A Benchmark Approach to Quantitative Finance · A Benchmark Approach to Quantitative Finance...
Transcript of A Benchmark Approach to Quantitative Finance · A Benchmark Approach to Quantitative Finance...
A Benchmark Approach to Quantitative Finance
Eckhard PlatenSchool of Finance and Economics and Department of Mathematical Sciences
University of Technology, Sydney
Lit: Platen, E. & Heath, D.: A Benchmark Approach to Quantitative Finance
Springer Finance, 700 pp., 199 illus., Hardcover, ISBN-10 3-540-26212-1 (2006).
Platen, E.: A benchmark approach to finance. Mathematical Finance16(1), 131-151 (2006).
Buhlmann, H. & Platen, E.: A discrete time benchmark approach for insurance and finance
ASTIN Bulletin33(2), 153-172 (2003).
Contents1 Asset Pricing 1
2 Continuous Financial Markets 55
3 Portfolio Optimization 91
4 Modeling Stochastic Volatility 107
5 Minimal Market Model 154
6 Stylized Multi-Currency MMM 214
7 Valuing Guaranteed Minimum Death Benefits 219
8 Markets with Event Risk 273
References 337
1 Asset Pricing
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.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 1
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time
ln(savings bond)ln(fair zero coupon bond)
ln(savings account)
Figure 1.1: Logarithms of savings bond, fair zero coupon bondand savings
account.c© Copyright Eckhard Platen 10 BA - Toronto C 1 2
Financial Market
• jth primary security account
Sjt
j ∈ 0, 1, . . . , d
• savings account
S0t
t ≥ 0
c© Copyright Eckhard Platen 10 BA - Toronto C 1 3
• strategy
δ = δt = (δ0t , δ1t , . . . , δ
dt )⊤, t ≥ 0
predictable
• portfolio
Sδt =
d∑
j=0
δjt S
jt
• self-financing
dSδt =
d∑
j=0
δjt dS
jt
c© Copyright Eckhard Platen 10 BA - Toronto C 1 4
• strictly positive portfolio
Sδ0 = x > 0
Sδt ∈ (0,∞) for all t ∈ [0,∞)
=⇒ Sδ ∈ V+x
c© Copyright Eckhard Platen 10 BA - Toronto C 1 5
Numeraire Portfolio
Definition 1.1 Sδ∗ ∈ V+x numeraire portfolioif
Et
Sδt+h
Sδ∗
t+h
Sδt
Sδ∗
t
− 1
≤ 0
for all nonnegativeSδ and t, h ∈ [0,∞).
• Sδ∗ “best” performing portfolio
Long (1990), Becherer (2001), Pl. (2002, 2006),
Buhlmann& Pl. (2003), Goll & Kallsen (2003), Karatzas & Kardaras (2007)
c© Copyright Eckhard Platen 10 BA - Toronto C 1 6
Main Assumption of the Benchmark Approach
Assumption 1.2 There exists a numeraire portfolioSδ∗ ∈ V+x .
c© Copyright Eckhard Platen 10 BA - Toronto C 1 7
EWI
MCI
DWI
WSI
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
1/1/73 28/9/76 25/6/80 22/3/84 18/12/87 15/9/91 12/6/95 9/3/99 4/12/02 31/8/06
$US
Figure 1.2: Constructed MCI, DWI, EWI and WSI. In the long term the WSI
and EWI outperform all other indices, Le& Platen (2006).
c© Copyright Eckhard Platen 10 BA - Toronto C 1 8
• benchmarked value
Sδt =
Sδt
Sδ∗
t
Supermartingale Property
Corollary 1.3 For nonnegativeSδ
Sδt ≥ Et
(
Sδs
)
0 ≤ t ≤ s < ∞
Sδ supermartingale
c© Copyright Eckhard Platen 10 BA - Toronto C 1 9
• long term growth rate
gδ = lim supt→∞
1
tln
(
Sδt
Sδ0
)
Theorem 1.4 For Sδ ∈ V+x
gδ ≤ gδ∗ .
• “pathwise best” in the long run
Karatzas & Shreve (1998), Pl. (2004a),
Karatzas & Kardaras (2007)
c© Copyright Eckhard Platen 10 BA - Toronto C 1 10
Pl. (2004a)
Definition 1.5 Nonnegative portfolioSδ outperforms systematicallySδ if
(i) Sδ0 = Sδ
0 ;
(ii) P(
Sδt ≥ Sδ
t
)
= 1
(iii) P(
Sδt > Sδ
t
)
> 0.
• “relative arbitrage” Fernholz & Karatzas (2005)
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time
ln(fair zero coupon bond)ln(savings account)
Figure 1.3: Logarithms of fair zero coupon bond and savings account.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 12
Theorem 1.6 Numeraire portfolio cannot be outperformed systematically.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 13
• portfolio ratio
Aδt,h =
Sδt+h
Sδt
• expected growth
gδt,h = Et
(
ln(
Aδt,h
))
t, h ≥ 0
c© Copyright Eckhard Platen 10 BA - Toronto C 1 14
Definition 1.7 Sδ growth optimal if
gδt,h ≤ g
δt,h
for all t, h ≥ 0 and nonnegativeSδ.
• expected log-utility
Kelly (1956)
Hakansson (1971a)
Merton (1973a)
Roll (1973)
Markowitz (1976)
c© Copyright Eckhard Platen 10 BA - Toronto C 1 15
Theorem 1.8 The numeraire portfolio is growth optimal.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 16
Strong Arbitrage
• market participants can only exploit arbitrage
• limited liability
=⇒ nonnegative total wealth of each market participant
Definition 1.9 A nonnegativeSδ is a strong arbitrageif Sδ0 = 0 and
P(
Sδt > 0
)
> 0.
Pl. (2002)-mathematical arguments (super-martingale property)
Loewenstein & Willard (2000)-economic arguments
c© Copyright Eckhard Platen 10 BA - Toronto C 1 17
Theorem 1.10 There isno strong arbitrage.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 18
• Delbaen & Schachermayer (1998)
free lunches with vanishing risk (FLVR)
• Loewenstein & Willard (2000)
free snacks& cheap thrills
=⇒ no-arbitrage pricing makes no sense under BA
c© Copyright Eckhard Platen 10 BA - Toronto C 1 19
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time
Figure 1.4:P (t, T ) minus savings account.
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-0.0004
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0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0.0018
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time
Figure 1.5:P (t, T ) minus savings account.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 21
• Existence of equivalent risk neutral probability measure is
more amathematical conveniencethan an economic necessity.
• Basing quantitative methods on classical risk neutral pricing is
restrictive.
• Trends are ignored under
Fundamental Theorem of Asset Pricing
Delbaen & Schachermayer (1998)
c© Copyright Eckhard Platen 10 BA - Toronto C 1 22
• 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)
All ignore trends !
c© Copyright Eckhard Platen 10 BA - Toronto C 1 23
• One needs consistent pricing concept that exploits trends
• All benchmarked nonnegative securities are supermartingales
(no upward trend)
c© Copyright Eckhard Platen 10 BA - Toronto C 1 24
Definition 1.11 Price is fair if, when benchmarked, forms martingale
(no trend)
Sδt = Et
(
Sδs
)
0 ≤ t ≤ s < ∞.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 25
Lemma 1.12 Theminimal nonnegative supermartingale that reaches a
given benchmarked contingent claimis a martingale.
see Pl.& Heath (2006)
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0
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time
benchmarked savings bondbenchmarked fair zero coupon bond
Figure 1.6: Benchmarked savings bond and benchmarked fair zero coupon
bond.c© Copyright Eckhard Platen 10 BA - Toronto C 1 27
Law of the Minimal Price
Pl. (2008)
Theorem 1.13 If a fair portfolio replicates a nonnegative payoff, then this
represents the minimal replicating portfolio.
• least expensive
• minimal hedge
• economically correct price in a competitive market
c© Copyright Eckhard Platen 10 BA - Toronto C 1 28
• contingent claim
HT
E0
(
HT
Sδ∗
T
)
< ∞
• SδH
t fair if
SδH
t =SδH
t
Sδ∗
t
= Et
(
HT
Sδ∗
T
)
Link with real economy !
c© Copyright Eckhard Platen 10 BA - Toronto C 1 29
Law of the Minimal Price =⇒
Corollary 1.14
Minimal price for replicableHT is fair and given by
real world pricing formula
SδH
t = Sδ∗
t Et
(
HT
Sδ∗
T
)
.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 30
• most pricing concepts are unified and generalized by real world pricing
=⇒
actuarial pricing
risk neutral pricing
pricing with stochastic discount factor
pricing with numeraire change
pricing with deflator
pricing with state pricing density
pricing with pricing kernel
pricing with numeraire portfolio
c© Copyright Eckhard Platen 10 BA - Toronto C 1 31
WhenHT independentof Sδ∗
T
=⇒ rigorous derivation of
• actuarial pricing formula
SδH
t = P (t, T )Et(HT )
with zero coupon bond
P (t, T ) = Sδ∗
t Et
(
1
Sδ∗
T
)
c© Copyright Eckhard Platen 10 BA - Toronto C 1 32
• real world pricing formula =⇒
SδH
0 = E0
(
ΛT
S00
S0T
HT
)
Λt =S0
t
S00
supermartingale
1 = Λ0 ≥ E0(ΛT )
=⇒
SδH
0 ≤E0
(
ΛTS0
0
S0T
HT
)
E0(ΛT )
similar for any numeraire (hereS0 savings account)
c© Copyright Eckhard Platen 10 BA - Toronto C 1 33
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50
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time
Figure 1.7: Discounted S&P500 total return index.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 34
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time
Radon-Nikodym derivativeTotal RN-Mass
Figure 1.8: Radon-Nikodym derivative and total mass of putative risk neutral
measure.c© Copyright Eckhard Platen 10 BA - Toronto C 1 35
• special case whensavings account is fair:
=⇒ ΛT = dQ
dPforms martingale; E0(ΛT ) = 1;
equivalent risk neutral probability measureQ exists;
Bayes’ formula =⇒risk neutral pricing formula
SδH
0 = EQ0
(
S00
S0T
HT
)
Harrison & Kreps (1979), Ingersoll (1987),
Constantinides (1992), Duffie (2001), Cochrane (2001),. . .
• otherwise “risk neutral price”≥ real world price
c© Copyright Eckhard Platen 10 BA - Toronto C 1 36
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9
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time
ln(index)ln(savings account)
Figure 1.9: ln(S&P500 accumulation index) and ln(savings account).
c© Copyright Eckhard Platen 10 BA - Toronto C 1 37
Extreme Maturity Bond
• short rate deterministic
• savings bond
P ∗(t, T ) =S0
t
S0T
• index S1 as numeraire portfolioSδ∗
c© Copyright Eckhard Platen 10 BA - Toronto C 1 38
• fair zero coupon bond
Law of the Minimal Price
=⇒ real world pricing formula
P (t, T ) = Sδ∗
t Et
(
1
Sδ∗
T
)
• benchmarked fair zero coupon bond
P (t, T ) =P (t, T )
Sδ∗
t
martingale
c© Copyright Eckhard Platen 10 BA - Toronto C 1 39
• discounted numeraire portfolio
Sδ∗
t =Sδ∗
t
S0t
numeraire portfolio for continuous market:
dSδ∗
t = αt dt+
√
Sδ∗
t αt dWt
time transformed squared Bessel process
c© Copyright Eckhard Platen 10 BA - Toronto C 1 40
• model the drift of discounted index as
αt = α expη t=⇒
Sδ∗
t time transformed squared Bessel process of dimension four
=⇒ Minimal Market Model
MMM, see Pl.& Heath (2006)
• net growth rate
η ≈ 0.054 with R2 of 0.89
c© Copyright Eckhard Platen 10 BA - Toronto C 1 41
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ln(discounted index)trendline
Figure 1.10: Logarithm of discounted S&P500 and trendline.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 42
• normalized index
Yt =Sδ∗
t
αt
dYt = (1 − η Yt) dt+√
Yt dWt
• quadratic variation of√Yt
d√
Yt = . . .+1
2dWt
it∑
ℓ=1
(
√
Ytℓ−√
Ytℓ−1
)2
≈[√Y]
t=t
4
• scaling parameter α ≈ 0.01468 with R2 of 0.995
c© Copyright Eckhard Platen 10 BA - Toronto C 1 43
0
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20
25
30
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time
[sqrt Y]trendline
Figure 1.11: Quadratic Variation of√Yt.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 44
• fair zero coupon bond
P (t, T ) = P ∗(t, T )
(
1 − exp
− 2 η Sδ∗
t
α (expη T − expη t)
)
• initial prices
P ∗(0, T ) = 0.0335
P (0, T ) = 0.00077
P (0,T )P ∗(0,T )
< 0.023 =⇒ 2.3%
no strong arbitrage
c© Copyright Eckhard Platen 10 BA - Toronto C 1 45
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time
benchmarked savings bondbenchmarked fair zero coupon bond
Figure 1.12: Benchmarked savings bond and benchmarked fair zero coupon
bond.c© Copyright Eckhard Platen 10 BA - Toronto C 1 46
0
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time
savings bondfair zero coupon bond
savings account
Figure 1.13: Savings bond, fair zero coupon bond and savings account.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 47
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ln(savings bond)ln(fair zero coupon bond)
Figure 1.14: Logarithms of fair zero coupon bond and savings account.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 48
• savings account isoutperformed systematically
by fair zero coupon bond
c© Copyright Eckhard Platen 10 BA - Toronto C 1 49
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ln(fair zero coupon bond)ln(savings account)
Figure 1.15: Logarithms of fair zero coupon bond and savings account.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 50
• self-financing hedge portfoliohedge ratio
δ∗t =
∂P (t, T )
∂Sδ∗
t
= P ∗(0, T ) exp
−2 η Sδ∗
t
α (expη T − expη t)
× 2 η
α (expη T − expη t)
c© Copyright Eckhard Platen 10 BA - Toronto C 1 51
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ln(zero coupon bond)ln(self financing hedge portfolio)
Figure 1.16: Logarithm of zero coupon bond and self-financinghedge port-
folio.c© Copyright Eckhard Platen 10 BA - Toronto C 1 52
-2e-005
-1e-005
0
1e-005
2e-005
3e-005
4e-005
5e-005
6e-005
7e-005
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time
Figure 1.17: Benchmarked profit and loss.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 53
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time
Figure 1.18: Fraction invested in the index.
