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Jump process, Monte Carlo Simulation variance reduction techniques, Greeks estimation,

American option, multi dimensional BS

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  1. 1. Giulio Laudani #12 Cod.20247 1 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 π‘₯ (𝑑) = π‘‰πœƒ(𝑑), where the replicatingstrategy is a self-financingone(European case) βˆ†π‘‰( 𝑑) = πœƒ1 βˆ†π‘†1 ( 𝑑) + πœƒ0 βˆ†π΅(𝑑), while the discounted one is βˆ†π‘‰Μ†( 𝑑) = πœƒ1 βˆ†π‘†Μ†1 ( 𝑑). The replicatingstrategy πœƒ1 π‘Žπ‘›π‘‘ πœƒ0 can be computed by a backward recursion that involves the conditional covarianceof the option valuewith the underlyingS1: πœƒ1 = π‘π‘œπ‘£(βˆ†π‘‰( 𝑑);βˆ†π‘†( 𝑑)) 𝑉(βˆ†π‘†( 𝑑)) , which is also called the delta of the portfolio and italso the regression coefficientbetween βˆ†π‘‰Μ†( 𝑑); βˆ†π‘†Μ†( 𝑑) πΆπ‘œπ‘£ ( 𝑉̆( 𝑑 + 1); βˆ†π‘†Μ†( 𝑑)) => πΆπ‘œπ‘£ (βˆ†π‘‰Μ†( 𝑑); βˆ†π‘†Μ†( 𝑑)) = πΆπ‘œπ‘£ ( πœƒ1 βˆ†π‘†Μ†1 ( 𝑑);βˆ†π‘†Μ†1 ( 𝑑)) = πœƒ1 πΆπ‘œπ‘£ (βˆ†π‘†Μ†1 ( 𝑑); βˆ†π‘†Μ†1 ( 𝑑)) 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.
  2. 2. Giulio Laudani #12 Cod.20247 2 American Option: The American option pricing is: 𝑉( 𝑑) = sup 𝑑<πœβ‰€π‘‡ 𝐸 𝑄 [ ( 𝑋( 𝜏) 𝐡( 𝜏) )𝑃𝑑 ] 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) β‰₯ 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]. βˆ†π‘‰Μ†( 𝑑) = πœƒ1 βˆ†π‘†Μ†1 ( 𝑑) βˆ’ βˆ†πΆΜ†( 𝑑), 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[ πœ‘( 𝑆( 𝑑 + 1)𝑃𝑑 )] = 𝐸 𝑄[ πœ‘( 𝑆( 𝑑 + 1)𝑆 𝑑 )] A look back American option payoffπ‘₯( 𝑑) = max 0β‰€π‘Žβ‰€π‘‘ 𝑆1 ( π‘Ž) βˆ’ 𝑆1 ( 𝑑) . 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( 𝑑) = max 0β‰€π‘Žβ‰€π‘‘ [𝑆1 ( π‘Ž)] or (to have a better understanding) 𝐹( 𝑑 + 1) = max 0β‰€π‘Žβ‰€π‘‘ [𝐹(𝑑); 𝑆1(𝑑 + 1)] runningmaximum 2. Construct the tree for S and F and this will bea Markovian process,so the process 𝐸 𝑄 [ πœ‘ (( 𝑆( 𝑑 + 1); 𝐹( 𝑑 + 1)) 𝑃𝑑 )] will depend only to F(t) and S(t), so we can rewrite the process as follow 𝐸 𝑄[ πœ‘( 𝑆( 𝑑 + 1); 𝐹( 𝑑 + 1) 𝑆 𝑑 𝐹( 𝑑))]. 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( 𝑑) = 𝐹( 𝑑) βˆ’ 𝑆( 𝑑),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
  3. 3. Giulio Laudani #12 Cod.20247 3 probability 1+π‘Ÿβˆ’π‘‘ π‘’βˆ’π‘‘ ) the updated runningmaximum is 𝐹( 𝑑) = max[𝐹( 𝑑); 𝑆( 𝑑) βˆ— 𝑒] 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 β€œt”,a Gaussiandistribution(for differenceofmotionwith differenttime interval) and each intervalis independentfrom thepreviously one
