Post on 18-Jan-2021
Algorithmic market making for options
Olivier Gueant (Universite Paris 1 Pantheon-Sorbonne)
joint work with Bastien Baldacci (Ecole Polytechnique) and Philippe Bergault
(Universite Paris 1 Pantheon-Sorbonne)
London, February 20th, 2020
1
Introduction
Market Making
What is a market maker?
• A market maker is a liquidity provider. He / she provides bid and
ask prices for a list of assets to other market participants.
• Today, market makers are often replaced by market making
algorithms.
A market maker faces a complex optimization problem
• Makes money out of buying low and selling high (bid-ask spread).
• Faces the risk that the price moves adversely without him/her being
able to unwind his position rapidly enough.
2
Market Making
What is a market maker?
• A market maker is a liquidity provider. He / she provides bid and
ask prices for a list of assets to other market participants.
• Today, market makers are often replaced by market making
algorithms.
A market maker faces a complex optimization problem
• Makes money out of buying low and selling high (bid-ask spread).
• Faces the risk that the price moves adversely without him/her being
able to unwind his position rapidly enough.
2
Literature
Literature: a bit of history
• Ho and Stoll (1981)
• Grossman and Miller (1988)
New interest 20 years later
• Avellaneda and Stoikov. High-frequency trading in a limit order
book. Quantitative Finance, 2008.
3
Literature
Literature: a bit of history
• Ho and Stoll (1981)
• Grossman and Miller (1988)
New interest 20 years later
• Avellaneda and Stoikov. High-frequency trading in a limit order
book. Quantitative Finance, 2008.
3
Avellaneda-Stoikov: a first model
Avellaneda-Stoikov modelling framework
• One asset with reference price process (mid-price) (St)t :
dSt = σdWt .
• Bid and ask prices of the MM denoted respectively
Sbt = St − δbt and Sa
t = St + δat .
• Point processes Nb and Na (indep. of W ) for the transactions (size
z = 1). Inventory (qt)t :
dqt = zdNbt − zdNa
t .
4
Avellaneda-Stoikov modelling framework
• One asset with reference price process (mid-price) (St)t :
dSt = σdWt .
• Bid and ask prices of the MM denoted respectively
Sbt = St − δbt and Sa
t = St + δat .
• Point processes Nb and Na (indep. of W ) for the transactions (size
z = 1). Inventory (qt)t :
dqt = zdNbt − zdNa
t .
4
Avellaneda-Stoikov modelling framework
• One asset with reference price process (mid-price) (St)t :
dSt = σdWt .
• Bid and ask prices of the MM denoted respectively
Sbt = St − δbt and Sa
t = St + δat .
• Point processes Nb and Na (indep. of W ) for the transactions (size
z = 1). Inventory (qt)t :
dqt = zdNbt − zdNa
t .
4
Avellaneda-Stoikov modelling framework (continued)
• The intensities of Nb and Na depend on the distance to the
reference price:
λbt = Λb(δbt ) and λat = Λa(δat ).
Λb, Λa decreasing. Avellaneda and Stoikov suggested
Λb(δ) = Λa(δ) = Ae−kδ.
• Cash process (Xt)t :
dXt = zSat dN
at − zSb
t dNbt = −Stdqt + δat zdN
at + δbt zdN
bt .
Three state variables: X (cash), q (inventory), and S (price).
5
Avellaneda-Stoikov modelling framework (continued)
• The intensities of Nb and Na depend on the distance to the
reference price:
λbt = Λb(δbt ) and λat = Λa(δat ).
Λb, Λa decreasing. Avellaneda and Stoikov suggested
Λb(δ) = Λa(δ) = Ae−kδ.
• Cash process (Xt)t :
dXt = zSat dN
at − zSb
t dNbt = −Stdqt + δat zdN
at + δbt zdN
bt .
Three state variables: X (cash), q (inventory), and S (price).
5
Avellaneda-Stoikov modelling framework (continued)
• The intensities of Nb and Na depend on the distance to the
reference price:
λbt = Λb(δbt ) and λat = Λa(δat ).
Λb, Λa decreasing. Avellaneda and Stoikov suggested
Λb(δ) = Λa(δ) = Ae−kδ.
• Cash process (Xt)t :
dXt = zSat dN
at − zSb
t dNbt = −Stdqt + δat zdN
at + δbt zdN
bt .
Three state variables: X (cash), q (inventory), and S (price).
5
Avellaneda-Stoikov objective function and HJB equation
CARA objective function
sup(δat )t ,(δbt )t∈A
E [− exp (−γ(XT + qTST ))] ,
where γ is the absolute risk aversion parameter, T a time horizon, and Athe set of predictable processes bounded from below.
