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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.............................................................................................................................................8
What is about?......................................................................................................................................8
A passage through Bias and Efficiency: ...................................................................................................8
Discretization procedure:.......................................................................................................................9
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 firstcornerstone of finance is the equivalencein valuebetween the priceof an assetand the replicatingportfolio
𝑉 𝑥
(𝑡) = 𝑉𝜃(𝑡), where the replicatingstrategy is a self-financingone(European case) ∆𝑉( 𝑡) = 𝜃1 ∆𝑆1
( 𝑡) + 𝜃0 ∆𝐵(𝑡), while
the discounted one is ∆𝑉̆( 𝑡) = 𝜃1 ∆𝑆̆1
( 𝑡). The replicatingstrategy 𝜃1 𝑎𝑛𝑑 𝜃0 can be computed by a backward recursion
that involves the conditional covarianceof the option valuewith the underlyingS1: 𝜃1 =
𝑐𝑜𝑣(∆𝑉( 𝑡);∆𝑆( 𝑡))
𝑉(∆𝑆( 𝑡))
, which is also
called the delta of the portfolio and italso the regression coefficientbetween ∆𝑉̆( 𝑡); ∆𝑆̆( 𝑡)
𝐶𝑜𝑣 ( 𝑉̆( 𝑡 + 1); ∆𝑆̆( 𝑡)) => 𝐶𝑜𝑣 (∆𝑉̆( 𝑡); ∆𝑆̆( 𝑡)) = 𝐶𝑜𝑣 ( 𝜃1 ∆𝑆̆1
( 𝑡);∆𝑆̆1
( 𝑡)) = 𝜃1 𝐶𝑜𝑣 (∆𝑆̆1
( 𝑡); ∆𝑆̆1
( 𝑡))
The second cornerstone if that the conditional expected valueunder Q of the option payoff is equal to the today price
itself 𝑉(0) = 𝐸 𝑄[ 𝑉( 𝑇)].
The backward recursion formula exploits (and is equivalentto) the Q-martingality of the discounted valueof the
European derivativeX. Startingfrom the terminal value VX
(T) = X(T); we determine by backward induction V X(t) from
the valueof VX
(t) at the step before, i.e. att + 1; for t = T -1; …; 0. This approach is preciouswhen dealingwith American
options,because itcan be generalized to accountfor the early exercisepremium.
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American Option:
The American option pricing is: 𝑉( 𝑡) = sup
𝑡<𝜏≤𝑇
𝐸 𝑄
[ (
𝑋( 𝜏)
𝐵( 𝜏)
)𝑃𝑡 ] where 𝜏 is a random variable, representing the optimal
investor time to early exercise the option before maturity, usingthe info availableup to time Pt. This expectation is called
Snell envelope and its properties are:
1. 𝑉(0) ≥ 𝐸 𝑄[ 𝑉( 𝑇)] hence it must have a decreasingmean, sincethe early exercisepremium will losevalue
2. The concept of super-matingality mustbe associated with the lowest one among all thepossibleavailable,this is
an important condition from the seller prospective
3. The 𝜏 variableis chosen as theminimum time valuethat ensure that the option valueis equal to the immediate
payoff, this condition is to state that waitingis equal to losemoney
The American option algorithmuses in the binomial model is equivalentto the free boundary solution,basically wewill
look after the maximum valuebetween the expected present value and the immediate payoff. The consequence of this
pricingformula is notto have a self-financingreplicating strategy;sincethe option payoff may have intermediate cash
flow (this consideration isimportantfor hedging purpose).
To solvethis problemtogether with the usual replicatingstrategy 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 howmuch he will consumeat
the beginningof the period,and this strategy is equivalentto the optimal buyer’s strategy]. ∆𝑉̆( 𝑡) = 𝜃1 ∆𝑆̆1
( 𝑡) − ∆𝐶̆( 𝑡),
hence the consumption variation is equal to the decrease in the expected valueof the option (those money represent the
valuethat the writer earn if the buyer do not early exercisewhen he is supposed to do so).
Markovianity is an useful feature of a priceprocess.It allows to price derivativesecurities,whosepayoff depends only on
the current underlying stock price, in a fastway. Hence, instead of computing the entire information structurefor a price
process S1; we can compute only the tree that describes the evolution of S1, hence T+1 nodes , basically theevolution till
t-1 plus the two new possibleevolution(Binomial case) instead of 2 𝑇
.
𝐸 𝑄[ 𝜑( 𝑆( 𝑡 + 1)𝑃𝑡
)] = 𝐸 𝑄[ 𝜑( 𝑆( 𝑡 + 1)𝑆 𝑡
)]
A look back American option payoff𝑥( 𝑡) = max
0≤𝑎≤𝑡
𝑆1
( 𝑎) − 𝑆1
( 𝑡) . It is not a Markovian process (better, we cannot simply
use the simplified tree method seen above), we need to Markovianizethe process by proceeding trough the following
procedure:
1. Introduce a State vector variabledefined as 𝐹( 𝑡) = max
0≤𝑎≤𝑡
[𝑆1
( 𝑎)] or (to have a better understanding) 𝐹( 𝑡 + 1) =
max
0≤𝑎≤𝑡
[𝐹(𝑡); 𝑆1(𝑡 + 1)] runningmaximum
2. Construct the tree for S and F and this will bea Markovian process,so the process 𝐸 𝑄 [ 𝜑 (( 𝑆( 𝑡 + 1); 𝐹( 𝑡 + 1))
𝑃𝑡 )] will depend only to F(t) and S(t), so we can rewrite the process as follow 𝐸 𝑄[ 𝜑( 𝑆( 𝑡 + 1); 𝐹( 𝑡 + 1)
𝑆 𝑡 𝐹( 𝑡))]. This method allows to F(t) to not be a recombiningfunction,however the node to be considered are
much more than the simpler one, we have a quadratic function
𝑇2
4
(still manageable).