  4. 4. Giulio Laudani #12 Cod.20247 4 𝐡( 𝑑) = π‘’βˆ« π‘Ÿ( 𝑠) 𝑑𝑠 𝑑𝑆( 𝑑) 𝑆( 𝑑) = π‘Ž( 𝑑) 𝑑𝑑 + 𝑏( 𝑑) π‘‘π‘Š( 𝑑) 𝑆( 𝑑) = 𝑆(0) + ∫ 𝑆(𝑑)πœ‡π‘‘π‘‘ + ∫ 𝑆(𝑑)πœŽπ‘‘π‘Š(𝑑) ==> 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( 𝑑; 𝑋( 𝑑)) 𝑑𝑑 βˆ— 𝑑𝑑 + 𝑓( 𝑑; 𝑋( 𝑑)) 𝑑π‘₯ βˆ— 𝑑𝑋( 𝑑) + 1 2 βˆ— 2𝑓( 𝑑; 𝑋( 𝑑)) 𝑑π‘₯2 βˆ— 𝑏2 (𝑑; 𝑋(𝑑)) 𝑑𝑑 Thanks to this trick we can solve the SDE and obtain the PDE of the security dynamics as following: 𝑆( 𝑑) = 𝑆(0) βˆ— 𝑒 ( πœ‡βˆ’ 1 2 βˆ—πœŽ2 ) 𝑑𝑑+πœŽπ‘‘π‘Š πΆπ‘œπ‘Ÿπ‘Ÿ ( 𝑑𝑆1 𝑆1 ; 𝑑𝑆2 𝑆2 ) = 𝜎1 𝜎2 𝑑𝑑 𝜎1βˆšπ‘‘π‘‘πœŽ2βˆšπ‘‘π‘‘ = 1 𝐸[ 𝑆( 𝑑)] = 𝑆0 𝑒 πœ‡π‘‘ ∫ 𝑒 βˆ’ 1 2 𝜎2 𝑑+πœŽβˆšπ‘‘ 𝑧 βˆ— 1 √2πœ‹ 𝑒 βˆ’ 𝑧2 2 𝑑𝑧 = ∫ 1 √2πœ‹ 𝑒 βˆ’ 1 2 ( π‘§βˆ’πœŽβˆšπ‘‘) 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. π‘π‘œπ‘£( 𝑆(1), 𝑆(2)) = Σ𝑖,𝑗 = πœŽπ‘– 𝑇 πœŽπ‘— = ( 𝜎1,1 𝜎2,1 + 𝜎1,2 𝜎2,2) 𝑑𝑑 𝑑𝑆𝑖( 𝑑) 𝑆𝑖( 𝑑) = πœ‡ 𝑖(𝑆(𝑑); 𝑑)𝑑𝑑 + πœŽπ‘– 𝑇 ( 𝑆( 𝑑); 𝑑) π‘‘π‘Š 𝑃 (𝑑) 𝑉 π‘Žπ‘Ÿ ( 𝑑𝑆1( 𝑑) 𝑆1( 𝑑) ) = (𝜎1,1 2 + 𝜎1,2 2 )𝑑𝑑 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 β€œP” into the motion under β€œQ” by the usual drift transformation. 𝑣𝑖 = πœ‡ 𝑖 βˆ’ π‘Ÿ πœŽπ‘– β†’ π‘Š 𝑃 = π‘Š 𝑄 βˆ’ 𝑣𝑑𝑑 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 (𝑑) = 𝑆𝑖 (0) 𝑒 ( π‘Ÿ1βˆ’ 1 2 πœŽπ‘– 𝑇 πœŽπ‘– ) 𝑑𝑑+πœŽπ‘– 𝑇 π‘Š 𝑄( 𝑑) The Ito’s formula in the multi and the change of drift are: 3 We can achievethesameresultofmodeling thecorrelationby assuming a correlatedBrownianmotion
  5. 5. Giulio Laudani #12 Cod.20247 5 π‘‘π‘Œ( 𝑑) = 𝑑𝑓( 𝑑, 𝑋( 𝑑)) 𝑑𝑑 + βˆ‘ 𝑑𝑓( 𝑑, 𝑋( 𝑑)) 𝑑π‘₯ 𝑖 𝑑𝑋𝑖 (𝑑) + 1 2 βˆ‘ 𝑑2 𝑓( 𝑑, 𝑋( 𝑑)) 𝑑π‘₯ 𝑖 𝑑π‘₯𝑗 𝑏𝑖 𝑑 𝑏𝑗 𝑑𝑑 π‘‘π‘Š 𝑄 = π‘‘π‘Š 𝑃 + 𝑣𝑑𝑑 π‘€β„Žπ‘’π‘Ÿπ‘’ πœ‡ 𝑖 βˆ’ π‘Ÿ = πœŽπ‘– 𝑇 𝑣 The parameters are the one of β€œπ‘“( 𝑑, 𝑋( 𝑑))”. The hedging strategy in this case is similar to the one-dimensional case, πœƒ0 = 𝐹( 𝑑, 𝑆( 𝑑)) βˆ’ βˆ‘ β„Žπ‘– 𝑆𝑖 π‘Žπ‘›π‘‘ πœƒπ‘– = 𝑑𝐹( 𝑑,𝑆( 𝑑)) 𝑑 𝑆 𝑖 β†’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{ π‘Š: π‘Ÿ( 𝑀) < 𝑑} ∈ β„±(π‘Ÿ), 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( 𝑑;𝑆) 𝑑𝑑 + 𝐹( 𝑑;𝑆) 𝑑𝑆 βˆ— π‘†π‘Ÿ + 1 2 βˆ— 𝐹2( 𝑑;𝑆) 𝑑2 𝑆 βˆ— 𝑆2 𝜎2 βˆ’ 𝐹( 𝑑; 𝑆) π‘Ÿ ≀ 0 π‘“π‘œπ‘Ÿ π‘Žπ‘›π‘¦ ( 𝑑; 𝑆) ∈ [0; 𝑇] 𝑋 β„›+ 2. ( 𝐹( 𝑑;𝑆) 𝑑𝑑 + 𝐹 ( 𝑑;𝑆) 𝑑𝑆 βˆ— π‘†π‘Ÿ + 1 2 βˆ— 𝐹2( 𝑑;𝑆) 𝑑2 𝑆 βˆ— 𝑆2 𝜎2 βˆ’ 𝐹( 𝑑; 𝑆) π‘Ÿ) (𝐹( 𝑑; 𝑆) βˆ’ 𝑓(𝑆)) = 0 π‘“π‘œπ‘Ÿ π‘Žπ‘›π‘¦ (𝑑; 𝑆) ∈ [0; 𝑇] β„›+ 3. 