An (a priori) awful Hamilton-Jacobi-Bellman
(HJB) 0 = ∂tu(t, x , q,S) +1
2σ2∂2
SSu(t, x , q,S)
+ supδb
Λb(δb)[u(t, x − zS + zδb, q + z ,S)− u(t, x , q,S)
]+ sup
δaΛa(δa) [u(t, x + zS + zδa, q − z ,S)− u(t, x , q,S)]
with final condition:
u(T , x , q,S) = − exp (−γ(x + qS)) .
6
Avellaneda-Stoikov objective function and HJB equation
CARA objective function
sup(δat )t ,(δbt )t∈A
E [− exp (−γ(XT + qTST ))] ,
where γ is the absolute risk aversion parameter, T a time horizon, and Athe set of predictable processes bounded from below.
An (a priori) awful Hamilton-Jacobi-Bellman
(HJB) 0 = ∂tu(t, x , q,S) +1
2σ2∂2
SSu(t, x , q,S)
+ supδb
Λb(δb)[u(t, x − zS + zδb, q + z ,S)− u(t, x , q,S)
]+ sup
δaΛa(δa) [u(t, x + zS + zδa, q − z ,S)− u(t, x , q,S)]
with final condition:
u(T , x , q,S) = − exp (−γ(x + qS)) .
6
A rigorous analysis
Solution of the Avellaneda-Stoikov model
• Gueant, Lehalle, and Fernandez-Tapia. Dealing with the Inventory
Risk: A solution to the market making problem. MAFE, 2013.
When risk limits are set, solving the AS model with exponential intensities
boils down to solving a system of linear ordinary differential equations!
7
A rigorous analysis
Solution of the Avellaneda-Stoikov model
• Gueant, Lehalle, and Fernandez-Tapia. Dealing with the Inventory
Risk: A solution to the market making problem. MAFE, 2013.
When risk limits are set, solving the AS model with exponential intensities
boils down to solving a system of linear ordinary differential equations!
7
Market making: an interesting
research strand
Literature
Many extensions of the initial one-asset model
• Multi-asset framework.
• General intensities (e.g. logistic).
• Variable RFQ sizes.
• Different objective functions (mean-variance-like criterion).
• Client tiering.
• Adverse selection.
• Drift / signal / alpha.
• Access to liquidity pools (exchange / IDB - for some asset classes).
• Market and limit orders (not relevant for all asset classes).
• ...
8
Literature
Papers by Cartea, Jaimungal et al.
• Cartea, Jaimungal, and Ricci. Buy low, sell high: A high frequency
trading perspective. SIAM Journal on Financial Mathematics, 2014.
• Cartea, Donnelly, and Jaimungal. Algorithmic trading with model
uncertainty. SIAM Journal on Financial Mathematics, 2017.
9
Literature
Figure 1: A nice book dealing with market making
10
Literature
Papers by Guilbaud and Pham
• Guilbaud and Pham. Optimal High-Frequency Trading with limit
and market orders. Quantitative Finance, 2013.
• Guilbaud and Pham. Optimal High-Frequency Trading in a Pro-Rata
Microstructure with Predictive Information. Mathematical Finance,
2015.
11
Multi-asset market making
Extensions to multi-asset portfolios
• Gueant. The Financial
Mathematics of Market Liquidity.
From Optimal Execution to Market
Making. CRC Press, 2016.
• Gueant. Optimal market making.
Applied Mathematical Finance,
2017.
Figure 2: Another nice book
12
Multi-asset market making
The problem
The number of equations to solve typically grows exponentially with the
number of assets.
Attempts to solve it
• Dimensionality reduction with risk factors: Bergault and Gueant,
Size matters for OTC market makers: general results and
dimensionality reduction techniques, 2020.
• Reinforcement learning techniques to go beyond the curse of
dimensionality: Gueant and Manziuk, Deep reinforcement learning
for market making in corporate bonds: beating the curse of
dimensionality, Applied Mathematical Finance, 2019.
• Other works in progress.
Our paper on options is inspired by the first approach.
13
Multi-asset market making
The problem
The number of equations to solve typically grows exponentially with the
number of assets.
Attempts to solve it
• Dimensionality reduction with risk factors: Bergault and Gueant,
Size matters for OTC market makers: general results and
dimensionality reduction techniques, 2020.
• Reinforcement learning techniques to go beyond the curse of
dimensionality: Gueant and Manziuk, Deep reinforcement learning
for market making in corporate bonds: beating the curse of
dimensionality, Applied Mathematical Finance, 2019.
• Other works in progress.
Our paper on options is inspired by the first approach.
13
Multi-asset market making
The problem
The number of equations to solve typically grows exponentially with the
number of assets.
Attempts to solve it
• Dimensionality reduction with risk factors: Bergault and Gueant,
Size matters for OTC market makers: general results and
dimensionality reduction techniques, 2020.
• Reinforcement learning techniques to go beyond the curse of
dimensionality: Gueant and Manziuk, Deep reinforcement learning
for market making in corporate bonds: beating the curse of
dimensionality, Applied Mathematical Finance, 2019.
• Other works in progress.
