1
Derivatives Performance Attribution
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d
C=f(S,t)
Derivatives Performance
Attribution
Mark Rubinstein
Paul Stephens Professor of Applied Investment Analysis
University of California at Berkeley
rubinste@haas.berkeley.edu
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Derivatives Performance Attribution
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Decomposition of Option Returns
d
C=f(S,t)
Two principal sources of profit from options: Ct - C
option relative mispricing at purchase (V - C)
+
subsequent underlying asset price changes (Ct - V)
-- only the first is due to the option itself, as distinct from what
would be possible from an investment in the underlying asset
-- important to distinguish between these two sources of
profit, since the second is more likely due to luck, or at best
much harder to measure
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Derivatives Performance Attribution
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d
Three Formulas
C=f(S,t)
 “true formula”: the option valuation formula based on the
actual risk-neutral stochastic process followed by the
underlying asset
 “market’s formula”: the option valuation formula used by
market participants to set market prices
 benchmark formula: the option valuation formula used in
the process of performance attribution to
(1) help determine the “true relative value” of the option
(2) decompose option mispricing profit into components
We distinguish between these formulas and their riskless return and
volatility inputs, for which there are also three estimates -- true, market,
and benchmark. Essentially, by the “formula” we mean the levels of all
the other higher moments of the risk-neutral distribution, where each
moment may possibly depend on the input riskless return and volatility.
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Derivatives Performance Attribution
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Decomposition of Option Returns
d
C=f(S,t)
option relative mispricing at purchase
volatility profit
formula profit
subsequent underlying asset price changes
asset profit
pure option profit
realized volatility cost
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Derivatives Performance Attribution
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d
[1] Asset Profit
C=f(S,t)
The net payoff of a call (realized horizon date payoff minus
purchase price) can be decomposed into 5 pieces:
Ct - C
 Payoff from an otherwise identical forward contract:
St - S(r/d)t
Positive results indicate that the investor may be good at
selecting the right underlying asset.
(In an efficient market with risk neutrality, asset profit will tend to be
zero. Thus, if it tends to be positive or negative, either this must be
compensation for risk or indicative of an inefficient asset market -- a
distinction we must leave to others.)
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Derivatives Performance Attribution
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[2] Pure Option Profit
d
C=f(S,t)
 Difference in payoff of a call and a forward on the asset:
[max(0, Std-(T-t) - Kr-(T-t)) - max(0, Sd-T - Kr-T)] - [St - S(r/d)t]
Positive results indicate that the investor was able to
achieve additional profits from using options rather than
forward contracts on the underlying asset itself, if no
consideration is given to the additional cost of the option
due to the volatility of the underlying asset.
This can also be interpreted as the profit from the option had there
been zero-volatility, minus the profit from an otherwise identical
forward contract. This component is model-free and under zerovolatility and an efficient market has zero present value.
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Derivatives Performance Attribution
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[3] Realized Volatility Cost
d
C=f(S,t)
 Portion of net payoff of option due to realized volatility:
[V - max(0, Sd-T - Kr-T)] - [Ct - max(0, Std-(T-t) - Kr-(T-t))]
This will normally be a nonnegative number since the
premium over parity of an option tends to shrink to zero as
the expiration date approaches. Indeed, at expiration (T = t):
Ct = max(0, St - K) = max(0, Std-(T-t) - Kr-(T-t))
so that the realized volatility cost becomes just
V - max(0, Sd-t - Kr-t)
This can be interpreted as today’s correct payment for the
pure option profit.
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u
d
C=f(S,t)
Derivatives Performance Attribution
Decomposition of Profit due to Fortuitous
Underlying Asset Price Changes
In summary, we have (assessed at expiration):
[1] asset profit: St - S(r/d)t
[2] pure option profit: [Ct - max(0, Sd-t - Kr-t)] - [St - S(r/d)t]
[3] realized volatility cost: V - max(0, Sd-t - Kr-t)
[1] + [2] - [3] = Ct - V
If [1] > 0, good at selecting underlying asset (or compensated for risk).
