A Semi-Analytic Approach To Valuing Auto-Callable Accrual Notes
AA Semi-Analytic Approach To ValuingAuto-Callable Accrual Notes
V. G. Filev ∗ , P. Neykov † , and G. S. Vasilev ‡ R&D, CloudRisk Ltd , Narodno Subranie 9, 1000 Sofia, Bulgaria HQ, CloudRisk Ltd , 308 Spice Quay Heights, 32 Shad Thames, London,SE1 2YL, United Kindom School of Theoretical Physics, Dublin Institute for Advanced Studies,10 Burlington Road, Dublin 4, Ireland Department of Physics, Sofia University, James Bourchier 5 blvd,1164 Sofia, Bulgaria
Abstract
We develop a semi-analytic approach to the valuation of auto-callable structures with accrual features subject to barrier conditions.Our approach is based on recent studies of multi-assed binaries, presentin the literature. We extend these studies to the case of time-dependentparameters. We compare numerically the semi-analytic approach andthe day to day Monte Carlo approach and conclude that the semi-analytic approach is more advantageous for high precision valuation. ∗ veselin.fi[email protected]; vfi[email protected] † [email protected] ‡ [email protected]; [email protected]fia.bg a r X i v : . [ q -f i n . P R ] A ug INTRODUCTION Contents
Auto-callable structures are quite popular in the world of structured prod-ucts. On top of the auto-callable structure it is common to add featuresrelated to interest payments. Hence, combining range accrual instrumentsand auto-call options not only leads to interesting conditional dynamics, butgives an illustrative example of a typical structured product ref. [1]. In ad-dition to the strong path dependence of the coupons the instrument’s finalredemption becomes path dependent too. Intriguingly, within the Black–Scholes world one can obtain a closed form expression for the payoff of sucha derivative. On the other side one can also rely on a straightforward MonteCarlo (MC) approach ref. [2]. Often the interest payment features embeddedin the instrument accrue a fixed amount daily, related to some trigger levelsof the underlyings. The standard approach for valuation of such instrumentsis a daily MC simulation. The goal of this paper is to propose an alterna-tive semi-analytic approach (SA), which in some cases performs significantlybetter than the brute force day to day MC evaluation ref. [3]. As we aregoing to show, the complexity of the evaluation of the auto-call probabilitiesgrows linearly with the number of observation times of the instrument andone may expect that at some point the MC approach would become more
THE INSTRUMENT The paper is structured as follows: In section two we begin with a briefdescription of the type of derivative instrument that we are studying.In section three we develop the quasi-analytic approach, extending theresults of ref. [4] to the case of time dependent deterministic parametersobtaining an expression for the probability of an early redemption in terms ofthe multivariate cumulative normal distribution. Building on this approachwe obtain similar expression for the payoff at maturity, subject to elaborateconditions. In addition we calculate the payoff of the coupons as a sumover multivariate barrier options ref. [5], using the developed SA approach torepresent the pay-off of the latter in terms of multivariate cumulative normaldistribution ref. [6].Finally, in section four we apply our approach to concrete examples. Weimplement numerically both the SA and MC approaches and demonstratethe advantage of applying the SA approach to lower dimensional systems,especially when a high precision valuation is required.
In this paper we analyse a type of instrument which combines the featuresof range accrual coupons with auto-call options.–The instrument is linked to the performance of two correlated assets S and S .–The instrument has a finite number M of observation times T , T , . . . , T M .If at the observation time T k both assets S i are simultaneously above certainbarriers b i, k the instrument redeems at 100%. This is the auto-call condi-tion. To shift the valuation time at zero we define τ = T − t and discuss theobservation times τ , τ , . . . , τ M . To the best of our knowledge, a closed formula for time-dependant parameters havenot been presented in the literature.
SEMI-ANALYTIC APPROACH in which both assets S i were above certain barriers c i .–If the instrument reaches maturity, it redeems at 100% if both assets S i are above certain percentage κ of their spot prices at issue time ¯ S i . If at leastone of the assets is bellow κ ¯ S i the instrument pays only a part proportionalto the minimum of the ratios S i / ¯ S i . In this section we outline our semi-analytic approach. We begin by providinga formula for the auto-call probability.
