1. Giulio Laudani #12 Cod. 20247
APPLIED NUMERICAL FINANCE
Discrete time framework:.....................................................................................................................................1
How to compute Excepted value: ....................................................................................................................1
American Option: .............................................................................................................................................2
Lattice approach: ..............................................................................................................................................3
Continuous time Framework: ...............................................................................................................................3
A brief review of Original Black’s:.....................................................................................................................3
Modeling more than one security: ...................................................................................................................4
American Option: .............................................................................................................................................5
Jump diffusion process: ....................................................................................................................................6
Monte Carlo ..........................................................................................................................................................7
What is about?..................................................................................................................................................7
A passage through Bias and Efficiency: ............................................................................................................8
Discretization procedure: .................................................................................................................................8
Variance reduction technique: ...................................................................................................................... 10
Discrete time framework:
This section is basically the Ortu's part. We spend few words only on new, or remarkable part.
How to compute excepted value:
The first cornerstone of finance is the equivalence in value between the price of an asset and the replicating portfolio
, where the replicating strategy is a self-financing one (European case) , while
the discounted one is .The replicating strategy can be computed by a backwardrecursion that
involves the conditional covariance of the option value withthe underlying S1: , which is also called the
delta of the portfolio and it also the regression coefficient between
The second cornerstone if that the conditional expected value under Q of the option payoff is equal to the today price
itself .
The backward recursion formula exploits (and is equivalent to) the Q-martingality of the discounted value of the
European derivative X. Starting from the terminal value ; we determine by backward induction V X(t) from
the value of at the step before, i.e. at t + 1; for t = T -1; …; 0. This approach is precious when dealing with American
options, because it can be generalized to account for the early exercise premium.
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2. Giulio Laudani #12 Cod. 20247
American Option:
The American option pricing is: where is a random variable, representing the optimal
investor time to early exercise the option before maturity, using the info available up to time Pt. This expectation is called
Snell envelope and its properties are:
1. hence it must have a decreasing mean, since the early exercise premium will lose value
2. The concept of super-matingality must be associated with the lowest one among all the possible available, this is
an important condition from the seller prospective
3. The variable is chosen as the minimum time value that ensure that the option value is equal to the immediate
payoff, this condition is to state that waiting is equal to lose money
The American option algorithm uses in the binomial model is equivalent to the free boundary solution, basically we will
look after the maximum value between the expected present value and the immediate payoff. The consequence of this
pricing formula is not to have a self-financing replicating strategy; since the option payoff may have intermediate cash
flow (this consideration is important for hedging purpose).
To solve this problem together with the usual replicating strategy we need to introduce a consumption process C(t) which
is an increasing (no strictly) function [the writer of the option decide thanks to this process how much he will consume at
the beginning of the period, and this strategy is equivalent to the optimal buyer’s strategy]. ,
hence the consumption variation is equal to the decrease in the expected value of the option (those money represent the
value that the writer earn if the buyer do not early exercise when he is supposed to do so).
Markovianity is an useful feature of a price process. It allows to pricederivative securities, whose payoff depends only on
the current underlyingstock price, in a fast way.Hence, instead of computing the entire information structure for a
priceprocess S1; we can compute only thetree that describes the evolution of S1, hence T+1 nodes , basically the evolution
till t-1 plus the two new possible evolution(Binomial case) instead of .
Alook back American optionpayoff . It is not a Markovian process (better, we cannot
simply use the simplified tree method seen above), we need to Markovianize the process by proceeding trough the
following procedure:
1. Introduce a State vector variable defined as or (to have a better understanding)
running maximum
2. Construct the tree for S and F and this will be a Markovian process, so the process
will depend only to F(t) and S(t), so we can rewrite the process as follow
.This method allows to F(t) to not be a recombining function, however the node to be considered are
much more than the simpler one, we have a quadratic function (still manageable).
