), . . .
• otherwise “risk neutral price” ≥ real world price
188
-1
0
1
2
3
4
5
6
7
8
9
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(index)ln(savings account)
ln(S&P500 accumulation index) and ln(savings account)
189
Risk Neutral Pricing under BS Model
θt = θ, at = a, rt = r, σt = σ
Drifted Wiener Process
• drifted Wiener process Wθ
Wθ(t) = Wt + θ t
• market price of risk
θ =a− rσ
190
• underlying security
dSt = (a− σ θ)St dt+ σ St (θ dt+ dWt)
= (a− σ θ)St dt+ σ St dWθ(t)
= r St dt+ σ St dWθ(t)
St = S0 exp
(r −
1
2σ2
)t+ σWθ(t)
Wθ is not a Wiener process under the real world probability P ,
however, will be under some Pθ
191
0
5
10
15
20
25
0 5 10 15 20 25 30 35
time
Simulated primary security accounts.
192
• Radon-Nikodym derivative
Λθ(T ) =dPθ
dP=S0T
S00
• risk neutral measure Pθ
Pθ(A) =
∫Ω
1A dPθ(ω) =
∫A
dPθ(ω)
dP (ω)dP (ω)
=
∫A
Λθ(T ) dP (ω)
for all subsetsA ∈ Ω
193
Definition P is equivalent to P if both have the same sets
of events that have probability zero.
• equivalence of P and Pθ
fundamental for risk neutral pricing
Key Risk Neutral assumption:
Λθ is (A, P ) martingale
194
• under BS model
Radon-Nikodym derivative
Λθ(t) =S0t
S00
is (A, P )-martingale,
=⇒ there exists equivalent risk neutral probability measure Pθ
for BS model
195
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0 5 10 15 20 25 30 35
time
Benchmarked savings account under BS model, martingale.
196
Risk Neutral Measure Transformation
dPθ
dP
∣∣∣∣At
= Λθ(t) = exp
−
1
2θ2 t− θWt
• total risk neutral measure at t = 0:
dPθ
dP
∣∣∣∣A0
= E(Λθ(T ) | A0) = Λθ(0) = 1
=⇒ Pθ is a probability measure
197
Let us show thatWθ is Wiener process under Pθ:
For fixed y ∈ ℜ,
t ∈ [0, T ] and s ∈ [0, t]
letA be the event
A = ω ∈ Ω : Wθ(t, ω)−Wθ(s, ω) < y
=⇒
A = ω ∈ Ω : Wt(ω)−Ws(ω) < y − θ (t− s)
198
Eθ denoting expectation under Pθ =⇒
Pθ(A) = Eθ(1A|A0) = E
(dPθ
dP
∣∣∣∣AT
1A
∣∣∣∣A0
)
= E(Λθ(T ) 1A|A0) = E
(Λθ(t) 1A
Λθ(T )
Λθ(t)
∣∣∣∣A0
)
= E
(Λθ(t) 1AE
(Λθ(T )
Λθ(t)
∣∣∣∣At
) ∣∣∣∣A0
)
= E (Λθ(t) 1A|A0) = E
(Λθ(s)
Λθ(t)
Λθ(s)1A
∣∣∣∣A0
)
= E (Λθ(s)|A0) E
(Λθ(t)
Λθ(s)1A
∣∣∣∣A0
)
= E
(Λθ(t)
Λθ(s)1A
∣∣∣∣A0
)199
since y = Wt −Ws is Gaussian ∼ N (0, t− s) under P =⇒
Pθ(A) = E
(Λθ(t)
Λθ(s)1A
)
=
∫ y−θ(t−s)
−∞exp
−θ2
2(t− s)− θ y
1√
2π (t− s)
exp
−
y2
2 (t− s)
dy
=
∫ y−θ(t−s)
−∞
1√2π (t− s)
exp
−
(y + θ (t− s))2
2 (t− s)
dy
=
∫ y
−∞
1√2π (t− s)
exp
−
z2
2 (t− s)
dz
for z = y + θ(t− s)
200
=⇒
Theorem (Girsanov) Under the above BS model the process
Wθ = Wθ(t) = Wt + θ t, t ≥ 0
is a standard Wiener process in the filtered probability space (Ω,AT ,A, Pθ).
• Wθ is (A, Pθ)-Wiener process
• simply a transformation of variable
(not always possible, exploits here BS model)
201
Real World Pricing =⇒ Risk Neutral Pricing
V (0, S0) = Sδ∗0 E
(H(ST )
Sδ∗T
∣∣∣∣A0
)
= E
(S0T
S00
H(ST )
S0T
∣∣∣∣A0
)
= E
(Λθ(T )
(H(ST )
S0T
) ∣∣∣∣A0
)
≤E(Λθ(T )
H(ST )S0
T
∣∣A0
)E(Λθ(T )
∣∣A0
)
202
=⇒ Risk Neutral Pricing if Λθ is (A, P )-martingale
V (0, S0) = E
(exp
−θ2
2T − θWT
exp−r TH(ST )
∣∣∣A0
)= exp−r T
∫ ∞
−∞H
(S0 exp
(a−
1
2σ2
)T + σ y
)× exp
−θ2
2T − θ y
1√TN ′(y√T
)dy
= exp−r T∫ ∞
−∞
(H
(S0 exp
(r −
1
2σ2
)T + σ z
))×
1√TN ′(z√T
)dz
= exp−r TEθ
(H(ST )
∣∣A0
)= Eθ
(H(ST )
S0T
∣∣∣A0
)z = y + θ T =⇒ Eθ - expectation with respect to Pθ
203
Risk Neutral SDEs under BS model
• discounted underlying security price
dSt = σ St dWθ(t)
driftless under Pθ, σS ∈ L2T =⇒
(A, Pθ)-martingale
204
• discounted option price
dV (t, St) =∂V (t, St)
∂Sσ St dWθ(t)
driftless under Pθ, ∂V∂SσS ∈ L2
T =⇒
(A, Pθ)-martingale
• discounted savings account S0t = 1 =⇒ (A, Pθ)-martingale
205
• discounted portfolio
dSδt = d(Sδ∗
t Sδt )
= Sδt π
1δ(t)σ (θ dt+ dWt)
= Sδt π
1δ(t)σ dWθ(t)
(A, Pθ)-local martingale
• if Sδ π1δ δ = δ1 S σ ∈ L2
T =⇒ Sδ is an (A, Pθ)-martingale,
=⇒ risk neutral pricing formula holds=⇒
Sδt = Eθ
(SδT
∣∣At
)= Eθ
(SδT
S0T
∣∣∣∣At
)• Sδ∗ is (A, Pθ)-martingale under BS model
206
• strict supermartingale portfolio under BS model
if Zt strict (A, Pθ)-supermartingale, independent of Sδ∗t and
Sδt = Sδ∗
t Zt
=⇒ Sδt strict (A, Pθ)-supermartingale
• risk neutral pricing formula does not hold for such Sδt
Sδt > Eθ
(SδT
∣∣At
)for T > t
=⇒ not all discounted portfolios are (A, Pθ)-martingales
even when Radon-Nikodym derivative Λθ is (A, P )-martingale
207
Change of Variables
• we performed a change of variables fromWt toWθ(t)
W Wiener processes under P
Wθ Wiener processes under Pθ
• relies on existence of an
equivalent risk neutral probability measure Pθ
208
General Girsanov Transformation
• m-dimensional standard Wiener process
W = W t = (W 1t , . . . ,W
mt )⊤, t ∈ [0, T ]
(Ω,AT ,A, P )
• A-adapted, predictablem-dimensional stochastic process
θ = θt = (θ1t , . . . , θmt )⊤, t ∈ [0, T ]
∫ T
0
m∑i=1
(θit)2 dt <∞
209
• assume Radon-Nikodym derivative
Λθ(t) = 1−m∑i=1
∫ t
0
Λθ(s) θis dW
is
= exp
−∫ t
0
θ⊤s dW s −1
2
∫ t
0
θ⊤s θs ds
<∞
forms (A, P )-martingale
210
• Λθ (A, P )-martingale
=⇒E(Λθ(t) | As) = Λθ(s)
for t ∈ [0, T ] and s ∈ [0, t]
=⇒E(Λθ(t) | A0) = Λθ(0) = 1
211
• define a candidate risk neutral measure Pθ with
dPθ
dP
∣∣∣∣AT
= Λθ(T )
by setting
Pθ(A) = E(Λθ(T ) 1A)
= Eθ(1A)
• since Λθ is an (A, P )-martingale
Pθ(Ω) = E(Λθ(T )) = E(Λθ(T )
∣∣A0
)= Λθ(0) = 1
=⇒ Pθ is a probability measure
212
Girsanov Theorem
Theorem (Girsanov) If a given strictly positive Radon-Nikodymderivative process Λθ is an (A, P )-martingale, then the m-dimensionalprocess Wθ = Wθ(t), t ∈ [0, T ], given by
Wθ(t) = W t +
∫ t
0
θs ds
for all t ∈ [0, T ], is an m-dimensional standard Wiener process on thefiltered probability space (Ω,AT ,A, Pθ).
• W is (A, P )-Wiener process
Wθ is (A, Pθ)-Wiener process
213
Novikov Condition
When is the strictly positive (A, P )-local martingale Λθ an (A, P )-martingale?
