Computational Finance:

Opportunities and challenges for AD

Mike Giles
mike.giles@maths.ox.ac.uk

Oxford University Mathematical Institute

Oxford-Man Institute for Quantitative Finance

Computational finance – p. 1
 Outline
Monte Carlo methods
PDE methods
challenges with lack of differentiability
assorted comments about the finance community

Computational finance – p. 2
 Computational Finance
biggest growth area in scientific computing,
roughly 20% of Top 500 “supercomputers”
biggest employer of Oxford mathematicians and
theoretical physicists, often as “quants”
traders – make/lose the money
quants – develop the models and codes
IT – organise execution on large distributed systems,
and connect to data and external world
two main kinds of computations
Monte Carlo (60% ?)
PDE (30% ?)
sensitivities are very important, so a good application
area for AD
Computational finance – p. 3
 SDEs in Finance
In computational finance, stochastic differential equations
are used to model the behaviour of
stocks
interest rates
exchange rates
weather
electricity/gas demand
crude oil prices
...

The stochastic term accounts for the uncertainty of

unpredictable day-to-day events.
Computational finance – p. 4
 SDEs in Finance
These models are then used to calculate “fair” prices for a
huge range of financial options:
an option to sell a stock portfolio at a specific price in 2
years time
an option to buy aviation fuel at a specific price in 6
months time
an option to sell US dollars at a specific exchange rate
in 3 years time

In most cases, the buyer of the financial option is trying to

reduce their risk.

Computational finance – p. 5
 SDEs in Finance
Examples:
Geometric Brownian motion (Black-Scholes model for
stock prices)
dS = r S dt + σ S dW
Cox-Ingersoll-Ross model (interest rates)
√
dr = α(b − r) dt + σ r dW

Heston stochastic volatility model (stock prices)

√
dS = r S dt + V S dW1
2
√
dV = λ (σ −V ) dt + ξ V dW2

with correlation ρ between dW1 and dW2

Computational finance – p. 6
 SDEs in Finance
Stochastic differential equation with general drift and
volatility terms:

dS(t) = a(S, t) dt + b(S, t) dW (t)

W (t) is a Wiener variable with the properties that for any

q < r < s < t, W (t)−W (s) is Normally distributed with mean 0
and variance t−s, independent of W (r)−W (q).
In many finance applications, we want to compute the
expected value of an option dependent on the terminal state

V = E[f (S(T ))].

Note: the drift and volatility functions are almost always

differentiable, but the payoff f (S) is often not.
Computational finance – p. 7
 Payoff functions
“Call” payoff:
f (S) = max(0, S −K)

“Put” payoff:
f (S) = max(0, K −S)

Digital call:
(
1, S > K
f (S) = H(S −K) =
0, otherwise

More complex payoffs have similar elements, but may

also depend on the underlying value at a number of
intermediate dates.

Computational finance – p. 8
 Monte Carlo Estimation
Euler discretisation with timestep h:

Sbn+1 = Sbn + a(Sbn , tn ) h + b(Sbn , tn ) ∆Wn

where ∆Wn are Normal with mean 0, variance h.

Simplest Monte Carlo estimator for expected payoff is an

average of M independent path simulations:
M
X
b(i)
M −1
f (ST /h )
i=1

Key idea is very simple, but in practice it gets much more

complicated through enhancements to reduce the variance
of the estimator.
Computational finance – p. 9
 Monte Carlo Estimation
In the simplest case of Geometric Brownian motion

dS(t) = r S dt + σ S dW (t),

the SDE can be solved analytically to give

S(T ) = S0 exp (r− 12 σ 2 ) T + σ W (T ) .