Our paper on options is inspired by the first approach.
13
Multi-asset market making
The problem
The number of equations to solve typically grows exponentially with the
number of assets.
Attempts to solve it
• Dimensionality reduction with risk factors: Bergault and Gueant,
Size matters for OTC market makers: general results and
dimensionality reduction techniques, 2020.
• Reinforcement learning techniques to go beyond the curse of
dimensionality: Gueant and Manziuk, Deep reinforcement learning
for market making in corporate bonds: beating the curse of
dimensionality, Applied Mathematical Finance, 2019.
• Other works in progress.
Our paper on options is inspired by the first approach.
13
Multi-asset market making
The problem
The number of equations to solve typically grows exponentially with the
number of assets.
Attempts to solve it
• Dimensionality reduction with risk factors: Bergault and Gueant,
Size matters for OTC market makers: general results and
dimensionality reduction techniques, 2020.
• Reinforcement learning techniques to go beyond the curse of
dimensionality: Gueant and Manziuk, Deep reinforcement learning
for market making in corporate bonds: beating the curse of
dimensionality, Applied Mathematical Finance, 2019.
• Other works in progress.
Our paper on options is inspired by the first approach.
13
Multi-asset market making
The problem
The number of equations to solve typically grows exponentially with the
number of assets.
Attempts to solve it
• Dimensionality reduction with risk factors: Bergault and Gueant,
Size matters for OTC market makers: general results and
dimensionality reduction techniques, 2020.
• Reinforcement learning techniques to go beyond the curse of
dimensionality: Gueant and Manziuk, Deep reinforcement learning
for market making in corporate bonds: beating the curse of
dimensionality, Applied Mathematical Finance, 2019.
• Other works in progress.
Our paper on options is inspired by the first approach.
13
Asset classes
Extensions to derivatives
• Stoikov and Saglam. Option market making under inventory risk,
Review of Derivatives Research, 2009.
• Abergel and El Aoud. A stochastic control approach to option
market making. Market Microstructure and Liquidity, 2015.
• Baldacci, Bergault and Gueant. Algorithmic market making
for options. 2020. (On ArXiv, in revision)
Relevant models should handle several option contracts and tackle the
question of ∆-hedging / trading in the underlying asset.
14
Asset classes
Extensions to derivatives
• Stoikov and Saglam. Option market making under inventory risk,
Review of Derivatives Research, 2009.
• Abergel and El Aoud. A stochastic control approach to option
market making. Market Microstructure and Liquidity, 2015.
• Baldacci, Bergault and Gueant. Algorithmic market making
for options. 2020. (On ArXiv, in revision)
Relevant models should handle several option contracts and tackle the
question of ∆-hedging / trading in the underlying asset.
14
Option market making: the
model
The market
Asset price dynamics under P{dSt = µStdt +
√νtStdW
St
dνt = aP(t, νt)dt + ξ√νtdW
νt .
Asset price dynamics under Q (pricing measure) - r = 0{dSt =
√νtStdW
St
dνt = aQ(t, νt)dt + ξ√νtdW
νt .
Another one-factor model can be chosen (e.g. Bergomi). Two-factor
models are also possible: they increase the dimensionality of the problem
by 1.
15
The market
Asset price dynamics under P{dSt = µStdt +
√νtStdW
St
dνt = aP(t, νt)dt + ξ√νtdW
νt .
Asset price dynamics under Q (pricing measure) - r = 0{dSt =
√νtStdW
St
dνt = aQ(t, νt)dt + ξ√νtdW
νt .
Another one-factor model can be chosen (e.g. Bergomi). Two-factor
models are also possible: they increase the dimensionality of the problem
by 1.
15
The market
Asset price dynamics under P{dSt = µStdt +
√νtStdW
St
dνt = aP(t, νt)dt + ξ√νtdW
νt .
Asset price dynamics under Q (pricing measure) - r = 0{dSt =
√νtStdW
St
dνt = aQ(t, νt)dt + ξ√νtdW
νt .
Another one-factor model can be chosen (e.g. Bergomi). Two-factor
models are also possible: they increase the dimensionality of the problem
by 1.
15
The market
The options
• We consider N ≥ 1 European options written on the above asset.
• For i = 1, . . . ,N:
• Maturity of the i-th option: T i
• Price process of the i-th option: (Oit)t∈[0,T i ]
Partial differential equation
Oit = O i (t,St , νt) where
0 = ∂tOi (t,S , ν) + aQ(t, ν)∂νO
i (t,S , ν) +1
2νS2∂2
SSOi (t,S , ν)
+ ρξνS∂2νSO
i (t,S , ν) +1
2ξ2ν∂2
ννOi (t,S , ν).
16
The market
The options
• We consider N ≥ 1 European options written on the above asset.