3. To reduce the time required to compute priceoption has been introduced various approximation.
a. The firstone is called the forward shooting grid approach1,where we are going to introduce an
auxiliary vector representing the runningmaximum.
b. Then we compute the immediate payoff for each node
c. The backward partconsiston usingthe backward pricingformula 𝑉( 𝑡) = 𝐹( 𝑡) − 𝑆( 𝑡),here we might
consider that the binomial treefeatures allowas to say that in caseof an upper movement (with
1 This method is suitablefor Asianoption as well
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probability
1+𝑟−𝑑
𝑢−𝑑
) the updated runningmaximum is 𝐹( 𝑡) = max[𝐹( 𝑡); 𝑆( 𝑡) ∗ 𝑢] in caseof an upper
movement and F(t) in caseof a down movement.
d. We will usethis statevector to compute the continuation valueof the option, however there could be a
mismatch between the updated runningmaxima different and the F(T+1), so we need to define a
selection procedure to proxy the result:
i. Chose the closestF(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 itis higher than the excepted value
The pricewill depend on the algorithmchosen 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 state vector.
The state vector is used to capture the specific path-dependent feature of the option contract. To enhance the accuracy of
the lattice methods without burdening the computational cost it is also possible to refine the tree representation of the
underlying 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 that
when 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 is to graft one or more small sections of the
fine high-resolution lattice onto a tree with coarser time and price steps to increase the computational accuracy only on
those regions where needed. The AMM approach can be adapted to a wide variety of contingent claims. For some
common problems, accuracy increases by several orders of magnitude with no increase i n execution time.
Discrete barrier options are often approximated with continuous barrier options (i.e. options where the barrier is
monitored continuously in time), by using the closed formula that can be derived in the continuous-time framework. Such
approximation overprices systematically the knock-in discrete option and underprices the knock-out discrete options. To
reduce this error one can apply a suitablecorrection 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 sincethe daily tradingactivity 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:
Firstof 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
component (diffusion) which is assumed to be a Brownian motion2.
2 The property of themotion are:zero mean,a timedependent volatility “t”,a Gaussiandistribution(for differenceofmotionwith differenttime
interval) and each intervalis independentfrom thepreviously one
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𝐵( 𝑡) = 𝑒∫ 𝑟( 𝑠) 𝑑𝑠
𝑑𝑆( 𝑡)
𝑆( 𝑡)
= 𝑎( 𝑡) 𝑑𝑡 + 𝑏( 𝑡) 𝑑𝑊( 𝑡)
𝑆( 𝑡) = 𝑆(0) + ∫ 𝑆(𝑡)𝜇𝑑𝑡 + ∫ 𝑆(𝑡)𝜎𝑑𝑊(𝑡) ==> 𝑆𝐷𝐸
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:
𝑓( 𝑡; 𝑋( 𝑡))
𝑑𝑡
∗ 𝑑𝑡 +
𝑓( 𝑡; 𝑋( 𝑡))
𝑑𝑥
∗ 𝑑𝑋( 𝑡) +
1
2
∗
2𝑓( 𝑡; 𝑋( 𝑡))
𝑑𝑥2
∗ 𝑏2
(𝑡; 𝑋(𝑡)) 𝑑𝑡
Thanks to this trick we can solve the SDE and obtain the PDE of the security dynamics as following:
𝑆( 𝑡) = 𝑆(0) ∗ 𝑒
( 𝜇−
1
2
∗𝜎2 ) 𝑑𝑡+𝜎𝑑𝑊
𝐶𝑜𝑟𝑟 (
𝑑𝑆1
𝑆1
;
𝑑𝑆2
𝑆2
) =
𝜎1 𝜎2 𝑑𝑡
𝜎1√𝑑𝑡𝜎2√𝑑𝑡
= 1
𝐸[ 𝑆( 𝑡)] = 𝑆0 𝑒 𝜇𝑡 ∫ 𝑒
−
1
2
𝜎2
𝑡+𝜎√𝑡 𝑧
∗
1
√2𝜋
𝑒
−
𝑧2
2 𝑑𝑧 = ∫
1
√2𝜋
𝑒
−
1
2
( 𝑧−𝜎√𝑡)
2
𝑑𝑧
Modeling more than one security:
In order to describe a given correlation structure among the log-returns of the 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 riskinessof the market. The classic approach allows only perfect correlated asset, hence we need something
more powerful to model non trivial Var-Cov matrix.
The model consists on defining k independent3 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.
𝑐𝑜𝑣( 𝑆(1), 𝑆(2)) = Σ𝑖,𝑗 = 𝜎𝑖
𝑇
𝜎𝑗 = ( 𝜎1,1 𝜎2,1 + 𝜎1,2 𝜎2,2) 𝑑𝑡
𝑑𝑆𝑖( 𝑡)
𝑆𝑖( 𝑡)
= 𝜇 𝑖(𝑆(𝑡); 𝑡)𝑑𝑡 + 𝜎𝑖
𝑇 ( 𝑆( 𝑡); 𝑡) 𝑑𝑊 𝑃
(𝑡)
𝑉 𝑎𝑟 (
𝑑𝑆1( 𝑡)
𝑆1( 𝑡)
) = (𝜎1,1
2
+ 𝜎1,2
2
)𝑑𝑡
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.
𝑣𝑖 =
𝜇 𝑖 − 𝑟
𝜎𝑖
→ 𝑊 𝑃
= 𝑊 𝑄
− 𝑣𝑑𝑡
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:
𝑆𝑖 (𝑡) = 𝑆𝑖
(0) 𝑒
( 𝑟1−
1
2
𝜎𝑖
𝑇
𝜎𝑖 ) 𝑑𝑡+𝜎𝑖
𝑇
𝑊 𝑄( 𝑡)
The Ito’s formula in the multi and the change of drift are:
3 We can achievethesameresultofmodeling thecorrelationby assuming a correlatedBrownianmotion
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𝑑𝑌( 𝑡) =
𝑑𝑓( 𝑡, 𝑋( 𝑡))
𝑑𝑡
+ ∑
𝑑𝑓( 𝑡, 𝑋( 𝑡))
𝑑𝑥 𝑖
𝑑𝑋𝑖 (𝑡) +
1
2
∑
𝑑2
𝑓( 𝑡, 𝑋( 𝑡))
𝑑𝑥 𝑖 𝑑𝑥𝑗
𝑏𝑖
𝑡
𝑏𝑗 𝑑𝑡
𝑑𝑊 𝑄
= 𝑑𝑊 𝑃
+ 𝑣𝑑𝑡 𝑤ℎ𝑒𝑟𝑒 𝜇 𝑖 − 𝑟 = 𝜎𝑖
𝑇
𝑣
The parameters are the one of “𝑓( 𝑡, 𝑋( 𝑡))”. The hedging strategy in this case is similar to the one-dimensional case, 𝜃0 =
𝐹( 𝑡, 𝑆( 𝑡)) − ∑ ℎ𝑖 𝑆𝑖 𝑎𝑛𝑑 𝜃𝑖 =
𝑑𝐹( 𝑡,𝑆( 𝑡))
𝑑 𝑆 𝑖
→it will change depending on the underling.
