Investing on behalf of a firm, a trader can feign personal skill by committing fraud that with high probability remains undetected and generates small gains, but that with low probability bankrupts the firm, offsetting ostensible gains. Honesty requires enough skin in the game: if two traders with isoelastic preferences operate in continuous-time and one of them is honest, the other is honest as long as the respective fraction of capital is above an endogenous fraud threshold that depends on the trader’s preferences and skill. If both traders can cheat, they reach a Nash equilibrium in which the fraud threshold of each of them is lower than if the other one were honest. More skill, higher risk aversion, longer horizons, and greater volatility all lead to honesty on a wider range of capital allocations between the traders.

1. Background Model Solution Conclusion
Rogue Traders
Huayuan Dong1
Paolo Guasoni1,2
Eberhard Mayerhofer3
Dublin City University1
Università di Bologna2
University of Limerick3
11th
Bachelier Congress
In memoriam Mark H.A. Davis
June 14th
, 2022

2. Background Model Solution Conclusion
Rogue Traders
• Rogue trader: “a market professional who engages in unauthorized purchases or sales of
securities, commodities or derivatives, often for a financial institution’s proprietary trading
account.” (Krawiec, 2000)
• Lucky bets generate small gains, disguised as personal skill to be rewarded.
• Unlucky bets reveal unauthorized activity and may bankrupt firm.
• Operational risk: “the risk of loss resulting from inadequate or failed internal processes, people
and systems or from external events”.
• Rogue trading risk: low-frequency, high-impact. Cannot be insured.
Capital charges under Basel II and III.
• “Operational-risk is unlike market and credit risk; by assuming more of it, a financial firm
cannot expect to generate higher returns.” (Crouhy et al., 2004)

3. Background Model Solution Conclusion
Rogue Trading Episodes
Name Country Year Loss ($ Millions) Institution
Joseph Jett USA 1994 75 Kidder, Peabody & Co
Nick Leeson UK 1995 1,300 Barings Bank
Toshihide Iguchi Japan 1995 1,100 Resona Holdings
Yasuo Hamanaka Japan 1996 2,600 Sumitomo Corporation
John Rusnak USA 2002 691 Allied Irish Banks
Chen Jiulin Singapore 2005 550 China Aviation Oil
Brian Hunter USA 2006 6,600 Amaranth Advisors LLC
Matthew Taylor USA 2007 118 Goldman Sachs
Boris Picano-Nacci France 2008 980 Groupe Caisse d’Epargne
Jerome Kerviel France 2008 6,900 Societe Generale
Alexis Stenfors UK 2009 456 Merrill Lynch
Kweku Adoboli UK 2011 2,300 UBS
Bruno Iksil UK 2012 6,200 JP Morgan

4. Background Model Solution Conclusion

5. Background Model Solution Conclusion
Literature
• Relatively sparse literature.
• Krawiec (2000, 2009) on legal aspects.
Wexler (2010) on psychology.
Moodie (2009) on regulation.
• Armstrong and Brigo (2019): risk measures insufficient to deter risk-seeking behavior.
• Gwilym and Ebrahim (2013): position limits do not restrain rogue trading.
• Xu, Zhu, Pinedo (2020): preventative vs. corrective controls.
• Operational risk events models as exogenous.
• But why do they do it?

6. Background Model Solution Conclusion
This Paper
• Structural model of rogue trading.
• Traders rational and risk averse.
• No risk premium for rogue trading.
• Some rogue trading optimal.
Incentive: keep the rewards, share the bankruptcy gains.
• Too little fraud leaves money on the table.
Too much wipes out wealth.
• Social interaction: if I know you cheat, I cheat. So do you.
• Dynamic Nash equilibrium in continuous time.
• Honesty with enough skin in the game.
• A trader starts cheating when own wealth share becomes too low.

