Stochastic methods in Finance
Nikos SKANTZOS
2010
Fair price


What is the fair price of an option?
Consider a call option on an asset
Delta hedging:




Δ1
Δ2
Δ3
Δ4
Δ5
For every time interval:
Buy/Sell the asset to make
the position:
Call – Spot * nbr Assets
insensitive to variations of
the Spot
Fair price is the amount
spent during delta-hedging:
Option Price =
=Δ1+ Δ2+ Δ3 + Δ4+ Δ5
It is fair because that is
how much we spent!
Black-Scholes: the mother model



Black-Scholes based option-pricing on
no-arbitrage & delta hedging
Previously pricing was based mainly on
intuition and risk-based calculations
Fair value of securities was unknown
Black-Scholes: main ideas
Assume a Spot Dynamics

S0  S1   St  St  t    ST
The rule for updating the spot has two terms:
 Drift : the spot follows a main trend
 Vol : the spot fluctuates around the main trend
 Black-Scholes assume as update rule
St  t  St  St    t  St    Wt


 
main trend
fluctuations
 This process is a “lognormal” process:
dSt    St  dt    St  dWt
σ: size of fluctuations
μ: steepness of main trend
ΔWt: random variable (pos/neg)
 “lognormal” means that drift and fluctuations are proportional to St
Black-Scholes: assumptions

No-arbitrage  drift  = risk-free rate


Impose no-arbitrage by requiring that
expected spot = market forward
Calculations simplify if

fluctuations are normal:
Wt
is Gaussian normal of zero mean, variance ~ T

Volatility (size of fluctuations)  is assumed constant

Risk-free rate is assumed constant

No-transaction costs, underlying is liquid, etc
Black-Scholes formula

Call option = e–rT ∙ E[ max(S(T)-K,0)]
= discounted average of the call-payoff
over various realizations of final spot
C  e  r1T  S  N (d1 )  e  r2 T  K  N (d 2 )
Solution
ln
d1 
S 
1 
  r2  r1   2   T
K 
2 
 T
ln
d2 
S 
1 
  r2  r1   2   T
K 
2 
 T
Interpretation of BS formula
 r1 T
C S e
 N (d1 )

Delta of the option:
number of shares to go long
e
 r2 T
K 
N (d 2 )

probability that spot
finishes in the money
Price = value of position at maturity – value of cash-flow at maturity
How much does the portfolio value
change when spot changes?
portfolio value
S
Delta neutral position,
∂Portfolio/∂S=0
S+dS
S
 Δ=0, Delta-neutral value:
if S  S+dS then portfolio value does not change
 Vega=0, Vega-neutral value:
if σ
 σ+dσ then portfolio value does not change
 “Greeks” measure sensitivity of portfolio value
Black-Scholes vs market
Comparison with market:




BS < MtM when in/out of the money
Plug MtM in BS formula to calculate volatility smile
Inverse calculation  “implied vol”
Call on EURUSD
80000
Smile
Black-Scholes
70000
15.00%
Black-Scholes
Market
60000
Market
30000
14.00%
20000
Volatility
40000
USD cash
14.50%
50000
13.50%
10000
0
1.2000
Strike
strike
1.2500
1.3000
1.3500
1.4000
1.4500
1.5000
1.5500
13.00%
1.2000
1.2500
1.3000
1.3500
1.4000
1.4500
1.5000
1.5500
1.6000
Spot probability density

Distribution of terminal spot
(given initial spot) obtained from
PST S0   e
r2 T
 2Call mkt

K 2
Market observable
Fat tails:
Market implies that the probability that
the spot visits low-spot values is higher
than what is implied by Black-Scholes
Main causes:
•Spot dynamics is not lognormal
•Spot fluctuations (vol) are not
constant
What information does the smile give ?

It represents the price of vanillas

Take the vol at a given strike

Insert it to Black-Scholes formula

Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics

even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility


“wrong quote into the wrong formula to give the right price” (R.Rebonato)
If there was an instantaneous volatility σ(t), the BS could be interpreted as
2
 BS
T
1
2
  d



T t t
the accumulated vol
Types of smile quotes


The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes?