• a sufficient condition is the Novikov condition
Novikov (1972)
E
(exp
1
2
∫ T
0
|θ⊤s θs| ds)
<∞
(is satisfied for the BS model)
• for other sufficient conditions see Revuz & Yor (1999)
214
Bayes’s Theorem
Theorem (Bayes) Assume that Λθ is an (A, P )-martingale, thenfor any given stopping time τ ∈ [0, T ] and any Aτ -measurable randomvariable Y , satisfying the integrability condition
Eθ(|Y |) <∞,
one has the Bayes rule
Eθ(Y | As) =E(Λθ(τ )Y | As)
E(Λθ(τ ) | As)
for s ∈ [0, τ ].
215
Proof of Bayes’s Theorem (*)
For eachAτ -measurable random variable Y and a setA ∈ As
with s ∈ [0, τ ], τ ≤ T
1AEθ (Y |As) = Eθ
(1A Y
∣∣As
)= E
(1A Y Λθ(T )
∣∣As
)= E (1A Y Λθ(τ )|As) = E (1AE (Y Λθ(τ )|As) |As)
= E
(Λθ(s)
1A
Λθ(s)E (Y Λθ(τ ) | As)
∣∣∣∣As
)
= Eθ
(1A
Λθ(s)E (Y Λθ(τ ) | As)
∣∣∣∣As
)
= Eθ
(1A
E (Λθ(τ )Y | As)
E (Λθ(τ ) | As)
∣∣∣∣As
)
= 1A
E (Λθ(τ )Y | As)
E (Λθ(τ ) | As).
216
Change of Numeraire Technique
Geman, El Karoui & Rochet (1995)
Jamshidian (1997)
• numeraireSδ = Sδ
t , t ∈ [0, T ]
normalizes all other portfolios
• may lead to simplifications in calculations
• relative price of a portfolioSδt
Sδt
217
Self-financing under Numeraire Change
Sδt = δ0t S
0t + δ1t St
dSδt = δ0t dS
0t + δ1t dSt
dSδt = δ0t dS
0t + δ1t dSt
Ito formula =⇒
218
d
(Sδt
Sδt
)=
1
Sδt
dSδt + Sδ
t d
(1
Sδt
)+ d
[1
Sδ, Sδ
]t
= δ0t
(1
Sδt
dS0t + S0
t d
(1
Sδt
))
+ δ1t
(1
Sδt
dSt + St d
(1
Sδt
)+ d
[1
Sδ, S
]t
)
[X,Y ]t ≈it∑
i=0
(Xti+1
−Xti
) (Yti+1
− Yti
)covariation
219
Ito formula for
S0t
Sδt
andSt
Sδt
=⇒
d
(Sδt
Sδt
)= δ0t d
(S0t
Sδt
)+ δ1t d
(St
Sδt
)
=⇒
• in denomination of another numeraire always self-financing
220
Numeraire Pairs
• numeraire pair (Sδ, Pθδ)
real world pricing formula:
St = E
(Sδ∗t
Sδ∗T
ST
∣∣∣∣At
)
existence of GOP
=⇒ numeraire pair (Sδ∗ , P )
221
if Λθ is (A, P )-martingale =⇒
• risk neutral pricing formula:
St = Eθ
(S0t
S0T
ST
∣∣∣∣At
)
=⇒ numeraire pair (S0, Pθ)
222
• for fair price process S
from real world pricing formula
St
Sδ∗t
= E
(ST
Sδ∗T
∣∣∣∣At
)
=⇒ for any strictly positive numeraire Sδ
St
Sδt
= E
(Sδ∗t
Sδt
SδT
Sδ∗T
ST
SδT
∣∣∣∣At
)
= E
(Λθδ
(T )
Λθδ(t)
ST
SδT
∣∣∣∣At
)
223
with benchmarked normalized numeraire
=⇒
Λθδ(t) =
Sδt
Sδ0
dΛθδ(t) = d
(Sδt
Sδ0
)
= Λθδ(t)(π1δ(t)σt − θt
)dWt
(A, P )-local martingale
=⇒ Radon-Nikodym derivative for candidate measure Pθδ
dPθδ
dP= Λθδ
(T )
224
• drifted Wiener process
dWθδ(t) = dWt + θδ(t) dt,
whereθδ(t) = θt − π1
δ(t)σt
• If Λθδis true (A, P )-martingale
=⇒ Pθδ equivalent to P and probability measure
allows to apply Girsanov transformation
=⇒ Wθδstandard Wiener process under Pθδ
=⇒ numeraire pair (Sδ, Pθδ)
225
Some Examples
• GOP Sδ∗ as numeraire =⇒ θδ∗(t) = 0
• savings account S0 as numeraire
π1δ(t) = 0 =⇒ θδ(t) = θt
• underlying security S as numeraire
π1δ(t) = 1 =⇒ θδ(t) = θt − σt
226
• unfair portfolio Sδt = Sδ∗
t Zt as numeraire
with Z an (A, P )-strict supermartingale, independent of Sδ∗
=⇒ Sδ is not an (A, P )-martingale
Girsanov Theorem can not be applied !
=⇒ If savings account is not fair,
then the change of numeraire approach is not applicable !
227
Pricing Formula for General Numeraire Sδ
V (0, S0) = E
(Sδ∗0
Sδ∗T
H(ST )
∣∣∣∣A0
)
= E
(Λθδ
(T )H(ST )
SδT
∣∣∣∣A0
)
the quantityΛθδ
(T )
SδT
=Sδ∗0
Sδ∗T
remains numeraire invariant
228
• if Λθδforms an (A, P )-martingale
Bayes’s Theorem =⇒
E
(Λθδ
(T )H(ST )
SδT
∣∣∣∣A0
)= Eθδ
(H(ST )
SδT
∣∣∣∣A0
)=⇒
pricing formula with general numeraire
V (0, S0) = Eθδ
(H(ST )
SδT
∣∣∣∣A0
)
• represents a transformation of variables in integration
• may provide computational advantages
229
Utility Indifference Pricing
discounted payoff H = HS0
T(not perfectly hedgable)
vδε,V = E
(U
((S0 − ε V )
SδT
S0
+ ε H
) ∣∣∣∣A0
)
• assume S0 is scalar diffusion
Definition Utility indifference price V s.t.
limε→0
vδε,V − vδ0,Vε
= 0
e.g. Davis (1997)
230
• Taylor expansion
vδε,V ≈ E
(U(SδT
)+ εU ′
(SδT
) (H −
V SδT
S0
) ∣∣∣∣A0
)
= vδ0,V + εE(U ′(SδT
)H∣∣∣A0
)− V εE
(U ′(SδT
) SδT
S0
∣∣∣∣A0
)
231
=⇒
• utility indifference price
V =E(U ′(SδT
)H∣∣∣A0
)E
(U ′(SδT
)Sδ
T
S0
∣∣∣A0
) =E(U ′(U ′−1
(λ
Sδ∗T
))H∣∣∣A0
)E
(U ′(U ′−1
(λ
Sδ∗T
))Sδ
T
S0
∣∣∣A0
)
=E(
H
Sδ∗T
∣∣∣A0
)1S0
S0
Sδ∗0
independent of U and independent of particular model =⇒
• real world pricing formula
V = Sδ∗0 E
(H
Sδ∗T
∣∣∣A0
)
232
Pricing via Hedging
• underlying security
S = St, t ∈ [0, T ]
dSt = at St dt+ σt St dWt
t ∈ [0, T ], S0 > 0
233
appropriate stochastic appreciation rate at
and volatility σt
Wiener processW = Wt, t ∈ [0, T ]
• savings account
dS0t = rt S
0t dt
t ∈ [0, T ], S00 = 1
appropriate stochastic short rate
234
• replicate payofff(ST ) ≥ 0
assumingE(f(ST )) <∞
• hedge portfolio
V (t, St) = δ0t S0t + δ1t St
δ0t units of savings account
δ1t units of underlying security
235
• hedging strategy
δ = δt = (δ0t , δ1t )
⊤, t ∈ [0, T ]
δ0 = δ0t , t ∈ [0, T ]
δ1 = δ1t , t ∈ [0, T ]
predictable processes
236
• replicating portfolio
V (T, ST ) = f(ST )
• SDE for hedge portfolio
dV (t, St) = δ0t dS0t + δ1t dSt
+S0t dδ
0t + St dδ
1t + d
[δ1, S
]t
237
• self-financing
portfolio changes are caused by gains from trade
dV (t, St) = δ0t dS0t + δ1t dSt
=⇒
self-financing condition:
S0t dδ
0t + St dδ
1t + d
[δ1, S
]t= 0
238
• discounted underlying security
St =St
S0t
t ∈ [0, T ]
Ito formula =⇒
dSt = (a− r) St dt+ σt St dWt
t ∈ [0, T ], S0 = S0
• discounted value function
V : [0, T ]× [0,∞)→ [0,∞)
V (t, St) =V (t, St)
S0t
239
• profit and loss (P&L)
Ct = price – gains from trade – initial price
• discounted P&L
Ct =Ct
S0t
= V (t, St)− Iδ1,S(t)− V (0, S0)
240
• discounted gains from trade
Iδ1,S(t) =
∫ t
0
δ1u dSu
=
∫ t
0
δ1u (a− r) Su du
+
∫ t
0
δ1u σ Su dWu
Iδ0,S0(t) =
∫ t
0
δ0u dS0u = 0
241
• initial payment
V (0, S0) = V (0, S0),
=⇒
C0 = 0
242
• identify hedging strategy δ
for which discounted P&L zero
Ct = 0
for all t ∈ [0, T ]
=⇒ perfect hedge
V (t, St) - replicating portfolio
243
• increments of discounted P&L
0 = Ct − Cs = V (t, St)− V (s, Ss)−∫ t
s
δ1u dSu
=
∫ t
s
[∂V (u, Su)
∂t+
1
2σ2u S
2u
∂2V (u, Su)
∂S2
+(au − ru) Su
(∂V (u, Su)
∂S− δ1u
)]du
+
∫ t
s
σu Su
(∂V (u, Su)
∂S− δ1u
)dWu
=⇒
244
• hedge ratio
δ1u =∂V (u, Su)
∂S
• discounted PDE
∂V (u, S)
∂t+
1
2σ2u S
2 ∂2V (u, S)
∂S2= 0
u ∈ [0, T ), S ∈ (0,∞)
with terminal condition
V (T, S) =f(S S0
T )
S0T
=f(S)
S0T
245
=⇒
• PDE
∂V (u, S)
∂t+ ru S
∂V (u, S)
∂S
+1
2σ2u S
2 ∂2V (u, S)
∂S2− ru V (u, S) = 0
u ∈ [0, T ), S ∈ (0,∞)
with terminal condition
V (T, S) = f(S)
• special boundary condition not specifiedPDE may not provide minimal possible price
246
=⇒
• units in savings account
δ0t =V (t, St)− δ1t St
S0t
= V (t, St)− δ1t St
• option price
V (t, St) = V (t, St)S0t
= δ0t S0t + δ1t St
t ∈ [0, T ]
247
=⇒ strategyδt = (δ0t , δ
1t )
⊤
such that0 = dCt = dV (t, St)− δ1t dSt
=⇒dV (t, St) = δ0t dS
0t + δ1t dSt
=⇒ self-financing
P&L vanishesCt = Ct S
0t = 0
248
for all t ∈ [0, T ]
If V (t, St) is not fair, then there may exist less expensive hedge portfo-lio!