7. Background Model Solution Conclusion
Investments
• Two traders manage a firm’s capital.
• Zero safe rate for brevity. Use discounted quantities.
• Each trader i starts with some wealth xi .
• Wealth of i-th trader invested in asset class with expected return µi , volatility σi , and shocks Bi
.
dYi,x
t = µi Yi,x
t dt + σi Yi,x
t dBi
t , Yi,x
0 = xi > 0,
• This is wealth dynamics in the absence of fraud.
• F filtration generated by Brownian motions.

8. Background Model Solution Conclusion
Fraud
• Each trader can engage in varying amounts of fraud.
• Intuition: a small fraud as a coin toss that yields 1/(1 − ε) with high probability 1 − ε, while
bankrupting the firm with probability ε. Fraud: risk without return.
• How many coins to toss? At each point in time?
• Continuous-time formulation: Each trader chooses a (cumulative) fraud process Ai
from
A :=
n
A = (At )t≥0 : F-adapted, right-continuous, non-decreasing, with A0− = 0 ∈ R
o
• Total fraud (from all traders) is AS
t =
PN
k=1 Ak
t . Bankruptcy time τA = inf
t ≥ 0 : AS
t ≥ θ ,
where θ is independent, exponential random variable with unit rate.
• Similar construction to doubly-stochastic models of credit risk.
• Wealth dynamics including fraud (Ãi
t := Ai,c
t +
P
0≤s≤t
e∆Ai
s − 1
):
dYi,x
t = µi Yi,x
t dt + σi Yi,x
t dBi
t + YS
t−dÃi
t , Yi,x
0− = xi 0, 1 ≤ i ≤ N,
• At bankruptcy, the wealth of all traders is annihilated: Xi,x
t = 1{tτA}Yi,x
t .

9. Background Model Solution Conclusion
Objective
• Each trader aims at maximizing expected power utility from terminal wealth
Ui
(xi ) =
x1−γi
i
1 − γi
, with 0 γi 1.
• Individual risk aversion γi .
• 0 γi 1 guarantees that Ui
(0) = 0.
• Terminal time τ: the earlier of bankruptcy and an independent random horizon with rate λ.
• Ji
(x; Ai
, Aj
) := E
Ui
Xi,x
τ (Ai
, Aj
)
• Vi
(x; Aj
) := supAi ∈A Ji
(x; Ai
, Aj
)

10. Background Model Solution Conclusion
Honesty in Solitude
Proposition
Let τ1 be an F-measurable, a.s. finite random horizon τ1 (independent of F∞ and θ) such that
E
h
e(1−γ1)(µ1−γ1σ2
1 /2)τ1
i
∞. If there is only one trader who maximizes
E
U1
(X1,x1
τ1
)
over all strategies A1
∈ A, then a strategy A1,∗
is optimal if and only if A1,∗
t = 0 a.s. for any t ≥ 0
such that P[τ1 ≥ t] 0. In particular:
1 If τ1 is unbounded, then A1,∗
t = 0 a.s. for all t ≥ 0.
2 If τ1 ≤ T1
a.s. for some T1
0, then A1,∗
T1
−
= 0. If P[τ1 = T1
] 0, then also A1,∗
T1 = 0 a.s.
• A sole trader rewarded on all the firm’s wealth does not cheat.
• Cannot share bankruptcy losses with anyone.

11. Background Model Solution Conclusion
Well Posedness
Assumption
λ (1 − γa ∧ γb) (µa ∨ µb).
Lemma
For any i ̸= j ∈ {a, b} and x ∈ R2
+,
1 Ji
(x; Ai
, Aj
) = E
Z ∞
0
e−λt−AS
t Ui
(Yi,x
t (Ai
, Aj
))dt
.
2 For any Aj
∈ A, 0 Vi
(x; Aj
) ≤
Ui
(xa + xb)
λ − (1 − γi )(µa ∨ µb)
.
• Sufficiently high impatience required, as in consumption-investment problems.
• Condition depends on minimum risk aversion and maximum expected return.