Sticky delta: if spot changes, implied vol of a given
“moneyness” doesn’t change
Sticky strike: if spot changes, implied vol of a
given strike doesn’t change
Moneyness: Δ=DF1·N(d1)
Spotladders: price, delta & gamma
0.4
1
0.8
0.3
3.5
3
2.5
2
1.5
1
0.5
0
0.6
0.6y
1y
0.2
0.4
0.1
0.2
spot
0
spot
0
0.8
1
1.2
1.8
0.6y
1y
spot
0.8
0.8
1
Knock-out
price
0.0045
0.004
0.0035
0.003
0.0025
0.002
0.0015
0.001
0.0005
0
1.6
1.2
1.4
1.6
1.8
0.9
1
1.1
1.2
1.3
1.4
0.03
0.02
0.01
0
-0.01 0.8
-0.02
-0.03
-0.04
-0.05
-0.06
Spot is far from barrier and far from OTM:
risk is minimum, price is maximum
0.6y
1y
spot
0.8
1
1.2
1.4
1.6
1.8
spot=1.28 strike=1.25 barrier=1.5
0.4
delta

1.4
0.6y
1y
gamma
price
0.5
1.2
Linear regime: S-K
0.2
0.9
1
spot
1.1
1.2
1.3
0.6y
1y
0.6y
1y
0
-0.2
-0.4
-0.6
Δ<0, price gets smaller
if spot increases
gamma
0.6
Vanilla
delta

Sensitivity of Delta to
spot is maximum
1 underlying is
needed to hedge
0.8
1 spot
1.2
Spotladders: vega, vanna & volga
Vanna: Sensitivity of Vega with respect to Spot
Vanilla
Volga: Sensitivity of Vega with respect to Vol
2
0.00004
0.6y
1y
0.00001
0
0.8
1
1.2
1.4
1.6
0.000004
0.000002
spot
0
vega
-0.000004
1.8
0.8
1
1.2
-2
1
1.2
0.6y
1y
1.4
1.6
1.8
spot
Knock-out

-0.000002 0.8
0
-1
spot
0.6y
1y
0.6y
1y
spot
0.8
1.5
2
1
1.5
0.5
0.6y
1y
1
0.8
-0.000006
-0.000008
-1
1.2
1.4
1.6
0.9
1
1.1
0.6y
1y
0.5
spot
0
-0.5
1
spot=1.28 strike=1.25 barrier=1.35
vanna
0.00002
1
vega
0.00003
3.5
3
2.5
2
1.5
1
0.5
0
volga
0.00005
vanna
3
0.00006
volga

1.2
1.3
0
-0.5 0.8
-1
1
1.2
spot
1.8
Simple analytic techniques: “moment matching”

Average-rate option payoff with N fixing dates
1
Asian  max 
N


Si  K ,0 

i 1

N
Basket option with two underlyings
 S1 T 

S 2 T 
Basket  max  a1
 a2
 K ,0 
S 2 t 
 S1 t 



TV pricing can be achieved quickly via “moment matching”
Mark-to-market requires correlated stochastic processes for
spots/vols (more complex)
“Moment matching”

To price Asian (average option) in TV we consider that




The spot process is lognormal
The sum of all spots is lognormal also
Note: a sum of lognormal variables is not lognormal. Therefore this
method is an approximation (but quite accurate for practical
purposes)
Central idea of moment matching


Find first and second moment of sum of lognormals:
E[Σi Si] , E[ (Σi Si)2 ],
Assume sum of lognormals is lognormal (with known moments
from previous step) and obtain a Black-Scholes formula with
appropriate drift and vol
Asian options analytics (1)

Prerequisites for the analysis: statistics of random increments
Increments of spot process have 0 mean and variance T
(time to maturity)

E[Wt]=0, E[Wt2]=t




If t1<t2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt12] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally, E[Wt1∙Wt2] = min(t1,t2)
From this and with some algebra it follows that
E[St1 ∙ St2] = S02 exp[r ∙(t1+t2) + σ2 ∙ min(t1,t2)]
Asian options analytics (2)


Asian payoff contains sum of spots
1 N
X   Si
N i 1
What are its mean (first moment) and variance?
1
E X   E 
N
 
EX


2
 1
S

i 
i 1
 N
N
 1
 E 2
N
N
1


E
S


i
N
i 1
 1
Si   S j   2

i 1
j 1
 N
N
N
N
 E S0  e

r 


N
i 1
N
1 2
2
t 
i
1
E
S

S


i
j
N2
i 1 j1
ti   N ( 0 ,1)
N


1

N
E S e
i , j 1
2
0

N
r ti
S

e
 0
i 1

r  ti  t j  2 min ( ti ,t j )
Looks complex but on the right-hand side all quantities are known and can
be easily calculated !
Therefore the first and second moment of the sum of spots can be
calculated

Asian options analytics (3)

Now assume that X follows lognormal process, with λ the (flat) vol, μ the drift
dX t    X t  dt    X t  dWt

Has solution (as in standard Black-Scholes)
X T  S0  e

   T   W
1 2
2
Take averages in above and obtain first and second moment in terms of μ,λ
EX T   S 0  e  T
  S
EX

T
2
T
2
0
e


2   12 2 T

E e
2 WT
Solving for drift and vol produces
1
E X T 
   log
T
S0
  E  X  e
2
T
 
1
E X T2

 log 2
T
E X T 
2T



Asian options analytics (4)