If [2] > 0, gained from use of an option in place of the forward, ignoring
the volatility cost.
If [2] - [3] > 0, unambiguous gain from use of the option, assuming it
were purchased at its “true relative value”.
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Derivatives Performance Attribution
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d
[4] Volatility Profit
C=f(S,t)
 Payoff attributed to difference between the realized (s)
and implied volatility at purchase based on benchmark
formula:
C(s) - C
Positive results indicate that the investor was clever enough
to buy options in situations where the market
underestimated the forthcoming volatility.
(Although we can know that an option is mispriced, we will not be able to
tell why it is mispriced if the benchmark formula used to calculate C(s) is
not a good approximation of the formula used by the market to set the
option price C.)
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Derivatives Performance Attribution
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d
[5] Formula Profit
C=f(S,t)
 Profit attributed to using a formula superior to the
benchmark formula, assuming realized volatility were known
in advance:
V - C(s)
Positive results indicate that the investor was clever enough
to buy options for which the benchmark formula, even with
foreknowledge of the realized volatility, undervalued the
options.
(Although we can know that an option is mispriced, we will not be able to
tell why it is mispriced if the benchmark formula used to calculate C(s) is
not a good approximation of the formula used by the market to set the
option price C.)
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Derivatives Performance Attribution
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d
Definition of “True Relative Value”
C=f(S,t)
V  C + r-t [Ct - Ct(S...St)]
where:
Ct(S…St) is the amount in an account after elapsed time t
of investing C on the purchase date in a self-financing
dynamic replicating portfolio, where the implied volatility is
used in the benchmark formula to estimate delta.
Ct - Ct(S...St) measures the extent by which the benchmark
formula with implied volatility fails to replicate the option.
 If the benchmark formula and implied volatility were
correct, then Ct = Ct(S…St) and V would equal C.
 If Ct > Ct(S…St), then the replicating strategy would not
start with enough money, so that V would be greater C.
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u
d
C=f(S,t)
Derivatives Performance Attribution
Definition of “True Relative Value”
(continued)
V = r-tE(Ct)
(assuming risk-neutrality)
 One way to approximate V is to measure r-tCt.
This is unbiased but will converge to V slowly.
 Instead use control variate C(S…St).
V becomes Ct, C becomes Ct(S…St) along same path
instead
V  r-tCt + [C - r-tCt(S…St)]
 If r-tCt > V, and Ct and Ct(S…St) are highly correlated,
Ct(S…St) > C creating an offsetting correction.
Comment: We do better, the closer the benchmark is to the true formula.
Comment: Under risk-aversion, r may be a reasonable discount rate.
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Derivatives Performance Attribution
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The Monte-Carlo Logic
d
C=f(S,t)
V  r-tCt + [C - r-tCt(S…St)]
simplifying : V*  r-tCt
and
C*  r-tCt(S…St)
Var(V) = Var(V*) + Var(C*) - 2 Cov(V*,C*)
Suppose that Var(V*) = Var(C*), then
Var(V) = 2 [Var V*] [1 - (V*,C*)]
Suppose that (V*,C*) = .9 (a fair benchmark) then
Var(V) = .2[Var(V*)]
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Derivatives Performance Attribution
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d
An Additional Benefit
C=f(S,t)
(1) V  r-tCt
vs
(2) V  r-tCt + [C - r-tCt(S…St)]
In an inefficient asset market, we hope that V will still serve
to separate volatility and formula profit from asset profit.
Definition (1) does not do this. But definition (2) does.
To see this, if S (or r) is too low, then Ct will tend to
include asset mispricing effects and be high. But Ct(S…St)
will also be high (since it requires buying the asset and
borrowing).
This will tend to offset leaving V - C
unchanged.
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Derivatives Performance Attribution
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d
Adding-Up Constraint (at expiration)
C=f(S,t)
Ct - C =
[1] St - S(r/d)t
(asset profit)
+ [2] Ct - max(0, Sd-t - Kr-t) - (St - S(r/d)t) (pure option profit)
- [3] V - max(0, Sd-t- Kr-t)
(realized volatility cost)
+ [4] C(s) - C
(volatility profit)
+ [5] V - C(s)
(formula profit)
V  C + r-t [Ct - Ct(S...St]
For correctly priced and benchmarked options:
C = E[V] (= r-tE(Ct))
E[C(s) - C] = E[V - C(s)] = 0
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Derivatives Performance Attribution
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Simulation Tests
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C=f(S,t)
 Common features of all simulations:





European call, S = K = 100, t = 60/360, d = 1.03
True annualized volatility = 20%, true annualized riskless rate = 7%
Performance evaluated on expiration date
Benchmark formula: standard binomial formula
10,000 Monte-Carlo paths
 Efficient (risk-neutral) market simulations:



“continuous” correct benchmark trading
“discrete” correct benchmark trading
wrong benchmark formula
 Inefficient (risk-neutral) option market simulations:



market makes wrong volatility forecast but uses “true formula”
market uses wrong formula but makes true volatility forecast
market uses wrong formula and wrong volatility forecast
 Inefficient (risk-averse) asset and option market simulation:

market uses wrong asset price, wrong volatility and wrong formula
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Derivatives Performance Attribution
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d
Generalized Binomial Simulation
C=f(S,t)
 Step 0: buy  shares of the underlying asset and invest
dollars in cash, where (u,d) is not known in advance
C - S
 Step 1u (up move): portfolio is then worth uS + (C - S)r  Cu; next
buy u shares and invest (Cu - uSu) dollars in cash, where (uu, du) is
not known in advance, or
 Step 1d (down move): portfolio is then worth dS + (C - S)r  Cd;
next buy d shares and invest (Cd - dSd) dollars in cash, where (ud, dd)
is not known in advance
 Step 2: depending on the sequence of
replicating portfolio will be worth either:
up-up:
uuuSu + (Cu - uSu)r
up-down:
uduSu + (Cu - uSu)r
down-up:
dudSd + (Cd - dSd)r
down-down: dddSd + (Cd - dSd)r
up and down moves, the




Cuu
Cud
Cdu
Cdd
( max[0, uuuS - K])
( max[0, uduS - K])
( max[0, dudS - K])
( max[0, dddS - K])
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Derivatives Performance Attribution
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d
C=f(S,t)
Simulation Test 1 (efficient risk-neutral market
with “continuous” and correct benchmark trading)
 true, market and benchmark formula: standard binomial
 true and market volatility/riskless rate = 20%/7%
 benchmark formula uses “continuous” trading
[1] asset profit
[2] pure option profit
[3] realized volatility cost
[4] volatility profit
[5] formula profit
-0.05
2.96
2.91
0.00
0.00
option value/price = $3.54
(8.21)
(4.37)
(0.00)
(0.00)
(0.00)
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Derivatives Performance Attribution
u
d
C=f(S,t)
Simulation Test 2 (efficient risk-neutral market
with “discrete” but correct benchmark trading)
 true, market and benchmark formula: standard binomial
 true and market volatility/riskless rate = 20%/7%
 benchmark formula uses “discrete” trading (once a day)
move every 1/2 day move every 1/8 day
[1] asset profit
-0.10 (8.15)
0.08 (8.28)
[2] pure option profit
2.98 (4.39)
2.95 (4.38)
[3] realized volatility cost 2.92 (0.25)
2.93 (0.33)
[4] volatility profit
-0.005 (0.21)
-0.011 (0.28)
[5] formula profit
0.008 (0.15)
0.013 (0.17)
option value/price = $3.54
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Derivatives Performance Attribution
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Alternative Risk-Neutral Distributions
d
C=f(S,t)
0.14
0.12
Normal
Skewed/Kurtic
0.08
0.06
0.04
0.02
standardized logarithmic returns
42%
38%
34%
30%
25%
21%
17%
13%
8%
4%
0%
-4%
-8%
-13%
-17%
-21%
-25%
-30%
-34%
-38%
0
-42%
probability
0.1
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Derivatives Performance Attribution
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Simulation Test 3 (efficient risk-neutral market
d
C=f(S,t)
with wrong benchmark formula)
 true and market formula: implied binomial tree

