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Analytic solution exists for vanillas !

Nikos SKANTZOS2010

Stochastic methods in Finance

What is the fair price of an option?

Consider a call option on an asset

Delta hedging:

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 fairbecause that is how much we spent!

Fair priceΔ1

Δ2

Δ3

Δ4

Δ5

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

- 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
- This process is a “lognormal” process:
- “lognormal” means that drift and fluctuations are proportional to St

σ: size of fluctuations

μ: steepness of main trend

ΔWt: random variable (pos/neg)

Black-Scholes: assumptions

- No-arbitrage drift = risk-free rate
- Impose no-arbitrage by requiring that
expected spot = market forward

- Impose no-arbitrage by requiring that
- Calculations simplify if
- fluctuations are normal:
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

- fluctuations are normal:

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

Solution

Interpretation of BS formula

Price = value of position at maturity – value of cash-flow at maturity

Δ=0, Delta-neutral value:

if S S+dS then portfolio value does not change

Vega=0, Vega-neutral value:

if σσ+dσ then portfolio value does not change

“Greeks” measure sensitivity of portfolio value

How much does the portfolio value change when spot changes?portfolio value

Delta neutral position,

∂Portfolio/∂S=0

S

S+dS

S

BS < MtM when in/out of the money

Plug MtM in BS formula to calculate volatility smile

Inverse calculation “implied vol”

Black-Scholes vs marketSpot probability density

- Distribution of terminal spot (given initial spot) obtained from

Market observable

Main causes:

- Spot dynamics is not lognormal
- Spot fluctuations (vol) are not constant

Fat tails:

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

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

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

- Vanilla
- Knock-out spot=1.28 strike=1.25 barrier=1.5

Sensitivity of Delta to spot is maximum

1 underlying is needed to hedge

Linear regime: S-K

Δ<0, price gets smaller if spot increases

Spot is far from barrier and far from OTM: risk is minimum, price is maximum

Spotladders: vega, vanna & volga

Vanna: Sensitivity of Vega with respect to Spot

- Vanilla
- Knock-out spot=1.28 strike=1.25 barrier=1.35

Volga: Sensitivity of Vega with respect to Vol

Simple analytic techniques: “moment matching”

- Average-rate option payoff with N fixing dates
- Basket option with two underlyings
- 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

- Find first and second moment of sum of lognormals:

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
- What are its mean (first moment) and variance?
- 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, μthe drift
- Has solution (as in standard Black-Scholes)
- Take averages in above and obtain first and second moment in terms of μ,λ
- Solving for drift and vol produces

Asian options analytics (4)

- Since we wrote Asian payoff as max(XT-K,0)
- We can quote the Black-Scholes formula
- With
- 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

- 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

- Or, equivalently, in terms of BS implied-vols:

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

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman & Kani)

Local Vol rule of thumbSfinal

Sinitial

Take smile quotes

Build local-vol

Use them in simulation and price vanillas

Compare resulting price of vanillas vs market quotes

(in smile terms)

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

Blue: 3 years maturity

Green: 5 years maturity

Analytic Local-Vol (2)

- Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

- Alternative: assume a form for the local-vol σ(St,t)
- Do that, for example, by:
- From historical market data calculate log-returns
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

- From historical market data calculate log-returns

Analytic Local-Vol (2)

- A popular choice is
- 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

- minimize
- Parameter space should not be
- too small: model cannot reproduce all market-quotes across tenors
- too large: more than one solution exists to calibration

- Lognormal for spot
- Mean-reverting for variance
- Correlated Brownian motions

- Coupled dynamics of underlying and volatility
- Interpretation of model parameters
- μ : drift of underlying
- κ : speed of mean-reversion
- ρ : correlation of Brownian motions
- ε : volatility of variance

- 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 skew:

- Correlation ρ
- Vol of variance ε

Affecting overall shift in vol:

- Speed of mean-reversion κ
- Long-run variance v∞

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

- 3 parameters to calibrate
- obtained from historical data:
- rmean (describes long-term mean)

- obtained from calibration:
- a: speed of mean reversion
- σ : volatility

- obtained from historical data:
- Has analytic solution for the bond price P = E[ e-∫r(t)dt ]

spot + domestic/foreign rates

To replicate FX volatilities match

FX,mktwith FX,model

Θ(s) is a function of all model parameters: FX,d,f,ad,af

Three-factor model in FOREXHull-White is often coupled to another underlying

- 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
- 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