American Option:
The most common analytic approaches to state and solve 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{ 𝑊: 𝑟( 𝑤) < 𝑡} ∈ ℱ(𝑟), 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
1.
𝐹( 𝑡;𝑆)
𝑑𝑡
+
𝐹( 𝑡;𝑆)
𝑑𝑆
∗ 𝑆𝑟 +
1
2
∗
𝐹2( 𝑡;𝑆)
𝑑2 𝑆
∗ 𝑆2
𝜎2
− 𝐹( 𝑡; 𝑆) 𝑟 ≤ 0 𝑓𝑜𝑟 𝑎𝑛𝑦 ( 𝑡; 𝑆) ∈ [0; 𝑇] 𝑋 ℛ+
2. (
𝐹( 𝑡;𝑆)
𝑑𝑡
+
𝐹 ( 𝑡;𝑆)
𝑑𝑆
∗ 𝑆𝑟 +
1
2
∗
𝐹2( 𝑡;𝑆)
𝑑2 𝑆
∗ 𝑆2
𝜎2
− 𝐹( 𝑡; 𝑆) 𝑟) (𝐹( 𝑡; 𝑆) − 𝑓(𝑆)) = 0 𝑓𝑜𝑟 𝑎𝑛𝑦 (𝑡; 𝑆) ∈ [0; 𝑇] ℛ+
3. 𝐹( 𝑡; 𝑆) ≥ 𝑓(𝑆) 𝑓𝑜𝑟 𝑎𝑛𝑦 (𝑡; 𝑆) ∈ [0; 𝑇] ℛ+
4. 𝐹(𝑇; 𝑆) = 𝑓(𝑆) 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑆 ∈ ℛ+
The firstequation 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 ℎ =
𝑑𝐹( 𝑡)
𝑑𝑆
.
In the variational inequality problem, as we have seen, the description of the continuation region and the early exercise
region is implicit. The variational inequality 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 describe the continuation region and the early exercise
region and then impose the Black-Scholes PDE only on the continuation region. This approach leads to the free boundary
problem. The free boundary is the linedividing between the continuation region and the early exercise region. Its features
depend on the payoff you are considering, and on the parameters of the model.
1.
𝐹( 𝑡;𝑆)
𝑑𝑡
+
𝐹( 𝑡;𝑆)
𝑑𝑆
∗ 𝑆𝑟 +
1
2
∗
𝐹2( 𝑡;𝑆)
𝑑2 𝑆
∗ 𝑆2
𝜎2
− 𝐹( 𝑡; 𝑆) 𝑟 = 0 𝑓𝑜𝑟 𝑎𝑛𝑦 ( 𝑡; 𝑆) ∈ [0; 𝑇], 𝑆 > 𝑆∗( 𝑡)
2. 𝐹( 𝑡; 𝑆) = ( 𝐾 − 𝑆)+
𝑓𝑜𝑟 𝑎𝑛𝑦 𝑡 ∈ [0; 𝑇], 𝑆 = 𝑆∗( 𝑡)
3.
𝑑𝐹( 𝑡; 𝑆)
𝑑𝑆
= −1 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑡 ∈ [0; 𝑇], 𝑆 = 𝑆∗( 𝑡) smooth pasting
4. 𝐹(𝑇; 𝑆) = 𝑓(𝑆) 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑆 ∈ ℛ+
We focus on the 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 the convexity with respect to the underlying S and the decreasing monotonicity property
with respect to S from the payoff function f. Moreover F is decreasing with respect to t:
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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
threshold under which it is optimal to exercise the option at "t". Unfortunately, no analytical formula4 is available to
compute S* as a function of time (and the other parameters of the American option problem). This is why no closed
formula is available in the finite maturity case for plain vanilla put options. 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:
𝐹∞
( 𝑡) = {
𝐴𝑆 𝑎
𝑓𝑜𝑟 𝑆 > 𝑆∞
𝐾 − 𝑆 𝑓𝑜𝑟 𝑆 ≤ 𝑆∞}
Where the elements in the basket are respectively: 𝑎 =
(−( 𝑟−𝜎2 )−√ 𝑟2+𝜎2 𝑟+
𝜎4
4
)
𝜎2
< 0 and 𝑆∞
∗
= −1−𝑎
𝑎
𝐾 < 𝐾 and 𝐴 =
𝑆∞
∗ (1−𝑎)
−𝑎
. The solution comes from applying the Ito’s formula applied to 𝐴𝑆 𝑎
𝑓𝑜𝑟 𝑆 > 𝑆∞
and condition 1, which leads to the
solution “a” and we should take the negative value since we want a decreasing function in mean.