Since we wrote Asian payoff as max(XT-K,0)
We can quote the Black-Scholes formula

Asian  DF  e T  S 0  N (d1 )  K  N (d 2 )


With
S 
1 
ln 0     2   T
K 
2 
d1 
 T
ln
d2 

S0 
1 
    2   T
K 
2 
 T
And μ, λ are written in terms of E[X], E[X2] which we have calculated as
sums over all the fixing dates

The “averaging” reduces volatility: we expect lower price than vanilla

Basket is based on similar ideas
Smile-dynamics models
 Large number of alternative models:






Volatility becomes itself stochastic
Spot process is not lognormal
Random variables are not Gaussian
Random path has memory (“non-markovian”)
The time increment is a random variable (Levy processes)
And many many more…
 A successful model must allow quick and exact pricing of vanillas to
reproduce smile
 Wilmott: “maths is like the equipment in mountain climbing: too much
of it and you will be pulled down by its weight, too few and you won’t
make it to the top”
Dupire Local Vol


Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process, but…


Vol depends on underlying at each time and time itself
It is therefore indirectly stochastic
dSt  St    dt  St   St , t   dWt


Local vol is a time- and spot-dependent vol
(something the BS implied vol is not!)
No-arbitrage fixes drift μ to risk-free rate
Local Vol

Technology invented independently by:



B. Dupire Risk (1994) v.7 pp.18-20
E. Derman and I. Kani Fin Anal J (1996) v.53 pp.25-36
They expressed local vol in terms of market-quoted vanillas
and its time/strike derivatives
 2 St , t  

CT  r1  C  K  r2  r1   C K
2
1
K
 C KK
2
K  S t ,T t
Or, equivalently, in terms of BS implied-vols:
 2 St , t  
1
2
 BS
T  t0

 BS

 K  r2  r1   BS
T
K
2
2







1
d
d1d 2 


2
BS
BS
BS
1
1
K


2






2
2
2



K
T

t

K

K

K


K
T

t

 BS 
 BS
0
BS
0
K  S t ,T t
Dupire Local Vol

Contains derivatives of mkt quotes with respect to:
Maturity, Strike

The denominator can cause numerical problems



CKK<0 (smile is locally concave), σ2<0, σ is imaginary
The Local-vol can be seen as an instantaneous volatility
 depends on where is the spot at each time step
Can be used to price path-dependent options
  S1 ,t1 
  S 2 ,t 2 
  ST 1 ,tT 1 
S1  S2 
  ST
Local Vol rule of thumb
Rule of thumb:
Local vol varies with
index level twice as fast
as implied vol varies with
strike
Sfinal
Sinitial
(Derman & Kani)
Local-Vol and vanillas

By design the local-vol model reproduces automatically vanillas
 No further calibration necessary, only market quotes needed
EURUSD market
Lines: market quotes
Markers: LV pricer

Example:


Blue: 3 years maturity

Green: 5 years maturity

Take smile quotes
Build local-vol
Use them in simulation and
price vanillas
Compare resulting price of
vanillas vs market quotes
(in smile terms)
Analytic Local-Vol (2)



Estimating the numerical derivatives of the Dupire Local-Vol can
be time-consuming
Alternative: assume a form for the localvol σ(St,t)
Do that, for example, by:

From historical market data calculate logreturns
log
S t  t
  S t , t 
St
These equal to the volatility



Make a scatter plot of all these
Pass a regression
The regression will give an idea of the
historically realised local-vol function
Analytic Local-Vol (2)

A popular choice is
2






Ft
Ft

 S , t    0  1      1      1   S

 F0 
 F0  



Ft the forward at time t
Three calibration parameters




σ0 : controlling ATM vol
α: controlling skew (RR)
β: controlling overall shift (BF)
Calibration is on vanilla prices

Solve Dupire forward PDE with initial condition C=(S0-K)+
Stochastic models
 Stochastic models introduce one extra source of randomness,
for example
 Interest rate dynamics
 Vol dynamics
 Jumps in vol, spot, other underlying
 Combinations of the above
Dupire Local Vol is therefore not a real stochastic model
 Main problem: Calibration
 minimize
(model output – market observable)2
 Example
(model ATM vol – market ATM vol)2
 Parameter space should not be
 too small: model cannot reproduce all market-quotes across tenors
 too large: more than one solution exists to calibration
Heston model

Coupled dynamics of underlying and volatility
dSt  St    dt  v t  St  dW1
dvt    v  vt   dt    vt  dW2
EdW1  dW2     dt

Interpretation of model parameters





Processes

Lognormal for spot

Mean-reverting for variance

Correlated Brownian motions
μ : drift of underlying
κ : speed of mean-reversion
ρ : correlation of Brownian motions
ε : volatility of variance
Analytic solution exists for vanillas !

