Economic Computation and Economic Cybernetics Studies and Research, Issue 2/2016, Vol. 50

Size: px
Start display at page:

Download "Economic Computation and Economic Cybernetics Studies and Research, Issue 2/2016, Vol. 50"

Transcription

1 Ecoomic Computatio ad Ecoomic Cyberetics Studies ad Research, Issue 2/216, Vol. 5 Kyoug-Sook Moo Departmet of Mathematical Fiace Gacho Uiversity, Gyeoggi-Do, Korea Yuu Jeog Departmet of Mathematics Korea Uiversity, Seoul, Korea Hogoog Kim Departmet of Mathematics, Korea Uiversity, Seoul, Korea hogoog@korea.ac.kr AN EFFICIENT BINOMIAL METHOD FOR PRICING ASIAN OPTIONS Abstract. We costruct a efficiet tree method for pricig pathdepedet Asia optios. The stadard tree method estimates optio prices at each ode of the tree, while the proposed method defies a iterval about each ode alog the stock price axis ad estimates the average optio price over each iterval. The proposed method ca be used idepedetly to costruct a ew tree method, or it ca be combied with other existig tree methods to improve the accuracy. Numerical results show that the proposed schemes show superiority i accuracy to other tree methods whe applied to discrete forward-startig Asia optios ad cotiuous Europea or America Asia optios. Keywords : biomial tree method, cell averagig, Asia optios. JEL Classificatio: G13, C63, C2 1. Itroductio A optio is a fiacial derivative which gives the ower the right, but ot the obligatio, to buy or sell a uderlyig asset for a give price o or before the expiratio date. From the semial papers of Black ad Scholes (1973) ad Merto (1973), the tradig volume of optios has bee icreased ad exotic optios with ostadard payoff patters have become more commo i the over-the-couter market. Amog them, a optio with the payoff determied by the average uderlyig price over some pre-defied period of time is called a Asia optio. A Asia optio has bee popular sice it could reduce the risk of market maipulatio of the uderlyig asset at maturity ad the volatility iheret i the optio. However, these Asia optios based o arithmetic averages caot be priced i a closed-form, ad oe eeds to rely o its umerical approximatio istead. There have bee may approaches to approximate the value of exotic optios, such as biomial tree method, Mote Carlo simulatio, fiite differece method for solvig Black-Scholes partial differetial equatios etc. Both the Mote Carlo method ad fiite differece method suffer from the difficulty to deal with early exercise without bias, whereas the biomial tree method by Cox, Ross ad 151

2 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim Rubistei (1979) are very popular due to its ease of implemetatio ad simple extesio to America type optios. However, due to the averagig ature of Asia optios, the umber of averagig odes i biomial tree grows expoetially. Therefore straightforward extesio of the stadard biomial method to Asia case is ot possible i practice. I order to solve this shortage of biomial method, Hull ad White (1993) cosidered a set of represetative averages at each ode icludig miimum ad maximum average values. Employig this set of represetative averages makes the biomial model feasible for pricig Asia optios, though it still suffers the lack of covergece, see Costabile, Massabo ad Russo (26) ad Forsyth, Vetzal ad Zva (22). For a discrete moitored Asia optio, Hsu ad Lyuu (211) proposed a quadratic-time coverget biomial method based o the Lagrage multiplier to choose the umber of states for each ode of a tree. I this paper, based o the cell averagig approach i Moo ad Kim (213), small bis o the asset price axis, called cells, are defied about each ode of the tree ad the average optio price over each cell has bee computed ad updated i time. See Sectio 2 for details. The biomial method of Hsu ad Lyuu (211) for discrete moitored Asia optios ad that of Hull ad White (1993) for cotiuous Asia optios have bee modified for improvemet with the help of cell averagig method. Numerical experimets i sectio 4 show that the proposed cell averagig biomial method gives more accurate results compared to other existig computatioal methods. The outlie of the paper is as follows. I sectio 2 we explai the problem ad itroduce the cell averagig biomial method. I sectio 3 we exted the cell averagig biomial method to discrete ad cotiuous moitored Asia optios. I sectio 4 we compare the accuracy ad efficiecy of the existig tree methods with those of the cell averagig biomial method. We fially summarize our coclusios i sectio Cell Averagig Biomial Methods Let us cosider the price of the uderlyig asset as a stochastic process { S( t), t [, T ]} which satisfies the followig stochastic differetial equatio: ds( t) S( t) dt + S( t) dw ( t), t T, (1) where is a expected rate of retur, is a volatility, T is a expiratio date, ad Wt () is a Browia motio. From the Ito formula i Øksedal (1998), X ( t) l( S( t)) satisfies 2 dx ( t) ( / 2) dt dw ( t), t. I the risk-eutral world, the value of the Europea optio ca be computed by the discouted coditioal expectatio of the termial payoff, ( ) (, ) r T t V x t e E ( X ( T )) X ( t) x, where ( XT ( )) is the payoff at t T. Without loss of geerality, we deote agai the risk eutral process to be Xt () with drift rate equal to the risk-free iterest rate r, istead of i (1). If we cosider a cotiuous divided yield q, the drift rate becomes r q. 152