c© Copyright Eckhard Platen 10 BA - Toronto C 1 54
2 Continuous Financial Markets
• d sources of continuoustraded uncertainty
independent standard Wiener processes
W 1,W 2, . . . ,W d, d ∈ 1, 2, . . .
c© Copyright Eckhard Platen 10 BA - Toronto C 2 55
Primary Security Accounts
Sjt - jth primary security account at timet,
j ∈ 0, 1, . . . , d
cum-dividend share value or savings account value,
dividends or interest are accumulated, reinvested
• vector process of primary security accounts
S = St = (S0t , . . . , S
dt )⊤, t ∈ [0, T ]
c© Copyright Eckhard Platen 10 BA - Toronto C 2 56
• assumeunique strong solutionof SDE
dSjt = Sj
t
(
ajt dt+
d∑
k=1
bj,kt dW k
t
)
t ∈ [0,∞), Sj0 > 0, j ∈ 1, 2, . . . , d
• predictable, integrable appreciation rate processesaj
• predictable, square integrable volatility processesbj,k
c© Copyright Eckhard Platen 10 BA - Toronto C 2 57
• savings account
S0t = exp
∫ t
0
rs ds
predictable, integrable short rate process
r = rt, t ∈ [0,∞)
c© Copyright Eckhard Platen 10 BA - Toronto C 2 58
• Market price of risk
The unique invariant of the market
θt = (θ1t , θ
2t , . . . , θ
dt )⊤
solution of relation
bt θt = at − rt 1
where
at = (a1t , a
2t , . . . , a
dt )
⊤
1 = (1, 1, . . . , 1)⊤
c© Copyright Eckhard Platen 10 BA - Toronto C 2 59
Assumption 2.1
Volatility matrixbt = [bj,kt ]dj,k=1 is invertible
=⇒• market price of risk equals
θt = b−1t (at − rt 1)
c© Copyright Eckhard Platen 10 BA - Toronto C 2 60
• rewritejth primary security account SDE
dSjt = Sj
t
rt dt+d∑
k=1
bj,kt (θk
t dt+ dW kt )
t ∈ [0,∞), j ∈ 1, 2, . . . , d
c© Copyright Eckhard Platen 10 BA - Toronto C 2 61
• strategy
δ = δt = (δ0t , . . . , δdt )⊤, t ∈ [0, T ]
predictable andS-integrable,
δjt number of units ofjth primary security account
• portfolio
Sδt =
d∑
j=0
δjt S
jt
c© Copyright Eckhard Platen 10 BA - Toronto C 2 62
• Sδ self-financing⇐⇒
dSδt =
d∑
j=0
δjt dS
jt
changes in the portfolio value only due to gains from trading,
self-financing in each denomination
c© Copyright Eckhard Platen 10 BA - Toronto C 2 63
• jth fraction
πjδ,t = δj
t
Sjt
Sδt
j ∈ 0, 1, . . . , d
d∑
j=0
πjδ,t = 1
c© Copyright Eckhard Platen 10 BA - Toronto C 2 64
• 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=1
πjδ,t b
j,kt
c© Copyright Eckhard Platen 10 BA - Toronto C 2 65
• logarithm of strictly positive portfolio
d ln(Sδt ) = gδ
t dt+
d∑
k=1
bkδ,t dW
kt
with growth rate
gδt = rt +
d∑
k=1
bkδ,t
(
θkt − 1
2bk
δ,t
)
c© Copyright Eckhard Platen 10 BA - Toronto C 2 66
Numeraire Portfolio Sδ∗ = Growth Optimal Portfolio
=⇒ maximize the growth rate
gδt = rt +
d∑
k=1
d∑
j=1
πjδ,t b
j,kt
(
θkt − 1
2
d∑
ℓ=1
πℓδ,t b
ℓ,kt
)
=⇒ for eachj ∈ 1, 2, . . . , d
0 =
d∑
k=1
bj,kt
(
θkt −
d∑
ℓ=1
πℓδ∗,t b
ℓ,kt
)
=⇒θ⊤
t = π⊤δ∗,t bt
c© Copyright Eckhard Platen 10 BA - Toronto C 2 67
Sincebt invertible=⇒
πδ∗,t = (π1δ∗,t, . . . , π
dδ∗,t)
⊤
= (b−1t )⊤ θt
=⇒ NP
dSδ∗
t = Sδ∗
t
([
rt +
d∑
k=1
(θkt )2
]
dt+
d∑
k=1
θkt dW
kt
)
finite NP exists
c© Copyright Eckhard Platen 10 BA - Toronto C 2 68
Benchmarked Portfolios
Sδt =
Sδt
Sδ∗
t
dSδt = Sδ
t
rt dt+
d∑
k=1
d∑
j=0
πjδ,t b
j,kt
(
θkt dt+ dW k
t
)
dSδ∗
t = Sδ∗
t
(
rt dt+d∑
k=1
θkt
(
θkt dt+ dW k
t
)
)
dSδt = −Sδ
t
d∑
k=1
d∑
j=0
πjδ,t s
j,kt dW k
t
driftless =⇒ local martingale =⇒ supermartingalewhen nonnegative
c© Copyright Eckhard Platen 10 BA - Toronto C 2 69
Theorem 2.2 Any nonnegative benchmarked portfolio is an
(A, P )-supermartingale(no upward trend).
c© Copyright Eckhard Platen 10 BA - Toronto C 2 70
=⇒ NP is numeraire portfolio
use NP asbenchmark for investment and asnumeraire for pricing
• markets have thousands of primary security accounts
c© Copyright Eckhard Platen 10 BA - Toronto C 2 71
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
c© Copyright Eckhard Platen 10 BA - Toronto C 2 72
• 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 overT = 32 yearsd = 50 risky primary security accounts
σ = 0.15, r = 0.05
c© Copyright Eckhard Platen 10 BA - Toronto C 2 73
0
5
10
15
20
25
0 5 10 15 20 25 30 35
time
Figure 2.1: Simulated primary security accounts.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 74
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPB
Figure 2.2: Simulated NP and savings account.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 75
Diversified Portfolios
Fundamental phenomenon ofdiversification leads naturally toNP:
diversified =⇒ fractions “small”
Definition 2.3 A strictly positive portfolioSδ is adiversified portfolio if
∣
∣
∣πj
δ,t
∣
∣
∣≤ K2
d12+K1
a.s. for allj ∈ 0, 1, . . . , d andt ∈ [0, T ] for K1,K2 ∈ (0,∞) and
K3 ∈ 1, 2, . . ., independent ofd ∈ K3,K3 + 1, . . ..
c© Copyright Eckhard Platen 10 BA - Toronto C 2 76
• jth benchmarked primary security account
Sjt =
Sjt
Sδ∗
t
dSjt = −Sj
t
d∑
k=1
sj,kt dW k
t
• (j, k)th specific volatility
sj,kt = θk
t − bj,kt
measuresspecific market risk of Sj with respect toW k
• θkt measuresgeneral market risk with respect toW k
c© Copyright Eckhard Platen 10 BA - Toronto C 2 77
• kth total specific absolute volatility
skt =
d∑
j=0
|sj,kt |
regular =⇒ variance of specific returns “bounded”
Definition 2.4 A market is calledregular if
E(
(
skt
)2)
≤ K5.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 78
Tracking Rate
Variance of benchmarked portfolio returns
Rdδ(t) =
d∑
k=1
d∑
j=0
πjδ,t s
j,kt
2
Rdδ(t) - quantifies distance betweenSδ and Sδ∗
benchmarked NP constant=⇒
Rdδ∗
(t) = 0
c© Copyright Eckhard Platen 10 BA - Toronto C 2 79
Approximate NP
tracking rate “small”
Definition 2.5 A strictly positive portfolioSδ is an approximate NP if
for all t ∈ [0, T ] and ε > 0
limd→∞
P(
Rdδ(t) > ε
)
= 0.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 80
Diversification Theorem
Theorem 2.6 In a regular market
any diversified portfolio is an approximate NP.
• model independent
• similarity to CLT
c© Copyright Eckhard Platen 10 BA - Toronto C 2 81
Equal Weighted Index (EWI)
SδEWI holds equal fractions
πjδEWI,t
=1
d+ 1
j ∈ 0, 1, . . . , d
• tracking rate for multi-asset BS example:
RdδEWI
(t) =
d∑
k=1
(
1
d+ 1
(
θkt − bk,k
t
)
)2
= d
(
1
d+ 1
(
σ√d
− σ
))2
≤ σ2
(d+ 1)→ 0
c© Copyright Eckhard Platen 10 BA - Toronto C 2 82
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPEWI
Figure 2.3: Simulated NP and EWI.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 83
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPindex
Figure 2.4: Simulated diversified accumulation index and NP.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 84
0
20,000
40,000
60,000
80,000
100,000
120,000
1/1/73 28/9/76 25/6/80 22/3/84 18/12/87 15/9/91 12/6/95 9/3/99 4/12/02 31/8/06
Figure 2.5: World industry sector stock market indices as primary security
accounts.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 85
EWI
MCI
DWI
WSI
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
1/1/73 28/9/76 25/6/80 22/3/84 18/12/87 15/9/91 12/6/95 9/3/99 4/12/02 31/8/06
$US
Figure 2.6: Constructed MCI, DWI, EWI and WSI. In the long term the WSI
and EWI outperform all other indices, Le& Platen (2006).
c© Copyright Eckhard Platen 10 BA - Toronto C 2 86
0
50
100
150
200
250
300
350
400
450
0 5 9 14 18 23 27 32
Figure 2.7: Primary security accounts under the MMM.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 87
0
10
20
30
40
50
60
0 5 9 14 18 23 27 32
Figure 2.8: Benchmarked primary security accounts.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 88
0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
EWI
GOP
Figure 2.9: NP and EWI.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 89
0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
Market index
GOP
Figure 2.10: NP and market index.
c© Copyright Eckhard Platen 10 BA - Toronto C 2 90
3 Portfolio Optimization
• NP best long term investment
Kelly (1956), Latane (1959), Breiman (1961),
Hakansson (1971b), Thorp (1972)
• optimal portfolio separated into two funds
Tobin (1958), Sharpe (1964)
c© Copyright Eckhard Platen 10 BA - Toronto C 3 91
• mean-variance efficient portfolio
Markowitz (1959)
• intertemporal capital asset pricing model
Merton (1973a)
c© Copyright Eckhard Platen 10 BA - Toronto C 3 92
Expected Utility Maximization
• utility function U(·), U ′(·) > 0 U ′′(·) < 0
• examples
power utility
U(x) =1
γxγ
for γ 6= 0 andγ < 1
log-utility
U(x) = ln(x)
c© Copyright Eckhard Platen 10 BA - Toronto C 3 93
5 10 15 20x
-15
-10
-5
5
U
Figure 3.1: Examples for power utility (upper graph) and log-utility (lower
graph).