249
Numeraire Invariance
V self-financing
dV (t, St) = d(V (t, St)S0t )
= S0t dV (t, St) + V (t, St) dS
0t + d[S0, V (·, S·)]t
= S0t δ
1t dSt + (δ0t + δ1t St) dS
0t + δ1t d[S
0, S]t
= δ0t dS0t + δ1t
(S0t dSt + St dS
0t + d[S0, S]t
)= δ0t dS
0t + δ1t d(S
0t St)
= δ0t dS0t + δ1t dSt
=⇒ V also self-financing
250
The Black-Scholes Formula
• solution cT,K(t, S) of PDE
with call payoff function
f(S) = (S −K)+
• at = a, σt = σ, rt = r
Black & Scholes (1973)Merton (1973a)
251
Black-Scholes Formula
cT,K(t, St) = StN(d1(t))−KS0t
S0T
N(d2(t))
with
d1(t) =ln(St
K
)+(r + 1
2σ2)(T − t)
σ√T − t
d2(t) = d1(t)− σ√T − t
t ∈ [0, T )
cT,K(t, St) =cT,K(t, St)
Sδ∗t
(P,A)− martingale
minimal possible price
252
• N(·) standard Gaussian distribution function
with density
N ′(x) =1√2π
exp
−x2
2
253
0
2
4
6
8
t
0.5
1
1.5
2
S
0
0.5
1
0
2
4
6
8
t
0
0.5
1
Black-Scholes European call option price.
254
Delta
• sensitivity with respect to the underlying St
∆ =∂V (t, St)
∂S= δ1t
∆ = N(d1(t))
255
0
2
4
6
8
t
0.5
1
1.5
2
S
0
0.25
0.5
0.75
1
0
2
4
6
8
t
0
0.25
0.5
0.75
Delta as a function of t and St.
256
Gamma
• sensitivity of the hedge ratio with respect to underlying
Γ =∂∆
∂S=∂2V (t, St)
∂S2
Γ = N ′(d1(t))1
St σ√T − t
257
0
2
4
6
8
t
0.5
1
1.5
2
S
0
1
2
3
4
0
2
4
6
8
t
0
1
2
3
Gamma as a function of t and St.
258
Simulation of Delta Hedging
• simulate scenario along an equidistant time discretization
• price of call option
cT,K(t, St) = δ1t St + δ0t S0t
• hedging strategy
δ =δt = (δ0t , δ
1t )
⊤, t ∈ [0, T ]
259
• hedge ratio, units in the underlying
δ1t = N(d1(t))
= N
(ln(St
K
)+(r + 1
2σ2)(T − t)
σ√T − t
)
260
• units in domestic savings account
δ0t = −K
S0T
N(d2(t))
= −K
S0T
N
(ln(St
K
)+(r − 1
2σ2)(T − t)
σ√T − t
)
261
• discounted P&L
Ct = V (t, St)−∫ t
0
δ1s dSs − V (0, S0)
V (t, St) = V (0, S0) +
∫ t
0
δ1s dSs
=⇒ theoretically under continuous hedging
Ct = 0
for all t ∈ [0, T ]
262
In-the-Money Scenario
• security price ends up in-the-money ST > K
=⇒ hedge ratio converges to δ1T = 1
• self-financing strategy δt = (δ0t , δ1t )
⊤
• intrinsic valuef(St) = (St −K)+
• discounted P&L Ct remains almost perfectly zero
263
0.5
1
1.5
2
2.5
0 2 4 6 8 10
time
S(t)Delta
Underlying security and hedge ratio for in-the-money call.
264
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1 2 3 4 5 6 7 8 9 10
time
call priceintrinsic value
P&L
Price, intrinsic value and P&L for in-the-money call.
265
Out-of-the Money Scenario
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10
time
S(t)Delta
Underlying security and hedge ratio for out-of-the-money call. 266
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 1 2 3 4 5 6 7 8 9 10
time
call priceintrinsic value
P&L
Price, intrinsic value and P&L for out-of-the-money call.
267
Hedging a European Call Option on the S&P500
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time
normalized S&P500Delta
Normalized S&P500 and hedge ratio forK = 1.2. 268
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time
Quadratic variation of log-S&P500.
269
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time
call priceintrinsic value
P&L
Call price on S&P500, intrinsic value and P&L.
270
Benchmarked Risk Minimization
Risk Minimization
• Follmer and Sondermann (1986)
• Follmer and Schweizer (1989)
• Schweizer (1991, 2001)
• Biagini et al. (1999)
• Biagini and Cretarola (2009)
• Du & Pl. (2012)
271
Financial Market
• d primary security accounts
• Si,jt - jth primary security account in ith security denomination
272
• locally riskless savings account
Si,it = Si,i
0 exp
∫ t
0
risds
• portfolio
Si,δt =
d∑j=1
δjtSi,jt
273
• self-financing portfolio
Si,δt = Si,δ
0 +d∑
j=1
∫ t
0
δjsdSi,js
274
Numeraire Portfolio
• Si,δ∗t numeraire portfolio
Sδt =
Si,δt
Si,δ∗t
≥ Et(Sδs)
0 ≤ t ≤ s <∞
• supermartingale property
275
• benchmarked jth primary security account
Sjt =
Si,jt
Si,δ∗t
local martingale; supermartingale
276
• matrix valued optional covariation process
[S] = [S]t = ([Si, Sj]t)di,j=1, t ∈ [0,∞)
• benchmarked primary security accounts
S = St = (S1t , . . . , S
dt )
⊤, t ∈ [0,∞)
277
Definition A dynamic trading strategy v, initiated at time t = 0, v =
vt = (ηt, ϑ1t , . . . , ϑ
dt )
⊤, t ∈ [0,∞), where ϑ = ϑt = (ϑ1t , . . . ,
ϑdt )
⊤, t ∈ [0,∞) forms benchmarked self-financing partϑ⊤t St of bench-
marked price process
V vt = ϑ⊤
t St + ηt
ϑ is predictable ∫ t
0
ϑ⊤u d[S]uϑu <∞
278
η = ηt, t ∈ [0,∞) adapted with η0 = 0 monitors the benchmarkednon-self-financing part of
V vt = V v
0 +
∫ t
0
ϑ⊤s dSs + ηt
V vt forms a supermartingale.
279
In general, capital has to be added or removed to match desired price
V vt = Sδ
t =d∑
j=1
δjt Sjt
with
δjt = ϑjt + ηtδ
j∗,t .
280
• benchmarked contingent claim HT
dynamic trading strategy v delivers HT if
V vT = Sδ
T = HT
=⇒ replicable if self-financing
281
Proposition If for HT a self-financing benchmarked portfolio SδHT exists,satisfying
SδHTt = E(HT |Ft)
for all t ∈ [0, T ] P -a.s., then this provides least expensive hedge for HT .