3 A Efficiet Biomial Method for Pricig Asia Optios 2.1. The biomial model Let us first discretize the time period,t ito N itervals of the same legth t T / N, t t1... tn T. The stadard biomial method by Cox, Ross ad Rubistei (1979) assumes that the asset price St ( ) at t t moves either up to S( t) u for u exp( t) 1 or dow to S( t) d for rt d 1/ u ad, 1,, N 1 with probabilities p ( e d) / ( u d) or 1 p, respectively, or Xt ( ) i log at t t moves either up to X ( t) h or dow to X ( t) h where h l u. Let X X (2 ) h deote the values at t t t for, 1,,, with X() X. The the stadard N N biomial method calculates the payoffs of the optio at expiry, V ( X ) for,1,..., N, ad computes the optio price V V ( X,) by backward averagig, r t V ( x, t) e ( pv ( x h, t t) (1 p) V ( x h, t t)), (2) where x X,,,, ad N 1, N 2,,. The biomial method approximatio coverges to the Black-Scholes value as the umber of time steps, N, teds to ifiity, see the geeral theory i Kwok (1998), Clewlow ad Stricklad (1998), Lyuu (22) ad Higham (24). However, it is widely reported that the covergece is ot mootoe ad the sawtooth patter i the sequece of approximatios makes the biomial approximatio less attractive. 2.2 The cell averagig biomial model I order to reduce the saw-tooth patters i the sequece of approximatios i the stadard biomial method, we employ the cell averagig method. Let us first divide the iterval X ( N 1) h, X ( N 1) h o the X -axis ito N 1 o-overlappig equidistat itervals of legth 2h, called cells, cetered at poits X (2 N) h,,, N, the compute average optio payoffs o each cell cetered at X at expiry t tn, N 1 X h N V ( )d N 2h,, N, (3) X h where () is the payoff fuctio at expiry. If (2) is satisfied at every poit X h, X h i the cell at time t, the the average optio price V 1 X h (, )d 2h V t satisfies the followig backward averagig relatio X h rt 1 V e pv p V 1 (4) 1 1 See Figure 1. Appropriate modificatio will be eeded if (2) does ot hold at 153

4 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim every poit X h, X h. For istace, see Moo ad Kim (213) for the case of barrier optios. The cell averages of the optio values at expiry (3) ca be updated iteratively, which evetually leads to the average of the optio price V o X h, X h at t. Figure 1 : Compariso of backward averagig betwee the stadard tree method (Left) ad the cell averagig tree method (Right). Figure 2 compares the stadard biomial method (dash) ad the cell averagig biomial method (solid) for the Europea up-ad-out barrier put optio price. The figure shows that this cell averagig idea reduces ig-saw oscillatios. Figure 2 : Parameters: The iitial stock price S() 1, the risk-free iterest rate r.5, the volatility.3, strike price K 9, barrier H 15 ad the maturity T 1 for a Europea up-ad-out barrier put optio. Compariso betwee stadard biomial (dash) ad cell averagig biomial (solid) lattice models. Cell averagig produces smoother covergece. 3. Asia Optios Now we exted the cell averagig biomial method i sectio 2 to pathdepedet Asia optios. As it has bee kow, there do ot exist explicit closedform aalytical solutios for arithmetic Asia optios because the arithmetic average of a set of logormal radom variables is ot log-ormally distributed. For that reaso, may umerical approaches have bee proposed. We first cosider discrete moitored Asia optios i Sectio 3.1 ad modify the method of Hsu ad Lyuu (211) to price it. The we improve the method of Hull ad White (1993) i 154