c© Copyright Eckhard Platen 10 BA - Toronto C 3 94
• assumeS0 is scalar diffusion
• maximize expected utility
vδ = maxSδ∈V+
S0
E0
(
U(
SδT
))
V+S0
strictly positive, discounted, fair portfolios
Sδ0 = S0 > 0
c© Copyright Eckhard Platen 10 BA - Toronto C 3 95
Theorem 3.1
Benchmarked, expected utility maximizing portfolio:
Sδt = u(t, S0
t ) = Et
(
U ′−1(
λ S0T
)
S0T
)
,
λ s.t. S0 = Sδ0 S
δ∗
0 = u(0, S0)Sδ∗
0
two fund separation with risk aversion coefficient
J δt =
1
1 − S0t
u(t,S0t )
∂u(t,S0t )
∂S0t
=⇒ invest 1
J δt
of wealth in NP and remainder in savings account
c© Copyright Eckhard Platen 10 BA - Toronto C 3 96
• 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 ) = Et
(
U ′−1(
λ S0T
)
S0T
)
= Et
(
1
λ S0T
S0T
)
=1
λ
Lagrange multiplier
c© Copyright Eckhard Platen 10 BA - Toronto C 3 97
λ =Sδ∗
0
S0
risk aversion coefficient
J δt = 1
• expected log-utility
vδ = E0
(
ln(
Sδ∗
T
))
= ln(λ) + ln(S0) +1
2
∫ T
0
E0
(
(
θ(s, Sδ∗
s ))2)
ds
if local martingale part forms a martingale
c© Copyright Eckhard Platen 10 BA - Toronto C 3 98
• power utility
U(x) =1
γxγ
for γ < 1, γ 6= 0
U ′(x) = xγ−1
U ′−1(y) = y1
γ−1
U ′′(x) = (γ − 1)xγ−2
concave
u(t, S0t ) = Et
(
λ
Sδ∗
T
)1
γ−11
Sδ∗
T
= λ1
γ−1 Et
(
(
Sδ∗
T
)γ
1−γ
)
c© Copyright Eckhard Platen 10 BA - Toronto C 3 99
Sδ∗ geometric Brownian motion
u(t, S0t ) = λ
1γ−1
(
Sδ∗
t
)
γ1−γ
×Et
(
exp
γ
1 − γ
(
θ2
2(T − t) + θ (WT −Wt)
))
= λ1
γ−1(
Sδ∗
t
)
γ1−γ exp
θ2
2
γ
(1 − γ)2(T − t)
c© Copyright Eckhard Platen 10 BA - Toronto C 3 100
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 coefficient
J δt = 1 − γ
expected utility
vδ = E0
(
1
γ
(
SδT
)γ)
= exp
−θ2
2
γ
1 − γT
(S0)γ
c© Copyright Eckhard Platen 10 BA - Toronto C 3 101
Utility Indifference Pricing
discounted payoff H = HS0
T
(not perfectly hedgable)
vδε,V = E0
(
U
(
(S0 − ε V )Sδ
T
S0
+ ε H
))
• assumeS0 is scalar diffusion
Definition 3.2 utility indifference price V s.t.
limε→0
vδε,V − vδ
0,V
ε= 0
Davis (1997)
c© Copyright Eckhard Platen 10 BA - Toronto C 3 102
• Taylor expansion
vδε,V ≈ E0
(
U(
SδT
)
+ εU ′(
SδT
)
(
H − V SδT
S0
))
= vδ0,V + εE0
(
U ′(
SδT
)
H)
− V εE0
(
U ′(
SδT
) SδT
S0
)
SδT = Sδ
T /S0T = U ′−1
(
λ S0T
)
=⇒
vδε,V − vδ
0,V = ε
(
E0
(
λ S0T H
)
− V E0
(
λ S0T
SδT
S0
))
c© Copyright Eckhard Platen 10 BA - Toronto C 3 103
=⇒• utility indifference price
V =E0
(
λ
Sδ∗
T
H)
E0
(
λ
Sδ∗
T
SδT
S0
) =E0
(
H
Sδ∗
T
)
1S0
S0
Sδ∗
0
independent ofU and of given model =⇒
• real world pricing formula
V = Sδ∗
0 E0
(
H
Sδ∗
T
)
c© Copyright Eckhard Platen 10 BA - Toronto C 3 104
Hedging
• benchmarked derivative price
UH(t) = Et
(
H
Sδ∗
T
)
• real world martingale representation
H
Sδ∗
T
= UH(t) +
d∑
k=1
∫ T
t
xkH(s) dW k
s +MH(t)
MH -orthogonal martingale
E([
MH ,Wk]
t
)
= 0
Follmer-Schweizer decomposition, Follmer & Schweizer (1991)
least square projection
c© Copyright Eckhard Platen 10 BA - Toronto C 3 105
Theorem 3.3 Derivative price
SδH
t = Sδ∗
t Et
(
H
Sδ∗
T
)
fractions
πδH(t) =
(
bδH(t)⊤ b−1
t
)⊤
with portfolio volatilities
bkδH
(t) =1
UH(t)
d∑
j=0
δjH(t) Sj
t bj,kt =
xkH(t)
UH(t)+ θk
t .
c© Copyright Eckhard Platen 10 BA - Toronto C 3 106
4 Modeling Stochastic Volatility
• consider well diversified accumulation index≈ NP
dSδ∗
t = Sδ∗
t
(
rt dt+ |θt|(
|θt| dt+ dWt
))
d ln(
Sδ∗
t
)
=
(
rt dt+1
2|θt|2
)
dt+ |θt| dWt
• volatility
|θt| =
√
d
dt[ln (Sδ∗)]t
c© Copyright Eckhard Platen 10 BA - Toronto C 4 107
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
1982 1992 2002
time
Figure 4.1: Estimated volatility of WSI from 1973–2004.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 108
Log-Returns of a Stock Index
-10 -5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Figure 4.2: Histogram for S&P500 hourly deseasonalized log-returns.
fitted Gaussian density, much fatter tails
c© Copyright Eckhard Platen 10 BA - Toronto C 4 109
Implied Volatilities
• European call for Black-Scholes model
cτ,K(0, S, σ)
maturity τ ∈ [0, T ]
strike priceK > 0
underlying indexS > 0
volatility σ
c© Copyright Eckhard Platen 10 BA - Toronto C 4 110
• observed European call option price
Vc,τ,K(0, S0)
• implied volatility
σcallBS (0, S0, τ,K)
obtained such that
Vc,τ,K(0, S0) = cτ,K
(
0, S0, σcallBS (0, S0, τ,K)
)
by some root finding method
c© Copyright Eckhard Platen 10 BA - Toronto C 4 111
100
200
300
time 0.6
0.8
1
1.2
1.4
moneyness
20
40
60
100
200
300
time
Figure 4.3: Implied volatilities for S&P500 three month options.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 112
implied volatility surface
0.5
0.75
1
1.25
1.5
KS
0.20.4
0.60.8
11.2
1.41.6T
0.22
0.24
0.26
0.28
0.5
0.75
1
1.25KS
0.20.4
0.60.8
11.2
1.4
Figure 4.4: Average S&P500 implied volatility surface.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 113
A Modified Constant Elasticity of Variance Model
• CEV Model
Cox (1975), Cox & Ross (1976)
Andersen & Andreasen (2000)
Lewis (2000)
Lo, Yuen & Hui (2000)
Brigo & Mercurio (2005)
Heath& Pl. (2005b)
c© Copyright Eckhard Platen 10 BA - Toronto C 4 114
• classical risk neutral CEV
Beckers (1980)
Schroder (1989)
Delbaen & Shirakawa (2002)
certain problems in risk neutral pricing of derivatives
c© Copyright Eckhard Platen 10 BA - Toronto C 4 115
Modified CEV Model
Heath& Pl. (2005b)
• savings account
dS0t = r S0
t dt
total market price for risk process
θt ≥ 0, t ∈ [0, T ]
• drifted Wiener process
dWθ(t) = θt dt+ dWt
c© Copyright Eckhard Platen 10 BA - Toronto C 4 116
• NP
dSδ∗
t = Sδ∗
t (rt dt+ θt dWθ(t))
• NP volatility
θt = (Sδ∗
t )a−1 ψ
exponenta ∈ (−∞,∞)
scaling parameterψ > 0
stochastic fora 6= 1
c© Copyright Eckhard Platen 10 BA - Toronto C 4 117
• hypothetical risk neutral CEV dynamics
dSδ∗
t = Sδ∗
t rt dt+ (Sδ∗
t )a ψ dWθ(t)
Wθ - Wiener process under a hypothetical risk neutral measurePθ
• If a = 1 =⇒ BS model
=⇒ equivalent risk neutral probability measurePθ
with market price of riskθt = ψ
c© Copyright Eckhard Platen 10 BA - Toronto C 4 118
• real world CEV dynamics
dSδ∗
t =(
Sδ∗
t rt +(
Sδ∗
t
)2a−1ψ2)
dt+(
Sδ∗
t
)aψ dWt
modified CEV model remains fora < 1 strictly positive
this isnot the case for its “risk neutral” dynamics
c© Copyright Eckhard Platen 10 BA - Toronto C 4 119
• squared Bessel process
Xt =(
Sδ∗
t
)2 (1−a)
for a 6= 1 with SDE
dXt =(
2(1 − a) rtXt + ψ2 (1 − a) (3 − 2a))
dt
+2ψ (1 − a)√
Xt dWt
anddimension
δ =3 − 2 a
1 − a
c© Copyright Eckhard Platen 10 BA - Toronto C 4 120
• Radon-Nikodym derivative process
Λθ = Λθ(t), t ∈ [0, T ](state price density)
• hypothetical risk neutral measurePθ
dPθ
dP= Λθ(T )
Λθ(t) =Sδ∗
0
Sδ∗
t
S0t =
(
X0
Xt
)q
S0t
q =1
2 (1 − a)
c© Copyright Eckhard Platen 10 BA - Toronto C 4 121
• squared Bessel process
dXt =(
2 (1 − a) rtXt + ψ2 (1 − a)(1 − 2 a))
dt
+2ψ (1 − a)√
Xt dWθ(t)
dimension
δθ =1 − 2 a
1 − a
c© Copyright Eckhard Platen 10 BA - Toronto C 4 122
• dimension
risk neutral
δθ =1 − 2 a
1 − a
real world
δ =3 − 2 a
1 − a
• if a < 1 =⇒ δ > 2 andδθ < 2
if a > 1 =⇒ δ < 2 andδθ > 2
c© Copyright Eckhard Platen 10 BA - Toronto C 4 123
-1 0 1 2 3
-4
-2
0
2
4
6
8
Risk Neutral
Real World
Figure 4.5: Dimensionsδ andδθ as a function of the exponenta.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 124
• hypothetical risk neutral probability
Pθ(A) =
∫
A
dPθ(ω) =
∫
A
dPθ(ω)
dP (ω)dP (w)
=
∫
A
Λθ(T ) dP (ω)
=⇒
Pθ(Ω) =
∫
Ω
Λθ(T ) dP (ω) = E0 (Λθ(T ))
c© Copyright Eckhard Platen 10 BA - Toronto C 4 125
• for a < 1 =⇒ δθ < 2
=⇒ Λθ is strict supermartingale =⇒
Pθ(Ω) < Λθ(0) = 1
=⇒ Pθ is not a probability measure
c© Copyright Eckhard Platen 10 BA - Toronto C 4 126
• for a 6= 1 P andPθ are not equivalent
=⇒ modified CEV model doesnot haveequivalent risk neutral
probability measure
• consider onlya < 1 since fora > 1 NP explodes
c© Copyright Eckhard Platen 10 BA - Toronto C 4 127
Real World Pricing
• contingent claim
Hτ = Hτ (Sδ∗
τ ), E
(
Hτ
Sδ∗
τ
)
< ∞
• fair price
uHτ(t, Sδ∗
t ) = Sδ∗
t uHτ(t, Sδ∗
t )
benchmarked price
uHτ(t, Sδ∗
t ) = Et
(
Hτ (Sδ∗
τ )
Sδ∗
τ
)
c© Copyright Eckhard Platen 10 BA - Toronto C 4 128
Martingale Representation
Hτ
Sδ∗
τ
= uHτ(τ, Sδ∗
τ )
= uHτ(t, Sδ∗
t ) +
∫ τ
t
(
Sδ∗
s
)aψ∂uHτ
(s, Sδ∗
s )
∂Sδ∗
dWs
c© Copyright Eckhard Platen 10 BA - Toronto C 4 129
Stochastic Volatility Models
• volatility function of underlying diffusion=⇒ LV model
• volatility as separate, possibly correlated process
Hull & White (1987, 1988)Johnson & Shanno (1987)Scott (1987)Wiggins (1987)Chesney & Scott (1989)Melino & Turnbull (1990)Stein & Stein (1991)Hofmann, Pl.& Schweizer (1992)Heston (1993),. . .