282
Real World Pricing
• least expensive replication by a self-financing benchmarked portfolio=⇒real world pricing formula
SδHTt = Et(HT )
minimal possible price process
283
Definition For a dynamic trading strategy v which delivers
Sδt =
d∑j=1
δjt Sjt
benchmarked P&L
Cδt = Sδ
t −d∑
j=1
∫ t
0
ϑjudS
ju − S
δ0
284
Corollary Cδt = ηt for t ∈ [0,∞).
• usually fluctuating benchmarked P&L
• intrinsic risk
285
What criterion would be most natural?
• symmetric view with respect to all primary security accounts, includingthe domestic savings account
• pooling in large trading book=⇒ vanishing total hedge error
286
Proposition HT,l, V vl withCvl independent square integrable martingales with
E
((C
vlt
Vvl0
)2)≤ Kt <∞ for l ∈ 1, 2, . . . , t ∈ [0, T ], T ∈ [0,∞).
at initial time well diversified trading book holds equal fractions
• total benchmarked wealth Ut =U0
m
∑ml=1
Vvlt
Vvl0
• total benchmarked P&L
Rm(t) =U0
m
m∑l=1
Cvlt
V vl0
,
⇒
limm→∞
Rm(t) = 0
P -a.s.
287
Definition A dynamic trading strategy v, initiated at time zero, is calledlocally real-world mean-self-financing if its adapted process ηt = Cδ
t formsan local martingale.
288
• classical risk minimizationFollmer and Sondermann (1986)Follmer and Schweizer (1989)Schweizer (1991, 1995)Schweizer (1999)Follmer-Schweizer decomposition assuming a risk neutral probabilitymeasureBuckdahn (1993) Schweizer (1994) Stricker (1996)Delbaen et al. (1997) Pham et al. (1998)
289
• local risk minimizationBouleau and Lamberton (1989)Duffie and Richardson (1991)Schweizer (1994)Biagini et al. (1999)Schweizer (2001)
290
• Benchmarked P&L is orthogonal if
ηtSt
is vector local martingale.
- above product has no trend- benchmarked hedge error orthogonal to any benchmarked traded wealth
291
• VHTset of locally real-world mean-self-financing dynamic trading strate-
gies v initiated at time zero with
V vt = Sδ
t , t ≥ 0
which deliver HT
with orthogonal benchmarked P&L
292
• market participants prefer more for less
⇒ Benchmarked Risk Minimization
For HT strategy v ∈ VHT
benchmarked risk minimizing (BRM) if for all v ∈ VHTwith V v
t = Sδt
price Sδt is minimal
Sδt ≤ S
δt
P -a.s. for all t ∈ [0, T ].
293
Regular Benchmarked Contingent Claims
HT is called regular if
HT = Et(HT ) +
d∑j=1
∫ T
t
ϑj
HT(s)dSj
s + ηHT(T )− ηHT
(t)
ϑHT- predictable
ηHT- local martingale, orthogonal to S
294
Theorem A regular HT has a BRM strategy v = vt = (ηt, ϑ1t , . . . , ϑ
dt )
⊤, t ∈[0, T ] ∈ VHT
with
Sδt = V v
t = Et(HT ) ,
SδT = HT ,
and orthogonal benchmarked P&L: Cδt = ηHT
(t)
295
=⇒
Cδt - local martingale
Cδt St - vector local martingale
orthogonal
296
hedgeable part:
Et(HT )− Cδt = E0(HT ) +
∫ t
0
ϑ⊤HT
(s)dSs .
local martingale
297
• benchmarked price V vt = Et(HT ) minimal
• do not request square integrability
• no equivalent risk neutral probability measure required
• need only to check zero drift of ηHT
• and ηHT(t), St orthogonality
298
Hedging a Regular Claim
• benchmarked contingent claim HT driven by continuous local mar-tingalesW 1,W 2, . . . ,W d orthogonal to each other
• benchmarked primary security account Sjt , j ∈ 1, . . . , d
dSjt = −Sj
t
d−1∑k=1
θj,kt dW kt
• d× d matrix Φt = [Φi,kt ]di,k=1
Φi,kt =
θi,kt for k ∈ 1, . . . , d− 11 for k = d
for t ∈ [0, T ].
299
Proposition assume Φt invertible and Vt = E(HT |Ft) has representa-tion
Vt = V0 +d∑
k=1
∫ t
0
xksdW
ks +
∫ t
0
xdsdW
ds .
Then HT is a regular benchmarked contingent claim
VvHTt = Vt for all t ∈ [0, T ]
ϑHT(t) = diag(St)
−1(Φ⊤t )−1ξt
ηHT(t) =
∫ t
0
xdsdW
ds
with ξt = (x1t , . . . , x
d−1t , V0 +
∑d−1k=1
∫ t
0xksdW
ks )
⊤ .
300
Si,jt =
Sjt
Sit
dSi,jt = Si,j
t
d−1∑k=1
(θi,kt − θj,kt )(θi,kt dt+ dW k
t )
bi,j,kt = (θi,kt − θj,kt ) .
bdt = [bd,j,kt ]d−1,d−1j,k=1
301
Proposition The matrix Φt is invertible if and only if bdt is invertible.
Proof: elementary transform of bdt
Φt ←→
bdt 0
θd,1t . . . θd,d−1t 1
Φt has full rank if and only if bdt has full rank
302
Comparison with Quadratic Criterion
•∫ t
0ϑ⊤HT
(s)dSs and ηHT(t) independent, square integrable martingales
• S1t , . . . , S
dt and ηHT
(t) independent, square integrable martingales
• ηHTorthogonal to benchmarked primary security accounts
• HT square integrable
• V vt = Sδ
t delivers HT
303
Quadratic Criterion
E((Cδ
T )2)= E
((HT −
∫ T
0
ϑ⊤s dSs − Sδ
0)2
)⇒ min
304
E((Cδ
T )2)
= E
((E(HT ) − Sδ
0 +∫ T
0(ϑ⊤
HT(s) − ϑ⊤
s )dSs + ηHT(T )
)2)
= E(E(HT ) − Sδ
0
)2
+ E(∫ T
0(ϑ⊤
HT(s) − ϑ⊤
s )2d[S]s)+ E((ηHT
(T ))2)
305
=⇒
• Sδ0 = E(HT )
• ϑt = ϑHT(t), t ∈ [0, T ]
• ηHT(T ) = Cδ
T
• second moment of benchmarked P&L becomes minimal
306
307
Nonhedgeable Contingent Claim
• savings account S1,1t = 1
• numeraire portfolio S1,δ∗t
• Ht = Et(HT ), t ∈ [0, T ]
- independent from S1,δ∗t
- continuous
308
Benchmarked risk minimization
• real world pricing formula
SδHTt = Et(HT ) = Et(HT )Et(S
1T ) = HtP (t, T )
P (t, T ) = Et(S1T )
dP (t, T ) =∂P (t, T )
∂S1dS1
t
• benchmarked NP Sδ∗t =
S1,δ∗t
S1,δ∗t
= 1
• benchmarked P&L
dCδHTt = dS
δHTt − ϑ1
HT(t)dS1
t
=
(Ht
∂P (t, T )
∂S1− ϑ1
HT(t)
)dS1
t + P (t, T )dHt
309
• Assume orthogonal benchmarked primary security accounts
• product CδHT S has zero drift if
0 =d[S, CδHT ]t
dt=
(Ht
∂P (t, T )
∂S1− ϑ1
HT(t)
)d[S1]t
dt.
=⇒
ϑ1HT
(t) = Ht
∂P (t, T )
∂S1
Depends onHt !
310
• Benchmarked P&L
ηHT(t) = C
δHTt =
∫ t
0
P (s, T )dHs
local martingale, orthogonal
• self-financing benchmarked hedgeable part
SδHTt − ηHT
(t) = SδHT0 +
∫ t
0
ϑ1HT
(s)dS1s
• ϑ1HT
(t) units in S1t
• remaining wealth in NP
• minimizing ddt[CδHT ]t
311
• Regular benchmarked contingent claim
HT =HT
S1,δ∗T
= SδHT0 +
∫ T
0
Ht
∂P (t, T )
∂S1dS1
t +
∫ T
0
P (t, T )dHt
ϑ1HT
(t) = Ht
∂P (t, T )
∂S1
ηHT(t) =
∫ t
0
P (s, T )dHs
312
Evolving Information
• consider special case when S1 true martingale
• classical risk minimization can be applied
• real world pricing formula yields
SδHTt =
SδHTt
S1t
= Et
(S1T
S1t
HT
S1,1T
)= Et
(ΛT
Λt
HT
S1,1T
)= EQ
t
(HT
S1,1T
)• equivalent minimal martingale measureQ
ΛT =dQ
dP|FT
=S1T
S1t
• same initial price as benchmarked risk minimization
SδHT0 = H0 = S
δHT0 (S1
0)−1
313
• classical risk minimization invests total initial value in the savings ac-count
ϑ1CRM(t) = H0
• BRM strategy different
• P (t, T ) = S1t , ∂P (t,T )
∂S1t
= 1
=⇒
ϑ1HT
(t) = Ht = Et(HT )
• intuitive and practically appealing
• under classical risk minimization, evolving information ignored
• provides less fluctuations for benchmarked P&L
• easier diversified
314
Modeling the Dynamics of Diversified Indices
Pl. & Rendek (2012)
• conjecture for normalized aggregate wealth dynamicstime transformed square root process
• Naive Diversification Theorem⇒ equity index = proxy of numeraireportfolio
• empirical stylized facts⇒ falsify models
315
• ⇒ propose realistic one factor, two component long term index model
• realistic model outside classical theory⇒ benchmark approach
• exact, almost exact simulation⇒ verify empirical facts, effects of esti-mation techniques etc.