12. Background Model Solution Conclusion
Dynamic Strategies
• Strategy: a process that depends on current wealths, price fluctuations, and past fraud.
Λ =
n
Ψ = {Ψt : t ≥ 0} such that Ψt : R2
+ × C ([0, t])
2
× D↑
([0, t]) → [0, ∞)
measurable and the path t 7→ Ψt (x, f[0,t], g[0,t]) is in D↑
([0, ∞)) ,
for all x ∈ R2
+, f ∈ C ([0, ∞))
2
, g ∈ D↑
([0, ∞))
o
,
• How can a trader see the other’s fraud?
• Cannot see directly, but can infer what the other trader would have done.
• Similar definition in terms of portfolio weights
Λ(0,1)
=
n
ψ = {ψt : t ≥ 0} where ψt : (0, 1) × C ([0, t])
2
× D↑
([0, t]) → [0, ∞)
is measurable and the path t 7→ ψt (w, f[0,t), g[0,t]) is in D↑
([0, ∞)) ,
for all w ∈ (0, 1), f ∈ C ([0, ∞))
2
, g ∈ D↑
([0, ∞))
o
.

13. Background Model Solution Conclusion
Nash Equilibrium
For any i ̸= j ∈ {a, b}, trader j responds with Ψj
·
x, B[0,·), Ai
[0,·]
∈ A to trader i’s strategy Ai
∈ A.
Definition (Nash equilibrium)
A pair (Ψa
, Ψb
) ∈ Λ2
is a Nash equilibrium if, for all i ̸= j ∈ {a, b}:
1 The pair (Aa,∗
, Ab,∗
) =
Ψa
·
x, B[0,·), Ab,∗
[0,·]
, Ψb
·
x, B[0,·), Aa,∗
[0,·]
∈ A2
;
2 Non-cooperative optimality holds, i.e.,
Ji
x; Ai
, Ψj
·
x, B[0,·), Ai
[0,·]
≤ Ji
x; Ai,∗
, Aj,∗
for any Ai
∈ A.
• No one wants to deviate.
• Not necessarily an optimal response policy.

14. Background Model Solution Conclusion
Skorokhod Reflection
• For any x = (x1, . . . , xN) ∈ RN
+ and any i ∈ {1, . . . , N}, define the wealth shares ri (x) = xi
PN
k=1 xk
.
• Let Wi,wi
t (Ai
, Aj
) = ri Yx
t (Ai
, Aj
)
for any t ≥ 0 with Wi,wi
0− (Ai
, Aj
) = ri (x) = wi .
Definition
Let i ̸= j ∈ {a, b} and mi ∈ (0, 1). The functional ψi,mi ∈ Λ(0,1)
is the solution of the (one-sided)
Skorokhod reflection problem (henceforth SPi
mi+) if for any Aj
∈ A and any wi ∈ (0, 1), there exists
a unique process Ai,∗
∈ A, given by Ai,∗
· = ψi,mi
·
wi , B[0,·), Aj
[0,·]
that satisfies
1 mi ≤ W
i,wi
t (Ai,∗
, Aj
) 1 a.s. for all t ≥ 0;
2
Z
[0,∞)
1n
W
i,wi
t (Ai,∗,Aj )mi
odAi,∗
t = 0 a.s.
• Pathwise construction.
• dAi,∗
t increases only on Wi,wi
t (Ai,∗
, Aj
) = mi .

15. Background Model Solution Conclusion
Existence of Nash Equilibrium
Theorem (Nash equilibrium)
There exists (w̃a, w̃b) in unit simplex such that the pair (Ψa,w̃a
, Ψb,w̃b ) is a Nash equilibrium and the
corresponding game values satisfy for any i ̸= j ∈ {a, b} and any x ∈ R2
+,
Vi
(x; Aj,∗
) = (xa + xb)1−γi
φi
(ri (x)) , with
φi
(w) =





ci
0(1 − w)−γi , w ∈ (0, w̃i ),
ci
1wαi (1 − w)ai + ci
2wβi (1 − w)bi + Ui
(w)
qi
w ∈ [w̃i , 1 − w̃j ),
ci
3w−γi , w ∈ [1 − w̃j , 1),
where ci
0, ci
1, ci
2, ci
3 are positive constants.
• Value functions explicit.
• Thresholds w̃a, w̃b are free boundaries, to be identified as part of the optimization problem.
• Minimal fraud to keep shares above thresholds, as in Davis and Norman (1990). No-fraud
region.