c© Copyright Eckhard Platen 10 BA - Toronto C 4 130
Empirical Evidence on Log-Returns
• Markowitz & Usmen (1996)
daily S&P500 → Studentt(4.5)
• Hurst & Platen (1997)
daily stock market indices→ Studentt(4)
• Fergusson& Pl. (2006), Pl.& Rendek (2008)
daily WSI, EWI104s with high accuracy→ Studentt(4)
• Breymann, Dias & Embrechts (2003)
copula of joint distribution of log-returns of exchange rates →Studentt copula with roughly four degrees of freedom
c© Copyright Eckhard Platen 10 BA - Toronto C 4 131
−10 −5 0 5 10
10−4
10−3
10−2
10−1
Figure 4.6: Log-histogram of the EWI104s log-returns and Studentt density
with four degrees of freedom.c© Copyright Eckhard Platen 10 BA - Toronto C 4 132
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
Figure 4.7: Log-likelihood function based on the EWI104s.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 133
Country Student-t NIG Hyperbolic VG ν
Australia 0.000000 76.770817 150.202282 181.632971 4.281222
Austria 0.000000 39.289103 77.505683 102.979330 4.725907
Belgium 0.000000 31.581622 60.867570 83.648470 4.989912
Brazil 2.617693 5.687078 63.800349 60.078395 2.713036
Canada 0.000000 47.506215 79.917741 104.297607 5.316154
Denmark 0.000000 41.509921 87.199686 114.853658 4.512101
Finland 0.000000 28.852844 68.677271 88.553080 4.305638
France 0.000000 26.303544 57.639325 80.567283 4.722787
Germany 0.000000 27.290205 52.667918 71.120798 5.005856
Greece 0.000000 60.432172 104.789463 125.601499 4.674626
Hong.Kong 0.000000 42.066531 100.834255 122.965326 3.930473
India 0.000000 74.773701 163.594078 198.002956 3.998713
Ireland 0.000000 77.727856 136.505582 170.013644 4.761519
Italy 0.000000 25.196598 55.185625 75.481897 4.668983
Japan 0.000000 37.630363 77.163656 102.967380 4.649745
Korea.S. 0.000000 120.904983 304.829431 329.854620 3.289204
c© Copyright Eckhard Platen 10 BA - Toronto C 4 134
Malaysia 0.000000 79.714054 186.013963 221.061290 3.785195
Netherlands 0.000000 26.832761 51.625813 71.541627 5.084056
Norway 0.000000 42.243851 89.012090 115.059003 4.472349
Portugal 0.000000 61.177624 137.681039 165.689683 3.984860
Singapore 0.000000 36.379685 77.600590 98.124375 4.251472
Spain 0.000000 56.694545 109.533768 138.259224 4.517153
Sweden 0.000000 77.618384 143.420049 178.983373 4.546640
Taiwan 0.000000 41.162560 96.283628 115.186585 3.914719
Thailand 0.000000 78.250621 254.590254 267.508143 3.032038
UK 0.000000 26.693076 55.937248 80.678494 4.952843
USA 0.000000 40.678242 79.617362 100.901197 4.636661
Table 1:Ln test statistic of the EWI104s for different currency denominations
c© Copyright Eckhard Platen 10 BA - Toronto C 4 135
Index Log-returns
• Student t distributed with about 4 degrees of freedom
• clustering
Find corresponding stationary volatility processes !
Kessler (1997), Prakasa Rao (1999)
c© Copyright Eckhard Platen 10 BA - Toronto C 4 136
• normal mixing
log-return conditionally Gaussian distributed
with stochastic variance process
for small time step sizeh > 0
∆ ln(Sδ∗
t ) = ln
(
Sδ∗
t+h
Sδ∗
t
)
∼ N(
θ2t
2h, θ2
t h
)
whenθ2t has inverse gamma stationary density
=⇒ estimated log-return appears to be Studentt distributed
c© Copyright Eckhard Platen 10 BA - Toronto C 4 137
A Class of Continuous Stochastic Volatility Models
• discounted NP
Sδ∗
t =Sδ∗
t
S0t
dSδ∗
t = Sδ∗
t (θ2t dt+ θt dWt)
c© Copyright Eckhard Platen 10 BA - Toronto C 4 138
Particular Stochastic Volatility Models
• Hull and White’87 model
Hull & White (1987)
dθ2t = µ θ2
t dt+ ξ θ2t dWt
squared volatility does not have a stationary dynamics
c© Copyright Eckhard Platen 10 BA - Toronto C 4 139
• Heston model
Hull & White (1988), Heston (1993)
mean reverting dynamics
k, θ andγ are positive constants
=⇒
dθ2t = k (θ2 − θ2
t ) dt+ γ |θt| dWt
θ20 > 0, κ
γ2 ≥ 12
=⇒ has as stationary density gamma density
c© Copyright Eckhard Platen 10 BA - Toronto C 4 140
• Scott model
Scott (1987)
Stein & Stein (1991)
Ornstein-Uhlenbeck process
k, θ, γ positive constants
dθt = k (θ − θt) dt+ γ dWt
θ0 ∈ ℜ
=⇒ θ2t has as stationary density a gamma density
c© Copyright Eckhard Platen 10 BA - Toronto C 4 141
• Wiggins model
Wiggins (1987)
Chesney & Scott (1989)
Melino & Turnbull (1990)
d ln(θt) = k(
ln(θ) − ln(θt))
dt+ γ dWt
θ0 > 0
θ2t has as stationary density a log-normal density
c© Copyright Eckhard Platen 10 BA - Toronto C 4 142
• inverted gamma distribution
pθ2(y) =(12ν)
12 ν
ε2 Γ(12ν)
(
y
ε2
)− 12 ν−1
exp
−12ν ε2
y
y > 0, degrees of freedomν > 0, scaling parameterε
c© Copyright Eckhard Platen 10 BA - Toronto C 4 143
• SDE for squared volatility
dθ2t = k θ
4(ξ−1)t (θ2 − θ2
t ) dt+ γ θ2 ξt dWt
degree of freedom
ν =2 (2 ξ − 1)
1 − ε2
θ2
=⇒ stationary density inverted gamma
c© Copyright Eckhard Platen 10 BA - Toronto C 4 144
•32
volatility model
ξ = 32
Pl. (1997), Lewis (2000), Pl. (2001)
dθ2t = k θ2
t (θ2 − θ2t ) dt+ γ θ3
t dWt
c© Copyright Eckhard Platen 10 BA - Toronto C 4 145
• ARCH diffusion
ξ = 1
Engle (1982), Nelson (1990) and Frey (1997)
dθ2t = k (θ2 − θ2
t ) dt+ γ θ2t dWt
continuous time limit of innovation process
GARCH(1,1) and NGARCH(1,1)
c© Copyright Eckhard Platen 10 BA - Toronto C 4 146
Implied Volatilities for the ARCH Diffusion Model
Hurst (1997), Lewis (2000), Heath, Hurst& Pl. (2001)
GARCH(1,1) of Bollerslev (1986) whenW andW are independent
continuous time limit of NGARCH(1, 1) model of Engle & Ng (1993)
two-factor model
c© Copyright Eckhard Platen 10 BA - Toronto C 4 147
80
90
100
110
120K 0.2
0.4
0.6
0.8
1
T
0.2
0.21
0.22
80
90
100
110
120K
Figure 4.8: Implied volatility surface for zero correlation.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 148
80
90
100
110
120K -0.4
-0.2
0
0.2
0.4
rho
0.18
0.2
0.22
80
90
100
110
120K
Figure 4.9: Effect of changing correlation on implied volatilities.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 149
8090
100
110
120K
5
10
15
20
k
0.1950.1975
0.2
0.2025
0.205
8090
100
110
120K
Figure 4.10: Effect of changing speed of adjustment on implied volatilities.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 150
8090
100
110
120K
0.5
1
1.5
2
gamma
0.19
0.2
0.21
8090
100
110
120K
Figure 4.11: Effect of changing volatility of the squared volatility on implied
volatilities.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 151
Moneyness
Opt
ion
Pric
e D
iffer
ence
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
-0.1
5-0
.10
-0.0
50.
00.
05
GLMRHestonBlack-Scholes
Figure 4.12: Option price differences between the ARCH diffusion (GLMR),
Heston and Black-Scholes models, with time to maturity of three months.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 152
Moneyness
Impl
ied
Vol
atili
ty
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
0.14
0.16
0.18
0.20
GLMRHestonBlack-Scholes
Figure 4.13: Implied volatilities of the ARCH diffusion (GLMR), Heston
and Black-Scholes models, with time to maturity of three months.
c© Copyright Eckhard Platen 10 BA - Toronto C 4 153
5 Minimal Market Model
• search for parsimonious one-factor model
• 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 dW
kt
c© Copyright Eckhard Platen 10 BA - Toronto C 5 154
0
10
20
30
40
50
60
70
80
90
100
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 5.1: Discounted S&P500 total return index.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 155
-2
-1
0
1
2
3
4
5
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 5.2: Logarithm of discounted S&P500.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 156
• logarithm of discounted NP
• is not a transformed Wiener process
• is not a process with independent increments
• has somefeedback effect
c© Copyright Eckhard Platen 10 BA - Toronto C 5 157
Drift Parametrization
• discounted NP drift
αt = Sδ∗
t |θt|2
strictly positive, predictable
=⇒• volatility
|θt| =
√
αt
Sδ∗
t
leverage effect creates naturalfeedbackunder independent drift
c© Copyright Eckhard Platen 10 BA - Toronto C 5 158
• discounted NP
dSδ∗
t = αt dt+
√
Sδ∗
t αt dWt
drift has economic meaning,
fundamental value
• transformed time
ϕt =1
4
∫ t
0
αs ds
c© Copyright Eckhard Platen 10 BA - Toronto C 5 159
• squared Bessel process of dimension four
Xϕt= Sδ∗
t
dW (ϕt) =√αt dWt
dXϕ = 4 dϕ+ 2√
Xϕ dW (ϕ)
Revuz & Yor (1999)
economically founded dynamics inϕ-time
still no specific dynamics int-time
c© Copyright Eckhard Platen 10 BA - Toronto C 5 160
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 time
ϕt =[√
Sδ∗
]
t
observable
c© Copyright Eckhard Platen 10 BA - Toronto C 5 161
0
1
2
3
4
5
6
7
8
9
10
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 5.3: Empirical quadratic variation of the square rootof the discounted
NP.c© Copyright Eckhard Platen 10 BA - Toronto C 5 162
Stylized Minimal Market Model
Pl. (2001, 2002, 2006)
• assumediscounted NP drift as
αt = α exp η t
• initial parameter α > 0
• net growth rate η
c© Copyright Eckhard Platen 10 BA - Toronto C 5 163
• transformed time
ϕt =α
4
∫ t
0
exp η z dz
=α
4 η(exp η t − 1)
c© Copyright Eckhard Platen 10 BA - Toronto C 5 164
0
1
2
3
4
5
6
7
8
9
10
1930 1940 1950 1960 1970 1980 1990 2000
time
[sqrt(discounted index)]theoretical variation
Figure 5.4: Fitted and observed transformed time.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 165
• normalized NP
Yt =Sδ∗
t
αt
dYt = (1 − η Yt) dt+√
Yt dWt
square root process of dimension four
=⇒ parsimonious model, long term viability
c© Copyright Eckhard Platen 10 BA - Toronto C 5 166
=⇒• discounted NP
Sδ∗
t = Yt αt
• NP
Sδ∗
t = S0t S
δ∗
t = S0t Yt αt
• scaling parameterα0 = 0.01468
• net growth rateη = 0.054
• reference level1η
= 18.5
• speed of adjustmentη
• half life time of major displacement ln(2)η
≈ 13 years
c© Copyright Eckhard Platen 10 BA - Toronto C 5 167
0
10
20
30
40
50
60
70
80
90
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 5.5: Normalized NP.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 168
• logarithm of discounted NP
ln(Sδ∗
t ) = ln(Yt) + ln(α0) + η t
• no need for extra volatility process in MMM
• realistic long term dynamics
c© Copyright Eckhard Platen 10 BA - Toronto C 5 169
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
Platen (1997), Lewis (2000)
c© Copyright Eckhard Platen 10 BA - Toronto C 5 170
0.1
0.15
0.2
0.25
0.3
0.35
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 5.6: Volatility of the NP under the stylized MMM.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 171
Transition Density of Stylized MMM
• transition density ofdiscounted 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
c© Copyright Eckhard Platen 10 BA - Toronto C 5 172
• non-central chi-squaredistributed random variable:
y
ϕt − ϕs
=Sδ∗
t
ϕt − ϕs
with δ = 4 degrees of freedom andnon-centrality parameter:
x
ϕt − ϕs
=Sδ∗
s
ϕt − ϕs
c© Copyright Eckhard Platen 10 BA - Toronto C 5 173
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
Figure 5.7: Transition density of squared Bessel process forδ = 4.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 174
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
c© Copyright Eckhard Platen 10 BA - Toronto C 5 175
δ = 4 =⇒
P (t, T ) = P ∗T (t)
(
1 − exp
− Sδ∗
t
2 (ϕT − ϕt)
)
< P ∗T (t)
for t ∈ [0, T ), Platen (2002)
with time transform
ϕt =α
4 η(expη t − 1)
P (0, T ) = 0.00077
P ∗T (0) = 0.0335
P (0,T )P ∗
T (0)≈ 0.023
c© Copyright Eckhard Platen 10 BA - Toronto C 5 176
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1930 1940 1950 1960 1970 1980 1990 2000
time
savings bondfair zero coupon bond
savings account
Figure 5.8: Savings bond, fair zero coupon bond and savings account.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 177
-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)
Figure 5.9: Logarithms of savings bond and fair zero coupon bond.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 178
Forward Rates under the MMM
• forward rate
f(t, T ) = − ∂
∂Tln(P (t, T ))
= rT + n(t, T )
c© Copyright Eckhard Platen 10 BA - Toronto C 5 179
• 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. (2005a), Miller& Pl. (2005)
c© Copyright Eckhard Platen 10 BA - Toronto C 5 180
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
Figure 5.10: Market price of risk contribution in dependence on η andT .