Without loss of generality, it is assumed that the auto-callable structure hastwo underlyings. On the set of dates are imposed trigger conditions relatedto the auto-call feature. If the auto-call triggers have never been breached atthe observation dates the auto-callable structure matures at its final maturitydate. On the opposite case, if one of the auto-call triggers have been breachedthe instrument auto-calls at this particular date and has its maturity.Let us denote with P k the probability to auto-call at observation time τ k .Note that this implies that at previous observation times the spot prices of thetwo assets where never simultaneously above the barriers b i . We introduce thefollowing notations: X i, k labels the spot value of the assets S i at observationtime τ k .Using the standard notations, if probability space (Ω , (cid:122) , P ) is given, and A ∈ (cid:122) , than the indicator function is defined as E ( A ) = P ( A ) . Using the above definition, the auto-call probability at time τ k , for thegeneral case with n underlying indices is then given by the expectation relatedto some probability measure Q of the indicator function: P k = E Q (cid:0) ( X , < b , ) ∪ ( X , < b , ) , ( X , < b , ) ∪ ( X , < b , ) , ..., ( X , k < b , k ) ∩ ( X , k < b , k ) (cid:1) . (1)In order to simplify the notation, hereafter we will omit the probability mea-sure Q. For the case of two underlyings, we can also define also the probability Or the valuation day for the first observation time.
SEMI-ANALYTIC APPROACH k observation times:¯ P k = E (cid:0) ( X , < b , ) ∪ ( X , < b , ) , ( X , < b , ) ∪ ( X , < b , ) , ..., ( X , k < b , k ) ∪ ( X , k < b , k ) (cid:1) . (2)Note that at each observation time we have more than one possibilities re-flected in the ∪ operation.For example the event ( X , < b ) ∪ ( X , < b ) can be split into thethree scenarios ( X , < b ) ∩ ( X , < b ), ( X , < b ) ∩ ( X , > b ),( X , > b ) ∩ ( X , < b ).We could do a bit better if we define ˜ X , s = X , s /b , s and ˜ X , s = X , s /b , s . Then the condition ( X , < b , ) ∪ ( X , < b , ) can be split intothe two conditions ( ˜ X , < ∩ ( ˜ X , ˜ X − , < X , ˜ X − , < ∩ ( ˜ X , < P k we need to sum over 2 k possible scenarios, eachscenario containing 2 k conditions. This requires summing over 2 k differ-ent 2 k -dimensional cumulative multivariate normal distributions [4], whichis computationally overwhelming for large values of k . Fortunately, using deMorgan rules we can substantially reduce the computational cost.Let us denote by E i the event ( X , i < b ) ∪ ( X , i < b ), then the event¯ E i is written as the single scenario ( X , i > b ) ∩ ( X , i > b ).Using the well known probability relation P (cid:16)(cid:92) ni =1 E i (cid:17) = (cid:88) i P ( E i ) − (cid:88) i,j P ( E i ∪ E j ) + (cid:88) i,j,k P ( E i ∪ E j ∪ E k ) + ... + ( − n P (cid:16)(cid:91) ni =1 E i (cid:17) and DeMorgan’s law (cid:16)(cid:91) ni =1 E i (cid:17) = (cid:92) ni =1 E i can be shown that¯ P k = P (cid:32) k (cid:92) s =1 E s (cid:33) = 1 + k (cid:88) s =1 (cid:88) σ s ∈ C ks ( − s P (cid:32) s (cid:92) j =1 ¯ E σ s ( j ) (cid:33) . (3)where the second sum is over all (sorted in ascending order) combinationsof k elements s − th class, C ks . Note that there are again 2 k different terms,however only the last term is 2 k -dimensional. In general the number of 2 s -dimensional terms is (cid:0) ks (cid:1) . SEMI-ANALYTIC APPROACH P k : P k = k − (cid:88) s =0 (cid:88) σ s ∈ C k − s ( − s P (cid:32) s (cid:92) j =1 ¯ E σ s ( j ) ∩ ¯ E k (cid:33) , (4)where we have used a convention: ∩ j =1 ¯ E σ ( j ) ∩ ¯ E k = ¯ E k . Equations (3) and(4) can be rewritten in terms of indicator functions. For compactness it isconvenient to adopt the notations of ref. [4]. We introduce a multi-indexnotation denoting by X I the element X i, s , where I = 1 , . . . , n and n is thenumber of all observed assets’ prices. In the case considered in equation (1)we have n = 2 k . Using lexicographical order we can make the map explicit:( i, s ) → I = I [ i, s ] = 2 ∗ ( s −
1) + i (5)where we have used that i = 1 ,
2. Next we define the following notation:( X A ) j = X A j . . . X A jn n j = 1 , . . . , m , (6)where m is the number of barrier conditions and A is an n × m matrix. Withthese notations a general indicator function can be written as: m ( S X A > S a ) (7)where a is a vector of barriers and to allow for different types of inequalitieswe have introduced the m × m diagonal matrix S whose diagonal elementstake the values ± > ’ and ’ − ’ for ’ < ’). Equations (3),(4) nowbecome: ¯ P k = E k (cid:88) s =1 (cid:88) σ s ∈ C ks ( − s s ( X A ( σ s ) > b ( σ s )) , (8) P k = E k − (cid:88) s =0 (cid:88) σ s ∈ C k − s ( − s s +2 ( X ˜ A ( σ s ) > ˜b ( σ s )) , (9)where A ( σ s ), b ( σ s ), ˜A ( σ s ), ˜b ( σ s ), are 2 k × s , 1 × s , 2 k × s +1), 1 × s +1)matrices, respectively. Their non-zero entries are: A ( σ s ) I [ i, σ s ( j )] , I [ i, j ] = 1 , ( b ( σ s )) I [ i,j ] = b i, σ s ( j ) , (10)˜ A ( σ s ) I [ i, σ s ( j )] , I [ i, j ] = 1 , ( ˜b ( σ s )) I [ i,j ] = b i, σ s ( j ) , (11)for i = 1 , j = 1 , . . . , s . ˜ A ( σ ) I [ i, k ] , i = 1 , ( ˜b ( σ )) i = b i, k , for i = 1 , . (12)In equation (10) we have used the map (5). Note that it is crucial that thecombinations σ s are sorted in ascending order. SEMI-ANALYTIC APPROACH If we restrict ourselves to time independent deterministic parameters (interestrate, dividend yield, volatility) we can directly apply the formula derived inref. [4] to calculate the indicator functions in equations (8) and (9). However,this is a very crude approximation when dealing with long instruments this iswhy we extend the results of ref. [4] to the time dependent case. The startingpoint is to model the dynamics of the asset S i with a geometric Brownianmotion: dS i S i = ( r ( s ) − q i ( s )) ds + σ i ( s ) dW i ( s ) , (13)where W i are correlated Brownian motions with correlation coefficient ρ ij .Indeed the integrated form of equation (13) is: S i ( τ ) = S (0) i exp τ (cid:90) (cid:18) r ( s ) − q i ( s ) − σ i ( s ) (cid:19) ds + τ (cid:90) σ i dW i ( s ) (14)For the asset i at time T k we can write:log ˜ X i,k = log x i + (cid:18) ¯ r i,k − ¯ q i,k −
12 ¯ σ i,k (cid:19) τ k + ¯ σ i,k √ τ k Z i,k , (15)where Z i,k is given by: Z i,k = 1¯ σ i,k √ τ k τ k (cid:90) σ i ( s ) dW i ( s ) (16)and ¯ r i,k = 1 τ k τ k (cid:90) ds r i ( s ) , ¯ q i,k = 1 τ k τ k (cid:90) ds q i ( s ) , (17)¯ σ i,k = 1 τ k τ k (cid:90) ds σ i ( s ) . Following ref. [4] we define the quantities: µ = (cid:18) ¯ r i,k − ¯ q i,k −