S L Heston "A Closed form solution for options with stochastic volatility" Rev
Fin Stud (1993) v.6 pp.327-343
Effect of Heston parameters on smile
Affecting overall shift in vol:


Speed of mean-reversion κ
Long-run variance v∞
Affecting skew:


Correlation ρ
Vol of variance ε
Local-vol vs Stochastic-vol

Dupire and Heston reproduce vanillas perfectly
But can differ dramatically when pricing exotics!

Rule of thumb:



skewed smiles: use Local Vol
convex smiles: use Heston
Hull-White model

It models mean-reverting underlyings such as
 Interest rates
 Electricity, oil, gas, etc
drt  rmean  a  rt  dt    dWt

3 parameters to calibrate
 obtained from historical data:


obtained from calibration:



rmean (describes long-term mean)
a: speed of mean reversion
σ : volatility
Has analytic solution for the bond price P = E[ e-∫r(t)dt ]
Three-factor model in FOREX
Hull-White is often coupled to another underlying
Three factor model in FOREX:
spot + domestic/foreign rates
dS  rd  rf   S  dt   FX  S  dW

 r

 dt  
drd  rdmean  ad  rd  dt   d  dWd
drf
mean
f
 af  rf
f
 dWf
To replicate FX volatilities match
FX,mkt with FX,model
2
 FX,
model
T
1
2
s  ds



T t t
Θ(s) is a function of all model
parameters: FX,d,f,ad,af
 Common calibration issue: "Variance squeeze“:
FX vol + IR vols up to a certain date have exceeded the FX-model vol.
 Solution (among other possibilities):
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities

Commodity models introduce the “convenience yield” (termed δ)
δ = benefit of direct access – cost of carry
Not observable but related to physical ownership of asset

No-arbitrage implies Forward: F(t,T) = St ∙ E [ e∫(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St, rt, δt ?

Problem:



δt is unobserved
Spot is not easy to observe

for electricity it does not exist

For oil, the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model

Classic commodities model
dSt  St  rt   t  dt   t  St  dWt1
d t       t  dt    dWt 2




Spot is lognormal (as in Black-Scholes)
Convenience yield is mean-reverting

Very similar to interest rate modeling
(although δt can be pos/neg)

Fluctuation of δ is in practise an order of magnitude higher
than that of r
 no need for stochastic interest rates
Analysis based on combining techniques
Calculate implied convenience yield from observed future prices

Miltersen extension:
Time-dependent parameters
Merton jump model


This model adds a new element to the
stochastic models: jumps in spot
Motivated by real historic data
Advantages



Can produce smile
Adds a realistic element
to dynamics
Has exact solution for
vanillas
Disadvantages


Risk cannot be eliminated
by delta-hedging as in BS
Hedging strategy is not
clear
Merton jump model
Extra term to the Black-Scholes process:


If jump does not occur
dS t
   dt    dWt
St
dSt
If jump occurs
   dt    dWt  Y  1
St
Then,
St  Stafter jump  Stbefore jump  Stbefore jump  Y  1
 Stafter jump  Stbefore jump  Y
Therefore, Y: size of the jump

Model has two extra parameters:
 size of the jump, Y
 frequency of the jump, λ

Jump size & jump times:
Random variables
Merton model solution

Merton assumed that

The jump size Y is lognormally-distributed,




Can be sampled as Y=eη+γ∙g; g is normal ~N(0,1) and η,γ are real
Jump times: Poisson-distributed with mean λ, Prob(n jumps)=e-λT(λT)n /n!
Jump times: independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas

Call price   e
- T
n 0
   T n  BSS
n!
0
, K , T ,  n , rn 




S
S




2
2
 log 0  rn  12  n T 
 log 0  rn  12  n T 
K
K
  e rn  r1 T K

BSS0 , K , T ,  n , rn   e r1T S
n T
n T













σn , rn , λ’ are functions of σ,r and the jump-statistics given by η, γ
  12 
    e
  
2
n
2
 2 n
T

  12  2
rn  r2  r1    e

1 

n
   12  2
T

Merton model properties

The model is able to produce a smile effect
Vanna-Volga method

Which model can reproduce market dynamics?