c© Copyright Eckhard Platen 10 BA - Toronto C 5 181
Free Snack from Savings Bond
• potential existence of weak form of arbitrage?
borrow amountP (0, T ) from savings account
Sδt = P (t, T ) − P (0, T ) expr t
such thatSδ0 = 0
SδT = 1 − P (0, T ) expr T > 0
lower bound
Sδt ≥ −P (0, T ) expr t
c© Copyright Eckhard Platen 10 BA - Toronto C 5 182
• sinceSδt may become negative
allowed by strong arbitrage concept
• cheap thrills may exist
Loewenstein & Willard (2000)
MMM doesnot admit an equivalent risk neutral probability measure
=⇒there is afree lunch with vanishing risk
Delbaen & Schachermayer (2006)
c© Copyright Eckhard Platen 10 BA - Toronto C 5 183
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
=S0
t
S0T
• Radon-Nikodym derivative
Λt =S0
t
S00
=Sδ∗
0
Sδ∗
t
strict (A, P )-supermartingaleunder MMM
c© Copyright Eckhard Platen 10 BA - Toronto C 5 184
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
Figure 5.11: Radon-Nikodym derivative and total mass of putative risk neu-
tral measure.c© Copyright Eckhard Platen 10 BA - Toronto C 5 185
• hypothetical risk neutral measure
total mass:
Pθ,T (Ω) = E0 (ΛT ) = 1 − exp
− Sδ∗
0
2ϕT
< Λ0 = 1
Pθ is not a probability measure
c© Copyright Eckhard Platen 10 BA - Toronto C 5 186
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
c© Copyright Eckhard Platen 10 BA - Toronto C 5 187
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)
Hulley & Pl. (2008)
Miller & Pl. (2008)
c© Copyright Eckhard Platen 10 BA - Toronto C 5 188
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
Figure 5.12: Implied volatility surface for the stylized MMM.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 189
• 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
c© Copyright Eckhard Platen 10 BA - Toronto C 5 190
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)
c© Copyright Eckhard Platen 10 BA - Toronto C 5 191
• put-call paritybreaks down if one uses the savings bondP ∗T (t)
pT,K(t, Sδ∗
t ) < cT,K(t, Sδ∗
t ) − Sδ∗
t +K exp−r (T − t)
for t ∈ [0, T )
c© Copyright Eckhard Platen 10 BA - Toronto C 5 192
• excellent hedge performance for extreme maturities
Hulley & Pl. (2008)
European calls
European puts
barrier options
c© Copyright Eckhard Platen 10 BA - Toronto C 5 193
Comparison to Hypothetical Risk Neutral Prices
• hypothetical risk neutral price c∗T,K(t, Sδ∗
t )
of a European call option on the NP
• 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
c© Copyright Eckhard Platen 10 BA - Toronto C 5 194
=⇒ 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 )
c© Copyright Eckhard Platen 10 BA - Toronto C 5 195
• hypothetical risk neutral put-call parity
p∗T,K(t, Sδ∗
t ) = c∗T,K(t, Sδ∗
t ) − Sδ∗
t +K P ∗T (t)
sinceP ∗T (t) > P (t, T ) =⇒
pT,K(t, Sδ∗
t ) < p∗T,K(t, Sδ∗
t )
t ∈ [0, T )
c© Copyright Eckhard Platen 10 BA - Toronto C 5 196
Difference in Asymptotic Put Prices
• hypothetical risk neutral prices can become extreme if NP value tends
towards zero
• asymptotic fair zero coupon bond
limS
δ∗
t →0PT (t, T )
a.s.= 0
for t ∈ [0, T )
• asymptotic fair European call
limS
δ∗
t →0cT,K(t, Sδ∗
t )a.s.= lim
Sδ∗
t →0Sδ∗
t cT,K(t, Sδ∗
t ) = 0
c© Copyright Eckhard Platen 10 BA - Toronto C 5 197
• asymptotic fair putfair put-call parity =⇒
limS
δ∗
t →0pT,K(t, Sδ∗
t )a.s.= 0
• asymptotic hypothetical risk neutral puthypothetical risk neutral put-call parity =⇒
limS
δ∗
t →0p∗
T,K(t, Sδ∗
t )
a.s.= lim
Sδ∗
t →0
(
pT,K(t, Sδ∗
t ) +KS0
t
S0T
exp
− Sδ∗
t
2(ϕT − ϕt)
)
a.s.= K
S0t
S0T
> 0
dramatic differences can arise
c© Copyright Eckhard Platen 10 BA - Toronto C 5 198
0
2
4
6
8
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(K*savings bond)ln(risk neutral put)
ln(put)
Figure 5.13: Logarithms of savings bond timesK, risk neutral put and fair
put.c© Copyright Eckhard Platen 10 BA - Toronto C 5 199
MMM with Random Market Activity
• squared Bessel process with dimensionδ = 4
Sδ∗ = Sδ∗
t , t ∈ [0,∞)
dSδ∗
t = αt dt+
√
αt Sδ∗
t dWt
c© Copyright Eckhard Platen 10 BA - Toronto C 5 200
• NP
Sδ∗
t = S0t S
δ∗
t
• volatility
|θt| =
√
αt
Sδ∗
t
• discounted NP drift
αt to be modeled
c© Copyright Eckhard Platen 10 BA - Toronto C 5 201
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
Figure 5.14: Implied volatility surface for the stylized MMM.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 202
0.5
0.75
1
1.25
1.5
KS
0.20.4
0.60.8
11.2
1.41.6T
0.22
0.24
0.26
0.28
0.5
0.75
1
1.25KS
0.20.4
0.60.8
11.2
1.4
Figure 5.15: Average S&P500 implied volatility surface.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 203
100
200
300
time 0.6
0.8
1
1.2
1.4
moneyness
20
40
60
100
200
300
time
Figure 5.16: Implied volatilities for S&P500 three month options.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 204
100
200
300
time 0.6
0.8
1
1.2
1.4
moneyness
1020304050
100
200
300
time
Figure 5.17: Implied volatilities for S&P500 one year options.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 205
• market activity mt
Breymann, Kelly& Pl. (2006)
Pl. & Rendek (2008)
αt = ξtmt
with
ξt = ξ0 exp η t
dmt = k(mt)β2 dt+ βmt
(
dWt +√
1 − 2 dWt
)
- correlation parameter
activity volatility β > 0
W - independent Wiener process
c© Copyright Eckhard Platen 10 BA - Toronto C 5 206
Example:
• drift
k(mt) = (p− gmt)mt
2
• stationary density
pm(y) =gp−1
Γ(p− 1)yp−2 exp−g y
gamma density with meanp−1g
and variance1g
for p > 1 andg > 0
c© Copyright Eckhard Platen 10 BA - Toronto C 5 207
0.5
1
1.5y
10
20
30
40
50
g
0
1
2
0.5
1
1.5y
Figure 5.18: Stationary density of market activitymt = y as function ofy
and speed of adjustment parameterg.c© Copyright Eckhard Platen 10 BA - Toronto C 5 208
Zero Coupon Bond
• benchmarked zero coupon bond
PT (t, Sδ∗
t ,mt) = Et
(
1
Sδ∗
T
)
= Et
(
1
S0T S
δ∗
T
)
• zero coupon bond
PT (t, Sδ∗
t ,mt) = Sδ∗
t PT (t, Sδ∗
t ,mt) = S0t S
δ∗
t PT (t, Sδ∗
t ,mt)
c© Copyright Eckhard Platen 10 BA - Toronto C 5 209
European Options on a Market Index
• put option price
pT,K(t, Sδ∗
t ,mt) = S0t S
δ∗
t Et
(
K
S0T S
δ∗
T
− 1
)+
effect of random market time on implied volatilities
curvature for the short dated implied volatility surface
Heath& Pl. (2005a)
c© Copyright Eckhard Platen 10 BA - Toronto C 5 210
80
90
100
110
120
K 0.2
0.4
0.6
0.8
1
T
0.30.32
0.34
0.36
80
90
100
110K
Figure 5.19: Implied volatilities for put options on index asa function of
strikeK and maturityT .c© Copyright Eckhard Platen 10 BA - Toronto C 5 211
80
90
100
110
120
K
2
4
6
8
10
g
0.3
0.32
0.34
80
90
100
110K
Figure 5.20: Implied volatilities for put options as a function of strikeK and
speed of adjustmentg.
c© Copyright Eckhard Platen 10 BA - Toronto C 5 212
80
100
120K 2
4
6
8
10
T
0.30.3250.35
0.375
0.4
80
100
120K
Figure 5.21: Implied volatilities for long dated put optionsas a function of
strikeK and maturityT .
c© Copyright Eckhard Platen 10 BA - Toronto C 5 213
6 Stylized Multi-Currency MMM
Pl. (2001)
Heath& Pl. (2005a)
d+ 1 currencies
Sδ∗
i (t) NP in ith currency
rit - ith short rate
θki (t) - market price of risk
Sδ∗
i (t) = αit Y
it S
ii(t)
c© Copyright Eckhard Platen 10 BA - Toronto C 6 214
αit = αi
0 expηi t
Sii(t) = expri t
• ith normalized NP
dY it =
(
1 − ηi Y it
)
dt+
√
Y it
d+1∑
k=1
qi,k dW kt
scaling levelsqi,k
d+1∑
k=1
(qi,k)2 = 1
c© Copyright Eckhard Platen 10 BA - Toronto C 6 215
• (i, j)th exchange rate
Xi,jt =
Sδ∗
i (t)
Sδ∗
j (t)=Y i
t αit S
ii(t)
Y jt α
jt S
jj (t)
dXi,jt = Xi,j
t
(ri − rj) dt+
d+1∑
k=1
qi,k
√
Y it
− qj,k
√
Y jt
(
qi,k
√
Y it
dt+ dW kt
)
c© Copyright Eckhard Platen 10 BA - Toronto C 6 216
• jth savings account inith currency
Sji (t) = Xi,j
t Sjj (t)
dSji (t) = Sj
i (t)
ri dt+
d+1∑
k=1
qi,k
√
Y it
− qj,k
√
Y jt
(
qi,k
√
Y it
dt+ dW kt
)
c© Copyright Eckhard Platen 10 BA - Toronto C 6 217
• stochastic market price of risk
θki (t) =
qi,k
√
Y it
• (j, k)th volatility
bj,ki (t) = θk
i (t) − θkj (t)
equity prices
commodity prices
c© Copyright Eckhard Platen 10 BA - Toronto C 6 218
7 Valuing Guaranteed Minimum Death Benefits
• variable annuitiesfund-linked
tax-deferred
guarantees
• guaranteed minimum death benefits(GMDBs)
roll-ups:
original investment accrued at a pre-defined interest rate
ratchets:
death benefit based upon the highest anniversary account value
embedded options:
hedging against market downturn occurrence of death
c© Copyright Eckhard Platen 10 BA - Toronto C 7 219
• GMDB put, floating put and/or look-back put option
long maturities of contracts
standard option pricing theory ?