12 ¯ σ i,k (cid:19) τ k , Σ = diag (¯ σ i,k √ τ k ) . (18) SEMI-ANALYTIC APPROACH R defined as: R ( i,k )( j,l ) ≡ (cid:104) Z i,k , Z j,l (cid:105) . (19)Using equation (16) and the formula: (cid:42) τ (cid:90) σ i ( s ) dW i ( s ) , τ (cid:90) σ j ( r ) dW j ( r ) (cid:43) = ρ ij min( τ ,τ ) (cid:90) σ i ( τ ) σ j ( τ ) dτ , (20)we obtain: R ( i,k )( j,l ) = ρ ij √ τ k τ l ¯ σ i,k ¯ σ j,l min( τ k ,τ l ) (cid:90) σ i ( τ ) σ j ( τ ) dτ . (21)Next following ref. [4] we define:Γ = Σ R Σ (cid:48) , (22) D = (cid:112) diag ( A Γ A (cid:48) ) ,C = D − ( A Γ A (cid:48) ) D − , d = D − (cid:2) log( x A / a ) + A µ (cid:3) . Here it is used that x i,k = x i for all k = 1 , . . . , M . In therms of thesequantities the indicator function is given by the same expression as in ref. [4],but the underlying variables are given in eq. (22) and due to the time-dependence thay are different from those given in the work ref. [4], m ( S ˜ X A ( ω ) > S a ) = N m ( S d ( ω ) , S C ( ω ) S ) , (23)where N m is the cumulative multivariate normal distribution (centred aroundzero).Note that equations (18)–(23) are valid for any n × m matrix A and anypositive barrier vector a . Applying equation (23) to calculate the auto-call probability P k we obtain: P k = k − (cid:88) s =0 (cid:88) σ s ∈ C k − s ( − s N s +2 ( d ( σ s ) , C ( σ s )) , k = 1 . . . M − , (24) SEMI-ANALYTIC APPROACH d ( σ s ) and C ( σ s ) are obtained by substituting ˜ A ( σ s ) and b ( σ s ) fromequation (10) into equation (22). Note that the index k in equation (24)runs from one to M −
1. The reason is that the last observation time is thematurity.Let us denote by P mat the probability to reach maturity . Clearly wehave: P mat = 1 − M − (cid:88) k =1 P k , (25)The probability P mat can be split into two contributions: P mat = P up + P down (26)Where P up is the probability to reach maturity with both assets simultane-ously above the barrier κ ¯ S i , and P down is the probability at least one fo theassets to be bellow the barrier. In fact the probability P up is exactly P M ,hence we can write: P up = M − (cid:88) s =0 (cid:88) σ s ∈ C k − s ( − s N s +2 ( d ( σ s ) , C ( σ s )) . (27)Clearly this also determines P down as P down = P mat − P up . To calculate thepayoff at maturity we also need the average performance of the assets subjectto the condition that the worst performing asset is bellow the barrier κ ¯ S i .The probability for this to happen is exactly P down , which is a function ofthe parameter κ .Let us denote ˆ X i = S i / ¯ S i and define ˆ X = min( ˆ X , ˆ X ), the probability P down can be written as: P down = P ( ˆ X < κ ) . (28)The average performance of the assets provided that at least one of the assetsis bellow the barrier κ is then proportional to the conditional expectationvalue (cid:104) ˆ X (cid:105)| ˆ X<κ : (cid:28) min (cid:18) S ¯ S , S ¯ S (cid:19)(cid:29) (cid:12)(cid:12)(cid:12) ˆ X<κ = − P down κ (cid:90) dκ κ dP up dκ , (29)where we have used that dP mat /dκ = 0. Therefore, the payoff at maturity isgiven by: V maturty = P up + P down (cid:28) min (cid:18) S ¯ S , S ¯ S (cid:19)(cid:29) (cid:12)(cid:12)(cid:12) ˆ X<κ = P up − κ (cid:90) dκ κ dP up dκ , (30) Note also that P mat = ¯ P M − SEMI-ANALYTIC APPROACH
To obtain the total payoff we have to evaluate the contribution of the coupons.This can be done by summing over a type of two-asset binary (cash-or-nothing) options, conditional on the survival of the instrument to the ap-propriate accrual period. Indeed the probability at time τ both assets to beabove the barrier is given by the probability for such an option to pay. In thecase of the first accrual period this reduced to the standard two-asset binaryoption [7]. To write down a closed form expression for this probability weneed to add one more