Market psychology is not subject to rigorous math models…

Brute force approach: Capture main features by a mixture model combining jumps,
stochastic vols, local vols, etc

But…




Vanna-Volga is an alternative pricing “recipie”




Difficult to implement
Hard to calibrate
Computationally inefficient
Easy to implement
No calibration needed
Computationally efficient
But…


It is not a rigorous model
Has no dynamics
Vanna-Volga main idea

The vol-sensitivities
 2 Price
Price
Vega
Vanna

S
Volga
 2 Price
 2
are responsible the smile impact

Practical (trader’s) recipie:



Construct portfolio of 3 vanilla-instruments which
zero out the Vega,Vanna,Volga of exotic option at hand
Calculate the smile impact of this portfolio
(easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic
+ Smile impact of portfolio of vanillas
Vanna-Volga hedging portfolio

Select three liquid instruments:



At-The-Money Straddle (ATM) =½ Call(KATM) + ½ Put(KATM)
25Δ-Risk-Reversal (RR) = Call(Δ=¼) - Put(Δ=-¼)
25Δ-Butterfly (BF) = ½ Call(Δ=¼) + ½ Put(Δ=-¼) – ATM
KATM
KATM
K25ΔP
ATM Straddle
KATM
K25ΔC
25Δ Risk-Reversal
RR carries mainly Vanna
K25ΔP
K25ΔC
25Δ Butterfly
BF carries mainly Volga
Vanna-Volga weights

Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF ∙ BF

What are the appropriate weights wATM ,, wRR, wBF?

Exotic option at hand X and portfolio of vanillas P are calculated
using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X:
 X vega   ATM vega

 
 X vanna    ATM vanna
X
  ATM
volga
volga

 

solve for the weights:
RR vega
RR vanna
RR volga
BFvega   w vega 
 

BFvanna    w vanna 
BFvolga   w volga 


-1
w  A X
Vanna-Volga price

Vanna-Volga market price is
XVV = XBS
+ wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct
price when spot is near
barrier

It reproduces vanilla smile
accurately
Vanna-Volga vs market-price


Can be made to fit the market price of exotics
More info in:

F Bossens, G Rayee, N Skantzos and G Delstra
"Vanna-Volga methods in FX derivatives: from theory to market practise“
Int J Theor Appl Fin (to appear)
Models that go the extra mile



Local Stochastic Vol model
Jump-vol model
Bates model
Local stochastic vol model

Model that results in both a skew (local vol) and a convexity (stochastic
vol)
dSt  St    dt   t St , t  Vt  dWt1
dVt      Vt   dt   Vt  dWt 2

For σ(St,t) = 1 the model degenerates to a purely stochastic model

For ξ=0

Calibration: hard

Several calibration approaches exist, for example:


the model degenerates to a local-volatility model


2
2
Construct σ(St,t) that fits a vanilla market,  Dupire LV St , t     St , t  Vt
Use remaining stochastic parameters to fit e.g. a liquid exotic-option market
2
Jump vol model

Consider two implied volatility surfaces

Bumped up from the original

Bumped down from the original

These generate two local vol surfaces σ1(St,t) and σ2(St,t)

Spot dynamics
dSt  St    dt  St   St , t   dWt
p
 1 St , t  with prob
 St , t   
 2 St , t  with prob 1 - p

Calibrate to vanilla prices using the bumping parameter and the
probability p
Bates model

Stochastic vol model with jumps
dSt  St    dt   t  dWt1  dZ t
d t       t   dt    Vt  dWt 2

Has exact solution for vanillas

Analysis similar to Heston based on deriving the
Fourier characteristic function

More info:

D S Bates “Jumps and Stochastic Volatility: Exchange
rate processes implicit in Deutsche Mark Options“
Rev Fin Stud (1996) v.9 pp.69-107
Which model is better?
Local Vol
Pros
Good for Skew
smiles
Good for simple
exotics
Heston
Vanna-Volga
Good for convex Fast + accurate
smiles
for simple
exotics
OT,KO,DKO,…
Allows fat-tails
Good for barrier
options <1y
Cons
Not good for
convex smiles
Not good for
Skew smiles
Approximates
numerical
derivatives
outside mkt
quotes
Often needs
time-dependent
params to fit
term structure
Cannot be used
for pathdependent
options
TARF,LKB,…
Multifactor
Good for
maturities>1y
Good if product
has spot & rates
as underlying
Not useful if
rates are
approx.
constant
Local-Stoch Vol
Can price most
types of
products (in
theory)
Often unstable
Choice of model



Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across
various tenors (term structure)
No easy answer to which model to use!
W. Schoutens, E. Simons, and J. Tistaert,
"A Perfect calibration! Now what?“ Wilmott Magazine, March 2004
One-touch tables





OT tables measure model success vs market price
OT price ≈ probability of touching barrier (discounted)
Collect mkt prices for TV in the range:
0%-100% (away-close to barrier)
Calculate model price – market price
The better model gives model-mkt≈0
OT tables depend on
OT table
3.00%
2.00%

nbr barriers

Type of underlying

Maturity

mkt conditions
mkt - model
1.00%
0.00%
-1.00%
0
0.2
0.4
0.6
0.8
-2.00%
-3.00%
-4.00%
VannaVolga
-5.00%
LocalVol
-6.00%
-7.00%
Heston
TV price
1
Numerical Methods
Monte Carlo
PDE