no obvious best choice for price:
IFRS Phase II
CFO-Forum (2008)
CRO-Forum (2008)
Solvency II
Verheugen & Hines (2008)
c© Copyright Eckhard Platen 10 BA - Toronto C 7 220
• pricing and hedging of GMDBs
benchmark approachin
Pl. (2002, 2004b), Pl.& Heath (2006)
Buhlmann& Pl. (2003)
best performing portfolio isbenchmark
doesnot require
existence of an equivalent risk neutral probability measure
uniquefair price is theminimal price
may provide significantly lower prices
c© Copyright Eckhard Platen 10 BA - Toronto C 7 221
Financial Model
• underlying risky security (unit)
dSt = (µt − γ)Stdt+ σtSt dWt
γ ≥ 0 management fee rates
Wt - Brownian motion
• savings account
dBt = rtBt dt
c© Copyright Eckhard Platen 10 BA - Toronto C 7 222
• market price of risk
θt =µt − γ − rt
σt
• risky asset
dSt
St
= rt dt+ σtθt dt+ σt dWt
c© Copyright Eckhard Platen 10 BA - Toronto C 7 223
• self-financing portfolio
Vt = δ0tBt + δ1tSt
dVt = δ0t dBt + δ1t dSt
• fractions
π0t = δ0t
Bt
Vt
, π1t = δ1t
St
Vt
π0t + π1
t = 1
c© Copyright Eckhard Platen 10 BA - Toronto C 7 224
• instantaneous portfolio return
dVt
Vt
= π0t
dBt
Bt
+ π1t
dSt
St
= rt dt+ π1tσt(θt dt+ dWt)
c© Copyright Eckhard Platen 10 BA - Toronto C 7 225
• numeraire portfolio (NP) as benchmark
Long (1990)
V ∗ is best performing in several ways
• is growth optimal portfolio
Kelly (1956)
V ∗ = maxE(log VT )
π1∗t =
µt − γ − rt
σ2t
=θt
σt
dV ∗t
V ∗t
= rt dt+ θt(θt dt+ dWt)
Merton (1992)
c© Copyright Eckhard Platen 10 BA - Toronto C 7 226
• NP best performing portfolio
lim supT →∞
1
Tlog
(
V ∗T
V ∗0
)
≥ lim supT →∞
1
Tlog
(
VT
V0
)
global well diversified index≈ NP
Pl. (2005b)
c© Copyright Eckhard Platen 10 BA - Toronto C 7 227
0
10
20
30
40
50
60
70
80
90
100
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 7.1: Discounted S&P500 total return index.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 228
-2
-1
0
1
2
3
4
5
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 7.2: Logarithm of discounted S&P500.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 229
• benchmarked price
Ut = Ut
V ∗
t
U nonnegative =⇒
Ut ≥ Et
(
Us
)
t ≤ s
supermartingales (no upward trend)
Pl. (2002)
no strong arbitrage
c© Copyright Eckhard Platen 10 BA - Toronto C 7 230
• benchmarked securities that form martingales are calledfair.
Ut = Et
(
Us
)
t ≤ s
c© Copyright Eckhard Platen 10 BA - Toronto C 7 231
• Law of the Minimal Price
Pl. (2008)
Fair prices are minimal prices
V nonnegative fair portfolio
V ′ nonnegative portfolio
VT = V ′T
supermartingale property
=⇒Vt ≤ V ′
t
t ∈ [0, T ],
c© Copyright Eckhard Platen 10 BA - Toronto C 7 232
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1930 1940 1950 1960 1970 1980 1990 2000
time
benchmarked savings bondbenchmarked fair zero coupon bond
Figure 7.3: Benchmarked savings bond and benchmarked fair zero coupon
bond.c© Copyright Eckhard Platen 10 BA - Toronto C 7 233
• real world pricing formulafor
Et
(
HT
V ∗T
)
< ∞
UH(t) = V ∗t Et
(
HT
V ∗T
)
t ∈ [0, T ]
no risk neutral probability needs to exist
c© Copyright Eckhard Platen 10 BA - Toronto C 7 234
• actuarial pricing formula
whenHT is independent ofV ∗T
=⇒UH(t) = P (t, T )Et(HT )
zero coupon bond
P (t, T ) = V ∗t Et
(
1
V ∗T
)
c© Copyright Eckhard Platen 10 BA - Toronto C 7 235
• standard risk neutral pricing
Ross (1976), Harrison & Pliska (1983)
complete market
candidate Radon-Nikodym derivative
Λt =dQ
dP
∣
∣
∣
∣
At
=BtV
∗0
B0V∗
t
supermartingale =⇒
1 = Λ0 ≥ E0 (ΛT )
c© Copyright Eckhard Platen 10 BA - Toronto C 7 236
=⇒
UH(0) = E
(
V ∗0
V ∗T
HT
)
= E
(
ΛT
B0
BT
HT
)
≤E(
ΛTB0
BTHT
)
E(ΛT )
c© Copyright Eckhard Platen 10 BA - Toronto C 7 237
Only in the special case whenΛT martingale
=⇒ risk neutral pricing formula:
UH(0) = EQ
(
B0
BT
HT
)
Ignores any real trend !
c© Copyright Eckhard Platen 10 BA - Toronto C 7 238
• Trends ignored also when using:
stochastic discount factor, Cochrane (2001);
deflator, Duffie (2001);
pricing kernel , Constantinides (1992);
state price density, Ingersoll (1987);
NP as Long (1990)
c© Copyright Eckhard Platen 10 BA - Toronto C 7 239
• Example zero coupon bond
benchmarked fair zero coupon
P (t, T ) =P (t, T )
V ∗t
= Et
(
1
V ∗T
)
martingale (no trend)
for rt - deterministic
P (t, T ) = V ∗t Et
(
1
V ∗T
)
= exp
−T∫
t
rs ds
Et
(
V ∗t
V ∗T
)
,
V ∗t =
V ∗
t
Bt- discounted NP
c© Copyright Eckhard Platen 10 BA - Toronto C 7 240
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1930 1940 1950 1960 1970 1980 1990 2000
time
benchmarked savings bondbenchmarked fair zero coupon bond
Figure 7.4: Benchmarked savings bond and benchmarked fair zero coupon
bond.c© Copyright Eckhard Platen 10 BA - Toronto C 7 241
=⇒ downward trend reflects equity premium
=⇒ Λ strict supermartingale (downward trend), then
P (t, T ) < exp
−T∫
t
rs ds
=Bt
BT
c© Copyright Eckhard Platen 10 BA - Toronto C 7 242
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
Figure 7.5: Radon-Nikodym derivative and total mass of putative risk neutral
measure.c© Copyright Eckhard Platen 10 BA - Toronto C 7 243
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1930 1940 1950 1960 1970 1980 1990 2000
time
savings bondfair zero coupon bond
savings account
Figure 7.6: Savings bond, fair zero coupon bond and savings account.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 244
-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)
Figure 7.7: Logarithms of savings bond and fair zero coupon bond.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 245
-2
-1
0
1
2
3
4
5
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 7.8: Logarithm of discounted S&P500.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 246
• discounted NP
V ∗t =
V ∗t
Bt
dV ∗t = V ∗
t θt (θt dt+ dWt)
= αt dt+
√
αt V∗
t dWt
c© Copyright Eckhard Platen 10 BA - Toronto C 7 247
• discounted NP drift
αt := V ∗t θ
2t
=⇒ volatility
θt =
√
αt
V ∗t
reflects leverage effect
c© Copyright Eckhard Platen 10 BA - Toronto C 7 248
• minimal market model
Pl. (2001, 2002)
MMM
discounted NP drift
assume
αt = α0 expηt
α0 > 0
net growth rateη > 0
=⇒ MMM
c© Copyright Eckhard Platen 10 BA - Toronto C 7 249
-2
-1
0
1
2
3
4
5
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(discounted index)trendline
Figure 7.9: Logarithm of discounted S&P500 and trendline.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 250
• normalized NP
Yt =V ∗
t
αt
dYt = (1 − ηYt) dt+√
Yt dWt
square root process of dimension four
Only one parameter !
c© Copyright Eckhard Platen 10 BA - Toronto C 7 251
• volatility of NP
θt =1√Yt
=
√
αt
V ∗t
c© Copyright Eckhard Platen 10 BA - Toronto C 7 252
• Zero coupon bond under the MMM
rt - deterministic
P (t, T ) = exp
−T∫
t
rs ds
(
1 − exp
− V ∗t
2(ϕ(T ) − ϕ(t))
)
Explicit formula !
c© Copyright Eckhard Platen 10 BA - Toronto C 7 253
-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)
Figure 7.10: Logarithms of savings bond and fair zero coupon bond.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 254
0
0.2
0.4
0.6
0.8
1
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 7.11: Fraction invested in the savings account.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 255
-2e-005
-1e-005
0
1e-005
2e-005
3e-005
4e-005
5e-005
6e-005
7e-005
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 7.12: Benchmarked profit and loss.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 256
since
Et
(
ΛT
Λt
)
= Et
(
V ∗t
V ∗T
)
=
(
1 − exp
− V ∗t
2(ϕ(T ) − ϕ(t))
)
=⇒ P (t, T ) < Bt
BT
Overpricing by savings bond
no equivalent risk neutral probability measure
c© Copyright Eckhard Platen 10 BA - Toronto C 7 257
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
Figure 7.13: Radon-Nikodym derivative and total mass of putative risk neu-
tral measure.c© Copyright Eckhard Platen 10 BA - Toronto C 7 258
• European put option under the MMM
Hulley, Miller & Pl. (2005)
p(t, V ∗t , T,K, r) = V ∗
t Et
(
(K − V ∗τ )+
V ∗τ
)
p(t, V ∗t , T,K, r) = −V ∗
t χ2(d1; 4, l2)
+Ke−r(T −t)(
χ2(d1; 0, l2) − exp −l2/2)
c© Copyright Eckhard Platen 10 BA - Toronto C 7 259
with
d1 =4ηK exp−r(T − t)
Btαt(expη(T − t) − 1)
and
l2 =4ηV ∗
t
Btαt(expη(T − t) − 1)
c© Copyright Eckhard Platen 10 BA - Toronto C 7 260
χ2(x;n, l) non-central chi-square distribution function
n ≥ 0 degrees of freedom
non-centrality parameterl > 0
χ2(x;n, l) =
∞∑
k=0
exp− l
2
(
l2
)k
k!
1 −Γ(
x2; n+2k
2
)
Γ(
n+2k2
)
c© Copyright Eckhard Platen 10 BA - Toronto C 7 261
• putative risk neutral price
p(t, V ∗t , T,K, r) = p(t, V ∗
t , T,K, r)+Ke−r(T −t) exp
− V ∗t
2(ϕ(T ) − ϕ(t))
risk neutral is overpricing
for V ∗t → 0 =⇒ p(t, V ∗
t , T,K, r) → 0
and
p(t, V ∗t , T,K, r) → K e−r(T −t) > 0
risk neutral ignores trends
c© Copyright Eckhard Platen 10 BA - Toronto C 7 262
0
2
4
6
8
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(K*savings bond)ln(risk neutral put)
ln(put)
Figure 7.14: Logarithms of savings bond timesK, risk neutral put and fair
put.c© Copyright Eckhard Platen 10 BA - Toronto C 7 263
Guaranteed Minimum Death Benefit (GMDB)
• payout to the policyholder
max(egτV0, Vτ )
time of deathτ
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 τ
c© Copyright Eckhard Platen 10 BA - Toronto C 7 264
• insurance chargesξ ≥ 0
=⇒ policyholder’s unit value
Vt = e−ξtV ∗t
=⇒
max(egτV0, Vτ ) = max(egτV0, e−ξτV ∗
τ )
= e−ξτ max(e(g+ξ)τV0, V∗
τ )
c© Copyright Eckhard Platen 10 BA - Toronto C 7 265
insurance company invests the entire fund valueV in the NP V ∗
=⇒ payoff
HT = GMDBT = e−ξT[
(e(g+ξ)TV ∗0 − V ∗
T )+ + V ∗T
]
c© Copyright Eckhard Platen 10 BA - Toronto C 7 266
fair valueGMDB0 of the total claim
real world pricing formula
GMDB0 = V ∗0 E
(
GMDBT
V ∗T
)
= e−ξT(
p(0, V ∗0 , T, e
(g+ξ)TV ∗0 , r) + V ∗
0
)
c© Copyright Eckhard Platen 10 BA - Toronto C 7 267
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
Figure 7.15: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 andY0 = 20.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 268
• lifetime τ is stochastic
GMDB0 = V ∗0 E
(
E
(
GMDBτ
V ∗τ
∣
∣
∣
∣
Fτ
))
GMDB0 =
∫ T
0
(
p(0, V ∗0 , t, e
(g+ξ)tV ∗0 , r) + V ∗
0
)
e−ξtfτ (t) dt
fτ (·) - mortality density
c© Copyright Eckhard Platen 10 BA - Toronto C 7 269
0 20 40 60 80 100 12010
−5
10−4
10−3
10−2
10−1
100
Age x
log(
q x)
malefemale
Figure 7.16: German mortality data
c© Copyright Eckhard Platen 10 BA - Toronto C 7 270
rational investors will lapse when embedded put-option outof the moneyGMDBs have stochastic maturityTitanic options, Milevsky & Posner (2001)
• roll-up GMDB payoff
Ht =
(1 − βt)V∗
t , if lapsed at timet,
max(egτV0, V∗
τ ), if death occurs at timet = τ
surrender chargeβt
βt =
(8 − ⌈t⌉)%, t ≤ 7,
0, t > 7
no credit riskno accumulation phase
c© Copyright Eckhard Platen 10 BA - Toronto C 7 271
20 30 40 50 60 70 801
1.05
1.1
1.15
Age x
GM
DB
0
malefemale
Figure 7.17: Value of the GMDB under the MMM for male and female pol-
icyholders agedx, assuming an irrational lapsation ofl = 1%.