16. Background Model Solution Conclusion
Notation
ŵi :=
−αi (1 − γi )
γi − αi
,
αi :=
1
σ2
ki −
q
k2
i + 2σ2pi
, σ2
:= σ2
a + σ2
b,
pi := λ − (1 − γi ) µj −
γi σ2
j
2
!
, ki := µj − µi − γi σ2
j +
σ2
2
.
qi := λ − (1 − γi )
µi −
γi σ2
i
2
, ai := 1 − γi − αi ,
βi :=
1
σ2
ki +
q
k2
i + 2σ2pi
, bi := 1 − γi − βi ,

17. Background Model Solution Conclusion
Solo Rogue Trader
Theorem
For any i ̸= j ∈ {a, b}, if Aj
≡ 0, then the optimal fraud process for trader i is Ai,∗
= Ψi,ŵi
· x, B[t,·), 0
and the corresponding value function satisfies
Vi
(x; 0) = (xa + xb)1−γi
φ̄i
(ri (x))
for any x ∈ R2
+, where
φ̄i
(w) =
(
si
0(1 − w)−γi w ∈ (0, ŵi ),
si
1wαi (1 − w)ai + Ui
(w)
qi
w ∈ [ŵi , 1),
si
0 =
ai (1 − ŵi )γi
qi (ŵi − αi )
Ui
(ŵi ) 0 and si
1 =
1 − γi − ŵi
(1 − γi )qi (ŵi − αi )
ŵi
1 − ŵi
ai
0.
• Only one trader cheats. The other one is honest. Useful for comparison of thresholds.
• w̃i
threshold when both can cheat. ŵi
threshold when only i can cheat.

18. Background Model Solution Conclusion
Fraud vs. Skill (µb = 10%, σa = σb = 20%, γa = γb = 0.5, λ = 1/3)
0% 20% 40% 60% 80%
a
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0% 20% 40% 60% 80%
a
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
• Parasite-host relationship. The unskilled feeds off the skilled and wants to keep him alive.

19. Background Model Solution Conclusion
Fraud vs. Volatility (µa = µb = 10%, σb = 20%, γa = γb = 0.5, λ = 1/3)
20% 40% 60% 80% 100%
a
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20% 40% 60% 80% 100%
a
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
• Background risk increases propensity for fraud. Qualitatively similar to lower skill.

20. Background Model Solution Conclusion
Risk Aversion as Fraud Aversion (µa = µb = 10%, σa = σb = 20%, γb = 0.5, λ = 1/3)
0 0.2 0.4 0.6 0.8 1
a
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2 0.4 0.6 0.8
a
0
0.2
0.4
0.6
0.8
1
1.2
1.4

21. Background Model Solution Conclusion
Fraud and Patience (µa = µb = 10%, σa = σb = 20%, γb = 0.5, λ = 1/3)
5 10 15 20 25 30
1/ (Average horizon)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
5 10 15 20 25 30
1/ (Average horizon)
0
0.005
0.01
0.015
0.02
0.025
0.03
• Fraud increases and then decreases with average horizon. Present vs. future gains.

22. Background Model Solution Conclusion
Conclusion
• Structural model of rogue trading.
• Rogue trades as bets on firm’s capital.
• Appropriate gains, share losses.
• Rational, risk-averse traders engage in fraud.
• Fraud occurs when skin in the game too low.
• Tradeoff between diversification and operational risk.
• Less fraud with more skill, more risk-aversion, and long horizons.

23. Background Model Solution Conclusion
Thank You!
Questions?