• For i = 1, . . . ,N:
• Maturity of the i-th option: T i
• Price process of the i-th option: (Oit)t∈[0,T i ]
Partial differential equation
Oit = O i (t,St , νt) where
0 = ∂tOi (t,S , ν) + aQ(t, ν)∂νO
i (t,S , ν) +1
2νS2∂2
SSOi (t,S , ν)
+ ρξνS∂2νSO
i (t,S , ν) +1
2ξ2ν∂2
ννOi (t,S , ν).
16
Requests
Distribution of requests
• Requests on option i arrive with intensities λi,brequest and λi,arequest.
• Request sizes for option i are distributed according to probability
measures µi,b(dz) and µi,a(dz).
• Bid and ask prices answered for option i (transaction of size z) if the
transaction does not violate risk limits:
Oit − δ
i,bt (z) and Oi
t + δi,at (z).
• Probabilities of trading:
f i,b(δi,bt (z)) and f i,a(δi,at (z)).
17
Inventory in the options
Resulting dynamics of the inventory process
dqit =
∫R∗+
zN i,b(dt, dz)−∫R∗+
zN i,a(dt, dz),
where N i,b and N i,a are marked point processes with kernels:
ν i,bt (dz) = λi,brequestfi,b(δi,bt (z))︸ ︷︷ ︸
Λi,b(δi,bt (z))
1{qt−+ze i∈Q}µi,b(dz),
ν i,at (dz) = λi,arequestfi,a(δi,at (z))︸ ︷︷ ︸
Λi,a(δi,at (z))
1{qt−−ze i∈Q}µi,a(dz).
18
Inventory in the underlying asset
∆-hedging
The market maker ensures perfect ∆-hedging where
∆t =N∑i=1
∂SOi (t,St , νt)qit .
Continuous trading is our real assumption:
• The assumption of perfect ∆-hedging can in fact be relaxed.
• One can hedge part of the vega by trading the underlying asset.
→ See the appendix of our paper on ArXiv.
19
Inventory in the underlying asset
∆-hedging
The market maker ensures perfect ∆-hedging where
∆t =N∑i=1
∂SOi (t,St , νt)qit .
Continuous trading is our real assumption:
• The assumption of perfect ∆-hedging can in fact be relaxed.
• One can hedge part of the vega by trading the underlying asset.
→ See the appendix of our paper on ArXiv.
19
Cash dynamics and Mark-to-Market value of the portfolio
Cash dynamics
dXt =N∑i=1
(∫R∗+
z(δi,bt (z)N i,b(dt, dz) + δi,at (z)N i,a(dt, dz)
)−Oi
tdqit
)+Std∆t + d
⟨∆,S
⟩t.
Mark-to-Market value of the portfolio
Vt = Xt −∆tSt +N∑i=1
qitOit .
20
Cash dynamics and Mark-to-Market value of the portfolio
Cash dynamics
dXt =N∑i=1
(∫R∗+
z(δi,bt (z)N i,b(dt, dz) + δi,at (z)N i,a(dt, dz)
)−Oi
tdqit
)+Std∆t + d
⟨∆,S
⟩t.
Mark-to-Market value of the portfolio
Vt = Xt −∆tSt +N∑i=1
qitOit .
20
Dynamics of the Mark-to-Market value of the portfolio
Dynamics of the MtM value
dVt =N∑i=1
∫R∗+
z(δi,bt (z)N i,b(dt, dz) + δi,at (z)N i,a(dt, dz)
)+
N∑i=1
qitdOit −∆tdSt
=N∑i=1
∫R∗+
z(δi,at (z)N i,a(dt, dz) + δi,bt (z)N i,b(dt, dz)
)+
N∑i=1
qit∂νOi (t,St , νt)
(aP(t, νt)− aQ(t, νt)
)dt
+√νtξq
it∂νO
i (t,St , νt)dWνt .
21
Dynamics of the Mark-to-Market value of the portfolio
Dynamics of the MtM value
dVt =N∑i=1
∫R∗+
z(δi,bt (z)N i,b(dt, dz) + δi,at (z)N i,a(dt, dz)
)+
N∑i=1
qitdOit −∆tdSt
=N∑i=1
∫R∗+
z(δi,at (z)N i,a(dt, dz) + δi,bt (z)N i,b(dt, dz)
)+
N∑i=1
qit∂νOi (t,St , νt)
(aP(t, νt)− aQ(t, νt)
)dt
+√νtξq
it∂νO
i (t,St , νt)dWνt .
21
Introducing vegas
Vega of the i-th option
V it := ∂√νO
i (t,St , νt) = 2√νt∂νO
i (t,St , νt).
Simplified dynamics of the MtM value
dVt =N∑i=1
∫R∗+
z(δi,at (z)N i,a(dt, dz) + δi,bt (z)N i,b(dt, dz)
)+
N∑i=1
qitV it
aP(t, νt)− aQ(t, νt)
2√νt
dt +N∑i=1
ξ
2qitV i
tdWνt .