In this case, we can avoid the Euler discretisation and

directly sample W (T ) to get
M
X
V ≡ E [f (S(T ))] ≈ M −1 f (S (i) )
i=1

– will use to explain approaches to calculating sensitivities

Computational finance – p. 10
 Hedging
Casinos make money consistently by having lots of
customers on different games (statistically unlikely to all
be lucky) and a large house margin (expected profit)
Banks don’t have as many customers/markets, and the
margins are smaller, so to get consistent profits they
use hedging
Basic idea: hold a portfolio of options so that the value
of the portfolio isn’t affected (to leading order) by
changes in stock prices, interest rates, etc.
This risk management is very important and affects the
financial reserves which the banks are forced to
maintain

Computational finance – p. 11
 Hedging
Suppose a portfolio has amount cj of option with payoff fj :
X
P = cj fj
j

What is the expected sensitivity to changes in interest rate?

∂ X ∂
E[P ] = cj E[fj ]
∂r ∂r
j

By choosing the cj so this is zero, the bank eliminates its

risk to unexpected interest rate changes.
To eliminate its risk to other unpredictable changes the
bank needs first (and sometimes second) derivatives of the
expected value with respect to various parameters
Computational finance – p. 12
 MC sensitivities
The sensitivities are known collectively as “the Greeks”.

∂V
Delta: first derivative w.r.t. stock price ∆ =
∂S0

∂V
Vega: first derivative w.r.t. volatility
∂σ

∂ 2V
Gamma: second derivative w.r.t. stock price Γ =
∂S02
– usually don’t need as much accuracy for second
derivatives

Computational finance – p. 13
 MC sensitivities
Simplest approach is to use a finite difference
approximation,

∂V V (θ+∆θ) − V (θ−∆θ)
≈ ,
∂θ 2 ∆θ

∂ 2V V (θ+∆θ) − 2 V (θ) + V (θ−∆θ)

2
≈ 2
.
∂θ (∆θ)
– very simple, but expensive and inaccurate if ∆θ is too big
or too small

However, this is what is currently used by many (most?)

banks because of its simplicity!

Computational finance – p. 14
 MC sensitivities
For simple cases where we know the terminal probability
distribution and so
Z
V ≡ E [f (S(θ; T ))] = f (S) pS (θ; S) dS,

we can differentiate this to get

Z Z  
∂V ∂pS ∂(log pS ) ∂(log pS )
= f dS = f pS dS = E f .
∂θ ∂θ ∂θ ∂θ

This is the Likelihood Ratio Method – can handle

non-differentiable payoffs, but doesn’t generalise well to
cases where we have to simulate the whole path

Computational finance – p. 15
 MC sensitivities
Alternatively, for simple Geometric Brownian Motion
Z
V ≡ E [f (S(θ; T ))] = f (S(θ; T )) pW (W ) dW,

and differentiating this gives

Z  
∂V ∂f ∂S(T ) ∂f ∂S(T )
= pW dW = E ,
∂θ ∂S ∂θ ∂S ∂θ

with ∂S(T )/∂θ being evaluated at fixed W .

This is the pathwise sensitivity approach – by considering

the limit of a sequence of regularised payoffs, it can be
proved that this is also OK for payoffs which are continuous
and piecewise differentiable
Computational finance – p. 16
 MC sensitivities
It also generalises well to cases where we have to simulate
the whole path.
Differentiating the Euler path discretisation, holding fixed
the Brownian increments, we get
b   b
∂ Sn+1 ∂a ∂b ∂ Sn ∂a ∂b
= 1+ h+ ∆Wn + h+ ∆Wn
∂θ ∂S ∂S ∂θ ∂θ ∂θ

leading to

∂ Vb X ∂f (m) ∂ Sb(m)
N
= M −1 (SbN ) .
∂θ m
∂S ∂θ

Computational finance – p. 17
 MC sensitivities
This corresponds to standard forward mode AD, with the
random number generation held fixed.

The reverse mode is obviously more efficient when one

needs multiple sensitivities.

I became known in computational finance by doing this,

introducing adjoints in “Smoking Adjoints” paper with
Paul Glasserman in Risk magazine.

