optimal process approximation: application to delta hedging and

Optimal Process Approximation:

Application to Delta Hedging and Technical Analysis

Bruno DupireBloomberg L.P. NY

[email protected]

Quantitative finance: Developments, Applications and Problems, Cambridge UK July 7, 2005

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?

Quantitative finance: Developments, Applications and Problems, Cambridge UK July 7, 2005 Bruno Dupire

2

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


3

Theoretical Results


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


5

Example

To simplify the analysis, both approximations are piecewise constant, denoted X∆t and Xα

96

98

100

102

spot pricetime based

move based


6

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


7

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


8

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


9

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


10

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)


11

∆ Hedging


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


13

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


14

First Attempts

• Time based

0

Wδ t2 −δ t

Wδ t

• Move based

0

α

−α

1

0

1

τ

Wτ2−τ


15

Second Attempts

• Truncate the move based

• Follow the break-even points

Nothing beats the move based. Is move based optimal?


16

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}


17

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.


18

Daily P&L Variation


19

Tracking Error Comparison


20

Technical Analysis


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


22

Binary Strings: Binary Coding


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


24

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.


25

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


26

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


27

Binary Strings: Algorithm


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


29

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


30

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)


31

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.


32

Binary Strings Predictive Power


Conditional Trees

Extra information regarding

strategy results can be com-

puted, such as a conditional

tree into the future as seen

today


34

Most predictive patterns

Each past sequence

votes for the next bit in

the current sequence.


35

Bloomberg Implementation


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


37

Outputs

• Initial time series with entry points

• Probability of up move

• Net exposure

• P&L comparison of Binary Strings strategy and Buy&Hold


38

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


40

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


41

optimal process approximation: application to delta hedging and

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