5 A Efficiet Biomial Method for Pricig Asia Optios Sectio 3.2 to price the cotiuous Asia optio. 3.1 The Discrete Moitored Asia optio The discretely moitored Asia optio is ofte foud i practice. The payoff of Asia call optio with strike price K at the expiry date T is give by the followig : 1 max S ti( ) K, (5) i 1 where is the umber of moitor poits ad the payoff of discrete type arithmetic average Asia call optio is moitored at time poits, t t t T. We assume each time iterval betwee two adacet 1 2 moitor poits is partitioed ito I time steps, ad I is called itraday. The we see that the moitor poits are at times, I, 2 I,, I ad the whole umber of time steps is N I. For the stadard Europea-style discrete Asia call, the payoff at expiry is 1 max SiI K, 1 i whereas the forward-startig discrete Asia optio omits the iitial S ad has the payoff 1 max SiI K, i 1 I order to be self-cotaied, we start with explaatio of the biomial method by Hsu ad Lyuu (211) which follows the stadard biomial method suggested by Cox, Ross ad Rubistei (1979). Let N( i, ) deote the ode at time i that results from dow moves ad i up moves. The the price sum to expiry date for a price path ( S, S1,, S i ), P, called the ruig sum, is computed by S SI S for stadard Asia optios i / I I P SI S2 I S for forward-startig Asia optios, i / I I where deotes floor fuctio ad i N. Sice the pricig of Asia optio usig biomial lattice produces 2 N possible paths for each time step N, Hsu ad Lyuu (211) suggested a discrete biomial method for Asia optio pricig, where they proposed the ruig sum P of the form ( 1) K 2( 1) K ( ki 1)( 1) K,,,,,( 1) K if i P ki ki ki (6) S if i= for stadard Asia optio 155

6 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim K 2K ( ki 1) K,,,,, K if i P ki ki ki (7) if i for forward-starig Asia optio where ki 1 represets the umber of states cosidered for each ode N( ii, ). Here 1 3 k i is computed by B( ii,, p) 2 2 ci i ki, (8) 1 2 si B( si, t, p) 3 2 s1 t s i where (,, ) i B i p p (1 p), ad c is the average umber of states per ode. If the 3-tuple ( ii, S, P ) deotes the curret state, the correspodig optio value V ( ii, S, P ii ) ca be computed by I 1 I 2l I 2l V ( ii, S, PiI ) pl V (( i 1) I, Su, PiI Su ) I, R l ii 2 where S Su, R exp( r t), ad the associated brachig probabilities are I I l l pl p (1 p) l for each brach l,, I, ad i,1,,( 1). Whe P ( 1) K, { PiI ( i) S}/ ( 1) K if R 1 V ( ii, S, P ) ( i) I ii ( i) I I 1 R R {( P ) / ( 1) } if R 1. ii SR K I 1 R For forward-startig discrete Asia optios, the similar formulas hold { PiI ( i) S}/ K if R 1 V ( ii, S, P ) ( i) I ii ( i) I I 1 R R {( P ) / } i f 1 ii SR K R I 1 R whe PiI K. Otherwise, liear iterpolatio is computed from the two bracketig ruig sums' correspodig optio values to obtai : I 2l I 2l V(( i 1) I, Su, P Su ) ii ( 1) K ( 1) K, I 2l I 2l lv ( i 1) I, Su,( sl 1) (1 l ) V ( i 1) I, Su, sl ki 1, l ki 1, l ii 156

7 A Efficiet Biomial Method for Pricig Asia Optios where l 1 for l,1,, I. Now let us apply cell averagig algorithm to this discrete model over the cells o the X -axis as i Sectio 2.2. For example, the cell-averaged payoffs at expiry for stadard discrete Asia call optio are computed as follows: * 1 x h 1 x max * P e K, dx 2h x h 1, * I 2l where x l Su for S, the stock price at t ( 1) I t. Algorithm 1 shows the cell averagig tree algorithm for pricig the Europea stadard discrete Asia call optio based o method of Hsu ad Lyuu (211) method The Cotiuous Asia optio Now we cosider the case of a cotiuously moitored Asia optio. The payoff for cotiuously moitored Asia call optio with strike price K at the expiry date T is give by the followig : 157