3. To reduce the time required to compute price option has been introduced various approximation.
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a. The first one is calledthe forward shooting gridapproach , where we are going to introduce an auxiliary
vector representing the running maximum.
b. Then we compute the immediate payoff for each node
c. The backward part consist on using the backward pricing formula , here we might
consider that the binomial tree features allow as to say that in case of an upper movement (with
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This method is suitable for Asian option as well
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3. Giulio Laudani #12 Cod. 20247
probability ) the updated running maximum is in case of an upper
movement and F(t) in case of a down movement.
d. We will use this state vector to compute the continuation value of the option, however there could be a
mismatch between the updated running maxima different and the F(T+1), so we need to define a
selection procedure to proxy the result:
i. Chose the closest F(t+1) to the updated F(t)
ii. Chose two F(t+1) which bound the update one and interpolate
e. Check than for immediate payoff value if it is higher than the excepted value
The price will depend on the algorithm chosen for both the forward and backward part
Lattice approach:
We are going to present a possible framework to develop tree analysis to price path dependent option(J. Cox S.A. Ross and
Rubinstein, 1979), where the usual methodology do not provide enough information. Our aim is to add to each node of
the tree more information by means of an auxiliary statevector.
The state vectoris used to capture the specific path-dependent feature of the option contract.To enhance the accuracy of
the lattice methods without burdening thecomputational cost it is also possible to refine the tree representation of
theunderlying in option-specific regions(S. Figlewski and B. Gao, 1999).
The Adaptive Mesh Model (AMM) sharply reduces the nonlinearity error. The non-linearity error refers to the fact
thatwhen the option value is highly nonlinear with respect to the underlying asset (for instance around the strike at
expiration), a uniform refinement of the step size does not efficiently increase accuracy, because much of the
computational effort is wasted on unimportant regions. The idea of the AMM isto graft one or more small sections of the
fine high-resolution lattice onto atree with coarser time and price steps to increase the computational accuracy only on
those regions where needed.The AMM approach canbe adapted to a wide variety of contingent claims. For some common
problems, accuracy increases by several orders of magnitude with no increase inexecution time.
Discrete barrier options are often approximated with continuous barrier options (i.e. options where the barrier is
monitored continuously in time), byusing the closed formula that can be derived in the continuous-time framework. Such
approximation overprices systematically the knock-in discreteoption and underprices the knock-out discrete options. To
reduce this errorone can apply a suitable correction for option barrier (Broadie, Mark, & Kou, 1997). Basically we will use a
suitable higher or lower (depending on the initial position) barrier.
Continuous time Framework:
Those models are the most used since the daily trading activity is on a continuous base. The models that will discuss in this
section are the base Black and the more advance topic regarding jump diffusion models, mean reverting and tailoring the
pricing to fat tails empirical evidence.
A brief review of Original Black’s:
First of all this model is based on Gaussian distribution assumption, with continuous payoff evolution and we are assuming
that the information comes into the market following a filtration rule. The dynamics used to model the risk free is simply
a time depend function, while the securities’ one is assumed to be defined by a deterministic drift plus a stochastic
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component(diffusion) which is assumed to be a Brownian motion .
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The property of the motion are: zero mean, a time dependent volatility “t”, a Gaussian distribution (for difference of motion with different time interval)
and each interval is independent from the previously one
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The presence of the diffusion element made the solving equation depending on a stochastic integral which do not allow
using the normal calculus solving methodology. To solve this equation we need to modify the payoff so that to eliminate
the dependency to S(t) of the drift and the diffusion; we can do that by applying the Ito Formula:
Thanks to this trick we can solve the SDE and obtain the PDE of the security dynamics as following:
Modeling more than one security:
In order to describe a given correlation structure among the log-returns ofthe risky securities, we are going to employ
many risk factors. In particular,a k-dimensional Brownian motion on the filtered probability space (W; F; P)is used to
represent the riskiness of the market. The classic approach allows only perfect correlated asset, hence we need something
more powerful to model non trivial Var-Cov matrix.