22
Introducing vegas
Vega of the i-th option
V it := ∂√νO
i (t,St , νt) = 2√νt∂νO
i (t,St , νt).
Simplified dynamics of the MtM value
dVt =N∑i=1
∫R∗+
z(δi,at (z)N i,a(dt, dz) + δi,bt (z)N i,b(dt, dz)
)+
N∑i=1
qitV it
aP(t, νt)− aQ(t, νt)
2√νt
dt +N∑i=1
ξ
2qitV i
tdWνt .
22
Option market making:
optimization problem,
assumptions, and approximations
Objective function
Objective function: risk-adjusted expectation
supδ∈A
E
VT −γ
2
∫ T
0
(N∑i=1
ξ
2qitV i
t
)2
dt
.for γ a risk aversion parameter and T a time horizon such that
T < mini Ti .
supδ∈A
E
∫ T
0
N∑i=1
∑j=a,b
∫R∗+
zδi,jt (z)Λi,j(δi,jt (z))1{qt−±jze i∈Q}µi,j(dz)
+ qitV i
t
aP(t, νt)− aQ(t, νt)
2√νt
dt − γξ2
8
∫ T
0
(N∑i=1
qitV it
)2
dt
,where ±b = + and ±a = −.
23
Objective function
Objective function: risk-adjusted expectation
supδ∈A
E
VT −γ
2
∫ T
0
(N∑i=1
ξ
2qitV i
t
)2
dt
.for γ a risk aversion parameter and T a time horizon such that
T < mini Ti .
supδ∈A
E
∫ T
0
N∑i=1
∑j=a,b
∫R∗+
zδi,jt (z)Λi,j(δi,jt (z))1{qt−±jze i∈Q}µi,j(dz)
+ qitV i
t
aP(t, νt)− aQ(t, νt)
2√νt
dt − γξ2
8
∫ T
0
(N∑i=1
qitV it
)2
dt
,where ±b = + and ±a = −.
23
Value function
Value function
u : (t,S , ν, q) ∈ [0,T ]× R+2
×Q 7→ u(t,S , ν, q)
given by
u(t, S , ν, q) = sup(δs )s∈[t,T ]∈At
E(t,S,ν,q)[∫ T
t
N∑i=1
((∑j=a,b
∫R∗+
zδi,js (z)Λi,j(δi,js (z))1{qs−±j zei∈Q}µ
i,j(dz))
+qisV i
saP(s, νs)− aQ(s, νs)
2√νs
)ds − γξ2
8
∫ T
t
( N∑i=1
qisV i
s
)2
ds
]
The problem is written in (space) dimension N + 2: it is a priori
untractable!
24
Value function
Value function
u : (t,S , ν, q) ∈ [0,T ]× R+2
×Q 7→ u(t,S , ν, q)
given by
u(t, S , ν, q) = sup(δs )s∈[t,T ]∈At
E(t,S,ν,q)[∫ T
t
N∑i=1
((∑j=a,b
∫R∗+
zδi,js (z)Λi,j(δi,js (z))1{qs−±j zei∈Q}µ
i,j(dz))
+qisV i
saP(s, νs)− aQ(s, νs)
2√νs
)ds − γξ2
8
∫ T
t
( N∑i=1
qisV i
s
)2
ds
]
The problem is written in (space) dimension N + 2: it is a priori
untractable!
24
Beating the curse of dimensionality
Assumption 1
We approximate the vega of each option over [0,T ] by its value at time
t = 0, namely V it = V i
0 =: V i , i = 1, . . . ,N.
Assumption 2
Authorized inventories correspond to vega risk limits:
Q =
{q ∈ RN
∣∣∣ N∑i=1
qiV i ∈ [−V,V]
}, with V ∈ R+∗.
25
Beating the curse of dimensionality
Assumption 1
We approximate the vega of each option over [0,T ] by its value at time
t = 0, namely V it = V i
0 =: V i , i = 1, . . . ,N.
Assumption 2
Authorized inventories correspond to vega risk limits:
Q =
{q ∈ RN
∣∣∣ N∑i=1
qiV i ∈ [−V,V]
}, with V ∈ R+∗.
25
Change of variables
Portfolio vega
Vπt :=N∑i=1
qitV i .
Optimal control problem
v (t, ν,Vπ) = sup(δs )s∈[t,T ]∈At
E(t,ν,Vπ)∫ T
t
N∑i=1
∑j=a,b
∫R∗+
zδi,js (z)Λi,j(δi,js (z))1|Vπs ±j zV i |≤Vµi,j(dz)
+Vπs
aP(s, νs)− aQ(s, νs)
2√νs
− γξ2
8Vπs 2
ds
.
26
Change of variables
Portfolio vega
Vπt :=N∑i=1
qitV i .