Advantages:




Easy to implement
Easy for multi-factor processes
Easy for complex payoffs
Disadvantages





Not accurate enough
CPU inefficient
Greeks not stable/accurate
American exercise: difficult
Depends on quality of
random number generator
Disadvantages




Hard to implement
Hard for multi-factor processes
Hard for complex payoffs
Advantages





Very accurate
CPU efficient
Greeks stable/accurate
American exercise: very easy
Independent of random numbers
Monte Carlo vs PDE
Monte Carlo
Based on discounted average payoff over realizations of spot:
Option Price  e  r T  Epayoff S T 
e
 r T
 
nbr Paths
1

  payoff ST(i )
nbr Paths i 1
Outline of Monte Carlo simulation

For each path:

At each time step till maturity




Draw a random number from Normal distribution N(0,T)
W
Update spot St  t  St  St    t  St    
t
Calculate payoff for this path
random number
Calculate average payoff across all paths
Monte Carlo vs PDE
Partial Differential Equation (PDE)
Based on alternative formulation of option price problem
P
P 1 2  2 P
  S 
   2  r  P
t
S 2
S
Idea is to rewrite it in discrete terms, e.g. with t+=t+Δt, S+=S+ΔS
P(t  )  P(t )
P ( S  )  P ( S ) 1 2 P ( S  )  2 P ( S )  P ( S )
  S 
  
 rP
2
t
2  S
2
S
Spot
Apply payoff at maturity and solve
PDE backwards till today
S0
K
today
time
maturity
Issues on simulations




Random numbers
Barriers and hit probability
Simulating american-exercise options
Likelihood ratio method
Random numbers


Simulations require at each time step a random number
Statistics: for example, normal-Gaussian (for lognormal process) mean=0
variance=1





This means that if we sum all random numbers we should get 0 and st.dev.=1
In practise, we draw uniform random numbers in [0,1] and convert them to
Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths & 102 steps: 107 random numbers
Deviations away from the required statistics produce unwanted bias in option
price
Random numbers do not fill in the space uniformly as they should !

This effect is more pronounced as the number of dimensions (=number of steps *
number of paths) increases
Pseudo-random number generators

RNG generate numbers in the interval [0,1]






With some transformations one then converts the sampling space [0,1] to
any other that is required (e.g. gaussian normal space)
Random numbers are not truly random (hence “pseudo”):
there is a formula behind taking as input the computer clock
After a while “random numbers” will repeat themselves
Good random numbers have a long period before repetition occurs
“Mersenne” random numbers have a period that is a Mersenne number,
i.e. can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to



They are quickly generated
Sequences are uncorrelated
Eventually (after many draws) they fill the space uniformly
“Low-discrepancy” random numbers
These numbers are not random at all !
“low discrepancy” = homogenous
LDRN fill the [0,1] space homogenously.
Passing uniform numbers through the cumulative of the probability
density will produce the correct density of points


homogenous numbers
form [0,1]

1
Gaussian cumulative
function
Gaussian probability
function
0
Non-homogenous numbers
in (-∞ ∞)
Higher density of points here
“Peak” implies that more points
should be sampled from here
Sobol’ numbers





Sobol’ numbers are low-discrepancy sequences
Quality depends on nbr of dimensions = nbr Paths x nbr Steps
Uniformity is good in low dimensions
Uniformity is bad in high dimensions
Are convenient because … they are not random !

Calculating the Greeks with finite difference requires the same
sequence of random numbers