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The model consists on defining k independent Brownian motion (one for each securities) and the related diffusion
coefficient that is a vector (in the simple case a constant) representing the sensitivity to each of the k Brownian motion.
From this vector we end up with the Var-Cov matrix of the whole market; note that the covariance is the product between
the diffusion coefficient and the variance is the sum of the square of the sensitivity coefficient.
The next step now is to define the EMM for all the securities involved, basically we need to find the unique vector which
defines the risk price. We achieve this result by applying the usual Girsanov’s Theorem and transforming the Brownian
motion under probability “P” into the motion under “Q” by the usual drift transformation.
To solve the SDE for each security we can still apply the Ito’s formula with the transformation to add the joint derivatives
terms to account the presence of more than one risky factor. The PDE is:
The Ito’s formula in the multi and the change of drift are:
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We can achieve the same result of modeling the correlation by assuming a correlated Brownian motion
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The parameters are the one of “ ”. The hedging strategy in this case is similar to the one-dimensional case,
it will change depending on the underling.
American Option:
The most common analytic approaches to state andsolve the American option problem in the continuous time framework
are the variational inequality and the free boundary problem.
As a preliminary step we need to define the concept of super-martingale, in fact in continuous time we cannot use the
backward recursive formula. To convert this concept in continuous time we need to formalize the stopping
time , if this event happens we won’t follow any more the continuous region and we won’t have the
martingale property, but instead we will be in the early exercise region, where the process will be a super martingale.
Hence the option is equal to the European option in the continuous region and equal to the immediate payoff elsewhere.
The first procedure consists on having a negative drift under the risk neutral measure for the super-martingale discount
payoff and that the terminal value is anchored to the final payoff/underlying value and that there could be only two
possible case (2) that the continuous region where the process is q-Martingale or the immediate payoff one
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2.
3.
4.
The first equation is the Ito’s formula applied to a portfolio short on the derivatives and long on “h” units of the underling,
by imposing to be a risk free portfolio, hence .
In the variational inequality problem, as we have seen, the description of thecontinuation region and the early exercise
region is implicit. The variationalinequality problem can be tackled with numerical techniques such as finite difference
schemes or finite elements techniques.
Another way to address the American option problem is to first describethe continuation region and the early exercise
region and then impose theBlack-Scholes PDE only on the continuation region. This approach leads tothe free boundary
problem. The free boundary is the line dividing between thecontinuation region and the early exercise region. Its features
depend on the payoff you are considering, and on the parameters of the model.
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2.
3.
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We focus onthe case of the put option, where the immediate payoff is f(S) = (K - S) .It can be proved that the American put
option price F(t; S) inherits theconvexity with respect to the underlying S and the decreasing monotonicity property with
respect to S from the payoff function f. Moreover F isdecreasing with respect to t:
Basically we want to find the critical value S(t)* which define the ends of the continuous region and the beginning of the
early exercise region for any given time. The value S*(t) is called the critical price of S at t and it can be defined as the
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thresholdunder which it is optimal to exercise the option at "t". Unfortunately, noanalytical formula is available to
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For infinite maturity it exist a closed formula
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compute S* as a function of time (andthe other parameters of the American option problem). This is why noclosed
formula is available in the finite maturity case for plain vanilla putoptions. The critical value evolution across time is an
increasing function of time (convex) and at maturity it coincides with the strike level K.
The infinite maturity solution we have that:
Where the elements in the basket are respectively: and and
. The solution comes from applying the Ito’s formula applied to and condition 1, which leads
to the solution “a” and we should take the negative value since we want a decreasing function in mean.
two possible solutions, only the negative is possible
Note that the value of this option will always dominate the value of the discrete formula
Jump diffusion process:
In this section we want to add to our motion jumps at random time with stochastic amplitude to model discontinuity in
the option payoff. The base of our study is the original works of Merton (Merton, 1976) later on generalized with the Levy
Process or marketed point process (Schonbucher, 2003).