8 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim 1 T max S ( )d -K, T (9) Let us cosider Hull ad White (1993) biomial model. As explaied above, pricig of Asia optio usig biomial tree produces 2 N possible paths for each time step N. Hull ad White (1993) proposed a biomial model for cotiuous Asia optios to solve this problem. They chose the represetative average values mh of the form Se for each ode, where h is a fixed costat, ad S S, is the kow iitial asset price. Let A i be the represetative averages, ad let mi max A i ad A i be the miimum ad maximum represetative average, respectively at time i t, i 1,2,,. The m is the smallest iteger chose to satisfy by followig iequalities : mi mh 1 mi Ai Se iai 1 dsi 1,, i 1 max mh 1 max Ai Se iai 1 usi 1, i1, i 1 See Costabile, Massabo ad Russo (26) for details. We ca also compute the kh averages of the form Se, where k m 1, m 2,, m 1 for each time step. Hull ad White model also follows the stadard lattice biomial method proposed by Cox, Ross ad Rubistei (1979) ad use the backward iductio procedure rt V ( i,, k) e pv ( i 1, 1, k ) qv ( i 1,, k ). V ( i 1, 1, k u ) is geerated by usig liear iterpolatio as follow. First, ( i 1) A us / ( i 2) i i, is computed ad let k u A be the smallest represetative average greater tha ( i 1) Ai us i, / ( i 2). The V ( i 1, 1, k u ) is the iterpolatio betwee two optio values associated to Ak 1 ad A k. The value V ( i 1, 1, k d ) is derived i a similar way. Now, let us modify Hull ad White model for improvemet by usig cell averages i Sectio 2.2. For istace, we replace the payoff i of Hull ad White model max( A K,) for the represetative averages A at the last time step with kh 1 Se max( x K,) dx kh 2 (1) Se where is fixed costat ad k represets all itegers betwee m ad m. Algorithm 2 shows the cell averagig tree algorithm for pricig the Europea cotiuous Asia call optio based o method of Hull ad White (1993). d 158

9 A Efficiet Biomial Method for Pricig Asia Optios 4. Numerical Experimets This sectio gives umerical results of cell averagig whe applied to exotic optios such as Bermuda optio or path-depedet optios icludig Discrete forward startig Asia optio ad Cotiuous Asia optio. We fid that the cellaveraged values are more accurate tha other schemes i the sese that these values fall i the iterval cotaiig the exact value faster (Bermuda optio) or coverge to a limitig value faster (path-depedet Asia optios). Sectio 4.1 shows that cell averagig ca be used idepedetly to derive a cell averagig tree method. Sectio 4.2 ad Sectio 4.3 show that cell averagig ca be easily combied with other existig tree methods as well. Sectio 4.2 shows that cell 159

10 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim averaged values are so smooth that the Richardso extrapolatio ca be used further to improve the accuracy. Sectio 4.3 shows that cell averagig ca be exteded to price eve America optios with ease Bermuda optio I this sectio, we implemet the cell averagig method to costruct a tree method to price the Bermuda call optio with the iitial stock price S 1, the risk-free iterest rate r.5, the divided q.1, the volatility.2, strike price K 1 ad the maturity 3. Whe there are m 2 exercisig poits, Aderso ad Broadie (24) provided usig a biomial method with 2 time steps a upper boud of 7.23 ad a lower boud of 7.8 for the optio price whose exact value is See Table 1. See also Aderso ad Broadie (24) for the computatioal results by Aderso ad Broadie ad the explaatios o them. The proposed cell-averaged values fell ito this boud with as low as 2 time steps. I additio, the boud of Aderso ad Broadie has the width of.15 with 2 time steps while the variatio of cellaveraged values for 4 2 is oly.3, which implies that the proposed cell averagig scheme coverges fast. Similar results are observed for the Bermuda optio with m 1 exercisig poits ad the America optio. Table 1 : Parameters: The iitial stock price S() 1, the risk-free iterest rate r.5, the divided q.1, the volatility.2, strike price K 1 ad the maturity T 3 for Bermuda call optio ad m is the umber of discrete exercisig poits. AB represets the lower ad upper bouds for the optio price by Aderso ad Broadie (24) usig a biomial lattice with 2 time steps ad f 1.6. f gives the ratio of the critical exercise price uder the suboptimal policy to the optimal critical exercise price (See Aderso ad Broadie (24) for details). Aderso ad Broadie also preseted exact values (MC-exact) i Aderso ad Broadie (24) Discrete Asia optio respectively. Sice cell averagig reduces oscillatios as explaied i Sectio 2.2, the cell averaged values ca be later improved by the Richardso extrapolatio. Table 2 shows the compariso of several umerical schemes by Večeř, Tavella ad Radall (TR), Curra, Hsu ad Lyuu (HL) with the cell 16