Price S  S   Price S  S 
2  S
The calculation of the Greeks should differ only in the “bumped” param
Random number quality
Plot pairs of columns
Draw (n x m) table of Sobol’ numbers
Nbr Steps
(1,2)
Nbr Paths
Pair( 10 , 20 )
1.000
1.000
1
0
0.5
0.25
0.75
0.875
0.375
0.125
0.625
0.6875
0.1875
0.4375
0.9375
0.8125
0.3125
0.0625
0.5625
0.59375
0.09375
0.34375
0.84375
0.96875
0.46875
0.21875
0.71875
0.65625
0.15625
0.40625
0.90625
2
0
0.5
0.75
0.25
0.875
0.375
0.625
0.125
0.8125
0.3125
0.5625
0.0625
0.6875
0.1875
0.9375
0.4375
0.96875
0.46875
0.71875
0.21875
0.59375
0.09375
0.84375
0.34375
0.65625
0.15625
0.90625
0.40625
3
0
0.5
0.25
0.75
0.125
0.625
0.875
0.375
0.8125
0.3125
0.0625
0.5625
0.4375
0.9375
0.6875
0.1875
0.34375
0.84375
0.59375
0.09375
0.96875
0.46875
0.21875
0.71875
0.03125
0.53125
0.78125
0.28125
4
0
0.5
0.75
0.25
0.625
0.125
0.875
0.375
0.1875
0.6875
0.4375
0.9375
0.0625
0.5625
0.3125
0.8125
0.90625
0.40625
0.65625
0.15625
0.78125
0.28125
0.53125
0.03125
0.34375
0.84375
0.09375
0.59375
(10,20)
5
0
0.5
0.25
0.75
0.375
0.875
0.625
0.125
0.0625
0.5625
0.8125
0.3125
0.9375
0.4375
0.1875
0.6875
0.78125
0.28125
0.03125
0.53125
0.15625
0.65625
0.90625
0.40625
0.34375
0.84375
0.59375
0.09375
6
0
0.5
0.75
0.25
0.375
0.875
0.125
0.625
0.6875
0.1875
0.9375
0.4375
0.3125
0.8125
0.0625
0.5625
0.84375
0.34375
0.59375
0.09375
0.21875
0.71875
0.46875
0.96875
0.90625
0.40625
0.65625
0.15625
Non-uniform filling for large dimensions!
7
0.900
0
0.800
0.5
0.700
0.25
0.75
0.600
0.625
0.500
0.125
0.375
0.400
0.875
0.300
0.5625
0.200
0.0625
0.3125
0.100
0.8125
0.000
0.6875
0.000
0.1875
0.4375
0.9375
0.03125
0.53125
1.000
0.78125
0.900
0.28125
0.15625
0.800
0.65625
0.700
0.90625
0.40625
0.600
0.09375
0.500
0.59375
0.400
0.84375
0.34375
0.300
,2 )
0.900
0.800
0.700
0.600
0.500
0.400
0.300
0.200
0.100
0.200
0.400
0.600
0.800
1.000
0.000
0.000
(13,40)
0.400
0.600
0.800
1.000
(20,881)
Pair( 20 , 881 )
Pair( 13 , 40 )
1.000
0.900
0.800
0.700
0.600
0.500
0.400
0.300
0.200
0.200
0.100
0.100
0.000
0.000
0.200
0.200
0.400
0.600
0.800
1.000
0.000
0.000
0.200
0.400
0.600
0.800
1.000
Barrier options




Consider a (slightly) complex barrier pattern
Payoff at maturity is alive if
Barrier A has not been hit
Barrier B has been hit
Barrier options


There is analytic expression for “survival probability”
=probability of not hitting
We rewrite the pattern in terms of “not-hitting” events:
rule
ProbB is hit AND A is not hit  Bayes'


 ProbB is hit GIVEN A is not hit   ProbA is not hit 
 1  ProbB is not hit GIVEN A is not hit   ProbA is not hit 
 ProbA is not hit   ProbB is not hit AND A is not hit 

This is equivalent to the replication formula: KIAKOB = KOB – DKOA,B

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
Barrier option replication
Prob(A is !hit) =
Prob(A is !hit in [t1,t2])∙
Prob(A is !hit in [t2,t3])
Prob(A is !hit AND B is !hit) =
=Prob(A is !hit in [t1,t2])∙
Prob(A AND B are !hit in [t2,t3]) ∙
Prob(B is !hit in [t3,t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo

When is it optimal to exercise the option?
S0
K
today

t
maturity
Naïve approach. If at any time t:


Spot is out-of-the-money, it is not optimal to exercise. Stop
Spot is in-the-money then



start new simulation from this spot
if (on average) final spot finishes more in-the-money, do not exercise now
if (on average) final spot finishes less in-the-money, exercise now
Least-squares Monte Carlo


Since this has to be done for every time step t:
Naïve Monte Carlo is clearly impractical
Methodology for american exercise provided by





Longstaff & Schwartz (2001) Rev Fin Studies v.14 pp.113-147
Method is not exact but quite accurate (versus e.g. PDE)
Is not hard to implement
But not as CPU-efficient as standard monte carlo
Central idea



Work backwards starting from maturity
At each step compare immediate exercise value with expected
cashflow from continuing
Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)



Generate spots for each path & for each time-step
Make an NpathsxNsteps table of spot paths (according to some dynamics)
Make an NpathsxNsteps empty table of cashflows (CF)
Cashflows
Spot Paths
Npaths












 


 

 

   

  

 


 
Nsteps
Out-of-the-money
In-the-money
0

0
0

0
0

0

0
0 0

0
0 0
0
0 0

0  0 0
0
0 0 
0
0 0

0
0 0
0
Least-squares Monte Carlo (2)

If spot at maturity is

in-the-money: assign for this path CF=payoff value,

out-of-the-money: assign for this path CF=0,
Cashflows
Spot Paths












 


 

 

   

  