The risk free asset is modeled as the usual Black’s world, the securities are instead model as following:
where Y is a random variable and N(t)is a Counting process of the number of jump up to t included and right continuous.
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This last process is distributed according to a Poisson distribution , with mean and variance equal to and marginal
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probability of occurrence , where is called intensity of the process. The J process is a compounded Poison process,
where the number of jump is still as the standard distribution, but the size is defined by the sequence of i.i.d random
variable, i.e. the Y.
Now we need to solve the SDE, first of all we need to have a better insight on the jump dynamics/effect: before the event
the price evolution is equal to the classic Black’s formula at the jump the price will be , so the jump effect
will prevail over all the other time evolution effect. Hence the PDE is:
All the process involved W, J and N are independent we can easily compute the first moment and the variance of the PDE:
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We have chosen this distribution since it ensure an independent, stationary increment equal 1, right continuous and non-decreasing
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It is possible to model the intensity as a function of time (inhomogeneous process) or as a stochastic function (Cox process)
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7. Giulio Laudani #12 Cod. 20247
It is easy to see that the variance in this case is higher than the simple lognormal process, this is a good property to better
fit the fatter tail of the empirical data. You need to note that E(y)-1=0 only if the probability to do not have any lose in
value for any jumps equal 1 (no jump effect). Note that those measures are under P.
This model grants NA, however it is incomplete, in fact there exist many super-martingale measure, from who will choose
the lower one. We need to apply Girsanov’s theorem to change the probability measure to the Jump process as following
7, where and jointlywe need to change the drift of the Wiener process .
So the SDE under Q of the discounted stock differential will be:
where .
If we compute the expected value we notice that the last term is equivalent to a pure jump martingale under Q
, hence the mean is zero.We end up with
and to be drift less (NA requirement) we do not have a unique solution since we have two parameters.
The two parameters cannot be uniquely define since we have one equation (imposing the drift to be zero), Merton
propose to choose as since in his opinion the jump risk can be perfectly hedged (in his mind), hence investor must
be neutral on it: by substituting the
The PDE , note that the drift under q is and that the
volatility is unchanged, so to compute the first and second moment we can simply change the drift, so that we have:
each traded security must earn the risk free rate under any EMM-Q
To price option we can use an intuitive approach base on the decision to choose a number “n” of jump during the tenor or
by applying the Ito’s formula:
The first one will be the intuitive one, besides the trick we assume that Y is log normal(a; ) which is equivalent to
, so the PDE will be if we modify the equation to use the
standardized distribution: .
Now we notice that the expected value of the present value option payoff is the product of the probability
and where the last term is the BS
formula and this last change has been made to change r with to fit the BS formula.
The solution of the SDE with Ito formula is made by a modified version of the standard one, in fact to the usual term will
add the for the dF(t), if we compute the integral of it will became
and all the other term are expressed as integral, since we are looking for the punctual estimate and not the
infinitesimal increment. We choose the usual log transformation we will have our PDE as seen above.
Monte Carlo
In this section we will speak about three main topics: what is a MC simulation, how to improve the efficiency and the
possible drawback and finally some comments on practical example.
What is about?
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The conditional distribution is unchanged, the size of the event is unaffected by the change of measure, the number of jump will be changed. There exist
other forms of the Girsonov’s theorem which allow changing the size of the jump as well.
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MC simulation are used to estimate not solvable equation with analytical solution, basically we are going to use an
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estimator based on “large number rules ”.
Since this is an estimate it is not a number but it carries with itself a distribution and an error, that’s why we have an IC for
that estimates that we need to minimize in order to improve our following consideration based on those results. The MC
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methods has a rate of convergence equal to , which is better than try to solve the integral where we are considering
high dimension problem (more than 4 elements).