Computational finance – p. 18
 MC sensitivities
Several (many?) major banks have adopted these ideas
One has licensed Tapenade
manual manual
(C −→ Fortran → Fortran −→ C !!!)
As far as I know, the others have done it by hand
C and C++ are the dominant languages, but a restricted
subset is maybe sufficient in practice
Operator overloading is great for verifying/debugging
hand-written code; and efficiency/ease-of-use tradeoff
may be attractive
MATLAB is also used increasingly for prototyping – a
reverse mode source transformation AD tool would be
very useful

Computational finance – p. 19
 MC sensitivities
Monte Carlo “engines” have two parts:
SDE path simulation
payoff evaluation

These are handled quite differently:

SDE path models change infrequently, so written by
quants in C/C++ for execution speed
want source transformation AD here (plus manual
optimisation?) for optimal performance?
payoffs change very frequently, so defined by traders
using flexible, inefficient, proprietary scripting
languages
forward mode operator overloading may be good
here, if payoff is differentiable Computational finance – p. 20
 MC sensitivities
AD can be used without problems for the path simulation
– everything is usually twice differentiable

Payoff evaluation can be a problem because of scripting

language and discontinuous payoffs.

Solutions:
use finite differences to estimate derivative ∂f /∂S
but variance = O(∆s−1 ) for digitals
use a hybrid pathwise / LRM method
Malliavin calculus (but no efficient adjoint?)
my new vibrato technique: var = O(h−1/2 ) for digitals
all of these require only payoff values, not derivatives
Computational finance – p. 21
 LIBOR Test Application
BGM model for behaviour of London InterBank
Overnight interest Rates
used to price various options which depend on current
and future interest rates
test problem performs N timesteps with a vector of
N +40 forward rates, and computes the sensitivity of a
portfolio of swaptions to changes in both initial forward
rates and volatilities
original code with hand-written sensitivities in C,
extended to C++ to use FADBAD++ (Ole Stauning);
results also for TAC++ (Michael Vossbeck) and
ADOL-C (Andreas Griewank)
involves quite a few exponentials, which are not
repeated in hand-coded forward/reverse code
Computational finance – p. 22
 LIBOR Test Application
Finite differences versus forward pathwise sensitivities:
250
finite diff delta
finite diff delta/vega
pathwise delta
200 pathwise delta/vega
relative cost

150

100

50

0
0 20 40 60 80 100
Maturity N

Computational finance – p. 23
 LIBOR Test Application
Hand-coded forward versus adjoint pathwise sensitivities:
40
forward delta
35 forward delta/vega
adjoint delta
adjoint delta/vega
30

25
relative cost

20

15

10

0
0 20 40 60 80 100
Maturity N

Computational finance – p. 24
 LIBOR Test Application

milliseconds/path for 80 sensitivities Gnu g++ Intel icc

original 0.37 0.10
hand-coded forward 0.97 0.52
hand-coded reverse 0.47 0.19
FADBAD++ forward 4.30 5.00
FADBAD++ reverse 6.20 4.86
hybrid (hand/FADBAD++) forward 1.02 0.63
hybrid (hand/FADBAD++) reverse 0.65 0.35
TAC++ forward 1.36 0.85
TAC++ reverse 1.28 0.45
ADOL-C reverse (scaled) 1.60

Computational finance – p. 25
 MC final comments
Monte Carlo is the main simulation tool in
computational finance
Application of AD to SDE path simulation is fairly
straightforward
For reverse mode, checkpoint every timestep
– storage requirements are minimal
In hand-written code, I also store the result of expensive
instructions – the rest I recompute
Challenges are in payoff evaluation:
use of scripting languages
lack of differentiability
solutions to these problems exist – see references

Computational finance – p. 26
 Financial PDEs
It can be shown that, given an SDE model for the
underlying stock, interest rate, exchange rate, etc.,
the value of a financial option satisfies a backward-time
convection-diffusion PDE.

For example, simple Geometric Brownian Motion leads to

the Black-Scholes equation for V (S, t)

Vt + r S VS + 21 σ 2 S 2 VSS = r V

to be solved backwards in time from the known payoff value.