observation time τ a , which will iterate over the accrualdates. Clearly the simplest case is when 0 ≤ t a ≤ τ , that is the first accrualperiod. In this case we apply formula (23), for just one observation time τ a ,with A = S = and a = c . In more details the probability the coupons topay at time τ a < T , P ( τ a ) is given by: P ( τ a ) = N ( d ( τ a ) , C ) ,d i = log( ¯ S i /c i ) + (¯ r − ¯ q i ( τ a ) − ¯ σ i ( τ a ) / τ a ¯ σ i ( τ a ) √ τ a , i = 1 , , (31) C = (cid:18) ρ ρ (cid:19) (32)where ¯ r, ¯ q i ( τ a ) and ¯ σ i ( τ a ) are given by equations (17) with τ k = τ a . The totalnumber of days in which coupons have been payed in the period 0 to τ , N is then given by: N = τ (cid:88) τ a = 1 P ( τ a ) . (33)In the same way we can obtain a formula for the number of coupon daysin the second accrual period. The only difference is that now in additionto the condition both assets to be above the accrual barrier we also havethe condition that the instrument did not auto-call at time τ . In generalthe probability the coupons to pay at time τ a in the k -th accrual period isthe joint probability that the instrument did not auto-call at the first k − τ a .Denoting by E Cτ a the event that the assets are above the accrual barrier at SEMI-ANALYTIC APPROACH τ a and using the notations from section 3.1, one can show that : P k − ,k ( τ a ) = k − (cid:88) s =0 (cid:88) σ s ∈ C k − s ( − s P (cid:32) s (cid:92) j =1 ¯ E σ s ( j ) ∩ E Cτ a (cid:33) , (34)where again we have used the convention: ∩ j =1 ¯ E σ ( j ) ∩ E Cτ a = E Cτ a . Equation(34) can be rewritten in analogy to equation (9) as: P k − ,k ( τ a ) = E k − (cid:88) s =0 (cid:88) σ s ∈ C k − s ( − s s +2 ( ˜X σ s > bc σ s ) , (35)where ˜X σ s is the vector: [ X ,σ s (1) , X ,σ s (1) , . . . , X ,σ s ( s ) , X ,σ s ( s ) , S ( τ a ) , S ( τ a )]and bc σ s is the vector: [ b ,σ s (1) , b ,σ s (1) , . . . , b ,σ s ( s ) , b ,σ s ( s ) , c , c ]. Denotingby ˜ C σ s ( τ a ) the covariant matrix constructed using equations (17)-(22) withtimes τ σ s (1) , . . . , τ σ s ( s ) , τ a and denoting by ˜d σ s the corresponding quantity inequation (22) constructed using the barrier vector bc σ s , we can write: P k − ,k ( τ a ) = k − (cid:88) s =0 (cid:88) σ s ∈ C k − s ( − s N s +2 ( ˜d σ s , ˜ C σ s ( τ a )) . (36)For the number of coupon paying days in the k -th accrual period we obtain: N k = τ k (cid:88) τ a = τ k − + 1 P k − ,k ( τ a ) . (37)To calculate the contribution of the coupons to the total payoff we needto take into account the discount factors, since we have assumed that thecoupons are payed at the observation times . Note that the probability thecoupons to pay already include the probability to reach that accrual period.Therefore, the total coupon contribution is given by: V C M = γ M (cid:88) s =1 e − ¯ r s τ s N s , (38)where γ is the daily rate of the coupon. The derivation is analogous to that of equation (4). Note that in practise there are a separate payment dates shortly after the correspond-ing observation date.
APPLICATIONS Assuming for simplicity that the instrument redeems at 100 % in the eventof an auto-call (which in reality is quite common), for the total payoff weobtain: V tot = V maturity + M − (cid:88) k =1 e − ¯ r k τ k P k + V C M (39)where we have substituted P mat from equation (26). In this section we outline some of the applications of the formalism developedabove. We begin with the simplest case of a pure accrual instrument.