316
The Affine Nature of Diversified WealthDynamics
Object: normalized units of wealth
Total wealth: Y ∆τi
Time steps: τi = i∆
Wealth unit value:√∆
317
Economic activity: during [τi, τi+1) ”projects”generate each independent wealth with variance increment v2∆3/2;consume each η∆ fraction of wealth;generate together on average β
√∆ new units.
318
Mean for increment of aggregate wealth:(β − ηY ∆
τi
)∆
319
Variance for increment aggregate wealth: , sum of variances
=⇒ proportional to number of wealth units:Y ∆
τi√∆
=⇒ proportional to aggregate wealth=⇒ variance of increment of aggregate wealth is v2Y ∆
τi∆
320
for ∆→ 0
Y ∆τi+1− Y ∆
τi=(β − ηY ∆
τi
)∆ + υ
√Y ∆τi
∆Wτi
E(∆Wτi) = 0, E((∆Wτi)2) = ∆
conjectures drift and diffusion terms
321
=⇒ Conjecture viaWeak convergenceKleoden & Pl. (1992), Alfonsi (2005), Diop (2003)for parameters: β = η = υ = 1
square root process:
dYτt = (1− Yτt) dτt +√YτtdWτt
322
• Quadratic variation:
[Yτ. ]t =
∫ t
0
Yτsdτs =
∫ t
0
YτsMsds
• Market activity:
Mt =dτt
dt
• integrate normalized index:can we findM =const. s.t.
M
∫ t
0
Yτsds ≈ [Yτ. ]t ?
323
Quadratic variation and integrated normalized S&P500monthly data, calendar time, Shiller data
1871 1921 1971 20210
0.5
1
1.5
2
2.5
3
[ Yτ ]
t
M ∫0τ
t Yτ
s
ds
M ≈ 0.0178
average long term fit
324
Market Activity: Mt =dτt
dtfrom model
1871 1921 1971 20210
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
325
Quadratic variation and integrated normalized S&P500monthly data, τ -time
1871 1921 1971 20210
0.5
1
1.5
2
2.5
3
[ Yτ ]
t
∫0τ
t Yτ
s
dτs
326
Eight Stylized Empirical Facts
• falsify potential models, Popper (1959)
• TOTMKWD in 26 currency denominations
about 1000 years of daily data
327
(i) uncorrelated log-returns
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Average autocorrelation function for log-returns
328
(ii) correlated absolute log-returns
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Average autocorrelation function for absolute log-returns
329
(iii) Student-t distributed log-returns
−20 −15 −10 −5 0 5 10 15 20 2510
−6
10−5
10−4
10−3
10−2
10−1
100
Empirical log−density
Theoretical log−density
Logarithm of empirical density of normalized log-returns with Student-tdensity
330
Results for log-returns of the EWI104s
Pl. & Rendek (2008)
0
0.5
1
1.5
2
−5−4
−3−2.15
−10
1 2
34
5
−2.94
−2.92
−2.9
−2.88
−2.86
−2.84
αλ
Estimated λ
Estimated LLF
× 105
SGH Student-t NIG Hyperbolic VG
σ 0.98 0.72 0.97 0.96 0.96
α 0.00 0.97 0.72
λ -2.16 1.49
ν 4.33
ln(L∗) -285796.39 -285796.39 -286448.94 -287152.08 -287499.83
Ln 0.0000004 1305.10 2711.38 3406.88
Ln = 0.0000004 < χ20.001,1 ≈ 0.000002
331
(iv) volatility clustering
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20130
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Estimated volatility
332
(v) long term exponential growth
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20133.5
4
4.5
5
5.5
6
6.5
4.178+0.048 t
Logarithm of index with trend line
333
(vi) leverage effect
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
Logarithms of normalized index and its volatility
334
(vii) extreme volatility at major downturns
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
Logarithms of normalized index and its volatility
335
(viii) Integrated Normalized Index and its Quadratic Variationmonthly data, calendar time
1871 1921 1971 20210
0.5
1
1.5
2
2.5
3
[ Yτ ]
t
M ∫0τ
t Yτ
s
ds
M ≈ 0.0178
average long term fit
336
⇒ Discounted Index Model generalizing conjecture
St = Aτt(Yτt)q,
Aτt = A expaτt
q > 0, a > 0,A > 0
conjecture expects q = 1
337
Normalized index: (Yτt)q = St
Aτt
dYτ =
δ4−
1
2
(Γ(δ2+ q
)Γ(δ2
) ) 1q
Yτ
dτ +√Yτ dW (τ )
Long term mean: limτ→∞1τ
∫ τ
0(Ys)
qds = 1 P-a.s
338
Market activity time: dτt = Mtdt
Inverse of market activity:
d
(1
Mt
)=
(ν
4γ − ϵ
1
Mt
)dt+
√γ
Mt
dWt,
where
dW (τt) =
√dτt
dtdWt =
√MtdWt
⋄ only oneWt
⋄ two component model
339
Discounted index SDE:
dSt = St (µtdt+ σtdWt)
Expected rate of return:
µt =
a
Mt
−q
2
(Γ(δ2+ q
)Γ(δ2
) ) 1q
+
(δ
4q +
1
2q(q − 1)
)1
MtYτt
Mt
Volatility:
σt = q
√Mt
Yτt
Pl. & Rendek (2012c)
340
Benchmark Approach
Bt – benchmark savings account
dBt = Bt
((−µt + σ2
t
)dt− σtdWt
)
σ2t ≤ µt ⇒ Bt is an (A, P )-supermartingale
⇒ no strong arbitrage; Pl. (2011)
341
Assumptions:
A1. δ = 2(q + 1) A2. q2
(Γ(2q+1)Γ(q+1)
) 1q ≤ a
=⇒ σ2t ≤ µt
342
Fitting the model to TOTMKWD
Step 1: Normalization of IndexAτt ≈ A exp 4aϵ
γ(ν−2)t⇒A = 65.21, 4aϵ
γ(ν−2)≈ 0.048
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20133.5
4
4.5
5
5.5
6
6.5
4.178+0.048 t
343
Normalized TOTMKWD
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20130.4
0.6
0.8
1
1.2
1.4
1.6
1.8
344
Step 2: Power q:Affine nature =⇒ conjecture: q = 1 = δ
2− 1 =⇒ δ = 4
• student-t with about four degree of freedomis considered and cannot be falsified,
=⇒ We set q = 1 and δ = 4
345
Step 3: Observing Market Activity:
d[√
Y ]τt
dt= 1
4dτt
dt= Mt
4
Qti ≈[√
Y ]τti+1−[
√Y ]τti
ti+1−ti
Qti+1= α√ti+1 − tiQti + (1− α√ti+1 − ti)Qti , α = 0.92
exponential smooth
robust
346
Market activity: Mt ≈ 4Qt
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20130
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
M0 = 0.0175
347
Step 4: Parameter γ:
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20130
500
1000
1500
2000
2500
3000
66.28 t
γ = 265.12
348
Step 5: Parameters ν and ϵ andStep 6: Long Term Average Net Growth Rate a:
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
10
20
30
40
50
60
ν ≈ 4, ϵ ≈ 2.18⇒ a = 2.55⇒ no strong arbitrage
349
Fitted model allows visualizingCalculated Volatility
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20130
0.1
0.2
0.3
0.4
0.5
0.6
0.7
σt ≈√
4Qt
Yτt, average volatility: 11.9%
350
Model applies also to proxies of numeraire portfolioS&P500 and VIX
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013−3
−2.5
−2
−1.5