𝐹( 𝑑; 𝑆) β‰₯ 𝑓(𝑆) π‘“π‘œπ‘Ÿ π‘Žπ‘›π‘¦ (𝑑; 𝑆) ∈ [0; 𝑇] β„›+ 4. 𝐹(𝑇; 𝑆) = 𝑓(𝑆) π‘“π‘œπ‘Ÿ π‘Žπ‘›π‘¦ 𝑆 ∈ β„›+ The firstequation is the Ito’s formula applied to a portfolio short on the derivatives and long on β€œh” units of the underling, by imposing to be a risk free portfolio, hence β„Ž = 𝑑𝐹( 𝑑) 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( 𝑑;𝑆) 𝑑𝑑 + 𝐹( 𝑑;𝑆) 𝑑𝑆 βˆ— π‘†π‘Ÿ + 1 2 βˆ— 𝐹2( 𝑑;𝑆) 𝑑2 𝑆 βˆ— 𝑆2 𝜎2 βˆ’ 𝐹( 𝑑; 𝑆) π‘Ÿ = 0 π‘“π‘œπ‘Ÿ π‘Žπ‘›π‘¦ ( 𝑑; 𝑆) ∈ [0; 𝑇], 𝑆 > π‘†βˆ—( 𝑑) 2. 𝐹( 𝑑; 𝑆) = ( 𝐾 βˆ’ 𝑆)+ π‘“π‘œπ‘Ÿ π‘Žπ‘›π‘¦ 𝑑 ∈ [0; 𝑇], 𝑆 = π‘†βˆ—( 𝑑) 3. 𝑑𝐹( 𝑑; 𝑆) 𝑑𝑆 = βˆ’1 π‘“π‘œπ‘Ÿ π‘Žπ‘›π‘¦ 𝑑 ∈ [0; 𝑇], 𝑆 = π‘†βˆ—( 𝑑) smooth pasting 4. 𝐹(𝑇; 𝑆) = 𝑓(𝑆) π‘“π‘œπ‘Ÿ π‘Žπ‘›π‘¦ 𝑆 ∈ β„›+ 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:
  6. 6. Giulio Laudani #12 Cod.20247 6 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∞ ( 𝑑) = { 𝐴𝑆 π‘Ž π‘“π‘œπ‘Ÿ 𝑆 > π‘†βˆž 𝐾 βˆ’ 𝑆 π‘“π‘œπ‘Ÿ 𝑆 ≀ π‘†βˆž} Where the elements in the basket are respectively: π‘Ž = (βˆ’( π‘Ÿβˆ’πœŽ2 )βˆ’βˆš π‘Ÿ2+𝜎2 π‘Ÿ+ 𝜎4 4 ) 𝜎2 < 0 and π‘†βˆž βˆ— = βˆ’1βˆ’π‘Ž π‘Ž 𝐾 < 𝐾 and 𝐴 = π‘†βˆž βˆ— (1βˆ’π‘Ž) βˆ’π‘Ž . The solution comes from applying the Ito’s formula applied to 𝐴𝑆 π‘Ž π‘“π‘œπ‘Ÿ 𝑆 > π‘†βˆž and condition 1, which leads to the solution β€œa” and we should take the negative value since we want a decreasing function in mean. 0 + 𝐴𝑆 π‘Žβˆ’1 βˆ— π‘Žπ‘†π‘Ÿ + 1 2 𝐴𝑆 π‘Žβˆ’2( π‘Ž βˆ’ 1) π‘Žπ‘†2 𝜎2 βˆ’ 𝐴𝑆 π‘Ž π‘Ÿ = 0 β†’ 𝐴𝑆 π‘Ž [ π‘Žπ‘Ÿ + 1 2 π‘Ž2 𝜎2 βˆ’ 1 2 π‘ŽπœŽ2 βˆ’ π‘Ÿ] = 0 π‘Ž ( π‘Ÿ βˆ’ 𝜎2 2 ) + π‘Ž2 𝜎2 2 βˆ’ π‘Ÿ = 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 π‘†βˆ—( 𝑑) > π‘†βˆž ( 𝑑) 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(tβˆ’)(ΞΌdt + ΟƒdW(t) + dJ(t)) 𝐽( 𝑑) = βˆ‘( π‘Œπ‘— βˆ’ 1) 𝑁(𝑑) 𝐽=1 𝑃 [ 𝑁( 𝑑) = 𝑛] = π‘’βˆ’πœ†π‘‘ ( πœ†π‘‘) 𝑛 𝑛! π‘€π‘–π‘‘β„Ž 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 πœ†π‘‘ and marginal probability of occurrence πœ†π‘‘π‘‘, where πœ†6 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( π‘Ÿπ‘‘ ) = 𝑆( π‘‘βˆ’ ) π‘Œπ‘‘, 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. 