0 + 𝐴𝑆 𝑎−1
∗ 𝑎𝑆𝑟 +
1
2
𝐴𝑆 𝑎−2( 𝑎 − 1) 𝑎𝑆2
𝜎2
− 𝐴𝑆 𝑎
𝑟 = 0 → 𝐴𝑆 𝑎 [ 𝑎𝑟 +
1
2
𝑎2
𝜎2
−
1
2
𝑎𝜎2
− 𝑟] = 0
𝑎 ( 𝑟 −
𝜎2
2
) +
𝑎2
𝜎2
2
− 𝑟 = 0 two possible solutions, only the negative is possible
Note that the value of this option will always dominate the value of the discrete formula 𝑆∗( 𝑡) > 𝑆∞
( 𝑡)
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 assetis modeled as the usual Black’s world,the securities areinstead model as following:
dS(t) = S(t−)(μdt + σdW(t) + dJ(t))
𝐽( 𝑡) = ∑( 𝑌𝑗 − 1)
𝑁(𝑡)
𝐽=1
𝑃 [ 𝑁( 𝑡) = 𝑛] = 𝑒−𝜆𝑡
( 𝜆𝑡) 𝑛
𝑛!
𝑤𝑖𝑡ℎ 𝑁(0) = 0
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.
This last process is distributed according to a Poisson distribution5, with mean and variance equal to 𝜆𝑡 and marginal
probability of occurrence 𝜆𝑑𝑡, where 𝜆6 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 i s 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 priceevolution is equal to the classic Black’s formula at the jump the price will be 𝑆( 𝑟𝑡
) = 𝑆( 𝑡− ) 𝑌𝑡, so the jump effect
will prevail over all the other time evolution effect. Hence the PDE is:
4 For infinite maturity it exist a closedformula
5 We have chosen this distribution sinceit ensurean independent, stationary increment equal1, right continuous andnon-decreasing
6 It is possible tomodeltheintensity as a function oftime (inhomogeneous process) or as a stochastic function(Cox process)
7.
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7
𝑆( 𝑡) = 𝑆(0) 𝑒
( 𝜇−
1
2
𝜎2 ) 𝑑𝑡+𝜎𝑊( 𝑡)
(∏( 𝑌𝑗)
𝑁( 𝑡)
𝑗=1
) 𝑤𝑖𝑡ℎ ∏( 𝑌𝑗) = 1
0
𝑗=1
𝑎𝑛𝑑 𝐸 [∏( 𝑌𝑗)
𝑁( 𝑡)
𝑗=1
] = 𝑒 𝜆𝑡( 𝐸( 𝑌1)−1)
All the process involved W, J and N are independent we can easily compute the first moment and the variance of the PDE:
𝐸( 𝑆( 𝑡)) = S(0)e 𝜇𝑡 𝑒 𝜆𝑡( 𝐸( 𝑌1)−1)
𝑉( 𝑆( 𝑡)) = 𝑆2(0)( 𝑒(2𝜇+𝜎2) 𝑡
𝑒 𝜆𝑡 ( 𝐸[ 𝑌2
]−1)
− e2𝜇𝑡 𝑒2𝜆𝑡( 𝐸( 𝑌1)−1)
)
It is easy to see that the variancein 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 proces s as following
λQ = ϕλP
7, where ϕ > 0 and jointly we need to change the drift of the Wiener process 𝑊𝑡
𝑄
= 𝑊𝑡
𝑃
+ 𝜃𝑡.
So the SDE under Q of the discounted stock differential will be: ( 𝜇 − 𝑟) 𝑑𝑡 + 𝜎𝑑𝑊(𝑡) 𝑃
+ 𝑑𝐽( 𝑡) ± ( 𝜇 𝑦 − 1) 𝜆 𝑄
𝑑𝑡 =
( 𝜇 − 𝑟 − 𝜎𝜃 + ( 𝜇 𝑦 − 1) 𝜆 𝑄) 𝑑𝑡 + 𝜎𝑑𝑊 𝑄 ( 𝑡) + ∆𝐽( 𝑡) − ( 𝜇 𝑦 − 1) 𝜆 𝑄
𝑑𝑡 where 𝜇 𝑦 = 𝐸 𝑃
(𝑌).
If we compute the expected value we notice that the last term is equivalent to a pure jump martingale under Q
𝐸 𝑄[ 𝑑𝐽( 𝑡)𝑃𝑡
] = 𝐸[ 𝑑𝐽( 𝑡)] ∗ 𝑄( 𝐽𝑢𝑚𝑝 𝑝𝑟𝑜𝑏) = ( 𝜇 𝑦 − 1) 𝜆 𝑄
𝑑𝑡, hence the mean is zero. We end up with ( 𝜇 − 𝑟 − 𝜎𝜃 +
( 𝜇 𝑦 − 1) 𝜆 𝑄) 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 ϕ = 1 since in his opinion the jump risk can be perfectly hedged (in his mind), hence investor must
be neutral on it: by substituting the 𝜃 =
( 𝜇−𝑟+( 𝜇 𝑦−1) 𝜆 𝑄)
𝜎
→ ( 𝑟 − (𝜇 𝑦 − 1)λQ −
1
2
𝜎2).
The PDE 𝑆( 𝑡) = 𝑆(0) 𝑒
( 𝑟−(𝜇 𝑦−1)λQ−
1
2
𝜎2) 𝑑𝑡+𝜎𝑊 𝑄( 𝑡)
(∏ ( 𝑌𝑗)𝑁( 𝑡)
𝑗=1
), note that the drift under q is 𝑟 − (𝜇 𝑦 − 1)λQ and that the
volatility is unchanged, so to compute the first and second moment we can simply change the drift, so that we have:
𝐸 𝑄( 𝑆( 𝑡)) = 𝑆(0) 𝑒 𝑟𝑡
each traded security must earn the risk free rate under any EMM-Q
To priceoption we can use an intuitive approach base on the decision to choose a number “n” of jump during the tenor o r
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;𝑏2
) which is equivalent to 𝑌 =
𝑒 𝑋~𝑁( 𝑎;𝑏2 )
, so the PDE will be 𝑆( 𝑡) = 𝑆(0) 𝑒
( 𝑟−(𝜇 𝑦−1)λQ−
1
2
𝜎2) 𝑑𝑡+𝜎𝑊 𝑄( 𝑡)
𝑒∑ 𝑋𝑖 if we modify the equation to use the
standardized distribution: 𝑆( 𝑡) = 𝑆(0) 𝑒
( 𝑟 𝑛 𝑇−
1
2
𝜎2
𝑇)+𝜎 𝑛 √𝑡𝑍
. 𝑟𝑛 = 𝑟 − 𝑚[= 𝜇 𝑦 − 1]λ +
nln(1+m)
T
; 𝜎 𝑛
2
𝑇 = 𝜎2
𝑇 + 𝑛𝑏2
Now we notice that the expected value of the present value option payoff is the product of the probability 𝑃[ 𝑁( 𝜃) = 𝑛]
and 𝑃[( 𝑆( 𝑡) − 𝐾)+
𝑒−𝑟𝑡
𝑁( 𝑡) = 𝑛] = ∑
𝑒λ′t(λ′
t)
n
𝑛!