To perform a simulation we need to know or to model the distribution of the underling random number which we are
going to compute, besides the theoretical consideration on what to use here we will speak on how we will use it. The
inverse function method is a sort of statement to allow getting all distribution starting from the Uniform , in
fact all PC application provides a random number generator which is based on the uniform distribution. This is an
important property that allowsretrying all continuous and discrete distribution:
For the first case no problem , just find the percentile as function of “U” form
For discrete case we need to define range in which any value of the “U” will be assign to the correct probability
measure, basically we will look for the right extreme inclusion is a convention [Generalize]
The proof of this relationship is based on the fact that since: , which can be proven by checking
that , so
A passage through Bias and Efficiency:
Now after that brief introduction we can describe the twin concept of bias and efficiency. Here we are speaking of bias
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referring to the discretization problem , in fact the estimator is by definition un-biased, and we are defining as efficiency
a multi-dimensional measure, in fact we are looking to both reduce the radius and the time needed to perform the
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simulation . Those two parameters play a contradictory rule, or better they are inversed influenced by the same
elements, that’s why we use the mean spare error measure to improve our estimate.
Thanks to this mathematical device we can jointly control for the bias and variance contribution to reduce the quality of
the estimate. We are usually interested in minimizing the variance, besides in the case of American option. The two cited
elements are the size of the discretization interval“h” and the number of sample used “n”, which contributes to the
radiusefficiency with the following dumb relationship for the discrete approximation case. See below
The time efficiency is considered as following: and the radius expressed in terms of time per simulation 12 is
where the variance is the one granted by the procedure by applying “h”.
Discretization procedure:
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On the convergence for big sample of the estimator to the correct value
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That is our original quantity that we want to guess
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This problem arise when we have to find the Greek of option, when we need to estimate the derivatives/marginal variation to given factors
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We have usually time constrain
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In case of stochastic time needed per simulation (barrier option case) we can use the expected value for simulation
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9. Giulio Laudani #12 Cod. 20247
This is a technique to both estimates path dependent payoff and to compute the option Greeks. We are going to simulate
both the payoff evolution and the marginal change for given change in some key factors.
Speaking about the Greeks there are three possible methodologies that can be used:
Finite discretization: we will look after the first derivatives respect to the given factor by approximating its limit
definition. This method is function of the size of the marginal increment considered and by the number of
simulation performed. This method is a non-consistent approach, since the discretization bias plays a big rule,
however it can be minimized by reducing the “h” size, however we need to control the variance explosion
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problem . There exist two possible methods:
o Forward Difference: . The bias in this case is reduced by a linear
function regardless the number of Taylor expansion terms in the proxy used, i.e. .
 hence the , so the bias
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 , still “h” is the higher order
o Central Difference: . The bias goes to zero faster than the forward
case , however this procedure is more time demanding since we need to compute two marginal
changes. If we assumed that the function is n-times continuously differentiable the bias is
 , so
the bias
o Speaking about the Variance effect we need to consider to possible estimation procedure:
 Independent sampling
 Same seed for both the sampling
The path wise method consists on determining the sensitivity, by deriving the payoff with respect to the
parameter you areinterested in, by swapping the expectation with the derivative operator:
, basically we will estimate the sample mean of that quantity .
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o This method is unbiased, however can be applied only under given hp, i.e. smoothness of the payoff .
o For practical use the estimator that we will use where S(T) is the underlying and
is the parameter on whom we are computing the derivatives.
 The first factor is computed by defining the value assumed by the payoff at maturity, in the case
of a European call we have: since
 The second factor is the derivatives of the underlining dynamics, in the case of the European
call with respect to S(0) [delta] is
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Given the estimates of the derivatives we need to analyze the Variance, in fact its estimates is reduced by the term for FD while for CD (if
the different draw are independent both for the marginal increase that for the original)
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You have to consider the higher order among all the variable
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Digital option do not allow using this methodology, note continuity is too much we need less.