−1
−0.5
0
log calculated volatility
log scaled VIX
A = 52.09, ϵ = 2.15, γ = 172.3, a = 1.5
351
Simulation StudyStep 1: Market activity:
1
Mti+1
=γ(1− e−ϵ(ti+1−ti))
4ϵ
χ23,i +
√ 4ϵe−ϵ(ti+1−ti)
γ(1− e−ϵ(ti+1−ti))
1
Mti
+ Zi
2
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
1.2
exact simulation
352
Step 2: τ -time:
τti+1− τti =
∫ ti+1
ti
Msds ≈Mti(ti+1 − ti)
0 5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
almost exact simulation
353
Step 3: Normalized index:
Yτti+1=
1− e−(τti+1−τti
)
4
χ23,i +
√√√√ 4e
−(τti+1−τti
)
1− e−(τti+1−τti
)Yτti
+ Zi
2
0 5 10 15 20 25 30 35 40
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
almost exact simulation
354
Model recovers stylized empirical facts:
Model is difficult to falsify: Popper (1934)
1. Uncorrelated returns
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
355
2. Correlated absolute returns
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
356
3. Student-t distributed returns
357
estimated statistics
Simulation Student-t NIG Hyperbolic VG ν
1 0.008934 37.474149 102.719638 131.240780 4.012850
2 11.485226 11.175028 96.457136 132.916256 3.450916
3 0.000000 100.928524 244.190151 294.719960 3.734148
4 9.002421 35.759464 347.060676 331.014904 2.579009
5 8.767003 11.551178 121.190482 144.084964 3.170449
6 0.401429 60.570898 205.788160 252.591737 3.432435
7 12.239056 4.354888 46.411554 78.273485 3.957696
8 1.693411 23.910523 94.408789 130.623174 3.849691
9 1.232454 47.830407 202.073144 237.168411 3.236322
10 0.000000 43.037206 128.807757 162.582353 3.774957
11 0.433645 47.782681 172.736397 208.847632 3.431803
12 0.000000 56.019354 146.077121 185.624888 3.899403
13 7.137154 48.219756 579.922931 477.383441 2.293363
14 5.873948 16.515390 107.770531 135.508299 3.388307
15 0.000000 54.718046 184.112794 217.304105 3.402049
16 6.982560 3.991610 29.192198 47.105125 4.268740
17 2.966916 22.914863 108.513143 138.044416 3.553629
18 0.000000 52.066364 129.790856 160.373085 3.959605
19 0.006909 39.568695 111.398645 143.914350 3.982892
20 0.000001 56.845664 169.915512 211.260626 3.651091
21 1.674578 17.681088 61.710576 90.679738 4.265834
22 14.010840 3.279722 47.433693 73.789313 3.770825
23 11.198940 12.074044 114.888817 143.800553 3.257146
24 0.455557 27.676102 86.841704 114.452947 4.006528
358
4. Volatility clustering
0 5 10 15 20 25 30 35 400
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
359
5. Long term exponential growth
0 5 10 15 20 25 30 35 403.5
4
4.5
5
5.5
6
6.5
4.16+0.05 t
360
6. Leverage effect and7. Extreme volatility at major market downturns
0 5 10 15 20 25 30 35 40−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
361
Conclusions:
⋄ equity index model: 3 initial parameters, 3 structural parameters and 1
Wiener process (nondiversifiable uncertainty)
⋄ model recovers 7 stylized empirical facts
⋄ allows long dated derivative pricing under benchmark approach
⋄ leads outside classical theory
⋄ if benchmarked savings account local martingale=⇒ only 2 structural parameters
362
Squared Bessel Processes (*)
Revuz & Yor (1999)
• squared Bessel process BESQδx
dimension δ ≥ 0
dXφ = δ dφ+ 2√|Xφ| dWφ
φ ∈ [0,∞) withX0 = x ≥ 0
• scaling property
Zφ =1
aXaφ
BESQδxa
, a > 0
363
• sum of squares
Xφ =δ∑
k=1
(wk +W kφ)
2
W kφ Wiener process
=⇒
dXφ = δ dφ+ 2
δ∑k=1
(wk +W k
φ
)dW k
φ
X0 =δ∑
k=1
(wk)2 = x
364
• setting
dWφ = |Xφ|−12
δ∑k=1
(wk +W k
φ
)dW k
φ
• quadratic variation
[W ]φ =
∫ φ
0
1
Xs
δ∑k=1
(wk +W k
s
)2ds = φ
Levy’s theorem
=⇒ W Wiener process in φ time
365
0
100
200
300
400
500
600
700
800
0 10 20 30 40 50 60 70 80 90
phi
Squared Bessel process of dimension δ = 4 in φ-time
366
• additivity property
Shiga & Watanabe (1973)
X = Xφ, φ ∈ [0,∞) BESQδx
Y = Yφ, φ ∈ [0,∞) independent BESQδ′
y
=⇒ Xφ + Yφ forms BESQδ+δ′
x+y
x, y, δ, δ′ ≥ 0
367
• δ > 2, strict positivity
withX0 = x > 0
P
(inf
0≤φ<∞Xφ > 0
)= 1
• δ = 2, no strict positivity
P
(inf
0≤φ<∞Xφ > 0
)= 0
• δ ∈ [0, 2), no strict positivityX0 = x > 0
P
(inf
0≤φ≤φ′Xφ = 0
)> 0
368
• transition density
δ > 0 , x > 0
pδ(ϱ, x;φ, y) =1
2(φ− ϱ)
(y
x
)ν2
exp
−
x+ y
2(φ− ϱ)
Iν
(√x y
φ− ϱ
)
Iν modified Bessel function of the first kind
index ν = δ2− 1
369
80
90
100
110
120
y0.1
0.2
0.3
0.4
0.5
t
0
0.05
0.1
0.15
80
90
100
110y
Transition density of squared Bessel process for δ = 4
370
• Y =Xφ
φnon-central chi-square distributed
dimension δnon-centrality parameter ℓ = x
φ
P
(Xφ
φ< u
)=
∞∑k=0
exp− ℓ
2
(ℓ2
)kk !
1−Γ(
u2; δ+2k
2
)Γ(
δ+2k2
)
Γ(·; ·) incomplete gamma function
371
• Moments for α > − δ2
, φ ∈ (0,∞) and δ > 2
E(Xα
φ
∣∣A0
)=
(2φ)α exp
−X0
2φ
∑∞k=0
(X0
2φ
)k Γ(α+k+ δ2)
k ! Γ(k+ δ2)
for α > − δ2
∞ for α ≤ − δ2
=⇒E(Xφ
∣∣A0
)= X0 + δ φ
372
• by monotonicity of the gamma function
E(Xα
φ
∣∣A0
)≤ (2φ)α exp
−X0
2φ
(Γ(α+ δ
2
)Γ(δ2
) + exp
X0
2φ
)
< ∞
• for δ = 4
E(X−1
φ
∣∣A0
)= X−1
0
(1− exp
−X0
2φ
)
373
A Strict Local Martingale
Zφ = X−1φ inverse of a squared Bessel processXφ of dimension 4
dZφ = −2Z 32φ dWφ
=⇒ local martingale
E(Zφ
∣∣A0
)= E
(X−1
φ
∣∣A0
)= Z0
(1− exp
−12Z0 φ
)< Z0
=⇒ strict local martingale
=⇒ strict supermartingale
374
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 10 20 30 40 50 60 70 80 90
phi
Inverse of a squared Bessel process of dimension δ = 4 in φ-time
375
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0 10 20 30 40 50 60 70 80 90
phi
Expectation of the inverse of the squared Bessel process for δ = 4 inφ-time
376
Time Transformation
• φ-time
φ(t) = φ(0) +1
4
∫ t
0
c2usudu
t ∈ [0,∞), s0 > 0
with
st = s0 exp
∫ t
0
bu du
377
• φ-time
for constant b < 0, c = 0
φ(t) = φ(0) +c2
4 b s0(1− exp−b t)
• time
t(φ) = −1
bln
(1−
4 b s0
c2(φ− φ(0))
)
378
0
5
10
15
20
25
30
0 10 20 30 40 50 60 70 80 90
phi
Time t(φ) against φ-time
379
0
100
200
300
400
500
600
700
800
0 5 10 15 20 25 30
t
Squared Bessel process in dependence on time t
380
100
150
200
250
300
350
400