11 A Efficiet Biomial Method for Pricig Asia Optios averagig modificatio of HL (HL-CA) usig the Algorithm 1, ad the Richardso extrapolatio of HL-CA method ( HL-CA ) for the iitial price S 95,1,15 ad various values of the umber of moitor poits. The umber of time steps N I. See Hsu ad Lyuu (211) for detailed explaatios o those schemes icludig the parameters used for the simulatios. I Table 2, for S 95, the variatios i HL-CA method from 25 to 5 ad from 5 to 1 are oly.2 ad.1, respectively. O the other had, correspodig variatios i other methods are about.2 ad.1. Similar differeces are observed for S 1 or S 15. Thus, the applicatio of cell averagig results i sufficietly smooth covergece ad the additioal applicatio of the Richardso extrapolatio ( HL-CA ) produces very rapid covergece. The proposed method is, i this sese, very competitive with may other existig methods. Table 2 : Parameters: A discrete forward startig Asia call optio is cosidered with the strike price K 1, the time to maturity 1, the volatility.4, the iterest rate r.1, ad the divided rate is ot cosidered. Parameters for umerical schemes: itraday period I 1 ad the average umber of states per ode c 5 for Hsu ad Lyuu (211) are cosidered. The table shows the 161

12 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim compariso of various umerical schemes by Večeř, Tavella ad Radall (TR), Curra, Hsu ad Lyuu (HL) with the cell averagig modificatio of HL (HL-CA) usig the Algorithm 1, ad the Richardso extrapolatio of HL-CA method (HL- ( CA for the iitial price S 95,1,15 ad various values of the umber of moitor poits (ad the umber of time steps N I ) 4.3. Cotiuous Asia optio I this sectio, we ow apply the cell averagig method to value aother pathdepedet optio, a cotiuous Asia call optio, based o the Hull ad White (1993) method ad the exted it to value a America optio. The iitial stock price S 5, strike price K 5, risk-free iterest rate r.1, volatility.3, expiry date T 1, ad divided rate q for Europea Asia call optio. The parameters for umerical scheme by is h.1. Figure 3 shows the errors i the Hull ad White biomial method ad the cell averagig Hull ad White method as the umber of time steps icreases, whe.5 is used for the Algorithm 2. The solutio computed by 4 Mote Carlo simulatios based o 1 8 time steps ad 1 simulatio rus is used whe the error is measured, which meas that Mote Carlo value has statistic 4 error of O (1 ). The figure shows that the error from the cell averaged values decreases to zero faster. Table 3 shows the optio values from Hull ad White biomial method (HW) ad the cell averagig HW method (HW-CA) usig the Algorithm 2 with.5,.6,.7, ad.8 as the umber of time steps,, icreases. The values i parethesis are the errors. We see that the covergece of the cell averagig HW-CA values is faster tha that of HW values. Figure 3 : Parameters: the iitial stock price S 5, strike price K 5, risk-free iterest rate r.1, volatility.3, expiry date T 1, ad divided rate q for Europea Asia call optio. The parameters for umerical scheme by Hull ad White (1993) ad Algorithm 2 are h.1 ad.5. The figure shows the errors i optio values from Hull ad White 162

13 A Efficiet Biomial Method for Pricig Asia Optios biomial method ad the cell averagig HW method usig the Algorithm 2 as the umber of time steps,, icreases. Table 3: Parameters: the iitial stock price S 5, strike price K 5, riskfree iterest rate r.1, volatility.3, expiry date T 1, ad divided rate q for Europea Asia call optio. The parameters for umerical scheme by Hull ad White (1993) ad Algorithm 2 are h.1 ad.5,.6,.7,.8. The table shows the optio values from Hull ad White biomial method (HW) ad the cell averagig HW method (HW-CA) usig the Algorithm 2 as the umber of time steps,, icreases. The values i parethesis are the errors. The solutio computed by Mote Carlo simulatios (MC) based o time steps ad 1 simulatio rus is used whe the errors are measured. A simple extesio of the Algorithm 2 for a exercise of the optio results i Table 3. Table 4: Parameters: the iitial stock price S 5, strike price K 5, riskfree iterest rate r.1, volatility.3, expiry date T 1, ad divided rate q for America Asia call optio. The parameters for umerical scheme by Hull ad White (1993) ad Algorithm 2 are h.1 ad.5,.6,.7,.8. The table shows the optio values from Hull ad White biomial method (HW) ad the cell averagig HW method (HW-CA) usig the Algorithm 2 as the umber of time steps,, icreases. The cell averagig method ca be easily exteded to America optios. Table 4 cosiders a America Asia call optio with the same parameters as those for a Europea Asia call optio above usig the Hull ad White method ad the cell averagig Hull ad White model. 5. Coclusios We propose the cell averagig method for pricig the exotic optios, i particular path-depedet Asia optios. Cell averagig reduces the oscillatios of 163