𝐸 𝑄
[𝑆(0) 𝑒
( 𝑟 𝑛 𝑇−
1
2
𝜎2
𝑇)+𝜎 𝑛 √𝑡𝑍
∗ 𝑒−𝑟 𝑛 𝑇] where the last term is the BS
formula and λ′
= (1 + 𝑚)λ this last change has been made to change r with 𝑟𝑛 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 ( 𝐹( 𝑡) − 𝐹( 𝑡− ))∆𝑁( 𝑡) for the dF(t), if we compute the integral of 𝐹( 𝑡) − 𝐹(0) it will became ∑ 𝐹( 𝑡, ∆𝑆( 𝑡− ) 𝑌𝑖
) −
𝐹( 𝑡, ∆𝑆( 𝑡− )) 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.
7 The conditional distributionis unchanged, thesize oftheeventis unaffected by the changeofmeasure,thenumber ofjump willbe changed. There
exist otherforms of theGirsonov’s theoremwhich allowchanging the sizeofthejump as well.
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8
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?
MC simulation are used to estimate not solvable equation with analytical solution, basically we are going to use an
estimator based on “large number rules8”.
𝑎̂ =
1
𝑛
∑ 𝑓(𝑥) 𝑤ℎ𝑒𝑟𝑒 lim
𝑛→∞
𝑎̂ = 𝑎 𝑎𝑛𝑑 𝐸( 𝑎̂) = 𝑎
| 𝑎̂ − 𝑎| =
𝜎̂
√𝑛
𝑧 𝛼 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑠𝑖𝑑𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑔𝑛 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛
Sincethis is an estimate itis not a number but itcarries 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
methods has a rate of convergence equal to
1
√𝑛
, which is better than try to solve the integral9 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 𝐹𝑥
−1
(𝑈)~𝑋, in
fact all PC application provides a random number generator which is based on the uniform distribution. This is an
important property that allows retrying all continuous and discrete distribution:
For the first case no problem , just find the percentile as function of “U” form 𝑢 = 𝐹𝑥 (𝑋)
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 𝑃[𝑞 𝑗−1 < 𝑈 ≤ 𝑞 𝑗] the right extreme inclusion is a convention [Generalize]
The proof of this relationship is based on the fact that since: 𝐹𝑥
−1( 𝑢) ≤ 𝑥 𝑖𝑖𝑓 𝑢 ≤ 𝐹𝑥 (𝑋), which can be proven by checking
that 𝐹𝑥
−1( 𝐹𝑥
−1( 𝑢)) ≤ 𝐹𝑥
( 𝑥) <=> 𝑢 ≤ 𝐹𝑥 (𝑋), so 𝑃[ 𝐹𝑥
−1( 𝑢) ≤ 𝑥] = 𝑃[ 𝑢 ≤ 𝐹𝑥
( 𝑋)] => 𝐹𝑥
−1
(𝑥)
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
referring to the discretization problem10,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
simulation11. 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.
𝑀𝑆𝐸 = 𝐸[( 𝑎̂ − 𝑎)2] = 𝐸[ 𝐸( 𝑎̂) − 𝑎]2
+ 𝐸[ 𝑎̂ − 𝐸( 𝑎̂)]2
= 𝐵𝑖𝑎𝑠( 𝑎̂)2
+ 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒(𝑎̂)
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
radius efficiency with the following dumb relationship 𝑛ℎ2
→ ∞ for the discrete approximation case. See below
8 On the convergence for big sampleoftheestimator tothecorrectvalue
9 That is our originalquantity thatwe want to guess
10 This problemarisewhen wehaveto findtheGreek ofoption, when weneed toestimate the derivatives/marginal variationto given factors
11 We have usually timeconstrain
9.
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9
The time efficiency is considered as following: 𝑛ℎ =
𝑇
𝑟ℎ
and the radius expressed in terms of time per simulation 𝑟ℎ
12 is
√𝑉ℎ 𝑟ℎ
√𝑇
where the variance is the one granted by the procedure by applying “h”.
Discretization procedure:
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 changefor given change in some key factors.
Speaking about the Greeks there are three possiblemethodologies that can be used:
Finite discretization: we will look after the firstderivatives respectto the given factor by approximatingits limit
definition.This method is function of the size of the marginal incrementconsidered and by the number of
simulation performed. This method is a non-consistentapproach,sincethe discretization biasplays a bigrule,
however itcan be minimized by reducing the “h” size, however we need to control the varianceexplosion
problem13. There existtwo possiblemethods:
o Forward Difference: 𝑎𝐼 ( 𝜃) ≈
𝑎( 𝜃+ℎ)−𝑎( 𝜃)
ℎ
𝑤𝑖𝑡ℎ ℎ → 0. The bias in this caseis reduced by a linear
function regardless the number of Taylor expansion terms in the proxy used, i.e. 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ∗ ℎ.