c© Copyright Eckhard Platen 10 BA - Toronto C 7 272
8 Markets with Event Risk
Jump Diffusion Markets
• filtered probability space(Ω,A,A, P )
• continuous trading uncertainty
m ∈ 1, 2, . . . , d
Wiener processesW k = W kt , t ∈ [0,∞), k ∈ 1, 2, . . . ,m
c© Copyright Eckhard Platen 10 BA - Toronto C 8 273
• events
counting processpk = pkt , t ∈ [0,∞)
intensity hk = hkt , t ∈ [0,∞)
hkt > 0
∫ t
0
hks ds < ∞
• jump martingale
dqkt =
(
dpkt − hk
t dt) (
hkt
)− 12
k ∈ 1, 2, . . . , d−m
c© Copyright Eckhard Platen 10 BA - Toronto C 8 274
E
(
(
qkt+∆ − qk
t
)2 ∣∣
∣At
)
= ∆
• trading uncertainty
W = W t = (W 1t , . . . , W
mt , q1
t , . . . , qd−mt )⊤, t ∈ [0,∞)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 275
Primary Security Accounts
dSjt = Sj
t−
(
ajt dt+
d∑
k=1
bj,kt dW k
t
)
j ∈ 1, 2, . . . , d
c© Copyright Eckhard Platen 10 BA - Toronto C 8 276
Assumption 8.1
bj,kt ≥ −
√
hk−mt
for all t ∈ [0,∞), j ∈ 1, 2, . . . , d andk ∈ m+ 1, . . . , d.
Assumption 8.2 bt is invertible
c© Copyright Eckhard Platen 10 BA - Toronto C 8 277
• market price of risk
θt = (θ1t , . . . , θ
dt )⊤ = b−1
t [at − rt 1]
at = (a1t , . . . , a
dt )
⊤
1 = (1, . . . , 1)⊤
=⇒
dSjt = Sj
t−
(
rt dt+
d∑
k=1
bj,kt (θk
t dt+ dW kt )
)
j ∈ 0, 1, . . . , d
c© Copyright Eckhard Platen 10 BA - Toronto C 8 278
• strategy
δ = δt = (δ0t , . . . , δdt )⊤, t ∈ [0,∞)
• portfolio
Sδt =
d∑
j=0
δjt S
jt
• self-financing if
dSδt =
d∑
j=0
δjt dS
jt
c© Copyright Eckhard Platen 10 BA - Toronto C 8 279
Growth Optimal Portfolio
• fraction
πjδ,t = δj
t
Sjt
Sδt
πδ,t = (π1δ,t, . . . , π
dδ,t)
⊤
dSδt = Sδ
t−
rt dt+ π⊤δ,t− bt (θt dt+ dW t)
strictly positive ⇐⇒
d∑
j=1
πjδ,t b
j,kt > −
√
hk−mt
k ∈ m+ 1, . . . , d
c© Copyright Eckhard Platen 10 BA - Toronto C 8 280
d ln(Sδt ) = gδ
t dt+
m∑
k=1
d∑
j=1
πjδ,t b
j,kt dW k
t
+d∑
k=m+1
ln
1 +d∑
j=1
πjδ,t−
bj,kt
√
hk−mt
√
hk−mt dW k
t
• growth rate
gδt = rt +
m∑
k=1
d∑
j=1
πjδ,t b
j,kt θk
t − 1
2
d∑
j=1
πjδ,t b
j,kt
2
+
d∑
k=m+1
d∑
j=1
πjδ,tb
j,kt
(
θkt −
√
hk−mt
)
+ln
1+
d∑
j=1
πjδ,t
bj,kt
√
hk−mt
hk−mt
c© Copyright Eckhard Platen 10 BA - Toronto C 8 281
Assumption 8.3√
hk−mt > θk
t
k ∈ m+ 1, . . . , d
c© Copyright Eckhard Platen 10 BA - Toronto C 8 282
• fractions of candidate benchmark
πδ∗,t = (π1δ∗,t, . . . , π
dδ∗,t)
⊤ =(
c⊤t b
−1t
)⊤
ckt =
θkt for k ∈ 1, 2, . . . ,m
θkt
1−θkt (h
k−mt )−
12
for k ∈ m+ 1, . . . , d
if θkt
√
hk−mt
≪ 1
=⇒ ckt ≈ θk
t
c© Copyright Eckhard Platen 10 BA - Toronto C 8 283
• candidate benchmark
dSδ∗
t = Sδ∗
t−(
rt dt+ c⊤t (θt dt+ dW t)
)
= Sδ∗
t−
(
rt dt+
m∑
k=1
θkt (θk
t dt+ dW kt )
+
d∑
k=m+1
θkt
1 − θkt (hk−m
t )− 12
(θkt dt+ dW k
t )
c© Copyright Eckhard Platen 10 BA - Toronto C 8 284
Definition 8.4 Sδ ∈ V+ NP if gδt ≤ g
δt for all Sδ ∈ V+.
=⇒
Sδ∗ is a NP.
=⇒
gδ∗
t = rt+1
2
m∑
k=1
(θkt )2−
d∑
k=m+1
hk−mt
ln
1 +θk
t√
hk−mt − θk
t
+θk
t√
hk−mt
c© Copyright Eckhard Platen 10 BA - Toronto C 8 285
for θkt
√
hk−mt
≪ 1 asymptotically
gδ∗
t ≈ rt +1
2
d∑
k=1
(θkt )2 = rt +
|θt|22
d ln(Sδ∗
t ) ≈ gδ∗
t dt+
d∑
k=1
θkt dW
kt
dSδ∗
t ≈ Sδ∗
t
(
rt +
d∑
k=1
θkt
(
θkt dt+ dW k
t
)
)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 286
• jump diffusion market (JDM) SJD(d) = S, a, b,~r, h,A, P
• benchmarked portfolio
Sδt =
Sδt
Sδ∗
t
dSδt =
m∑
k=1
d∑
j=1
δjt S
jt b
j,kt − Sδ
t θkt
dW kt
+
d∑
k=m+1
d∑
j=1
δjt S
jt− b
j,kt
1 − θkt
√
hk−mt
− Sδt− θ
kt
dW kt
driftless
c© Copyright Eckhard Platen 10 BA - Toronto C 8 287
• benchmarked savings account
dS0t = −S0
t−
d∑
k=1
θkt dW
kt
driftless
c© Copyright Eckhard Platen 10 BA - Toronto C 8 288
benchmarked portfolioSδ driftless =⇒ (A, P )-local martingale
Ansel & Stricker (1994)
=⇒
Theorem 8.5 In a JDM a nonnegativeSδ is an (A, P )-supermartin-
gale
Sδt ≥ E
(
Sδτ
∣
∣
∣At
)
τ ∈ [0,∞) andt ∈ [0, τ ].
c© Copyright Eckhard Platen 10 BA - Toronto C 8 289
A nonnegative portfolio is a strongarbitrage if it can get out of zero.
Corollary 8.6
A JDM does not allow nonnegative portfolios that permit strong arbitrage.
c© Copyright Eckhard Platen 10 BA - Toronto C 8 290
Law of the Minimal Price
Corollary 8.7 Aτ -measurable payoffH withE( H
Sδ∗
τ
|A0) < ∞. If
there exists a fair nonnegative portfolioSδ that replicates the payoff
Sδτ = H,
then this is the minimal nonnegative replicating portfolio.
=⇒• fair portfolios provide the cheapest hedge
c© Copyright Eckhard Platen 10 BA - Toronto C 8 291
H Aτ -measurable payoff, withE( H
Sδ∗
τ
) < ∞
=⇒
real world pricing formula
UH(t) = Sδ∗
t E
(
H
Sδ∗
τ
∣
∣
∣
∣
At
)
• equivalent to riskneutral pricing as long as candidate Radon-Nikodym
derivative S0t
S00
forms(A, P )-martingale
c© Copyright Eckhard Platen 10 BA - Toronto C 8 292
• zero coupon bond
P (t, T ) = E
(
Sδ∗
t
Sδ∗
T
∣
∣
∣
∣
At
)
• benchmarked zero coupon bond
P (t, T ) = P (t,T )
Sδ∗
t
dP (t, T ) = −P (t−, T )
d∑
k=1
σk(t, T ) dW kt
c© Copyright Eckhard Platen 10 BA - Toronto C 8 293
=⇒
ln(P (t, T )) = ln(P (0, T )) −m∑
k=1
(∫ t
0
σk(s, T )dW ks +
1
2
∫ t
0
(σk(s, T ))2 ds
)
+
d∑
k=m+1
(
∫ t
0
σk(s, T )√
hk−ms ds+
∫ t
0
ln
(
1 − σk(s, T )√
hk−ms
)
dpks
)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 294
• forward rate
f(t, T ) = − ∂
∂Tln(P (t, T )) = − ∂
∂Tln(P (t, T ))
=⇒• forward rate equation
f(t, T ) = f(0, T ) +
m∑
k=1
∫ t
0
(
∂
∂Tσk(s, T )
)
(
σk(s, T ) ds+ dW ks
)
+
d∑
k=m+1
∫ t
0
1
1 − σk(s,T )√h
k−ms
∂
∂Tσk(s, T )
(
σk(s, T ) ds+ dW ks
)
similar to HJM equation
c© Copyright Eckhard Platen 10 BA - Toronto C 8 295
Sequences of Diversified Portfolios
Sequence(Sδ(d))d∈N is a sequence of DPs if
∣
∣
∣πj
δ,t
∣
∣
∣≤ K2
d12+K1
for all j ∈ 0, 1, . . . , d, t ∈ [0,∞) and d ∈ d0, d0 + 1, . . ..
c© Copyright Eckhard Platen 10 BA - Toronto C 8 296
• benchmarked primary security account
dSj
(d)(t) = −Sj
(d)(t−)
d∑
k=1
σj,k
(d)(t) dWkt
• kth total specific volatility
σk(d)(t) =
d∑
j=0
|σj,k
(d)(t)|
c© Copyright Eckhard Platen 10 BA - Toronto C 8 297
Sequence of JDMsregular if
E
(
(
σk(d)(t)
)2)
≤ K5
for all t ∈ [0,∞), d ∈ N and k ∈ 1, 2, . . . , d.
c© Copyright Eckhard Platen 10 BA - Toronto C 8 298
Sequence of Approximate NPs
• tracking rate
Rδ(d)(t) =
d∑
k=1
d∑
j=0
πjδ,t σ
j,k
(d)(t)
2
Sδ(d) is NP ⇐⇒ Sδ
(d) ≡ constant
⇐⇒Rδ
(d)(t) = 0
c© Copyright Eckhard Platen 10 BA - Toronto C 8 299
(Sδ(d))d∈N is a sequence ofapproximate NPsif
limd→∞
Rδ(d)(t)
P= 0
• expected tracking rate
eδ(d)(t) = E
(
Rδ(d)(t)
)
(Sδ(d))d∈N has vanishing expected tracking rate, if
limd→∞
eδ(d)(t) = 0
=⇒
c© Copyright Eckhard Platen 10 BA - Toronto C 8 300
limd→∞
P(
Rδ(d)(t) > ε
)
≤ limd→∞
1
εeδ(d)(t) = 0
Lemma 8.8 For a sequence of JDMs, any sequence of strictly positive
portfolios with vanishing expected tracking rate is a sequence of approximate
NPs.
c© Copyright Eckhard Platen 10 BA - Toronto C 8 301
Diversification Theorem
Platen (2005a)
For a regular sequence of JDMs(SJD(d))d∈N , each sequence(Sδ
(d))d∈N
of DPs is a sequence of approximate NPs. Moreover
eδ(d)(t) ≤ (K2)
2K5
d2K1.