Each separate stock price, interest rate, etc, introduces an

extra dimension in the PDE. An option based on 20 stock
prices gives a 20-dimensional PDE!
Computational finance – p. 27
 Financial PDEs
In practice, PDEs are only used for dimensions 1-3;
beyond that Monte Carlo methods are used.

Numerical approximations usually use:

simple structured grids (no FE methods)
implicit time-marching (Crank-Nicolson plus
Rannacher startup)
either ADI or preconditioned iterative solution of
implicit equations at each timestep

Also, simple options lead to linear PDEs, but there are other
more complex options with optional early exercise features
(American, Bermudan options) which lead to nonlinear
PDEs
Computational finance – p. 28
 PDE sensitivities
Sensitivities are again needed for hedging.

Application of AD is straightforward, in principle, but can

require care to avoid loss of accuracy

O(∆S m ) accuracy for option value does not imply O(∆S m )

accuracy for sensitivities

Might only get O(∆S m−1 ) accuracy for first derivative,

and O(∆S m−2 ) accuracy for second derivative

This could be a problem since usually m ≤ 2.

Computational finance – p. 29
 PDE sensitivities
An extreme example: Black-Scholes PDE with
discontinuous digital call payoff:

V (S, T ) = H(S −K)

where K is the strike. If we use a fixed S grid and take the

numerical payoff pointwise, then the accuracy in V (S, 0) is
O(∆S), but AD will give ∂V /∂K = 0.

6
U (S0 , 0)

-
K
Computational finance – p. 30
 PDE sensitivities
How do we fix this?

tn
Use a grid which moves with K : Sjn = Sj0 + (K −K0 )
T
and then differentiate w.r.t. K
( (((
((( (
6 (( ( ( (((
((( ((( (
( ((( (( ( ( (((
( (
S (((((( ((((( ((
( (((
( ((( (( ((
((( (( ( ((( (
(( ( ((( (( ((
(( ( (
(( ( ((((
((((
-
t
This can significantly increase the programming complexity,
but is not needed for all derivatives.
Computational finance – p. 31
 PDE sensitivities
Adjoints (reverse mode AD) are again very useful when
large numbers of sensitivities are needed
multiple future interest rates
“knot” values in (bi-)cubic spline function defining local
volatility function σ(S, t)

Adjoints for financial PDE applications were introduced,

I think, by Achdou & Pironneau (SIAM, 2005), carrying over
ideas from CFD

Computational finance – p. 32
 PDE sensitivities
Alex Prideaux’s PhD research:
investigation of differentiability problems
high-level NLA adjoints for ADI and other implicit
time-marching schemes
adjoints for integro-PDE problems (jump-diffusion
models) with implementations involving FFTs
modular AD for applications involving several stages:
σ knot values → spline and evaluate at grid points
→ PDE solution
second derivatives and use of AD tools

Computational finance – p. 33
 Conclusions
computational finance is a rapidly growing field,
just waking up to the benefits of adjoints and AD
language needs are C/C++ and MATLAB, but a
restricted subset of each is sufficient
there are important challenges due to lack of
differentiability, but solutions exist
for Monte Carlo applications, apply AD to path
simulation, and use other techniques for payoff
evaluation
for PDE simulations, take care to avoid loss of one
order of accuracy with each derivative

Computational finance – p. 34
 Further Reading
P. Glasserman, Monte Carlo Methods in Financial
Engineering, Springer, 2004.
M.B. Giles & P. Glasserman, “Smoking Adjoints: fast Monte
Carlo Greeks”, Risk, January 2006.
M.B. Giles, “Monte Carlo evaluation of sensitivities in
computational finance”, HERCMA conference, 2007.
Y. Achdou & O. Pironneau, Computational Methods for
Option Pricing, SIAM, 2005
A. Prideaux, PhD thesis on adjoints with financial PDEs
(due end of 2008)

people.maths.ox.ac.uk/∼gilesm/
Email: mike.giles@maths.ox.ac.uk
Computational finance – p. 35