7. Giulio Laudani #12 Cod.20247 7 𝑆( 𝑑) = 𝑆(0) 𝑒 ( πœ‡βˆ’ 1 2 𝜎2 ) 𝑑𝑑+πœŽπ‘Š( 𝑑) (∏( π‘Œπ‘—) 𝑁( 𝑑) 𝑗=1 ) π‘€π‘–π‘‘β„Ž ∏( π‘Œπ‘—) = 1 0 𝑗=1 π‘Žπ‘›π‘‘ 𝐸 [∏( π‘Œπ‘—) 𝑁( 𝑑) 𝑗=1 ] = 𝑒 πœ†π‘‘( 𝐸( π‘Œ1)βˆ’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( 𝑑)) = S(0)e πœ‡π‘‘ 𝑒 πœ†π‘‘( 𝐸( π‘Œ1)βˆ’1) 𝑉( 𝑆( 𝑑)) = 𝑆2(0)( 𝑒(2πœ‡+𝜎2) 𝑑 𝑒 πœ†π‘‘ ( 𝐸[ π‘Œ2 ]βˆ’1) βˆ’ e2πœ‡π‘‘ 𝑒2πœ†π‘‘( 𝐸( π‘Œ1)βˆ’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 Ξ»Q = ϕλP 7, where Ο• > 0 and jointly we need to change the drift of the Wiener process π‘Šπ‘‘ 𝑄 = π‘Šπ‘‘ 𝑃 + πœƒπ‘‘. So the SDE under Q of the discounted stock differential will be: ( πœ‡ βˆ’ π‘Ÿ) 𝑑𝑑 + πœŽπ‘‘π‘Š(𝑑) 𝑃 + 𝑑𝐽( 𝑑) Β± ( πœ‡ 𝑦 βˆ’ 1) πœ† 𝑄 𝑑𝑑 = ( πœ‡ βˆ’ π‘Ÿ βˆ’ πœŽπœƒ + ( πœ‡ 𝑦 βˆ’ 1) πœ† 𝑄) 𝑑𝑑 + πœŽπ‘‘π‘Š 𝑄 ( 𝑑) + βˆ†π½( 𝑑) βˆ’ ( πœ‡ 𝑦 βˆ’ 1) πœ† 𝑄 𝑑𝑑 where πœ‡ 𝑦 = 𝐸 𝑃 (π‘Œ). If we compute the expected value we notice that the last term is equivalent to a pure jump martingale under Q 𝐸 𝑄[ 𝑑𝐽( 𝑑)𝑃𝑑 ] = 𝐸[ 𝑑𝐽( 𝑑)] βˆ— 𝑄( π½π‘’π‘šπ‘ π‘π‘Ÿπ‘œπ‘) = ( πœ‡ 𝑦 βˆ’ 1) πœ† 𝑄 𝑑𝑑, hence the mean is zero. We end up with ( πœ‡ βˆ’ π‘Ÿ βˆ’ πœŽπœƒ + ( πœ‡ 𝑦 βˆ’ 1) πœ† 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 Ο• = 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 πœƒ = ( πœ‡βˆ’π‘Ÿ+( πœ‡ π‘¦βˆ’1) πœ† 𝑄) 𝜎 β†’ ( π‘Ÿ βˆ’ (πœ‡ 𝑦 βˆ’ 1)Ξ»Q βˆ’ 1 2 𝜎2). The PDE 𝑆( 𝑑) = 𝑆(0) 𝑒 ( π‘Ÿβˆ’(πœ‡ π‘¦βˆ’1)Ξ»Qβˆ’ 1 2 𝜎2) 𝑑𝑑+πœŽπ‘Š 𝑄( 𝑑) (∏ ( π‘Œπ‘—)𝑁( 𝑑) 𝑗=1 ), note that the drift under q is π‘Ÿ βˆ’ (πœ‡ 𝑦 βˆ’ 1)Ξ»Q 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( 𝑑)) = 𝑆(0) 𝑒 π‘Ÿπ‘‘ 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 β€œn” 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 π‘Œ = 𝑒 𝑋~𝑁( π‘Ž;𝑏2 ) , so the PDE will be 𝑆( 𝑑) = 𝑆(0) 𝑒 ( π‘Ÿβˆ’(πœ‡ π‘¦βˆ’1)Ξ»Qβˆ’ 1 2 𝜎2) 𝑑𝑑+πœŽπ‘Š 𝑄( 𝑑) π‘’βˆ‘ 𝑋𝑖 if we modify the equation to use the standardized distribution: 𝑆( 𝑑) = 𝑆(0) 𝑒 ( π‘Ÿ 𝑛 π‘‡βˆ’ 1 2 𝜎2 𝑇)+𝜎 𝑛 βˆšπ‘‘π‘ . π‘Ÿπ‘› = π‘Ÿ βˆ’ π‘š[= πœ‡ 𝑦 βˆ’ 1]Ξ» + 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( πœƒ) = 𝑛] and 𝑃[( 𝑆( 𝑑) βˆ’ 𝐾)+ π‘’βˆ’π‘Ÿπ‘‘ 𝑁( 𝑑) = 𝑛] = βˆ‘ 𝑒λ′t(Ξ»β€² t) n 𝑛! 𝐸 𝑄 [𝑆(0) 𝑒 ( π‘Ÿ 𝑛 π‘‡βˆ’ 1 2 𝜎2 𝑇)+𝜎 𝑛 βˆšπ‘‘π‘ βˆ— π‘’βˆ’π‘Ÿ 𝑛 𝑇] where the last term is the BS formula and Ξ»β€² = (1 + π‘š)Ξ» this last change has been made to change r with π‘Ÿπ‘› 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( 𝑑) βˆ’ 𝐹( π‘‘βˆ’ ))βˆ†π‘( 𝑑) for the dF(t), if we compute the integral of 𝐹( 𝑑) βˆ’ 𝐹(0) it will became βˆ‘ 𝐹( 𝑑, βˆ†π‘†( π‘‘βˆ’ ) π‘Œπ‘– ) βˆ’ 𝐹( 𝑑, βˆ†π‘†( π‘‘βˆ’ )) 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.
