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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

Nikos SKANTZOS2010

Stochastic methods in Finance


Fair price

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 price

Δ1

Δ2

Δ3

Δ4

Δ5


Black scholes the mother model

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

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

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

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

Interpretation of BS formula

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


How much does the portfolio value change when spot changes

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


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”

Black-Scholes vs market


Spot probability density

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

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

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

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

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

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

“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

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 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

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

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

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

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

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 vol1

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 rule of thumb

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


Local vol and vanillas

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

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 21

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

  • 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

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

    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

    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

    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

    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

    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

    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


    Merton jump model

    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 model

    Advantages

    • Can produce smile

    • Adds a realistic element to dynamics

    • Has exact solution for vanillas


    Merton jump model1

    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 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

    Merton model properties

    • The model is able to produce a smile effect


    Vanna volga method

    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

    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


    Vanna volga hedging portfolio

    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

    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 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 price


    Vanna volga vs market price

    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

    Models that go the extra mile

    • Local Stochastic Vol model

    • Jump-vol model

    • Bates model


    Local stochastic vol 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

    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

    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


    Which model is better

    Which model is better?

    Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

    Pros

    Cons


    Choice of model

    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

    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


    Numerical methods

    Monte Carlo

    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 Methods


    Monte carlo vs pde

    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 pde1

    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

    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

    Random numbers


    Pseudo random number generators

    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

    “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

    • 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


    Random number quality

    , 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!


    Barrier options

    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 options1

    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)


    Barrier option replication

    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])


    Barrier options formula

    Barrier options formula


    Barrier option formula

    Barrier option formula


    American exercise in monte carlo

    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

    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

    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

    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

    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

    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

    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 51

    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

    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

    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

    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

    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

    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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