𝑎( 𝜃 + ℎ) − 𝑎( 𝜃) = 𝑎′
ℎ +
1
2
𝑎′′
ℎ2
+ 𝑜(ℎ2
) hence the 𝐸[∆𝑎] = 𝑎′
+
1
2
𝑎′′
ℎ + 𝑜(ℎ), so the bias
𝐸[∆𝑎] − 𝑎′
=
1
2
𝑎′′
ℎ + 𝑜(ℎ) ==> ℎ ∗ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡14
𝑎′
ℎ +
1
2
𝑎′′
ℎ2
+
𝑎′′′
ℎ3
1
6
+ 𝑜(ℎ3
), still “h”is the higher order
o Central Difference: 𝑎𝐼 ( 𝜃) ≈
𝑎( 𝜃+ℎ)−𝑎( 𝜃−ℎ)
2ℎ
𝑤𝑖𝑡ℎ ℎ → 0. The bias goes to zero faster than the forward
case 𝑜(ℎ), however this procedure is more time demanding sincewe need to compute two marginal
changes. If we assumed that the function is n-times continuously differentiablethe bias is 𝑜(ℎ2)
𝑎( 𝜃 + ℎ) − 𝑎( 𝜃) = 𝑎′
ℎ +
1
2
𝑎′′
ℎ2
+
1
6
𝑎′′′
ℎ3
+ 𝑜(ℎ3) − (−𝑎′
ℎ +
1
2
𝑎′′
ℎ2
−
𝑎′′′
1
6
ℎ3
+ 𝑜(ℎ3 )), so
the bias
2𝑎′ℎ+
2
3
𝑎′′′ℎ3 +𝑜(ℎ3)
2ℎ
− 𝑎′
=
1
3
𝑎′′′
ℎ2
+ 𝑜(ℎ2
)
o Speaking about the Varianceeffect we need to consider to possibleestimation procedure:
Independent sampling
𝑉𝑓
(∆𝑎) = ℎ−2
𝑉 (
∑ ∆𝑎
𝑛
) = ℎ𝑛−2 ∑[𝑉( 𝑦( 𝑡 + ℎ)) + 𝑉( 𝑦( 𝑡)) ] =
2
ℎ2 𝑛
∗ 𝐶𝑜𝑛𝑠𝑡
𝑉𝐶
(∆𝑎) =
ℎ−2
4
𝑉 (
∑ ∆𝑎
𝑛
) =
1
2ℎ2 𝑛
∗ 𝐶𝑜𝑛𝑠𝑡
Same seed for both the sampling
𝑉𝑓
(∆𝑎) = ℎ−2
𝑉 (
∑ ∆𝑎
𝑛
) = ℎ ∗
𝑐𝑜𝑛𝑠𝑡
ℎ2 𝑛
=
𝑐𝑜𝑛𝑠𝑡
ℎ𝑛
𝑉𝐶
(∆𝑎) =
ℎ−2
4
𝑉 (
∑ ∆𝑎
𝑛
) = ℎ ∗
𝐶𝑜𝑛𝑠𝑡
ℎ24𝑛
=
𝐶𝑜𝑛𝑠𝑡
ℎ4𝑛
The path wise method consists on determiningthe sensitivity, by deriving the payoff with respect to the
parameter you areinterested in, by swappingthe expectation with the derivativeoperator:
𝑑
𝑑𝜃
𝐸[ 𝑌( 𝜃)] =
𝐸 [
𝑑
𝑑𝜃
𝑌( 𝜃)] 𝑤ℎ𝑒𝑟𝑒
𝑑
𝑑𝜃
𝑌( 𝜃) = 𝑌 𝐼
(𝜃), basically wewill estimate the samplemean of that quantity
1
𝑛
∑ 𝑌 𝐼 ( 𝜃).
o This method is unbiased,however can be applied only under given hp, i.e. smoothness of the payoff15.
12 In case of stochastictime needed per simulation (barrier option case) we can usetheexpectedvaluefor simulation
13 Given the estimates ofthederivatives weneedto analyzetheVariance, infactits estimates is reduced bytheterm n ∗ h2
for FD while n ∗ h for CD (if
the different drawareindependent both for the marginalincrease thatfor theoriginal)
14 You have to consider thehigher order among allthevariable
15 Digital option do not allowusing this methodology, notecontinuity is too much weneed less.
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10
o For practical usethe estimator that we will use
𝑑
𝑑𝜃
𝑌( 𝜃) =
𝑑𝑌( 𝜃)
𝑑𝑆( 𝑇)
∗
𝑑𝑆( 𝑇)
𝑑𝜃
where S(T) is the underlying and
𝜃 is the parameter on whom we are computing the derivatives.
The firstfactor is computed by definingthe valueassumed by the payoff at maturity, in the case
of a European call wehave: 𝑒−𝑟𝑡
𝐼 𝑆( 𝑇)>𝐾 since 𝑌( 𝜃) = ( 𝑆 − 𝐾)+
→ 𝑆+
The second factor is the derivatives of the underliningdynamics,in thecaseof the European
call with respect to S(0) [delta] is
𝑆( 𝑇)
𝑆(0)
The European casethe 𝐸 [ 𝑒−𝑟𝑡
𝐼𝑆( 𝑇)>𝐾
𝑆( 𝑇)
𝑆(0)
] =
𝑒−𝑟𝑡
𝑆(0)
𝐸( 𝑆( 𝑇)) 𝐸( 𝐼 𝑆( 𝑇)>𝐾) =
𝑒−𝑟𝑡
𝑆(0)
∗ 𝑆(0) 𝑒 𝑟𝑡
𝑁( 𝑑1
) ,
if we want to estimate the Vega we need to change justthe second factor
o This method can be applied to any diffusion processby freezing path wise the coefficient(Euler
Discretization)
The likelihood ratio method has been introduced to overcome the limitof the previously method, hence it is a
more general one. It consistson simulatingthe payoff density,which is far more smooth than the original payoff,
hence will usethe continuous definition of expected value:
𝑑
𝑑𝜃
𝐸[ 𝑌( 𝜃)] = ∫
𝑑
𝑑𝜃
∗ 𝑦𝑔𝜃
( 𝑦) 𝑑𝑦 = ∫ 𝑦 ∗
𝑑𝑔𝜃
( 𝑦)
𝑑𝜃
𝑔𝜃
( 𝑦)
𝑔𝜃
( 𝑦) 𝑑𝑦
𝑑𝑔𝜃
( 𝑦)
𝑑𝜃
𝑔𝜃
( 𝑦)
= 𝑆𝑐𝑜𝑟𝑒( 𝑌) ==>
1
𝑛
∑ 𝑌𝑖 ∗ 𝑆𝑐𝑜𝑟𝑒(𝑌)
Where 𝑔𝜃
( 𝑦) =
𝑑
𝑑𝑥