450
0 5 10 15 20 25 30
t
Expectation of a squared Bessel process in dependence on time t
381
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0 5 10 15 20 25 30
t
Expectation of the inverse of a squared Bessel process in dependence ontime t
382
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 5 10 15 20 25 30
t
Inverse of squared Bessel process in dependence on time t
383
Square Root Process
Yt = stXφ(t)
dYt = st dXφ(t) +Xφ(t) dst
= st δ dφ(t) + st 2√Xφ(t) dWφ(t) +Xφ(t) st bt dt
=
(δ
4c2t + bt Yt
)dt+ ct
√Yt
√4 st
c2tdWφ(t)
384
dUt =
√4 st
c2tdWφ(t)
has quadratic variation
[U ]t =
∫ t
0
4 sz
c2zdφ(z) = t
U Wiener process
=⇒ SR process
dYt =
(δ
4c2t + bt Yt
)dt+ ct
√Yt dUt
385
10
15
20
25
30
35
40
45
50
0 5 10 15 20 25 30
t
Sample path of a square root process in dependence on time t
386
• moment
E(Y αt
∣∣A0
)= (2 φt st)
α exp
−Y0
2 φt
∞∑k=0
(Y0
2 φt
)kΓ(α+ k + δ2)
k! Γ(k + δ2)
δ > 2, α > − δ2
• moment estimate
E(Y αt
∣∣A0
)≤ (2 φt st)
α exp
−Y0
2 φt
(Γ(α+ δ
2)
Γ( δ2)
+ exp
Y0
2 φt
)
< ∞ for α ∈ (−δ
2, 0)
st =st
s0= exp
∫ t
0
bu du
φt = s0 (φ(t)− φ(0)) =
1
4
∫ t
0
c2usudu
387
• first moment of SR process
E(Yt
∣∣A0
)= E
(Y0
∣∣A0
)exp
∫ t
0
bs ds
+
∫ t
0
δ
4c2s exp
∫ t
s
bz dz
ds
• case δ = 4
for c2t = c2 > 0 and bt = b < 0
limt→∞
E(Yt
∣∣A0
)= −
c2
b
limt→∞
E(Y −1t
∣∣A0
)= −2
b
c2
388
• transition density
p(s, Ys; t, Yt) =pδ
(φ(s), Ys
ss;φ(t), Yt
st
)st
p(0, x; t, y) =1
2st φt
(y
x st
) ν2
exp
−x+ y
st
2 φt
Iν
√x y
st
φt
for 0 < t <∞, x, y ∈ (0,∞),
where ν = δ2− 1, st = expb t and φt =
c2
4 b(1− 1
st)
389
• stationary density
is gamma density
pY∞(y) =
(−2 bc2
) δ2
yδ2−1
Γ( δ2)
exp
2 b
c2y
390
Minimal Market ModelPl. (2001), special case of Pl.-Rendek model
Volatility Parametrization
• Diversification Theorem =⇒
well diversified stock market index approximates NP
• discounted NP
dSδ∗t = Sδ∗
t |θt| (|θt| dt+ dWt),
where
dWt =1
|θt|
d∑k=1
θkt dWkt
391
0
20
40
60
80
100
120
140
160
1926 1950 1975 2000
Discounted NP
392
0
1
2
3
4
5
1926 1950 1975 2000
ln(disc WSI)ln(A(t))
Logarithm of discounted NP
393
Drift Parametrization
• discounted NP driftαt = Sδ∗
t |θt|2
strictly positive, predictable=⇒
• volatility
|θt| =√αt
Sδ∗t
leverage effect creates natural feedbackunder reasonably independent drift
394
• discounted NP
dSδ∗t = αt dt+
√Sδ∗t αt dWt
drift has economic meaning
• transformed time
φt =1
4
∫ t
0
αs ds
395
• squared Bessel process of dimension four
Xφt = Sδ∗t
dW (φt) =
√αt
4dWt
dXφ = 4 dφ+ 2√Xφ dW (φ)
Revuz & Yor (1999)
economically founded dynamics in φ-time
still no specific dynamics in t-time assumed
396
Time Transformed Bessel Process
d
√Sδ∗t =
3αt
8
√Sδ∗t
dt+1
2
√αt dWt
• quadratic variation [√Sδ∗
]t=
1
4
∫ t
0
αs ds
• transformed timeφt =
[√Sδ∗
]t
observable
397
0
2
4
6
8
10
12
14
1926 1950 1975 2000
Empirical quadratic variation of the square root of the discounted S&P 500
398
Stylized Minimal Market Model
Pl. (2001, 2002, 2006), Pl. & Rendek (2012)
• assume discounted NP drift as
αt = α exp η t
• initial parameter α > 0
• net growth rate η
399
• transformed time
φt =α
4
∫ t
0
exp η z dz
=α
4 η(exp η t − 1)
400
0
2
4
6
8
10
12
14
1926 1950 1975 2000
observed transformed timefitted transformed time
Fitted and observed transformed time
401
• normalized NP
Yt =Sδ∗t
αt
dYt = (1− η Yt) dt+√Yt dWt
square root process of dimension four
=⇒ parsimonious model, long term viabilityconfirmation of stylized Pl.-Rendek modelPl. & Rendek (2012)
402
=⇒
• discounted NPSδ∗t = Yt αt
• NPSδ∗t = S0
t Sδ∗t = S0
t Yt αt
• scaling parameter α = 0.043
• net growth rate η = 0.0528
• reference level 1η= 18.93939
• speed of adjustment η
• half life time of major displacement ln(2)η≈ 13 years
403
20
30
40
50
60
70
80
90
100
1926 1950 1975 2000
Normalized NP
404
• logarithm of discounted NP
ln(Sδ∗t ) = ln(Yt) + ln(α) + η t
• no need for extra volatility process in MMM
• realistic long term dynamics
405
Volatility of NP under the stylized MMM
|θt| =1√Yt
• squared volatility
d|θt|2 = d
(1
Yt
)= |θt|2 η dt−
(|θt|2
) 32 dWt
3/2 volatility model
Pl. (1997), Lewis (2000)
406
0.1
0.12
0.14
0.16
0.18
0.2
0.22
1926 1950 1975 2000
Volatility of the NP under the stylized MMM
407
Transition Density of Stylized MMM
• transition density of discounted NP Sδ∗
p(s, x; t, y) =1
2 (φt − φs)
(y
x
) 12
exp
−
x+ y
2 (φt − φs)
× I1( √
x y
φt − φs
)
φt =α
4 η(expη t − 1)
I1(·) modified Bessel function of the first kind
408
• non-central chi-square distributed random variable:
y
φt − φs
=Sδ∗t
φt − φs
with δ = 4 degrees of freedom and non-centrality parameter:
x
φt − φs
=Sδ∗s
φt − φs
409
80
90
100
110
120
y0.1
0.2
0.3
0.4
0.5
t
0
0.05
0.1
0.15
80
90
100
110y
Transition density of squared Bessel process for δ = 4410
Zero Coupon Bond under the stylized MMM
• zero coupon bond
P (t, T ) = Sδ∗t Et
(1
Sδ∗T
)= P ∗
T (t)Et
(Sδ∗t
Sδ∗T
)
with savings bond
P ∗T (t) = exp
−∫ T
t
rs ds
411
δ = 4 =⇒
P (t, T ) = P ∗T (t)
(1− exp
−
Sδ∗t
2 (φT − φt)
)< P ∗
T (t)
for t ∈ [0, T ), Pl. (2002)
with time transform
φt =α
4 η(expη t − 1)
P (0, T ) = 0.00423
P ∗T (0) = 0.04947
P (0,T )P ∗
T (0)≈ 0.0855
412
0
0.2
0.4
0.6
0.8
1
1.2
1926 1950 1975 2000
savings bondP(t,T)
Zero coupon bond and savings bond
413
-6
-5
-4
-3
-2
-1
0
1
1926 1950 1975 2000
ln(savings bond)ln(hedge portfolio)
ln(P(t,T))
ln from Zero coupon bond, hedge portfolio and savings bond
414
Forward Rates under the MMM
• forward rate
f(t, T ) = −∂
∂Tln(P (t, T ))
= rT + n(t, T )
415
• market price of risk contribution
n(t, T ) = −∂
∂Tln
(1− exp
−
Sδ∗t
2(φT − φt)
)
=1(
exp
Sδ∗t
2(φT −φt)
− 1
) Sδ∗t
(φT − φt)2
αT
8
limT→∞
n(t, T ) = η
Pl. (2005), Miller & Pl. (2005)
416
0.020.04
0.06
0.08
0.1
eta
20
40
60
80
T
0
0.025
0.05
0.075
0.1
0.020.04
0.06
0.08eta
Market price of risk contribution in dependence on η and T417
Free Snack from Savings Bond
• potential existence of weak form of arbitrage?