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10. Giulio Laudani #12 Cod. 20247
 The European case the , if
we want to estimate the Vega we need to change just the second factor
o This method can be applied to any diffusion process by freezing path wise the coefficient (Euler
Discretization)
The likelihood ratio method has been introduced to overcome the limit of the previously method, hence it is a
more general one. It consists on simulating the payoff density, which is far more smooth than the original payoff,
hence will use the continuous definition of expected value:
Where is the density function of y for a fixed parameter This estimator is
consistent and unbiased and extendable to the multidimensional case. In concrete this method to be applied: first we
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need find the risk neutral probability of the derivatives payment occurrence . Here there is an example for the delta of a
call European:
At first compute the density function to respect of the parameter, which is S0, so the density is the
ln 0− − 22 , so = 1[= ]∗ ′[=1 ∗1 ]
Then we need to compute the derivatives of g(x) to respect to S0:
The score will be
This method can be used in a multidimensional world, as well as in a path dependent option estimation where the
is the vector of one dimensional random variable with the same density g(x).
Variance reduction technique:
The efficiency is an important goal, here we will describe the most important one:
Antithetic Variate, it is really easy, it consists on using for each simulation the given percentile and its opposite,
so that they have the same distribution but they are not independent, but negatively correlated.
. The variance is smaller
Control Variate is based on using the error in the estimate of known quantities to reduce the error in the
estimateof the unknown one. We will use the combination of the known variable and the unknown one
, which it will beused as estimator.
o This estimator is unbiased for
o So we need to choose a parameter “b” to minimize the new estimator variance to ensure
. This method allows to reduce the variance if the control variate is correlated to the unknown, the
sign do not matter, only size the higher the better with the trivial requirement
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In the case of European option it is the d1
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11. Giulio Laudani #12 Cod. 20247
o If we joint estimate b and X we will have a bias, in fact those variables will be correlated so
. To solve this issue we need to run two independent simulation, the first
regressing Y on X to obtain “b” ( it converges to the correct value b) and the second running the
simulation for the estimator itself
o The “b” comes from
, now we can compute the FOC or just notice that it is a parabola so the vertex is the
minimum as well. Note that
Matching underling asset: the key idea is to match the moments of the underlying asset to reduce the risk of
mispricing derivatives. There are two possibilities, both of them are assuming a Geometric Brownian motion:
o Simple Moment matching: ,(explaining ) this for the first matching (which grants
positive payoff), however it is hard for higher moment.Multiplicative correction.
Additive correction, however do not preserve positivity. Note that the first approach
do not grant to the new parameter to be distributed as the original one, while the second does.
o Weighted MC: The paths’ Weighs Si (T) for i = 1; …; n with weights such that the
moments of S are matched and then use the same weights to estimate the expected payoff:
. Those weights are chosen to maximize the (negative entropy) distance from the uniform
distribution: with the constrain
 Basically we are forcing the estimator to have same
 We need to write the Lagrangian and find the FOC[ ]and the result:
but we can rewrite the risk aversion coefficient v as so
, by exploiting
Importance sampling (Weighted MC): we want to change the paths importance of f (X) that havegreater impact
on determining the expected value. We proceed to choose the weight as following:
o At first we compute the continuous mean
o We apply the Ridon Nikodin derivatives to change the density measure: the new
measure will be
o The new target is by the strong law of large numbers, hence
it is unbiased.
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o Now we want to find the g(x) that minimize the variance we may chose the where “a” is
the expected value of f(x). However we cannot do that since we do not know the distribution ex-ante,
but we know that g(x) is proportional to . We can apply an exponential twisting the
, thisrescaling function which depends only on one parameter.
 First the function is a parabola (also first derivatives are equivalent) to
simplify the computation. It is the moment generating function of X and it is
distributed according to a Normal.It is made to allow a decreasing mean before a key time, and
increasing after to push the path closer to the significant path.
 The new function will be new target function. Note
that the multi-dimensional case is the one used, there will be an g(x) for each period considered
 The is chosen depending on the underling dynamics and it will change depending on the
event matching , this parameter is compute by doing the FOC for
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There will two equation one for , and by exploiting the property, and
we will have:
= +…+ ; = ′ − = + ′
The new variable x will be distrusted(under g measure) as a normal with same variance
but different mean
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