  8. 8. Giulio Laudani #12 Cod.20247 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 β€œlarge number rules8”. π‘ŽΜ‚ = 1 𝑛 βˆ‘ 𝑓(π‘₯) π‘€β„Žπ‘’π‘Ÿπ‘’ lim π‘›β†’βˆž π‘ŽΜ‚ = π‘Ž π‘Žπ‘›π‘‘ 𝐸( π‘ŽΜ‚) = π‘Ž | π‘ŽΜ‚ βˆ’ π‘Ž| = πœŽΜ‚ βˆšπ‘› 𝑧 𝛼 π‘€β„Žπ‘’π‘Ÿπ‘’ π‘‘β„Žπ‘’ π‘Ÿπ‘–π‘”β„Žπ‘‘ 𝑠𝑖𝑑𝑒 π‘œπ‘“ π‘‘β„Žπ‘’ 𝑠𝑖𝑔𝑛 𝑖𝑠 π‘π‘Žπ‘™π‘™π‘’π‘‘ π‘Ÿπ‘Žπ‘‘π‘–π‘’π‘  π‘œπ‘“ π‘‘β„Žπ‘’ π‘ π‘–π‘šπ‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘› 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 βˆšπ‘› , 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 𝐹π‘₯ βˆ’1 (π‘ˆ)~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: ο‚· For the first case no problem , just find the percentile as function of β€œU” form 𝑒 = 𝐹π‘₯ (𝑋) ο‚· For discrete case we need to define range in which any value of the β€œU” will be assign to the correct probability measure, basically we will look for 𝑃[π‘ž π‘—βˆ’1 < π‘ˆ ≀ π‘ž 𝑗] the right extreme inclusion is a convention [Generalize] The proof of this relationship is based on the fact that since: 𝐹π‘₯ βˆ’1( 𝑒) ≀ π‘₯ 𝑖𝑖𝑓 𝑒 ≀ 𝐹π‘₯ (𝑋), which can be proven by checking that 𝐹π‘₯ βˆ’1( 𝐹π‘₯ βˆ’1( 𝑒)) ≀ 𝐹π‘₯ ( π‘₯) <=> 𝑒 ≀ 𝐹π‘₯ (𝑋), so 𝑃[ 𝐹π‘₯ βˆ’1( 𝑒) ≀ π‘₯] = 𝑃[ 𝑒 ≀ 𝐹π‘₯ ( 𝑋)] => 𝐹π‘₯ βˆ’1 (π‘₯) 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[( π‘ŽΜ‚ βˆ’ π‘Ž)2] = 𝐸[ 𝐸( π‘ŽΜ‚) βˆ’ π‘Ž]2 + 𝐸[ π‘ŽΜ‚ βˆ’ 𝐸( π‘ŽΜ‚)]2 = π΅π‘–π‘Žπ‘ ( π‘ŽΜ‚)2 + π‘‰π‘Žπ‘Ÿπ‘–π‘Žπ‘›π‘π‘’(π‘ŽΜ‚) 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 β€œh” and the number of sample used β€œn”, which contributes to the radius efficiency with the following dumb relationship π‘›β„Ž2 β†’ ∞ 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. 9. Giulio Laudani #12 Cod.20247 9 The time efficiency is considered as following: π‘›β„Ž = 𝑇 π‘Ÿβ„Ž and the radius expressed in terms of time per simulation π‘Ÿβ„Ž 12 is βˆšπ‘‰β„Ž π‘Ÿβ„Ž βˆšπ‘‡ where the variance is the one granted by the procedure by applying β€œh”. 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: ο‚· 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 β€œh” size, however we need to control the varianceexplosion problem13. There existtwo possiblemethods: o Forward Difference: π‘ŽπΌ ( πœƒ) β‰ˆ π‘Ž( πœƒ+β„Ž)βˆ’π‘Ž( πœƒ) β„Ž π‘€π‘–π‘‘β„Ž β„Ž β†’ 0. The bias in this caseis reduced by a linear function regardless the number of Taylor expansion terms in the proxy used, i.e. π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ βˆ— β„Ž.  