Optimal control problem
v (t, ν,Vπ) = sup(δs )s∈[t,T ]∈At
E(t,ν,Vπ)∫ T
t
N∑i=1
∑j=a,b
∫R∗+
zδi,js (z)Λi,j(δi,js (z))1|Vπs ±j zV i |≤Vµi,j(dz)
+Vπs
aP(s, νs)− aQ(s, νs)
2√νs
− γξ2
8Vπs 2
ds
.
26
Low-dimensional HJB equation
Low-dimensional HJB equation
The associated Hamilton-Jacobi-Bellman equation is:
0 = ∂tv(t, ν,Vπ) + aP(t, ν)∂νv(t, ν,Vπ) +1
2νξ2∂2
ννv(t, ν,Vπ)
+Vπ aP(t, ν)− aQ(t, ν)
2√ν
− γξ2
8Vπ2
+N∑i=1
∑j=a,b
∫R∗+
z1|Vπ±j zV i |≤VHi,j
(v(t, ν,Vπ
)− v(t, ν,Vπ ±j zV i
)z
)µi,j(dz),
with final condition v(T , ν,Vπ) = 0, where
H i,j(p) := supδi,j≥δ∞
Λi,j(δi,j)(δi,j − p).
This equation in (space) dimension 2 can be solved numerically on a grid
with a Euler scheme and linear interpolation.
27
Low-dimensional HJB equation
Low-dimensional HJB equation
The associated Hamilton-Jacobi-Bellman equation is:
0 = ∂tv(t, ν,Vπ) + aP(t, ν)∂νv(t, ν,Vπ) +1
2νξ2∂2
ννv(t, ν,Vπ)
+Vπ aP(t, ν)− aQ(t, ν)
2√ν
− γξ2
8Vπ2
+N∑i=1
∑j=a,b
∫R∗+
z1|Vπ±j zV i |≤VHi,j
(v(t, ν,Vπ
)− v(t, ν,Vπ ±j zV i
)z
)µi,j(dz),
with final condition v(T , ν,Vπ) = 0, where
H i,j(p) := supδi,j≥δ∞
Λi,j(δi,j)(δi,j − p).
This equation in (space) dimension 2 can be solved numerically on a grid
with a Euler scheme and linear interpolation.
27
Optimal quotes
Once the value function is known, the optimal mid-to-bid and ask-to-mid
associated with the N options, are given by the following formula:
Optimal quotes
δi,j∗t (z) = max
(δ∞,
(Λi,j)−1
(−H i,j′
(v(t, νt ,Vπt−
)− v(t, νt ,Vπt− ±j zV i
)z
))).
28
Change of variables when aP = aQ
If aP = aQ, then v(t, ν,Vπ) = w(t,Vπ) where w is solution of the
simpler Hamilton-Jacobi-Bellman:
0 = ∂tw(t,Vπ)− γξ2
8Vπ2
+N∑i=1
∑j=a,b
∫R∗+
z1|Vπ±j zV i |≤VHi,j
(w(t,Vπ
)− w
(t,Vπ ±j zV i
)z
)µi,j(dz),
with final condition w(T ,Vπ) = 0.
The problem is now in (space) dimension 1.
29
Change of variables when aP = aQ
If aP = aQ, then v(t, ν,Vπ) = w(t,Vπ) where w is solution of the
simpler Hamilton-Jacobi-Bellman:
0 = ∂tw(t,Vπ)− γξ2
8Vπ2
+N∑i=1
∑j=a,b
∫R∗+
z1|Vπ±j zV i |≤VHi,j
(w(t,Vπ
)− w
(t,Vπ ±j zV i
)z
)µi,j(dz),
with final condition w(T ,Vπ) = 0.
The problem is now in (space) dimension 1.
29
Numerical results
Model parameters
. Stock price at time t = 0: S0 = 10 e.
. Instantaneous variance at time t = 0: ν0 = 0.0225 year−1.
. Heston model with aP(t, ν) = κP(θP − ν) where κP = 2 year−1 and
θP = 0.04 year−1, and aQ(t, ν) = κQ(θQ − ν) where κQ = 3 year−1
and θQ = 0.0225 year−1.
. Volatility of volatility: ξ = 0.2 year−1.
. Spot-variance correlation: ρ = −0.5.
30
Options
Strikes and maturities
K = {8 e, 9 e, 10 e, 11 e, 12 e}
T = {1 year, 1.5 years, 2 years, 3 years}.
Intensities
Λi,j(δ) =λrequest
1 + eα+ β
V i δ, i ∈ {1, . . . ,N}, j = a, b.
where λrequest = 17640 = 252× 30 year−1, α = 0.7, and β = 150 year12 .
Size of transactions
z i = 5·105
Oi0
contracts: µi,b and µi,a are Dirac masses.
31
Options
Strikes and maturities
K = {8 e, 9 e, 10 e, 11 e, 12 e}
T = {1 year, 1.5 years, 2 years, 3 years}.
Intensities
Λi,j(δ) =λrequest
1 + eα+ β
V i δ, i ∈ {1, . . . ,N}, j = a, b.
where λrequest = 17640 = 252× 30 year−1, α = 0.7, and β = 150 year12 .