𝑄[ 𝑆 > 𝐾] 𝐸𝑢𝑟𝑜𝑝𝑒𝑎𝑛 𝑐𝑎𝑙𝑙 is the density function of y for a fixed parameter 𝜃. This estimator is
consistentand unbiased and extendable to the multidimensional case.In concrete this method to be applied:firstwe
need find the risk neutral probability of the derivatives payment occurrence16. Here there is an example for the delta of a
call European:
At firstcompute the density function to respect of the parameter, which is S0, so the density is the
𝑑
𝑑𝑥
𝑁 ( 𝑑1 [=
ln( 𝐾
𝑆(0)
)−( 𝑟−
𝜎2
2
) 𝑡
𝜎√𝑇
]), so 𝑔( 𝑥) = 𝑁( 𝑑1
)[= 𝜖] ∗ 𝜖′
[=
1
𝜎√𝑇
∗
1
𝑥
]
Then we need to compute the derivatives of g(x) to respect to S0:
𝑑
𝑑𝑆0
𝑔( 𝑥) =
𝑑
𝑑𝑆0
𝑁( 𝜖(𝑥)) ∗
1
𝜎√𝑇
∗
1
𝑥
=
1
𝜎√𝑇
∗
1
𝑥
∗
(
1
√2𝜋
∗ 𝑒
−
𝜖(𝑥)2
2 ) ∗ −𝜖( 𝑥) ∗
𝑑𝜖( 𝑥)
𝑑 𝑆0
[= −
1
𝜎√𝑇
∗
1
𝑆0
] = 𝒈( 𝒙) ∗
𝝐( 𝒙)
𝑺 𝟎
∗
𝟏
𝝈√𝑻
The scorewill be
𝑔( 𝑥)∗
𝜖(𝑥)
𝑆0
∗
1
𝜎√𝑇
𝑔( 𝑥)
=
𝝐( 𝒙)
𝑺 𝟎
∗
𝟏
𝝈√𝑻
This method can be used in a multidimensional world,as well as in a path dependent option estimation where the
𝑓( 𝑋1 … 𝑋2
) 𝑎𝑛𝑑 𝑒𝑎𝑐ℎ 𝑋𝑖 is the vector of one dimensional randomvariablewith the same density g(x).
Variance reduction technique:
The efficiency is an important goal,here we will describe the most important one:
Antithetic Variate, itis really easy,itconsists on using for each simulation thegiven percentile and its opposite,
so that they have the same distribution butthey arenot independent, but negatively correlated. 𝐸[ 𝑌 𝐴𝑉] =
1
2
( 𝐸[ 𝑌] + 𝐸[ 𝑌̂]). The varianceis smaller
16 In the caseof European option it is thed1
11.
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11
Control Variate is based on usingthe error in the estimate of known quantities to reduce the error in the
estimate of the unknown one. We will usethe combination of the known variableand the unknown one 𝑌( 𝑏𝑖
) =
𝑌𝑖 − 𝑏( 𝑋𝑖 − 𝐸( 𝑋)), which itwill be used as estimator.
o This estimator is unbiased for 𝑛 → ∞ 𝐸( 𝑌) − 𝑏 ( 𝐸( 𝑋 − 𝐸( 𝑋))) = 𝐸( 𝑌) − 0
o So we need to choose a parameter “b” to minimize the new estimator varianceto ensure 𝑉 ( 𝑌̂( 𝑏)) <
𝑉(𝑌). This method allows to reduce the varianceif the control variateis correlated to the unknown, the
sign do not matter, only sizethe higher the better 𝜎 𝑦(1 − 𝜌𝑥 ,𝑦
2
) with the trivial requirement 𝜌𝑥 ,𝑦
2
≠ 0
o If we jointestimate b and X we will havea bias,in factthose variables will becorrelated so 𝐸(𝑏( 𝑋 −
𝐸( 𝑋)) ≠ 𝐸( 𝑏) 𝐸( 𝑋 − 𝐸( 𝑋)) . To solvethis issuewe need to run two independent simulation, the first
regressingY on X to obtain “b” (𝑛 → ∞ itconverges to the correct valueb) and the second running the
simulation for the estimator itself
o The “b” comes from 𝑉 ( 𝑌𝑖 − 𝑏( 𝑋𝑖 − 𝐸( 𝑋))) = 𝑉( 𝑌𝑖
)+ 𝑉 ( 𝑏( 𝑋𝑖 − 𝐸( 𝑋))) − 2𝑏𝐶𝑜𝑣( 𝑌𝑖; 𝑋𝑖
) = 𝜎 𝑦
2
+
𝑏2
𝜎 𝑥
2
− 2𝑏𝜌𝜎 𝑦 𝜎 𝑥, now we can compute the FOC or justnoticethat itis a parabola so thevertex is the
minimum as well.Note that 𝑉( 𝑌𝑖
( 𝑏)) = 𝜎 𝑦(1 − 𝜌𝑥,𝑦
2
)
Matching underling asset: the key idea is to match the moments of the underlying assetto reduce the risk of
mispricingderivatives.There aretwo possibilities,both of them are assuminga Geometric Brownian motion:
o Simple Moment matching: 𝑆𝑖
̌( 𝑇) =
𝑆 𝑖
( 𝑇) 𝐸[ 𝑆( 𝑇)]
𝑆̂𝑖( 𝑇)
, (explaining 𝑆̂𝑖
( 𝑇) ) this for the firstmatching(which
grants positivepayoff), however it is hard for higher moment. Multiplicative correction. 𝑆𝑖
̌( 𝑇) =
𝑆𝑖
( 𝑇) + 𝐸[ 𝑆( 𝑇)] − 𝑆̂ 𝑖
( 𝑇) 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,whilethe second
does.