π‘Ž( πœƒ + β„Ž) βˆ’ π‘Ž( πœƒ) = π‘Žβ€² β„Ž + 1 2 π‘Žβ€²β€² β„Ž2 + π‘œ(β„Ž2 ) hence the 𝐸[βˆ†π‘Ž] = π‘Žβ€² + 1 2 π‘Žβ€²β€² β„Ž + π‘œ(β„Ž), so the bias 𝐸[βˆ†π‘Ž] βˆ’ π‘Žβ€² = 1 2 π‘Žβ€²β€² β„Ž + π‘œ(β„Ž) ==> β„Ž βˆ— π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘14  π‘Žβ€² β„Ž + 1 2 π‘Žβ€²β€² β„Ž2 + π‘Žβ€²β€²β€² β„Ž3 1 6 + π‘œ(β„Ž3 ), still β€œh”is the higher order o Central Difference: π‘ŽπΌ ( πœƒ) β‰ˆ π‘Ž( πœƒ+β„Ž)βˆ’π‘Ž( πœƒβˆ’β„Ž) 2β„Ž π‘€π‘–π‘‘β„Ž β„Ž β†’ 0. The bias goes to zero faster than the forward case π‘œ(β„Ž), 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 π‘œ(β„Ž2)  π‘Ž( πœƒ + β„Ž) βˆ’ π‘Ž( πœƒ) = π‘Žβ€² β„Ž + 1 2 π‘Žβ€²β€² β„Ž2 + 1 6 π‘Žβ€²β€²β€² β„Ž3 + π‘œ(β„Ž3) βˆ’ (βˆ’π‘Žβ€² β„Ž + 1 2 π‘Žβ€²β€² β„Ž2 βˆ’ π‘Žβ€²β€²β€² 1 6 β„Ž3 + π‘œ(β„Ž3 )), so the bias 2π‘Žβ€²β„Ž+ 2 3 π‘Žβ€²β€²β€²β„Ž3 +π‘œ(β„Ž3) 2β„Ž βˆ’ π‘Žβ€² = 1 3 π‘Žβ€²β€²β€² β„Ž2 + π‘œ(β„Ž2 ) o Speaking about the Varianceeffect we need to consider to possibleestimation procedure:  Independent sampling ο‚· 𝑉𝑓 (βˆ†π‘Ž) = β„Žβˆ’2 𝑉 ( βˆ‘ βˆ†π‘Ž 𝑛 ) = β„Žπ‘›βˆ’2 βˆ‘[𝑉( 𝑦( 𝑑 + β„Ž)) + 𝑉( 𝑦( 𝑑)) ] = 2 β„Ž2 𝑛 βˆ— πΆπ‘œπ‘›π‘ π‘‘ ο‚· 𝑉𝐢 (βˆ†π‘Ž) = β„Žβˆ’2 4 𝑉 ( βˆ‘ βˆ†π‘Ž 𝑛 ) = 1 2β„Ž2 𝑛 βˆ— πΆπ‘œπ‘›π‘ π‘‘  Same seed for both the sampling ο‚· 𝑉𝑓 (βˆ†π‘Ž) = β„Žβˆ’2 𝑉 ( βˆ‘ βˆ†π‘Ž 𝑛 ) = β„Ž βˆ— π‘π‘œπ‘›π‘ π‘‘ β„Ž2 𝑛 = π‘π‘œπ‘›π‘ π‘‘ β„Žπ‘› ο‚· 𝑉𝐢 (βˆ†π‘Ž) = β„Žβˆ’2 4 𝑉 ( βˆ‘ βˆ†π‘Ž 𝑛 ) = β„Ž βˆ— πΆπ‘œπ‘›π‘ π‘‘ β„Ž24𝑛 = πΆπ‘œπ‘›π‘ π‘‘ β„Ž4𝑛 ο‚· 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 π‘‘πœƒ 𝐸[ π‘Œ( πœƒ)] = 𝐸 [ 𝑑 π‘‘πœƒ π‘Œ( πœƒ)] π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑑 π‘‘πœƒ π‘Œ( πœƒ) = π‘Œ 𝐼 (πœƒ), basically wewill estimate the samplemean of that quantity 1 𝑛 βˆ‘ π‘Œ 𝐼 ( πœƒ). 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 βˆ— h2 for FD while n βˆ— 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.
  10. 10. Giulio Laudani #12 Cod.20247 10 o For practical usethe estimator that we will use 𝑑 π‘‘πœƒ π‘Œ( πœƒ) = π‘‘π‘Œ( πœƒ) 𝑑𝑆( 𝑇) βˆ— 𝑑𝑆( 𝑇) π‘‘πœƒ where S(T) is the underlying and πœƒ 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: π‘’βˆ’π‘Ÿπ‘‘ 𝐼 𝑆( 𝑇)>𝐾 since π‘Œ( πœƒ) = ( 𝑆 βˆ’ 𝐾)+ β†’ 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 [ π‘’βˆ’π‘Ÿπ‘‘ 𝐼𝑆( 𝑇)>𝐾 𝑆( 𝑇) 𝑆(0) ] = π‘’βˆ’π‘Ÿπ‘‘ 𝑆(0) 𝐸( 𝑆( 𝑇)) 𝐸( 𝐼 𝑆( 𝑇)>𝐾) = π‘’βˆ’π‘Ÿπ‘‘ 𝑆(0) βˆ— 𝑆(0) 𝑒 π‘Ÿπ‘‘ 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) ο‚· 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 π‘‘πœƒ 𝐸[ π‘Œ( πœƒ)] = ∫ 𝑑 π‘‘πœƒ βˆ— π‘¦π‘”πœƒ ( 𝑦) 𝑑𝑦 = ∫ 𝑦 βˆ— π‘‘π‘”πœƒ ( 𝑦) π‘‘πœƒ π‘”πœƒ ( 𝑦) π‘”πœƒ ( 𝑦) 𝑑𝑦 π‘‘π‘”πœƒ ( 𝑦) π‘‘πœƒ π‘”πœƒ ( 𝑦) = π‘†π‘π‘œπ‘Ÿπ‘’( π‘Œ) ==> 1 𝑛 βˆ‘ π‘Œπ‘– βˆ— π‘†π‘π‘œπ‘Ÿπ‘’(π‘Œ) Where π‘”πœƒ ( 𝑦) = 𝑑 𝑑π‘₯ 𝑄[ 𝑆 > 𝐾] πΈπ‘’π‘Ÿπ‘œπ‘π‘’π‘Žπ‘› π‘π‘Žπ‘™π‘™ is the density function of y for a fixed parameter πœƒ. 