borrow amount P (0, T ) from savings account
Sδt = P (t, T )− P (0, T ) expr t
such that Sδ0 = 0
SδT = 1− P (0, T ) expr T > 0
lower bondSδt ≥ −P (0, T ) expr t
418
• since Sδt may become negative
not strong arbitrage
Fundamental Theorem of Asset Pricing
Delbaen & Schachermayer (1998) =⇒ the MMM does not
admit an equivalent risk neutral probability measure
=⇒there is a free lunch with vanishing risk
Delbaen & Schachermayer (2006)
419
Absence of an Equivalent Risk Neutral Probability Measure
• fair zero coupon bond has a lower price than the savings bond
P (t, T ) < P ∗T (t) = exp
−∫ T
t
rs ds
=S0t
S0T
• Radon-Nikodym derivative
Λt =S0t
S00
=Sδ∗0
Sδ∗t
strict supermartingale under MMM
420
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1926 1950 1975 2000
total massRadon Nikodym deriv
Radon-Nikodym derivative and total mass of candidate risk neutral measure
421
• hypothetical risk neutral measure
total mass:
Pθ,T (Ω) = E (ΛT ) = 1− exp
−Sδ∗0
2φT
< Λ0 = 1
Pθ is not a probability measure
422
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
1926 1950 1975 2000
P*^P^
Benchmarked zero coupon bond
423
Hedge Simulation
• delta units in the NP
δ∗(t) =∂P (t, T )
∂Sδ∗t
= exp
−∫ T
0
rs ds
exp
−
Sδ∗t
2(φT − φt)
1
2(φT − φt)
424
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1926 1950 1975 2000
Ratio in the NP
425
-8e-006
-6e-006
-4e-006
-2e-006
0
2e-006
4e-006
6e-006
8e-006
1926 1950 1975 2000
Benchmarked P&L for hedge portfolio
426
European Call Options under the MMM
real world pricing formula =⇒
cT,K(t, Sδ∗t ) = Sδ∗
t Et
((Sδ∗
T −K)+
Sδ∗T
)
= Et
(Sδ∗t −
K Sδ∗t
Sδ∗T
)+
= Sδ∗t
(1− χ2(d1; 4, ℓ2)
)−K exp−r (T − t) (1− χ2(d1; 0, ℓ2))
χ2(·; ·, ·) non-central chi-square distribution
427
with
d1 =4 ηK exp−r (T − t)
S0t αt (expη (T − t) − 1)
and
ℓ2 =2 η Sδ∗
t
S0t αt (expη (T − t) − 1)
Hulley, Miller & Pl. (2005)
Miller & Pl. (2008)
428
0.6
0.8
1
1.2K2
4
6
8
10
T
0.2
0.225
0.25
0.275
0.6
0.8
1
1.2K
Implied volatility surface for the stylized MMM
429
• implied volatility
in BS-formula adjust short rate to
r = −1
T − tln(P (t, T ))
otherwise put and call implied volatilities do not match
430
European Put Options under the MMM
• fair put-call parity relation
pT,K(t, Sδ∗t ) = cT,K(t, Sδ∗
t )− Sδ∗t +K P (t, T )
• European put option formula
pT,K(t, Sδ∗t ) = −Sδ∗
t
(χ2(d1; 4, ℓ2)
)+K exp−r (T − t)
(χ2(d1; 0, ℓ2)− exp−ℓ2
)
Hulley, Miller & Pl. (2005)
431
0
5000
10000
15000
20000
25000
30000
35000
Dec-2
5
Dec-2
8
Dec-3
1
Dec-3
4
Dec-3
7
Dec-4
0
Dec-4
3
Dec-4
6
Dec-4
9
Dec-5
2
Dec-5
5
Dec-5
8
Dec-6
1
Dec-6
4
Dec-6
7
Dec-7
0
Dec-7
3
Dec-7
6
Dec-7
9
Dec-8
2
Dec-8
5
Dec-8
8
Dec-9
1
Dec-9
4
Dec-9
7
Dec-0
0
Dec-0
3
index
K*savings bond
"risk neutral" put
fair put
Risk neutral and fair put on index
432
0
50
100
150
200
250
300
30
/03
/19
28
31
/07
/19
30
30
/11
/19
32
29
/03
/19
35
30
/07
/19
37
30
/11
/19
39
31
/03
/19
42
31
/07
/19
44
29
/11
/19
46
31
/03
/19
49
31
/07
/19
51
30
/11
/19
53
30
/03
/19
56
31
/07
/19
58
30
/11
/19
60
29
/03
/19
63
30
/07
/19
65
30
/11
/19
67
31
/03
/19
70
31
/07
/19
72
29
/11
/19
74
31
/03
/19
77
31
/07
/19
79
30
/11
/19
81
30
/03
/19
84
31
/07
/19
86
30
/11
/19
88
29
/03
/19
91
30
/07
/19
93
30
/11
/19
95
31
/03
/19
98
31
/07
/20
00
29
/11
/20
02
38
44
2
index
K*savings bond
"risk neutral" put
fair put
Benchmarked “risk neutral” and fair put on index
433
• put-call parity breaks down if one uses the savings bond P ∗T (t)
pT,K(t, Sδ∗t ) < cT,K(t, Sδ∗
t )− Sδ∗t +K exp−r (T − t)
for t ∈ [0, T )
434
Comparison to Hypothetical Risk Neutral Prices
• hypothetical risk neutral price c∗T,K(t, Sδ∗t )
of a European call option on the GOP
• benchmarked hypothetical risk neutral call price
c∗T,K(t, Sδ∗t ) =
c∗T,K(t, Sδ∗t )
Sδ∗t
local martingale
c∗T,K(·, ·) uniformly bounded
435
=⇒ c∗T,K - martingale =⇒
c∗T,K(t, Sδ∗t ) = cT,K(t, Sδ∗
t )
=⇒c∗T,K(t, Sδ∗
t ) = cT,K(t, Sδ∗t )
436
• hypothetical risk neutral put-call parity
p∗T,K(t, Sδ∗t ) = c∗T,K(t, Sδ∗
t )− Sδ∗t +K P ∗
T (t)
since P ∗T (t) > P (t, T ) =⇒
pT,K(t, Sδ∗t ) < p∗T,K(t, Sδ∗
t )
t ∈ [0, T )
437
Difference in Asymptotic Put Prices
• hypothetical risk neutral prices can become extreme if NP value tendstowards zero
• asymptotic fair zero coupon bond
limSδ∗
t →0PT (t, T )
a.s.= 0
for t ∈ [0, T )
• asymptotic fair European call
limSδ∗
t →0cT,K(t, Sδ∗
t )a.s.= lim
Sδ∗t →0
Sδ∗t cT,K(t, Sδ∗
t ) = 0
438
• asymptotic fair put
fair put-call parity =⇒
limSδ∗
t →0pT,K(t, Sδ∗
t )a.s.= 0
439
• asymptotic hypothetical risk neutral put
hypothetical risk neutral put-call parity =⇒
limSδ∗
t →0p∗T,K(t, Sδ∗
t )
a.s.= lim
Sδ∗t →0
(pT,K(t, Sδ∗
t ) +KS0t
S0T
exp
−
Sδ∗t
2(φT − φt)
)
a.s.= K
S0t
S0T
> 0
dramatic differences can arise
440
0
5000
10000
15000
20000
25000
30000
35000
Dec-2
5
Dec-2
8
Dec-3
1
Dec-3
4
Dec-3
7
Dec-4
0
Dec-4
3
Dec-4
6
Dec-4
9
Dec-5
2
Dec-5
5
Dec-5
8
Dec-6
1
Dec-6
4
Dec-6
7
Dec-7
0
Dec-7
3
Dec-7
6
Dec-7
9
Dec-8
2
Dec-8
5
Dec-8
8
Dec-9
1
Dec-9
4
Dec-9
7
Dec-0
0
Dec-0
3
index
K*savings bond
"risk neutral" put
fair put
“Risk neutral” and fair put on index
441
Pricing Annuities
Baldeaux and Pl. (2012)
Interest Indexed Payouts
Q(t, T ) = S∗t Et
(BT
S∗T
)S∗t =
S∗t
Bt
Bt - savings account
442
e.g.: Minimal Market Model (MMM), Pl. (2001)=⇒
Q(t, T ) = 1− exp
−
2 η S∗t
α (expη T − expη t)
443
Interest Indexed Life Annuity
θ-entitlement levelT -retirement dateT0 < T1 < T2 < · · ·ξx - remaining life time of individual aged x at time 0
Uθx,T (t) = S∗
t Et
∑Ti≥T
Iξx>Tiθ BTi
S∗Ti
= θ
∑Ti≥T
Pt (ξx > Ti) Q(t, Ti)
444
Threshold Life Table
P (ξx > T ) =
exp− b
ln(C)
(Cx+T − 1
)for T ≤ N − x
q(1 + λ
(x+T−N
ω
))− 1λ
for T > N − x
?)
445
60 70 80 90 100 110 1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ung
radu
ated
mor
talit
y ra
te
age
Figure 1: Mortality rates fitted to mortality data of the US population in 2007with threshold atN = 93.
446
2
4
6
8
10
12
14
16
18
10 20 30 40
Figure 2: Discounted fair interest indexed life annuity as function of time toretirement.
447
0
2
4
6
8
10
12
14
16
18
20
1920 1940 1960
discounted hedge portfoliodiscounted fair annuity
Figure 3: Discounted hedging portfolio and discounted fair interest indexedlife annuity evolving over the years.
448
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1920 1940 1960
Figure 4: Fraction of the hedging portfolio invested in the benchmark port-folio.
449
Targeted Pension
Γl(t)-account of l th member in units of interest indexed life annuity
• total discounted valueMti∑l=1
Γl(ti)Uθti
xl,Tl(ti) = δ0total(ti) + δ∗total(ti) S
∗ti
=⇒ entitlement level
θti =δ0total(ti) + δ∗total S
∗ti∑Mi
l=1 Γl(ti)∑
Tk≥TlPti(ξxl > Tk)Q(ti, Tk)
450
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1920 1940 1960
Figure 5: Entitlement level over the years.
451
0.99
0.995
1
1.005
1.01
1.015
1.02
1973 1980 1990 2000 2010
Figure 6: Entitlement level using fast growing portfolio and daily hedging.
452
Variable Annuity
Guaranteed Minimum Death Benefits
Marquardt, Pl. & Jaschke (2008)
• payout to the policyholder
max(egτV0, Vτ )
τ time of death
g ≥ 0 is the guaranteed instantaneous growth rate
V0 is the initial account value
Vτ is the unit value of the policyholder’s account at time of death τembedded put option
453
0 5 10 15 20 25 300.8
0.85
0.9
0.95
1
1.05
1.1
MMM
T
GM
DB
0
0 5 10 15 20 25 300.8
0.85
0.9
0.95
1
1.05
1.1
risk neutral
T
GM
DB
0
0 5 10 15 20 25 300.8
0.85
0.9
0.95
1
1.05
1.1
1.15Black Scholes
T
GM
DB
0
0 5 10 15 20 25 300.9
0.95
1
1.05
1.1
1.15MMM
T
GM
DB
0
0 5 10 15 20 25 300.9
0.95
1
1.05
1.1
1.15risk neutral
T
GM
DB
0
0 5 10 15 20 25 300.9
0.95
1
1.05
1.1
1.15Black Scholes
T
GM
DB
0g=0.025
g=0
g=0.025g=0.025
g=0g=0
Present value of the GMDB under the real world pricing formula (left), the risk
neutral pricing formula (middle) and the Black Scholes formula (right) for
η = 0.05, α0 = 0.05, r = 0.05, ξ = 0.01 and Y0 = 20.454
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