o Weighted MC: The paths’Weighs Si (T) for i = 1; …; n with weights “𝑤𝑖” 𝑓𝑜𝑟 𝑖 = 1; … ; 𝑛 such that the
moments of S are matched and then use the same weights to estimate the expected payoff: 𝑌 𝑊𝑀𝐶
=
∑ 𝑤𝑖 𝑌𝑖. Those weights are chosen to maximizethe (negative entropy) distancefrom the uniform
distribution:∑ 𝑤𝑖 ln( 𝑤𝑖
) with the constrain ∑ 𝑤𝑖 = 1 𝑎𝑛𝑑 ∑ 𝑤𝑖 𝑥 𝑖 = 𝜇 𝑥
Basically weare forcing the estimator to have same ∑ 𝑤𝑖 𝑆(𝑡 𝑚) = 𝑆(0) 𝑒 𝑟 𝑡 𝑚
We need to write the Lagrangian and find the FOC [𝑙𝑛𝑤𝑖 − 𝑣 − 𝜆𝑥 𝑖 + 1 = 0] and the result:
𝑤𝑖 = 𝑒−1+𝑣+𝜆𝑥𝑖 but we can rewrite the risk aversion coefficientv as 𝑣 = − ln(∑ 𝜆𝑥 𝑖
)so 𝑤𝑖 =
1
∑ 𝜆𝑥1
𝑒 𝜆𝑥𝑖, by exploiting 1 = ∑ 𝑒−1+𝑣+𝜆𝑥𝑖
Importance sampling (Weighted MC): we want to change the paths importanceof f (X) that have greater impact
on determining the expected value. We proceed to choosethe weight as following:
o At firstwe compute the continuous mean ∫ 𝑓( 𝑥) 𝑓𝑥
( 𝑋) 𝑑𝑥
o We apply the Ridon Nikodin derivatives to change the density measure: ∫
𝑓( 𝑥) 𝑓𝑥
( 𝑋)
𝑔( 𝑥)
𝑔( 𝑥) 𝑑𝑥 the new
measure will be 𝑔( 𝑥) 𝑑𝑥 ==> 𝐸 𝑔
(𝑓(𝑥))
o The new target is 𝐸 𝑔 (
𝑓( 𝑥) 𝑓𝑥
( 𝑋)
𝑔( 𝑥)
) , 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝐸( 𝑓( 𝑥)) by the stronglawof largenumbers, hence
it is unbiased.
o Now we want to find the g(x) that minimize the variance17 we may chosethe 𝑔( 𝑥) =
𝑓( 𝑥) 𝑓𝑥
( 𝑥)
𝑎
where “a”
is the expected valueof f(x). However we cannotdo that sincewe do not know the distribution ex-ante,
but we know that g(x) is proportional to 𝑓( 𝑥) 𝑓𝑥 (𝑥).We can apply an exponential twisting the
𝑔( 𝑥) = 𝑓𝑥
( 𝑥) 𝑒 𝜃𝑥−𝜑( 𝜃)
, this rescaling function which depends only on one parameter.
17 𝐸 𝑔 [
𝑓(𝑥) 𝑓𝑥(𝑋)
𝑔(𝑥)
]
2
? < 𝐸(( 𝑓( 𝑥))
2
) → ∫
𝑓(𝑥)2
𝑓𝑥(𝑥) 2
𝑔 (𝑥)
𝑑𝑥 [= 𝑎 ∫ 𝑓( 𝑥) 𝑓𝑥( 𝑥) 𝑑𝑥 = 𝑎2] < ∫ 𝑓( 𝑥)2
𝑓𝑥( 𝑥) 𝑑𝑥 𝑤𝑖𝑡ℎ 𝑔( 𝑥) =
𝑓(𝑥) 𝑓𝑥(𝑥)
𝑎
12.
Giulio Laudani #12 Cod.20247
12
Firstthe function 𝜑( 𝜃) is a parabola 𝜑( 𝜃−
) = 𝜑( 𝜃+
) (also firstderivatives areequivalent) to
simplify thecomputation. It is the moment generating function of X = 𝑙𝑛𝐸( 𝑒 𝑥𝜃 )and it is
distributed accordingto a Normal. It is made to allowa decreasingmean before a key time, and
increasingafter to push the path closer to the significantpath.
The new function will be∫
𝑓( 𝑥) 𝑓𝑥 (𝑥)
𝑓𝑥(𝑥)𝑒 𝜃𝑥−𝜑(𝜃)
𝑔( 𝑥) 𝑑𝑥 = ∫
1
𝑒 𝜃𝑥−𝜑(𝜃)
𝑔( 𝑥) 𝑑𝑥 new target function. Note
that the multi-dimensional caseis theone used, there will bean g(x) for each period considered
The 𝜃 is chosen depending on the underlingdynamics and itwill changedepending on the
event matching 𝜏, this parameter is compute by doingthe FOC for 𝑖 < 𝜏 𝑎𝑛𝑑 𝑖 > 𝜏
There will two equation one for 𝜃+ 𝑎𝑛𝑑 𝜃−, and by exploitingthe 𝜑( 𝜃) property, and
𝜑( 𝜃) = ( 𝑟 −
𝜎2
2
)∆𝑡𝜃 +
𝜎2
2
𝜃2
∆𝑡 we will have:
𝜃+,− =
(+
−
2𝑏+𝑐
𝑀
−( 𝑟−
𝜎2
2
)∆𝑡)
∆𝑡𝜎2
𝑤ℎ𝑒𝑟𝑒 𝑀 𝑖𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑 𝑎𝑛𝑑 − 𝑏 = ( 𝑥 𝑖 + ⋯ +
𝑥 𝜏
) 𝑎𝑛𝑑 𝑐 = ( 𝑥 𝑖 + ⋯+ 𝑥 𝑀
); 𝜏 =
𝑏
𝜑′( 𝜃)
𝑎𝑛𝑑 𝑀 − 𝜏 =
𝑐+𝑏
𝜑′( 𝜃)
The new variablex will bedistrusted(under g measure) as a normal with same variance
∆𝑡𝜎2
but different mean ( 𝑟 −
𝜎2
2
)∆𝑡 + ∆𝑡𝜎2
𝜃
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