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# Nikos SKANTZOS 2010 - PowerPoint PPT Presentation

Nikos SKANTZOS 2010. 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

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Nikos SKANTZOS 2010

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### 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!

Δ1

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

• 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

### 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

Comparison with market:

BS < MtM when in/out of the money

Plug MtM in BS formula to calculate volatility smile

Inverse calculation  “implied vol”

### Spot 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

### 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

Rule of thumb:

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

(Derman & Kani)

### Local Vol rule of thumb

Sfinal

Sinitial

Example:

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

### 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

• Parameter space should not be

• too small: model cannot reproduce all market-quotes across tenors

• too large: more than one solution exists to calibration

Processes

• Lognormal for spot

• Mean-reverting for variance

• Correlated Brownian motions

### Heston model

• 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

• 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 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

• Has analytic solution for the bond price P = E[ e-∫r(t)dt ]

Three factor model in FOREX:

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 FOREX

Hull-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

• 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

Risk cannot be eliminated by delta-hedging as in BS

Hedging strategy is not clear

### Merton jump model

• 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 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

• 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

K25ΔC

KATM

K25ΔP

K25ΔC

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:

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

• 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)

• 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

• 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 Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

### 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

Monte Carlo

Easy to implement

Easy for multi-factor processes

Easy for complex payoffs

Not accurate enough

CPU inefficient

Greeks not stable/accurate

American exercise: difficult

Depends on quality of

random number generator

PDE

Hard to implement

Hard for multi-factor processes

Hard for complex payoffs

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:

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

• 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

### 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

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 quality

Plot 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) =

Prob(A is !hit in [t1,t2])∙

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

### Barrier option replication

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])

### 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