14 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim the tree method ad thus improves the accuracy. It ca be used to derive a idepedet tree scheme or to be combied with existig methods. It ca be eve combied with the extrapolatio as i Sectio 4.2 to ehace the accuracy usig the fact that the correspodig result is smooth or it ca be easily exteded to value America path-depedet optios as i Sectio 4.3. Algorithms 1 ad 2 show that the itroductio of cell averagig does ot icrease computatioal loads much, while umerical experimets validate that cell averagig improves the accuracy of Hsu ad Lyuu method ad Hull ad White method pricig path-depedet Asia optios. For istace, cell averagig gives better represetative averages tha those proposed by Hull ad White. We are curretly workig o mathematical aalysis o the order of covergece of HL-CA ad HW-CA methods for path-depedet Asia optios. REFERENCES [1] Aderso, L. ad Broadie, M. (24),Primal-Dual Simulatio Algorithm for Pricig Multidimesioal America Optios. Maagemet Sciece 5, 9, ; [2] Black, F ad Sholes, M. (1973), The Pricig of Optios ad Corporate Liabilities. The Joural of Political Ecoomy 81, 3, ; [3] Clewlow, L. ad Stricklad, C. (1998),Implemetig Derivatives Models; Joh Wiley & Sos, Chichester, UK; [4] Costabile, M. Massabo, I. ad Russo, E. (26),A Adusted Biomial Model for Pricig Asia Optios. Rev Quat Fia Acc 27, ; [5] Cox, J. Ross, S. ad Rubistei, M. (1979), Optio Pricig : A Simplified Approach. Joural of Fiacial Ecoomics 7, ; [6] Forsyth, P. A., Vetzal, K. R. ad Zva, R. (22), Covergece of Numerical Methods for Valuig Path-Depedet Optios Usig Iterpolatio. Rev Derivatives Res 5, ; [7] Higham, D. J. (24), A Itroductio to Fiacial Optio Valuatio. Cambridge Uiversity Press; [8] Hsu, W. Y. ad Lyuu, Y. D. (211), Efficiet Pricig of Discrete Asia Optios. Applied Mathematics ad Computatio 217, ; [9] Hull, J. ad White, A. (1993), Efficiet Procedures for Valuig Europea ad America Path-Depedet Optios. Joural of Derivatives 1, 21-31; [1] Øksedal, B. (1998), Stochastic Differetial Equatios. Spriger, Berli; [11] Kwok, Y. K. (1998), Mathematical Models of Fiacial Derivatives. Spriger, Sigapore; [12] Lyuu,Y. D. (22), Fiacial Egieerig ad Computatio. Cambridge; [13] Merto, R. C. (1973), Theory of Ratioal Optio Pricig. Bell Joural of Ecoomics ad Maagemet Sciece 4, 1, ; [14] Moo, K. S. ad Kim, H. (213),A Multi-dimesioal Local Average Lattice Method for Multi-asset Models. Quatitative Fiace 13,

Hopscotch and Explicit difference method for solving Black-Scholes PDE

Hopscotch and Explicit difference method for solving Black-Scholes PDE Mälardale iversity Fiacial Egieerig Program Aalytical Fiace Semiar Report Hopscotch ad Explicit differece method for solvig Blac-Scholes PDE Istructor: Ja Röma Team members: A Gog HaiLog Zhao Hog Cui 0

More information

Binomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge

Binomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge Biomial Model Stock Price Dyamics The value of a optio at maturity depeds o the price of the uderlyig stock at maturity. The value of the optio today depeds o the expected value of the optio at maturity

