optimal process approximation: application to delta hedging and
TRANSCRIPT
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Optimal Process Approximation:
Application to Delta Hedging and Technical Analysis
Bruno DupireBloomberg L.P. NY
Quantitative finance: Developments, Applications and Problems, Cambridge UK July 7, 2005
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Process Approximation
• Financial data are recorded down to the tick transaction (virtually continuous time)
1985 1990 1995 20000
500
1000
1500S&P500
0 1 week 2 week3900
4000
4100 S&P500
0 1 hour 2 hour
3990
3995
4000
4005
S&P500
• How can we simplify this information into an exploitable format?
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Agenda
We mostly compare the classical ”time based” approximation to the ”move based”
approximation.
The plan is the following:
• Theoretical results
• Delta Hedging
• Technical Analysis
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Theoretical Results
Quantitative finance: Developments, Applications and Problems, Cambridge UK July 7, 2005 Bruno Dupire
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Move Based / Time Based
We assume a continuous path (t, Xt) which we want to approximate with a finite number
of points (ti, Xti)
A) Time based approximation: (i∆t, Xi∆t)
3990
3995
4000
4005
S&P500
B) Move based approximation:
fix α > 0, (τi, Xτi)
where τi ≡ inf{t > τi−1 : |Xt − Xτi−1| ≥ α}
0 1 hour 2 hour
3990
3995
4000
4005
S&P500
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Example
To simplify the analysis, both approximations are piecewise constant, denoted X∆t and Xα
96
98
100
102
spot pricetime based
move based
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Convergence Properties
• L∞ convergence
– Time based: no uniform L∞ convergence, just E[‖X − X∆t‖∞]
– Move based: by construction ‖X − Xα‖∞ ≤ α
The knowledge of the string of ups and downs locates the path up to an error δ.
• L1, L2 convergence for X a Brownian motion:
– Time based: E[‖X − X∆t‖1] =T
3
√∆t
√
4
πE[‖X − X∆t‖2
2] =T
2∆t
– Move based: E[‖X − Xα‖1] =T
3α E[‖X − Xα‖2
2] =T
6α2
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Convergence Properties II
• Pathwise convergence of variations of all orders for move and time based approxima-
tions
• Furthermore if dYt = µtdt + σtdWt:
– Time based: lim∆t→0
1√∆t
(〈Y ∆tt 〉 − 〈Yt〉) =
√2
∫ t
0
σ2sdBs
– Move based: limα→0
1
α(〈Y α
t 〉 − 〈Yt〉) =
√
2
3
∫ t
0
σsdBs
where B and W are two independent brownian motions
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Convergence Properties III
• Furthermore ifdYt
Yt= µ(t, Yt)dt + σ(t, Yt)dWt, for a European option with Γ(t, Yt):
– Time based: (Bertsimas, Kogan, Lo, 2000)
lim∆t→0
1√∆t
TEt =
√
1
2
∫ t
0
σ2sY
2s Γ(s, Ys)dBs
– Move based:
limα→0
1
αTEt =
√
1
6
∫ t
0
σsYsΓ(s, Ys)dBs
B and W are two independent brownian motions
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Expected Length
• X continuous martingale, X0 = 0
• τi date of ith crossing
• L(α, T ) = max{i : τi < T} = n
X2τn
=∑n
i=1(Xτi − Xτi−1
)2 + 2∑n
i=1Xτi−1
(Xτi − Xτi−1)
As (Xτi − Xτi−1)2 = α2, E[X2
τn] = E[n]α2
E[X2T ] = E[X2
T − X2τn
] + α2E[n]
τn ≤ T ≤ τn+1 ⇒ E[n]α2 = E[X2τn
] ≤ E[X2T ] ≤ E[X2
τn+1] = E[(n + 1)]α2
E[X2T ]
α2− 1 ≤ E[L(α, T )] ≤ E[X2
T ]
α2
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Bridges and Meanders
Alternate intervals of bridges and meanders. Sequence of ups and downs is the same
when computed FWD or BWD (Indeed crossing times are different)
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∆ Hedging
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Optimal Delta Hedging
• Optimal delta hedging is a complicated topic in the presence of transaction costs.
• Highly dependent on specific assumptions
• Simplified problem
– Bachelier dynamics
– Option with constant Γ: parabola
– Expected time between two rehedges imposed
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Stopping Times
X(W, T ) = W 2T −→ X(Wt, t) = Et[W
2T ] = W 2
t + T − t
• Rehedge at stopping time τ : δP&L = W 2τ − τ .
• As E[W 2τ − τ ] = 0 for τ of finite expectation,
we aim at minimizing :
E[(W 2τ −τ )2] subject to E[τ ] = 1
or
minτ
√
E[(W 2τ − τ )2]
E[τ ]= Ratio
0
P&L<0
P&L>0
t
W
break−evenpoints
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First Attempts
• Time based
0
Wδ t2 −δ t
Wδ t
• Move based
0
α
−α
1
0
1
τ
Wτ2−τ
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Second Attempts
• Truncate the move based
• Follow the break-even points
Nothing beats the move based. Is move based optimal?