Size of transactions
z i = 5·105
Oi0
contracts: µi,b and µi,a are Dirac masses.
31
Options
Strikes and maturities
K = {8 e, 9 e, 10 e, 11 e, 12 e}
T = {1 year, 1.5 years, 2 years, 3 years}.
Intensities
Λi,j(δ) =λrequest
1 + eα+ β
V i δ, i ∈ {1, . . . ,N}, j = a, b.
where λrequest = 17640 = 252× 30 year−1, α = 0.7, and β = 150 year12 .
Size of transactions
z i = 5·105
Oi0
contracts: µi,b and µi,a are Dirac masses.
31
Implied volatility surface
Time to Maturity
1.001.25
1.50 1.75 2.00 2.25 2.50 2.75 3.00
Strike
8.0 8.5 9.0 9.5 10.0 10.5 11.011.5
12.0
Implied Volatility
0.140
0.145
0.150
0.155
0.160
0.1400
0.1425
0.1450
0.1475
0.1500
0.1525
0.1550
Figure 3: Implied volatility surface associated with the above parameters.
32
Value function
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
100000
120000
140000
160000
180000
200000
Valu
e fu
nctio
n
Figure 4: Value function as a function of the portfolio vega for ν = 0.0225 –
γ = 10−3 e−1, t=0, T=0.2 days.33
Convergence to stationary values
0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008Time
0.02
0.00
0.02
0.04
0.06
0.08
0.10
Optim
al b
id q
uote
Figure 5: Optimal mid-to-bid quotes as a function of time for ν = 0.0225 –
γ = 10−3 e−1.34
Optimal bid quotes
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
0.0050
0.0025
0.0000
0.0025
0.0050
0.0075
0.0100
Optim
al m
id-to
-bid
div
ided
by
price
(K i,T i) = (8.0,1.0) -- price = 20.53, vega = 4.09(K i,T i) = (8.0,1.5) -- price = 21.17, vega = 4.67(K i,T i) = (8.0,2.0) -- price = 21.80, vega = 5.09(K i,T i) = (8.0,3.0) -- price = 22.60, vega = 4.82
Figure 6: Optimal mid-to-bid quotes divided by option price for K = 8 and
ν = 0.0225 – γ = 10−3 e−1.35
Optimal bid quotes
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
0.010
0.005
0.000
0.005
0.010
0.015
0.020
0.025
Optim
al m
id-to
-bid
div
ided
by
price
(K i,T i) = (9.0,1.0) -- price = 12.12, vega = 9.04(K i,T i) = (9.0,1.5) -- price = 13.18, vega = 8.31(K i,T i) = (9.0,2.0) -- price = 14.05, vega = 7.72(K i,T i) = (9.0,3.0) -- price = 15.67, vega = 6.60
Figure 7: Optimal mid-to-bid quotes divided by option price for K = 9 and
ν = 0.0225 – γ = 10−3 e−1.36
Optimal bid quotes
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Optim
al m
id-to
-bid
div
ided
by
price
(K i,T i) = (10.0,1.0) -- price = 5.87, vega = 12.55(K i,T i) = (10.0,1.5) -- price = 7.27, vega = 10.83(K i,T i) = (10.0,2.0) -- price = 8.32, vega = 9.45(K i,T i) = (10.0,3.0) -- price = 10.24, vega = 7.64
Figure 8: Optimal mid-to-bid quotes divided by option price for K = 10 and
ν = 0.0225 – γ = 10−3 e−1.37
Optimal bid quotes
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
0.02
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Optim
al m
id-to
-bid
div
ided
by
price
(K i,T i) = (11.0,1.0) -- price = 2.18, vega = 10.47(K i,T i) = (11.0,1.5) -- price = 3.36, vega = 9.83(K i,T i) = (11.0,2.0) -- price = 4.46, vega = 9.08(K i,T i) = (11.0,3.0) -- price = 6.22, vega = 7.51
Figure 9: Optimal mid-to-bid quotes divided by option price for K = 11 and
ν = 0.0225 – γ = 10−3 e−1.38
Optimal bid quotes
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
0.00
0.05
0.10
0.15
0.20
Optim
al m
id-to
-bid
div
ided
by
price
(K i,T i) = (12.0,1.0) -- price = 0.62, vega = 5.71(K i,T i) = (12.0,1.5) -- price = 1.32, vega = 6.63(K i,T i) = (12.0,2.0) -- price = 2.07, vega = 6.86(K i,T i) = (12.0,3.0) -- price = 3.66, vega = 6.68
Figure 10: Optimal mid-to-bid quotes divided by option price for K = 12 and
ν = 0.0225 – γ = 10−3 e−1.39
Optimal bid quotes in terms of IV
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
0.94
0.95
0.96
0.97
0.98
0.99
1.00
1.01
IV o
f opt
imal
bid
quo
te d
ivid
ed b
y in
itial
IV