- Very similar to interest rate modeling
- Analysis based on combining techniques
- Calculate implied convenience yield from observed future prices

Miltersen extension:

Time-dependent parameters

This model adds a new element to the stochastic models: jumps in spot

Motivated by real historic data

Disadvantages

Risk cannot be eliminated by delta-hedging as in BS

Hedging strategy is not clear

Merton jump modelAdvantages

- Can produce smile
- Adds a realistic element to dynamics
- Has exact solution for vanillas

Merton jump model

Extra term to the Black-Scholes process:

- If jump does not occur
- If jump occurs
Then,

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 jump size Y is lognormally-distributed,
- The model has solution a weighted sum of Black-Scholes formulas
- σn , rn , λ’ are functions of σ,r and the jump-statistics given by η, γ

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…
- Difficult to implement
- Hard to calibrate
- Computationally inefficient

- Vanna-Volga is an alternative pricing “recipie”
- 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
Vega Vanna Volga

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

- Construct portfolio of 3 vanilla-instruments which

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

K25ΔC

KATM

K25ΔP

K25ΔC

ATM Straddle

25ΔRisk-Reversal

25ΔButterfly

RR carries mainly Vanna 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:
- solve for the weights:

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 priceVanna-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)

- F Bossens, G Rayee, N Skantzos and G Delstra

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)
- For σ(St,t) = 1 the model degenerates to a purely stochastic model
- For ξ=0 the model degenerates to a local-volatility model
- Calibration: hard
- Several calibration approaches exist, for example:
- Construct σ(St,t) that fits a vanilla market,
- Use remaining stochastic parameters to fit e.g. a liquid exotic-option market

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
- Calibrate to vanilla prices using the bumping parameter and the probability p

Bates model

- Stochastic vol model with jumps
- 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

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

- nbr barriers
- Type of underlying
- Maturity
- mkt conditions

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

PDE

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

Numerical MethodsMonte Carlo vs PDE

Monte Carlo

Based on discounted average payoff over realizations of spot:

Outline of Monte Carlo simulation

- For each path:
- At each time step till maturity
- Draw a random number from Normal distribution N(0,T)
- Update spot

- Calculate payoff for this path

- At each time step till maturity
- Calculate average payoff across all paths

Monte Carlo vs PDE

Partial Differential Equation (PDE)

Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms, e.g. with t+=t+Δt, S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

Spot

S0

K

time

today

maturity

Issues on simulations

- Random numbers
- Barriers and hit probability
- Simulating american-exercise options
- Likelihood ratio method

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

Random numbersPseudo-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

Gaussian cumulative function

homogenous numbers form [0,1]

1

Gaussian probability function

0

Non-homogenous numbers in (-∞ ∞)

“Peak” implies that more points should be sampled from here

Higher density of points 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
- The calculation of the Greeks should differ only in the “bumped” param

, 2)

1.000

0.900

0.800

0.700

0.600

0.500

0.400

0.300

0.200

0.100

0.000

1.000

0.000

0.200

0.400

0.600

0.800

Random number qualityPlot pairs of columns

Draw (n x m) table of Sobol’ numbers

(1,2) (10,20)

Nbr Steps

Nbr Paths

(13,40) (20,881)

Non-uniform filling for large dimensions!

Payoff at maturity is alive if

Barrier A has not been hit

Barrier B has been hit

Barrier options- Consider a (slightly) complex barrier pattern

Barrier options

- There is analytic expression for “survival probability”
=probability of not hitting

- We rewrite the pattern in terms of “not-hitting” events:
- 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)

Prob(A is !hit in [t1,t2])∙

Prob(A is !hit in [t2,t3])

Barrier option replicationProb(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])

American exercise in Monte Carlo

- When is it optimal to exercise the option?
- 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

S0

K

today

today

t

maturity

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

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

* 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)

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
- 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

E(S)

Y

S

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)

- 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)

- 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,

- 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
- PDF: probability density function of the spot
- zi the Gaussian random number used to make the jump Si-1 Si

- Probsurv the total survival probability for the spot path (given some barrier levels)
- 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)
- This is simple derivatives over analytic functions (see previous slide)!
- For example,
- Delta becomes the new payoff
- 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

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