skewness = -.398
and
kurtosis = 4.86
 true and market volatility/riskless rate = 20%/7%
 benchmark formula: standard binomial (“continuous” trading)
[1] asset profit
[2] pure option profit
[3] realized volatility cost
[4] + [5] mispricing profit
value (V)
payoff (r-tCt)
-0.13
2.67
2.52
0.001
3.27
3.27
option value/price = $3.25
(8.07)
(4.66)
(0.26)
(0.26)
(0.26)
(5.01)
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Derivatives Performance Attribution
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Simulation Test 4 (inefficient risk-neutral option
d
C=f(S,t)
market because market uses wrong volatility)
 true, market and benchmark formula: standard binomial
 true and market riskless rate = 7%
 true volatility = 20%
market volatility = 15%/25%
market vol = 15%
[1] asset profit
-0.04
(8.12)
[2] pure option profit
2.92
(4.36)
[3] realized volatility cost
2.92
(0.32)
[4] volatility profit
0.801 (0.00)
[5] formula profit
0.007 (0.32)
market vol = 25%
-0.09
(8.30)
3.03
(4.37)
2.92
(0.30)
-0.802 (0.00)
-0.004 (0.30)
value (V)
payoff (r-tCt)
3.55
3.48
(0.33)
(5.08)
3.55
3.53
option value = $3.54
and
option price = $2.74/$4.34
(0.29)
(5.19)
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Derivatives Performance Attribution
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Simulation Test 5 (inefficient risk-neutral option
d
C=f(S,t)
market because market uses wrong formula)
 true formula: implied binomial tree

skewness = -.398
and
kurtosis = 4.86
 true and market volatility/riskless rate = 20%/7%
 benchmark and market formula: standard binomial
[1] asset profit
[2] pure option profit
[3] realized volatility cost
[4] volatility profit
[5] formula profit
value (V)
payoff (r-tCt)
-0.24
2.63
2.63
-0.000
-0.284
(7.99)
3.26
3.25
(0.33)
(4.96)
(4.64)
(0.33)
(0.53)
(0.27)
option value = $3.25 and option price = $3.54
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Derivatives Performance Attribution
u
Simulation Test 6 (inefficient risk-neutral option
d
C=f(S,t)
market because market uses wrong volatility/formula)
 true formula: implied binomial tree

skewness = -.398
and
kurtosis = 4.86
 true and market riskless rate = 7%
 true volatility = 20%
market volatility = 15%/25%
 benchmark and market formula: standard binomial
market vol = 15%
[1] asset profit
-0.13
(8.03)
[2] pure option profit
2.64
(4.75)
[3] realized volatility cost 2.62
(0.25)
[4] volatility profit
0.793 (0.54)
[5] formula profit
-0.282 (0.53)
value (V)
3.25
(0.25)
payoff (r-tCt)
3.25
(4.90)
option value = $3.25
and
market vol = 25%
-0.06
(8.19)
2.69
(4.75)
2.63
(0.54)
-0.802 (0.53)
-0.287 (0.36)
3.25
(0.54)
3.34
(5.10)
option price = $2.74/$4.34
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Derivatives Performance Attribution
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Simulation Test 7 (inefficient risk-averse market
d
C=f(S,t)
because market uses wrong asset price, volatility, formula)
 true formula: implied binomial tree

skewness = -.398
and
kurtosis = 4.86
 true volatility = 20%
market volatility = 25%
 true riskless rate = 7% market riskless rate = 5%/9%
 benchmark and market formula: standard binomial
riskless rate = 5%
[1] asset profit
0.38
(8.03)
[2] pure option profit
2.63
(4.64)
[3] realized volatility cost
2.77
(0.50)
[4] volatility profit
-0.810 (0.54)
[5] formula profit
-0.286 (0.24)
value (V)
3.09
(0.50)
payoff (r-tCt)
3.29
(4.99)
riskless rate = 9%
-0.36
(8.04)
2.69
(4.77)
2.49
(0.58)
-0.794 (0.54)
-0.281 (0.28)
3.42
(0.58)
3.22
(4.89)
option value = $3.25 ($3.07/$3.44) and option price = $4.19/$4.50
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Derivatives Performance Attribution
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Summary
d
C=f(S,t)
 Basic attribution: profit due to underlying asset price
changes vs profit due to option mispricing


robust to discrete trading and wrong benchmark formula
low standard error for option mispricing by using Monte-Carlo
analysis with dynamic replicating portfolio as control variate
 Decompose option mispricing into volatility and formula
profits


requires benchmark formula similar to market’s formula
low standard errors
 Unbiased estimate of asset profit in a risk-averse or
inefficient asset pricing market


can not distinguish between risk aversion and market inefficiency
high standard error