The pure accrual instrument that we consider in this subsection has thefollowing characteristics:–It pays a daily coupon at rate γ if at closing time both assets S i areabove the accrual barriers c i –At maturity (time τ m ), it redeems at 100% if both assets S i are abovecertain percentage κ of their spot prices at issue time ¯ S i . If at least one ofthe assets is bellow κ ¯ S i the instrument pays only a part proportional to theminimum of the ratios S i / ¯ S i .Clearly this is the general instrument that we considered with the auto-call option removed. In this simple case the semi-analytic approach of sec-tion 3 is particularly efficient. The coupons are calculated by the first periodformulas in equations (31), (33) with τ a = τ m , while the payoff at maturityis calculated using equation (30), with P up given by: P up = N ( ˜d ( τ m ) , C ) , (40)where C is given in equation (31) and ˜d ( τ m ) is given by: d i = log( ¯ S i /c i ) + (¯ r − ¯ q i ( τ m ) − ¯ σ i ( τ m ) / τ m ¯ σ i ( τ m ) √ τ m , i = 1 , , (41)where ¯ r, ¯ q i ( τ m ) and ¯ σ i ( τ m ) are given by equations (17) with τ k = τ m . APPLICATIONS In this section we compare the efficiency of our semi-analytic (SA) approachand that of a standard Monte Carlo (MC) approach. Since the dimensionalityof the SA problem increases linearly with the number of the auto-call dates,we consider the case of one auto-call date and two range accrual periods.Therefore, our problem is four dimensional and we would still need to relyon numerical methods to estimate the cumulative distributions.To simplify the analysis even further and facilitate the comparison, wesimplify the pay-off at maturity. The instrument pays 100% if both under-lyings perform above the final barrier κ (as before), but if this condition isnot satisfied, instead of redeeming a worse performance: min (cid:0) S / ¯ S , S / ¯ S (cid:1) fraction, the instrument redeems at κ ×
100 %. Equation (30) then simplifiesto: V maturty = P up + κ P down . (42)The description of the coupon payments remains the same as in section 3.4.The volatilities σ i , dividend yields q i , interest rate r and correlation corre-lation coefficient ρ used in the numerical example are presented in table 1.In addition the final barrier was set at 60% ( κ = 0 .
60) and the daily accrualrate used was (15/365)% ( γ = 0 . / τ = 1 and τ = 2. σ i q i r ρ Volatilities σ i , dividend yields q i , interest rate r and correlation ρ usedin the numerical example. To compare the efficiency of the algorithms we compared the runningtimes T (cid:15) as functions of the absolute error (cid:15) . The resulting plot is presentedin figure 1. The round dots correspond to the SA approach, while the squarepoints represent the MC data. As one may expect, the running time T (cid:15) forthe MC algorithm increases as ∼ /(cid:15) and while negligible for (cid:15) < .
01, itincreases rapidly to ∼ s , for (cid:15) = 5 . × − . On the other side the SAmethod has a steady computation time T (cid:15) ∼ s , for (cid:15) < . × − . TheSA and MC curves intersect at (cid:15) ≈ . × − . The advantage of using theSA method for higher precision (cid:15) < . × − is evident. For example acalculation with (cid:15) = 2 . × − would require running the MC simulation forroughly ∼ s , while the same accuracy can be achieved by the SA methodfor ∼ s , which is a factor of twelve. Clearly the comparison depends on the CONCLUSION ÊÊÊÊÊÊÊÊÊÊÊÊÊ ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ e T e H s L Ê SA ‡ MC Figure 1:
A plot of the running time T (cid:15) in seconds as a function of the absoluteerror (cid:15) . The round red dost represent the SA results, while the square blue dostcorrespond to the MC data. implementation and the choice of parameters. To make the comparison fairwe used MatLab for both methods. Using a vectorised MC algorithm for theMonte Carlo part and the built in MatLab cumulative distribution functionsfor the SA approach.Another obvious advantage of the SA approach is the higher precision inthe estimation of the sensitivities of the instrument. Semi-analytic expres-sions could be derided for most of the greeks, which enables their calculationwith a limited numerical effort. This is clearly not the case in the MC ap-proach, where one usually relies on a numerical differentiation.Finally, as we pointed out at the beginning of this section the dimension-ality of the problem increases linearly with the number of auto-call times.It is therefore expected that at some point the MC approach would becomemore efficient. Nevertheless, the SA approach could still be more efficient ifthe sensitivities are difficult to analyse in the MC approach. This paper makes several contributions to the related literature.Our main result is the development of a semi-analytical valuation method
PROOF OF THE VALUATION FORMULA
Acknowledgements:
We would like to thank Bojidar Ibrishimov for crit-ically reading the manuscript.