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: ο‚· At firstcompute the density function to respect of the parameter, which is S0, so the density is the 𝑑 𝑑π‘₯ 𝑁 ( 𝑑1 [= ln( 𝐾 𝑆(0) )βˆ’( π‘Ÿβˆ’ 𝜎2 2 ) 𝑑 πœŽβˆšπ‘‡ ]), so 𝑔( π‘₯) = 𝑁( 𝑑1 )[= πœ–] βˆ— πœ–β€² [= 1 πœŽβˆšπ‘‡ βˆ— 1 π‘₯ ] ο‚· Then we need to compute the derivatives of g(x) to respect to S0: 𝑑 𝑑𝑆0 𝑔( π‘₯) = 𝑑 𝑑𝑆0 𝑁( πœ–(π‘₯)) βˆ— 1 πœŽβˆšπ‘‡ βˆ— 1 π‘₯ = 1 πœŽβˆšπ‘‡ βˆ— 1 π‘₯ βˆ— ( 1 √2πœ‹ βˆ— 𝑒 βˆ’ πœ–(π‘₯)2 2 ) βˆ— βˆ’πœ–( π‘₯) βˆ— π‘‘πœ–( π‘₯) 𝑑 𝑆0 [= βˆ’ 1 πœŽβˆšπ‘‡ βˆ— 1 𝑆0 ] = π’ˆ( 𝒙) βˆ— 𝝐( 𝒙) 𝑺 𝟎 βˆ— 𝟏 πˆβˆšπ‘» ο‚· The scorewill be 𝑔( π‘₯)βˆ— πœ–(π‘₯) 𝑆0 βˆ— 1 πœŽβˆšπ‘‡ 𝑔( π‘₯) = 𝝐( 𝒙) 𝑺 𝟎 βˆ— 𝟏 πˆβˆšπ‘» This method can be used in a multidimensional world,as well as in a path dependent option estimation where the 𝑓( 𝑋1 … 𝑋2 ) π‘Žπ‘›π‘‘ π‘’π‘Žπ‘β„Ž 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: ο‚· 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[ π‘Œ 𝐴𝑉] = 1 2 ( 𝐸[ π‘Œ] + 𝐸[ π‘ŒΜ‚]). The varianceis smaller 16 In the caseof European option it is thed1
  11. 11. Giulio Laudani #12 Cod.20247 11 ο‚· 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 π‘Œ( 𝑏𝑖 ) = π‘Œπ‘– βˆ’ 𝑏( 𝑋𝑖 βˆ’ 𝐸( 𝑋)), which itwill be used as estimator. o This estimator is unbiased for 𝑛 β†’ ∞ 𝐸( π‘Œ) βˆ’ 𝑏 ( 𝐸( 𝑋 βˆ’ 𝐸( 𝑋))) = 𝐸( π‘Œ) βˆ’ 0 o So we need to choose a parameter β€œb” to minimize the new estimator varianceto ensure 𝑉 ( π‘ŒΜ‚( 𝑏)) < 𝑉(π‘Œ). 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 βˆ’ 𝜌π‘₯ ,𝑦 2 ) with the trivial requirement 𝜌π‘₯ ,𝑦 2 β‰  0 o If we jointestimate b and X we will havea bias,in factthose variables will becorrelated so 𝐸(𝑏( 𝑋 βˆ’ 𝐸( 𝑋)) β‰  𝐸( 𝑏) 𝐸( 𝑋 βˆ’ 𝐸( 𝑋)) . To solvethis issuewe need to run two independent simulation, the first regressingY on X to obtain β€œb” (𝑛 β†’ ∞ itconverges to the correct valueb) and the second running the simulation for the estimator itself o The β€œb” comes from 𝑉 ( π‘Œπ‘– βˆ’ 𝑏( 𝑋𝑖 βˆ’ 𝐸( 𝑋))) = 𝑉( π‘Œπ‘– )+ 𝑉 ( 𝑏( 𝑋𝑖 βˆ’ 𝐸( 𝑋))) βˆ’ 2π‘πΆπ‘œπ‘£( π‘Œπ‘–; 𝑋𝑖 ) = 𝜎 𝑦 2 + 𝑏2 𝜎 π‘₯ 2 βˆ’ 2π‘πœŒπœŽ 𝑦 𝜎 π‘₯, now we can compute the FOC or justnoticethat itis a parabola so thevertex is the minimum as well.Note that 𝑉( π‘Œπ‘– ( 𝑏)) = 𝜎 𝑦(1 βˆ’ 𝜌π‘₯,𝑦 2 ) ο‚· 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)] βˆ’ 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 β€œπ‘€π‘–β€ π‘“π‘œπ‘Ÿ 𝑖 = 1; … ; 𝑛 such that the moments of S are matched and then use the same weights to estimate the expected payoff: π‘Œ π‘Šπ‘€πΆ = βˆ‘ 𝑀𝑖 π‘Œπ‘–. Those weights are chosen to maximizethe (negative entropy) distancefrom the uniform distribution:βˆ‘ 𝑀𝑖 ln( 𝑀𝑖 ) with the constrain βˆ‘ 𝑀𝑖 = 1 π‘Žπ‘›π‘‘ βˆ‘ 𝑀𝑖 