c© Copyright Eckhard Platen 10 BA - Toronto C 8 302
Diversification in an MMM Setting
• savings account
S0(d)(t) = expr t
• discounted NP drift
αδ∗
t = α0 expη t
• jth benchmarked primary security account
Sj
(d)(t) =1
Y jt α
δ∗
t
c© Copyright Eckhard Platen 10 BA - Toronto C 8 303
• SR process
dY jt =
(
1 − η Y jt
)
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)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 304
0
50
100
150
200
250
300
350
400
450
0 5 9 14 18 23 27 32
Figure 8.1: Primary security accounts under the MMM.
c© Copyright Eckhard Platen 10 BA - Toronto C 8 305
0
10
20
30
40
50
60
0 5 9 14 18 23 27 32
Figure 8.2: Benchmarked primary security accounts.
c© Copyright Eckhard Platen 10 BA - Toronto C 8 306
0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
EWI
GOP
Figure 8.3: NP and EWI.
c© Copyright Eckhard Platen 10 BA - Toronto C 8 307
0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
Market index
GOP
Figure 8.4: NP and market index.
c© Copyright Eckhard Platen 10 BA - Toronto C 8 308
Real World Pricing for Two Market Models
Specifying a Continuous NP
market prices of event risks are zero
θkt = 0
NP
dSδ∗
t = Sδ∗
t
(
rt dt+
m∑
k=1
θkt
(
θkt dt+ dW k
t
)
)
Sδ∗
0 = 1
c© Copyright Eckhard Platen 10 BA - Toronto C 8 309
Benchmarked Primary Security Accounts
dSjt = −Sj
t−
d∑
k=1
σj,kt dW k
t
Sjt = Sj
0 exp
−1
2
∫ t
0
m∑
k=1
(
σj,ks
)2ds−
m∑
k=1
∫ t
0
σj,ks dW k
s
× exp
∫ t
0
d∑
k=m+1
σj,ks
√
hk−ms ds
d∏
k=m+1
pk−mt∏
l=1
1 −σj,k
τkl −
√
hk−m
τkl −
pivotal objects of study
c© Copyright Eckhard Platen 10 BA - Toronto C 8 310
• simplifying notation:
|σjt | =
√
√
√
√
m∑
k=1
(
σj,kt
)2
• aggregate continuous noise processes
W jt =
m∑
k=1
∫ t
0
σj,ks
|σjs|dW k
s
c© Copyright Eckhard Platen 10 BA - Toronto C 8 311
• generalized volatility matrix bt = [bj,kt ]dj,k=1 invertible
bj,kt = θk
t − σj,kt
for k ∈ 1, 2, . . . ,m
bj,kt = −σj,k
t
k ∈ m+ 1, . . . , d
c© Copyright Eckhard Platen 10 BA - Toronto C 8 312
• constant intensities
hkt = hk > 0
σj,kt = σj,k ≤
√hk−m
• benchmarkedjth primary security account:
Sjt = Sj,c
t Sj,dt
c© Copyright Eckhard Platen 10 BA - Toronto C 8 313
• continuous part
Sj,ct = Sj
0 exp
−1
2
∫ t
0
|σjs|2 ds−
∫ t
0
|σjs| dW j
s
• compensated jump part
Sj,dt = exp
d∑
k=m+1
σj,k√hk−m t
d∏
k=m+1
(
1 − σj,k
√hk−m
)pk−mt
c© Copyright Eckhard Platen 10 BA - Toronto C 8 314
Merton Model
rt = r, σj,kt = σj,k
Sj,ct = Sj
0 exp
−1
2|σj|2 t− |σj| W j
t
Girsanov’s theorem, see Section 9.5
c© Copyright Eckhard Platen 10 BA - Toronto C 8 315
A Minimal Market Model with Jumps
Hulley, Miller & Pl. (2005)
• discounted NP drift
αj(t) = αj0 expηjt
ηj - net growth rate
c© Copyright Eckhard Platen 10 BA - Toronto C 8 316
• jth square root process
dY jt =
(
1 − ηjY jt
)
dt+
√
Y jt dW
jt
• continuous part
Sj,ct =
1
αj(t)Y jt
c© Copyright Eckhard Platen 10 BA - Toronto C 8 317
• time transformation
ϕj(t) = ϕj0 +
1
4
∫ t
0
αj(s) ds
• squared Bessel process of dimension four
Xj
ϕj(t)= αj(t)Y j
t =1
Sj,ct
S0 is a strict local martingale
c© Copyright Eckhard Platen 10 BA - Toronto C 8 318
Zero Coupon Bonds
P (t, T ) = Sδ∗
t E
(
1
Sδ∗
T
∣
∣
∣
∣
At
)
=1
S0t
E
(
exp
−∫ T
t
rs ds
S0T
∣
∣
∣
∣
At
)
• MM case
P (t, T ) = exp−r(T−t) 1
S0t
E(
S0T
∣
∣
∣At
)
= exp−r(T−t)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 319
• MMM case
λjt =
1
Sjt (ϕ
j(t) − ϕj(T ))
S0T = S0,c
T
P (t, T ) = E
(
exp
−∫ T
t
rs ds
∣
∣
∣
∣
At
)
1
S0t
E(
S0T
∣
∣
∣At
)
= E
(
exp
−∫ T
t
rs ds
∣
∣
∣
∣
At
)
(
1 − exp
−1
2λ0
t
)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 320
Forward Contracts
Sδ∗
t E
(
F j(t, T ) − SjT
Sδ∗
T
∣
∣
∣
∣
At
)
= 0
forward price
F j(t, T ) =Sδ∗
t E(
SjT
∣
∣
∣At
)
Sδ∗
t E(
1
Sδ∗
T
∣
∣
∣At
) =
Sjt
P (t,T )1
Sjt
E(
SjT
∣
∣At
)
if Sjt > 0
0 if Sjt = 0
c© Copyright Eckhard Platen 10 BA - Toronto C 8 321
• MM caseF j(t, T ) = Sj
t expr(T − t)standard expression
• MMM case
1
Sjt
E(
SjT
∣
∣
∣At
)
=1
Sj,ct
E(
Sj,cT
∣
∣
∣At
) 1
Sj,dt
E(
Sj,dT
∣
∣
∣At
)
= 1 − exp
−1
2λj
t
forward price
F j(t, T ) = Sjt
1 − exp
−12λj
t
1 − exp−1
2λ0
t
(
E
(
exp
−∫ T
t
rs ds
∣
∣
∣
∣
At
))−1
c© Copyright Eckhard Platen 10 BA - Toronto C 8 322
Asset-or-Nothing Binaries
Ingersoll (2000), Buchen (2004) and Buchen & Konstandatos (2005)
rt = r
c© Copyright Eckhard Platen 10 BA - Toronto C 8 323
Aj,k(t, T,K) = Sδ∗
t E
(
1SjT ≥K
SjT
Sδ∗
T
∣
∣
∣
∣
At
)
=Sj
t
Sjt
E(
1SjT ≥K(S0
T )−1S0T S
jT
∣
∣
∣At
)
=Sj
t
Sj,ct
E
(
1Sj,cT ≥g(p
k−m
T −pk−mt )S0
T
× exp
σj,k√hk−m (T − t)
(
1 − σj,k
√hk−m
)pk−m
T −pk−mt
Sj,cT
∣
∣
∣
∣
At
c© Copyright Eckhard Platen 10 BA - Toronto C 8 324
Aj,k(t, T,K) =
=
∞∑
n=0
exp−hk−m(T − t)
(hk(T − t))n
n!exp
σj,k√hk−m(T − t)
×(
1 − σj,k
√hk−m
)nSj
t
Sj,ct
E(
1Sj,cT ≥g(n)S0
T Sj,cT
∣
∣
∣At
)
for all t ∈ [0, T ], where
g(n) =K
S0t S
j,dt
exp
−(
r + σj,k√hk−m
)
(T − t)
(
1 − σj,k
√hk−m
)−n
for all n ∈ N
c© Copyright Eckhard Platen 10 BA - Toronto C 8 325
MM case
Aj,k(t, T,K) =∞∑
n=0
exp−hk−m (T − t)
(hk−m (T − t))n
n!
× exp
σj,k√hk−m (T − t)
(
1 − σj,k
√hk−m
)n
Sjt N(d1(n))
for all t ∈ [0, T ], where
d1(n) =
ln(
Sjt
K
)
+
r + σj,k√hk−m + n
ln
(
1− σj,k√hk−m
)
T −t+ 1
2
(
σ0,j)2
(T − t)
σ0,j√T − t
c© Copyright Eckhard Platen 10 BA - Toronto C 8 326
σi,j =√
|σi|2 − 2 i,j |σi| |σj| + |σj|2
i,j - correlation
c© Copyright Eckhard Platen 10 BA - Toronto C 8 327
MMM case
S0 and Sj,c are independent
Aj,k(t, T,K) =
∞∑
n=0
exp−hk−m(T − t)
(hk−m(T − t))n
n!
× exp
σj,k√hk−m(T − t)
(
1 − σj,k
√hk−m
)n
×Sjt
(
G′′0,4
(ϕ0(T ) − ϕ0(t)
g(n);λj
t , λ0t
)
− exp
−1
2λj
t
)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 328
G′′0,4(x;λ, λ
′) equals probabilityP ( ZZ′
≤ x)
non-central chi-square distributed random variableZ ∼ χ2(0, λ)
non-central chi-square distributed random variableZ′ ∼ χ2(4, λ′)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 329
Bond-or-Nothing Binaries
MM case
Bj,k(t, T,K) =
∞∑
n=0
exp−hk−m (T − t)(hk−m (T − t))n
n!
×K exp−r(T − t)N(d2(n))
for all t ∈ [0, T ], where
d2(n) =
ln(
Sjt
K
)
+
(
r + σj,k√hk−m + n
ln(1− σj,k√hk−m
)
T −t− 1
2
(
σ0,j)2
)
(T − t)
σ0,j√T − t
= d1(n) − σ0,j√
T − t
c© Copyright Eckhard Platen 10 BA - Toronto C 8 330
MMM case
Bj,k(t, T,K)
=
∞∑
n=0
exp−hk−m(T − t)(hk−m (T − t))n
n!
×K exp−r(T − t)(
1 −G′′0,4
(
(ϕj(T ) − ϕj(t))g(n);λ0t , λ
jt
))
c© Copyright Eckhard Platen 10 BA - Toronto C 8 331
European Call Options
cj,kT,K(t) = Sδ∗
t E
(
SjT −K
)+
Sδ∗
T
∣
∣
∣
∣
At
= Sδ∗
t E
(
1SjT ≥K
SjT −K
Sδ∗
T
∣
∣
∣
∣
At
)
= Aj,k(t, T,K) −Bj,k(t, T,K)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 332
MM case
cj,kT,K(t) =
∞∑
n=0
exp−hk−m(T − t)(hk−m(T − t))n
n!
×(
exp
σj,k√hk−m (T − t)
(
1 − σj,k
√hk−m
)n
Sjt N(d1(n))
−K exp−r(T − t)N(d2(n))
)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 333
MMM case
cj,kT,K(t) =
∞∑
n=0
exp−hk−m(T − t)(hk−m(T − t))n
n!
×[
exp
σj,k√hk−m (T − t)
(
1 − σj,k
√hk−m
)n
×Sjt
(
G′′0,4
((
ϕ0(T ) − ϕ0(t)
g(n)
)
;λjt , λ
0t
)
− exp
−1
2λj
t
)
−K exp−r(T − t)(
1 −G′′0,4
(
(ϕj(T ) − ϕj(t))g(n);λ0t , λ
jt
))
]
c© Copyright Eckhard Platen 10 BA - Toronto C 8 334
Defaultable Zero Coupon Bonds
maturityT
defaults at the first jump timeτk−m1 of pk−m
zero recovery
P k−m(t, T ) = Sδ∗
t E
(
1τk−m1 >T Sδ∗
T
∣
∣
∣
∣
At
)
= Sδ∗
t E
(
1
Sδ∗
T
∣
∣
∣
∣
At
)
E(
1τk−m1 >T
∣
∣
∣At
)
= P (t, T )P (pk−mT = 0
∣
∣At)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 335
• conditional probability of survival
P(
pk−mT = 0
∣
∣At
)
= E(
1pk−mt =0 1p
k−m
T −pk−mt =0
∣
∣
∣At
)
= 1pk−mt =0 P
(
pk−mT − pk−m
t = 0∣
∣
∣At
)
= 1pk−mt =0E
(
exp
−∫ T
t
hk−ms ds
∣
∣
∣
∣
∣
At
)
c© Copyright Eckhard Platen 10 BA - Toronto C 8 336
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