(K i,T i) = (8.0,1.0) -- price = 20.53, vega = 4.09, IV = 0.1597(K i,T i) = (8.0,1.5) -- price = 21.17, vega = 4.67, IV = 0.1627(K i,T i) = (8.0,2.0) -- price = 21.80, vega = 5.09, IV = 0.1623(K i,T i) = (8.0,3.0) -- price = 22.60, vega = 4.82, IV = 0.1524
Figure 11: Optimal bid quotes in terms of implied volatility for K = 8 and
ν = 0.0225 – γ = 10−3 e−1.40
Optimal bid quotes in terms of IV
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
0.95
0.96
0.97
0.98
0.99
1.00
1.01
IV o
f opt
imal
bid
quo
te d
ivid
ed b
y in
itial
IV
(K i,T i) = (9.0,1.0) -- price = 12.12, vega = 9.04, IV = 0.1534(K i,T i) = (9.0,1.5) -- price = 13.18, vega = 8.31, IV = 0.1530(K i,T i) = (9.0,2.0) -- price = 14.05, vega = 7.72, IV = 0.1512(K i,T i) = (9.0,3.0) -- price = 15.67, vega = 6.60, IV = 0.1510
Figure 12: Optimal bid quotes in terms of implied volatility for K = 9 and
ν = 0.0225 – γ = 10−3 e−1.41
Optimal bid quotes in terms of IV
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
0.94
0.95
0.96
0.97
0.98
0.99
1.00
1.01
1.02
IV o
f opt
imal
bid
quo
te d
ivid
ed b
y in
itial
IV
(K i,T i) = (10.0,1.0) -- price = 5.87, vega = 12.55, IV = 0.1472(K i,T i) = (10.0,1.5) -- price = 7.27, vega = 10.83, IV = 0.1489(K i,T i) = (10.0,2.0) -- price = 8.32, vega = 9.45, IV = 0.1478(K i,T i) = (10.0,3.0) -- price = 10.24, vega = 7.64, IV = 0.1486
Figure 13: Optimal bid quotes in terms of implied volatility for K = 10 and
ν = 0.0225 – γ = 10−3 e−1.42
Optimal bid quotes in terms of IV
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
0.94
0.95
0.96
0.97
0.98
0.99
1.00
1.01
IV o
f opt
imal
bid
quo
te d
ivid
ed b
y in
itial
IV
(K i,T i) = (11.0,1.0) -- price = 2.18, vega = 10.47, IV = 0.1406(K i,T i) = (11.0,1.5) -- price = 3.36, vega = 9.83, IV = 0.1424(K i,T i) = (11.0,2.0) -- price = 4.46, vega = 9.08, IV = 0.1446(K i,T i) = (11.0,3.0) -- price = 6.22, vega = 7.51, IV = 0.1448
Figure 14: Optimal bid quotes in terms of implied volatility for K = 11 and
ν = 0.0225 – γ = 10−3 e−1.43
Optimal bid quotes in terms of IV
1.0 0.5 0.0 0.5 1.0Portfolio vega 1e7
0.94
0.95
0.96
0.97
0.98
0.99
1.00
1.01
IV o
f opt
imal
bid
quo
te d
ivid
ed b
y in
itial
IV
(K i,T i) = (12.0,1.0) -- price = 0.62, vega = 5.71, IV = 0.1357(K i,T i) = (12.0,1.5) -- price = 1.32, vega = 6.63, IV = 0.1380(K i,T i) = (12.0,2.0) -- price = 2.07, vega = 6.86, IV = 0.1396(K i,T i) = (12.0,3.0) -- price = 3.66, vega = 6.68, IV = 0.1435
Figure 15: Optimal bid quotes in terms of implied volatility for K = 12 and
ν = 0.0225 – γ = 10−3 e−1.44
Conclusive remarks
• Option market making is tractable using one- or two-factor
stochastic volatility models.
• It is possible to go beyond the constant-vega approximation using a
Taylor expansion around that approximation
→ see the appendix of our paper.
• A model with several underlying assets can easily be written. The
feasibility of numerical approximation with grids depend on the
global number of factors.
45
Conclusive remarks
• Option market making is tractable using one- or two-factor
stochastic volatility models.
• It is possible to go beyond the constant-vega approximation using a
Taylor expansion around that approximation
→ see the appendix of our paper.
• A model with several underlying assets can easily be written. The
feasibility of numerical approximation with grids depend on the
global number of factors.
45
Conclusive remarks
• Option market making is tractable using one- or two-factor
stochastic volatility models.
• It is possible to go beyond the constant-vega approximation using a
Taylor expansion around that approximation
→ see the appendix of our paper.
• A model with several underlying assets can easily be written. The
feasibility of numerical approximation with grids depend on the
global number of factors.
45
Questions
Thanks for your attention.
Do not hesitate to make remarks and ask questions.
46