A Proof of the valuation formula
For completeness we provide a proof of formula (23). Our proof follows thesteps outlined in reference [4]. Using the definitions (6), (18) and equation(15) it is easy to obtain:log ˜X A = log x A + A µ + A Σ Z . (43)Furthermore, the monotonicity of the logarithmic function implies: m ( S X A > S a ) = m ( S log X A > S log a ) = m ( B Z < b ) (44)where: B = − S A Σ , (45) b = S (log x A / a + A µ ) . (46)Now we use a Lemma from ref. [4] (which we will prove for completeness): Lemma 1. If B is an m × n matrix of rank m ≤ n and Z is a random unit PROOF OF THE VALUATION FORMULA variate vector of length n with correlation matrix R . Then: E { m ( B Z < b ) } = N m ( D − b , D − ( B R B T ) D − ) , (47) where: D = (cid:112) diag( B R B T ) . (48)Applying Lemma 1 for B and b given in equation (45), we obtain: D = diag( S A Σ R Σ T A T S ) = diag( A Σ R Σ A T ) = diag( A Γ A T ) ,D − ( S A
Σ ) R (Σ A T S ) D − = S D − A Σ R Σ A T D − S = S C S ,D − b = S D − (log x A / a + A.µ ) = S d , (49)where we have used that S and D are diagonal and commute and that S i,i =1. Substituting relations (49) into equation (47) we arrive at equation (23).Now let us prove Lemma 1: Proof:
Let us complete the m × n matrix B to an n × n non-singular matrix˜ B . We write: ˜ B = (cid:20) BB ⊥ (cid:21) , (50)where B ⊥ is an ( n − m ) × n matrix, which we are going to specify bellow.Consider the Cholesky decomposition of the correlation matrix R : R = U U T . (51)Next we transform the matrix ˜ B with U via ˜ B (cid:48) = ˜ B U , which implies: B (cid:48) = B U , (52) B (cid:48)⊥ = B ⊥ U . (53)Since B has rank m and U is invertible, B (cid:48) also has a rank m . We cantherefore think of B (cid:48) as m independent n − column vectors. Spanning an m -dimensional subspace L m . We are always free to choose B (cid:48)⊥ to be a matrixof n − m independent n − column vectors spanning the orthogonal completionof L m . Making this choice of B (cid:48)⊥ implies: B ⊥ R B T = B ⊥ U ( B U ) T = B (cid:48)⊥ B (cid:48) T = 0 , (54)Next we apply the transformation Y = ˜ B Z . The covariance matrix of therandom vector Y is given by: C = ˜ B R ˜ B T = (cid:20) B R B T B ⊥ R B T ⊥ (cid:21) , (55) EFERENCES Y || = B Z , Y ⊥ = B ⊥ Z , (56)the condition m ( B Z < b ) becomes m ( Y || < b ). Furthermore, the proba-bility density function of Y factorises: ρ ( Y ) = 1(2 π ) n/ (cid:113) det( ˜ B R ˜ B T ) exp (cid:18) − Y T ( ˜ B R ˜ B T ) − Y (cid:19) == 1(2 π ) ( n − m ) / (cid:112) det( B ⊥ R B T ⊥ ) exp (cid:18) − Y T ⊥ ( B ⊥ R B T ⊥ ) − Y ⊥ (cid:19) ×× π ) m/ (cid:112) det( B R B T ) exp (cid:18) − Y T || ( B R B T ) − Y || (cid:19) == ρ ⊥ ( Y ⊥ ) × ρ || ( Y || ) (57)Since there are no conditions imposed on Y ⊥ the integral over ρ ⊥ ( Y ⊥ ) issimply unity. What remains is the integral over ρ || ( Y || ), which upon thenormalisation: Y ⊥ → D − / Y ⊥ gives equation (47). References [1] M. Bouzoubaa and A. Osseiran (2010). ”Exotic Options and Hybrids -A Guide to Structuring, Pricing and Trading”, ,[2] P. Glasserman (2003). ”Monte Carlo Methods in Financial Engineering”, ,[3] R. Korn, E. Korn and G. Kroisandt (2010). ”Monte Carlo Methods andModels in Finance and Insurance”, ,[4] Max Skipper and Peter Buchen (2009). ”A valuation formula for multi-asset, multi-period binaries in a Black–Scholes economy”. The ANZIAMJournal, 50, pp 475-485. doi:10.1017/S1446181109000285,[5] J. Hull (2015). ”Options, Futures, and other Derivatives, 9ed.”, ,[6] P. Zhang (1998). ”Exotic Options, 2ed.”, ,[7] R. C. Heynen and H. M. Kat (1996). ”Brick by Brick”,