π‘₯ 𝑖 = πœ‡ π‘₯  Basically weare forcing the estimator to have same βˆ‘ 𝑀𝑖 𝑆(𝑑 π‘š) = 𝑆(0) 𝑒 π‘Ÿ 𝑑 π‘š  We need to write the Lagrangian and find the FOC [𝑙𝑛𝑀𝑖 βˆ’ 𝑣 βˆ’ πœ†π‘₯ 𝑖 + 1 = 0] and the result: 𝑀𝑖 = π‘’βˆ’1+𝑣+πœ†π‘₯𝑖 but we can rewrite the risk aversion coefficientv as 𝑣 = βˆ’ ln(βˆ‘ πœ†π‘₯ 𝑖 )so 𝑀𝑖 = 1 βˆ‘ πœ†π‘₯1 𝑒 πœ†π‘₯𝑖, by exploiting 1 = βˆ‘ π‘’βˆ’1+𝑣+πœ†π‘₯𝑖 ο‚· 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( π‘₯) 𝑓π‘₯ ( 𝑋) 𝑑π‘₯ o We apply the Ridon Nikodin derivatives to change the density measure: ∫ 𝑓( π‘₯) 𝑓π‘₯ ( 𝑋) 𝑔( π‘₯) 𝑔( π‘₯) 𝑑π‘₯ the new measure will be 𝑔( π‘₯) 𝑑π‘₯ ==> 𝐸 𝑔 (𝑓(π‘₯)) o The new target is 𝐸 𝑔 ( 𝑓( π‘₯) 𝑓π‘₯ ( 𝑋) 𝑔( π‘₯) ) , π‘€β„Žπ‘–π‘β„Ž 𝑖𝑠 π‘’π‘žπ‘’π‘Žπ‘™ π‘‘π‘œ 𝐸( 𝑓( π‘₯)) 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( π‘₯) = 𝑓( π‘₯) 𝑓π‘₯ ( π‘₯) π‘Ž where β€œa” 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( π‘₯) 𝑓π‘₯ (π‘₯).We can apply an exponential twisting the 𝑔( π‘₯) = 𝑓π‘₯ ( π‘₯) 𝑒 πœƒπ‘₯βˆ’πœ‘( πœƒ) , this rescaling function which depends only on one parameter. 17 𝐸 𝑔 [ 𝑓(π‘₯) 𝑓π‘₯(𝑋) 𝑔(π‘₯) ] 2 ? < 𝐸(( 𝑓( π‘₯)) 2 ) β†’ ∫ 𝑓(π‘₯)2 𝑓π‘₯(π‘₯) 2 𝑔 (π‘₯) 𝑑π‘₯ [= π‘Ž ∫ 𝑓( π‘₯) 𝑓π‘₯( π‘₯) 𝑑π‘₯ = π‘Ž2] < ∫ 𝑓( π‘₯)2 𝑓π‘₯( π‘₯) 𝑑π‘₯ π‘€π‘–π‘‘β„Ž 𝑔( π‘₯) = 𝑓(π‘₯) 𝑓π‘₯(π‘₯) π‘Ž
  12. 12. Giulio Laudani #12 Cod.20247 12  Firstthe function πœ‘( πœƒ) is a parabola πœ‘( πœƒβˆ’ ) = πœ‘( πœƒ+ ) (also firstderivatives areequivalent) to simplify thecomputation. It is the moment generating function of X = 𝑙𝑛𝐸( 𝑒 π‘₯πœƒ )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( π‘₯) 𝑓π‘₯ (π‘₯) 𝑓π‘₯(π‘₯)𝑒 πœƒπ‘₯βˆ’πœ‘(πœƒ) 𝑔( π‘₯) 𝑑π‘₯ = ∫ 1 𝑒 πœƒπ‘₯βˆ’πœ‘(πœƒ) 𝑔( π‘₯) 𝑑π‘₯ new target function. Note that the multi-dimensional caseis theone used, there will bean g(x) for each period considered  The πœƒ is chosen depending on the underlingdynamics and itwill changedepending on the event matching 𝜏, this parameter is compute by doingthe FOC for 𝑖 < 𝜏 π‘Žπ‘›π‘‘ 𝑖 > 𝜏 ο‚· There will two equation one for πœƒ+ π‘Žπ‘›π‘‘ πœƒβˆ’, and by exploitingthe πœ‘( πœƒ) property, and πœ‘( πœƒ) = ( π‘Ÿ βˆ’ 𝜎2 2 )βˆ†π‘‘πœƒ + 𝜎2 2 πœƒ2 βˆ†π‘‘ we will have: ο‚· πœƒ+,βˆ’ = (+ βˆ’ 2𝑏+𝑐 𝑀 βˆ’( π‘Ÿβˆ’ 𝜎2 2 )βˆ†π‘‘) βˆ†π‘‘πœŽ2 π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑀 𝑖𝑠 π‘‘β„Žπ‘’ π‘›π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘π‘’π‘Ÿπ‘–π‘œπ‘‘ π‘Žπ‘›π‘‘ βˆ’ 𝑏 = ( π‘₯ 𝑖 + β‹― + π‘₯ 𝜏 ) π‘Žπ‘›π‘‘ 𝑐 = ( π‘₯ 𝑖 + β‹―+ π‘₯ 𝑀 ); 𝜏 = 𝑏 πœ‘β€²( πœƒ) π‘Žπ‘›π‘‘ 𝑀 βˆ’ 𝜏 = 𝑐+𝑏 πœ‘β€²( πœƒ) ο‚· The new variablex will bedistrusted(under g measure) as a normal with same variance βˆ†π‘‘πœŽ2 but different mean ( π‘Ÿ βˆ’ 𝜎2 2 )βˆ†π‘‘ + βˆ†π‘‘πœŽ2 πœƒ