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Optimality of the Move Based
• By 2 successive Ito formulas, transform the problem to Kurtosis minimization of Wτ :
E[(W 2τ − τ )2] =
2
3E[W 4
τ ]
• As E[τ ] = E[W 2τ ] = 1, our new problem is:
find a density with fixed variance and minimum kurtosis. This is achieved by a
binomial distribution.
• The only integrable τ s.t. Wτ follows a binomial density is
inf{t : |Xt| ≥ δ}
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Links with Business Time Delta Hedge
• Business Time delta hedge is based on the DDS time change
• Important to link P&L to realized variance
• Cross points of move based correspond to business activity
• The sequence ταi converges to the time change
• Assume you have a correct estimate of the time based vol and of the move based vol.
Hedging prescription: decrease the variance parameter by α2 at each cross point.
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Daily P&L Variation
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Tracking Error Comparison
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Technical Analysis
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Technical Analysis
PAST
Economic Theory
Prices follow a random walk Anything ———————>
Technical Analysis
Past may influence future Pattern, Indicator————>
FUTURE
50%
50%
UP
DOWN
60%
40%
UP
DOWN
If Technical Analysis is valid, we need to
1. Identify patterns
2. Predict from patterns
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Binary Strings: Binary Coding
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Optimal Coding of Price History
• Many possible approaches (PCA, wavelet analysis, time based, move based)
• Move based: define set of levels, and each time a level different from the last one is
hit, record it
96
98
100
102
spot pricetime based
move based
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Digitizing the Price History
• Information compression: condenses the recent past into a sequence of binary digits.
1 0 1 0 1 1 0 0 0 1 0 180
90
100
110
120 – up −→ 1
– down −→ 0
Sequence:
1 0 1 0 1 1 0 0 0 1 0 1
• Keeps track of sequences of levels that have been reached.
• Extracts the relevant moves regardless of the precise timing.
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Move Based Coding Properties
• Theory of processes approximation: move-based coding
– Captures relevant price movements
– Filters out irrelevant periods
– Linked to optimal ∆ hedging
– Good volatility estimate
– Similar to Renko-Kagi coding
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Deciphering the Financial Genome
• Akin to finding genes in a DNA sequence
. . . GAATCGACTTGAGGCTACG . . .
Gene coding for a protein
. . . 0010111011101100100001110010 . . .
Pattern predicting a up
• In both cases, extracting meaning from noise
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Binary Strings: Algorithm
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Cracking the code
• Identify past strings similar to the current one
• Average their following bit (up/down) to get a trading signal
• Convert it into a trading rule
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Distance between Strings
old ————>new
A= 0 1 1 0 0 1
B= 1 1 1 0 0 1
C= 0 1 1 0 0 0
• A: current string
• B and C differ from A by one bit
• B is closer to A than C is because most recent bits match
• Distance between strings X and Y
d(X, Y ) =
n∑
i=1
(xi − yi)2αn−i
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Voting Scheme
• The bit to come is “voted” by each of the past strings.
• Each past string Xi is followed by bit ǫi
Xi︷ ︸︸ ︷0 1 1 0 0 1
ǫi︷︸︸︷
1
• Each past string Xi votes ǫi with a weight that depends on di = distance(xi, A)
• The result is the estimated probability of an up-move given by:
p =
∑e−λdiǫi
∑e−λdi
(most similar strings have highest weights)
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Trading strategy
The indicator p (probability of an up move) defines the buy/sell rule
• If p < L :
sell
• If p > H :
buy
0 100 200 300 400 500 600 700 800 900 10000.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
BUY
SELL
H
L
Filters out L < p < H
Exit rule: Unwind each position when next rung (up or down) is reached.
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Binary Strings Predictive Power
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Conditional Trees
Extra information regarding
strategy results can be com-
puted, such as a conditional
tree into the future as seen
today
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Most predictive patterns
Each past sequence
votes for the next bit in
the current sequence.
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Bloomberg Implementation
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Input Parameters
• period: defines the trading period
• delta: defines the levels. Levels are delta/100 ∗ S apart from each other.
• buy threshold, sell threshold(H and P): if p > H, buy, p < L, sell
• position size: number of shares traded at each signal
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Outputs
• Initial time series with entry points
• Probability of up move
• Net exposure
• P&L comparison of Binary Strings strategy and Buy&Hold
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Binary Strings Summary
• Binary Strings scrutinizes historical data in search of violations of Efficient Market
Hypothesis
• It is based on a very compact encoding
• It makes use of non parametric estimation
• It is opportunistic, not dogmatic: it does not favor trend following nor range trading
per se; it just exploits the data optimally
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Conclusion
• How to compress information depends on the objective
• Superiority of move based over time based
• Better delta hedging
• Good way to compact information for technical Analysis
• Better volatility estimates
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