 


 
Out-of-the-money
In-the-money
0

0
0

0
0

0

0
0 

0
0 
0
0 0

0  0 
0
0 0 
0
0 

0
0 
0
* CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)

Go one time-step backwards. If spot is


in-the-money: option holder must decide whether to exercise now or continue.
Calculate Y=discounted cashflow at next step if option is not exercised now
out-of-the-money: assign for this path CF=0
Cashflows
Spot Paths












 


 

 

   

  

 


 
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-Δt,T) ∙ CF(T)
0

0
0

0
0

0

0
0 

0
0 
0
0 0

0  0 
0
0 0 
0
0 

0
0 
0
Least-squares Monte Carlo (4)


On the pairs {Spath i,Ypath i} pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing
to hold the option from this time point on
Y
E(S)
S

If E(Spath(T-Δt)) < (Spath(T-Δt)-K)+ :



exercise the option at this time step
Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) > (Spath(T-Δt)-K)+ :


Do not exercise the option at this time step
Maintain same value of cashflow at next steps
Least-squares Monte Carlo (5)


Proceed similarly till the first time step
and populate the matrix of cashflows
There should be one non-zero cashflow
per path!
(the option can be exercised only once)
Amer 

1
N paths
N paths

0

0
0

0
0

0

0
0 0 0  0 0

0 0 0 0  0
0 0  0 0 0

0 0  0 0 0
0 0 0 0  0 
0  0 0 0 0

 0 0 0 0 0
   
  DF today, t iexer  CF Si t iexer
i 1
Callables are priced with the same idea
Least-squares Monte Carlo (5)


Proceed similarly till the first time step
and populate the matrix of cashflows
There should be one non-zero cashflow
per path!
(the option can be exercised only once)
Amer 

1
N paths
N paths

0

0
0

0
0

0

0
0 0 0  0 0

0 0 0 0  0
0 0  0 0 0

0 0  0 0 0
0 0 0 0  0 
0  0 0 0 0

 0 0 0 0 0
   
  DF today, t iexer  CF Si t iexer
i 1
Callables are priced with the same idea
Greeks in Monte Carlo

To calculate Greeks with Monte Carlo:





Bump sensitivity parameter (spot, vol, etc)
Recalculate market data with the bumped parameter (smile, curves, etc)
Re-run Monte Carlo
Calculate Greeks as finite difference
For example,
Price S  S   Price S  S 

2  S
Vega 



Price S  S   2  Price S   Price S  S 
S 2
Price      Price  

This requires at least 12 Monte Carlo runs for all Greeks !
Not ideal for impatient traders
Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea:
Express Greeks as payoffs
Price the new “payoffs” with the same simulation



Note:
The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS)
The LR greeks will not be in general the same as the finite difference
greeks !!

This is because of the modification of the market data when using the finite
difference method
Likelihood ratio method (2)


Consider an exotic option with a path-dependent payoff
Its price will depend on all spots in the path
Exotic  DF   dS1  dS m  PDFS1 ,, S m   Prob surv S1 ,, S m   Payoff

PDF: probability density function of the spot
m
m
i 1
i 1
PDFS1 ,, S m    PDFSi   

1  12 zi2
e
2
S
2i  ti i
zi the Gaussian random number used to make the jump Si-1
log
zi 

1
Si
Si 
1 
  r   i2   ti
Si 1 
2 
 i ti
Probsurv the total survival probability for the spot path (given some barrier levels)
m
i 1 t i
Prob surv   Prob tsurv
i 1

For explicit expressions for the surv.prob. of KO or DKO see previous slides
Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot, vol, etc)
 PDFS1m   Prob surv S1m 
Exotic
 DF   dS1m  Payoff 



PDFS1m 
Prob surv S1m  
1
1
 DF   dS1m  Payoff  





Prob surv S1m 

 PDFS1m 



This is simple derivatives over analytic functions (see previous slide)!
For example,

Delta becomes the new payoff

  DF   Payoff



0 t1

PDFS1m 
Prob tsurv
1
1
 

0 t1


PDF
S

S
S 0
Prob tsurv
1m
0




To be priced with the same spot path as the Payoff itself
Similarly for other Greeks: more lengthy expressions but doable!
References

Options



“Options, Futures & other derivatives” John C Hull, (2008)
Prentice Hall
“Paul Wilmott on Quantitative Finance 3 Vol Set” Paul
Wilmott, (2000) Wiley
Numerical methods:



PDE: "Pricing Financial Instruments: The Finite Difference
Method", D Tavella and C Randall, (2000) Wiley
Monte Carlo: “Monte Carlo methods in Finance", P Jäckel,
(2003) Wiley
Monte Carlo: “Monte Carlo methods in Financial
Engineering", P Glasserman, (2000) Springer