More information

Minhyun Yoo, Darae Jeong, Seungsuk Seo, and Junseok Kim

Minhyun Yoo, Darae Jeong, Seungsuk Seo, and Junseok Kim Hoam Mathematical J. 37 (15), No. 4, pp. 441 455 http://dx.doi.org/1.5831/hmj.15.37.4.441 A COMPARISON STUDY OF EXPLICIT AND IMPLICIT NUMERICAL METHODS FOR THE EQUITY-LINKED SECURITIES Mihyu Yoo, Darae

More information

0.1 Valuation Formula:

0.1 Valuation Formula: 0. Valuatio Formula: 0.. Case of Geeral Trees: q = er S S S 3 S q = er S S 4 S 5 S 4 q 3 = er S 3 S 6 S 7 S 6 Therefore, f (3) = e r [q 3 f (7) + ( q 3 ) f (6)] f () = e r [q f (5) + ( q ) f (4)] = f ()

More information

1 Estimating sensitivities

1 Estimating sensitivities Copyright c 27 by Karl Sigma 1 Estimatig sesitivities Whe estimatig the Greeks, such as the, the geeral problem ivolves a radom variable Y = Y (α) (such as a discouted payoff) that depeds o a parameter

More information

Stochastic Processes and their Applications in Financial Pricing

Stochastic Processes and their Applications in Financial Pricing Stochastic Processes ad their Applicatios i Fiacial Pricig Adrew Shi Jue 3, 1 Cotets 1 Itroductio Termiology.1 Fiacial.............................................. Stochastics............................................

More information

Proceedings of the 5th WSEAS Int. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 17-19, 2005 (pp )

Proceedings of the 5th WSEAS Int. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 17-19, 2005 (pp ) Proceedigs of the 5th WSEAS It. Cof. o SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 7-9, 005 (pp488-49 Realized volatility estimatio: ew simulatio approach ad empirical study results JULIA

More information

AMS Portfolio Theory and Capital Markets

AMS Portfolio Theory and Capital Markets AMS 69.0 - Portfolio Theory ad Capital Markets I Class 6 - Asset yamics Robert J. Frey Research Professor Stoy Brook iversity, Applied Mathematics ad Statistics frey@ams.suysb.edu http://www.ams.suysb.edu/~frey/

More information

Chapter 13 Binomial Trees. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull

Chapter 13 Binomial Trees. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull Chapter 13 Biomial Trees 1 A Simple Biomial Model! A stock price is curretly $20! I 3 moths it will be either $22 or $18 Stock price $20 Stock Price $22 Stock Price $18 2 A Call Optio (Figure 13.1, page

More information

Itroductio he efficiet ricig of otios is of great ractical imortace: Whe large baskets of otios have to be riced simultaeously seed accuracy trade off

Itroductio he efficiet ricig of otios is of great ractical imortace: Whe large baskets of otios have to be riced simultaeously seed accuracy trade off A Commet O he Rate Of Covergece of Discrete ime Cotiget Claims Dietmar P.J. Leise Staford Uiversity, Hoover Istitutio, Staford, CA 945, U.S.A., email: leise@hoover.staford.edu Matthias Reimer WestLB Pamure,

More information

Subject CT1 Financial Mathematics Core Technical Syllabus

Subject CT1 Financial Mathematics Core Technical Syllabus Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig

More information

ON THE RATE OF CONVERGENCE

ON THE RATE OF CONVERGENCE ON THE RATE OF CONVERGENCE OF BINOMIAL GREEKS SAN-LIN CHUNG WEIFENG HUNG HAN-HSING LEE* PAI-TA SHIH This study ivestigates the covergece patters ad the rates of covergece of biomial Greeks for the CRR

More information

A Hybrid Finite Difference Method for Valuing American Puts

A Hybrid Finite Difference Method for Valuing American Puts Proceedigs of the World Cogress o Egieerig 29 Vol II A Hybrid Fiite Differece Method for Valuig America Puts Ji Zhag SogPig Zhu Abstract This paper presets a umerical scheme that avoids iteratios to solve

More information

Estimating Proportions with Confidence

Estimating Proportions with Confidence Aoucemets: Discussio today is review for midterm, o credit. You may atted more tha oe discussio sectio. Brig sheets of otes ad calculator to midterm. We will provide Scatro form. Homework: (Due Wed Chapter

More information

Statistics for Economics & Business

Statistics for Economics & Business Statistics for Ecoomics & Busiess Cofidece Iterval Estimatio Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for the mea ad the proportio How to determie

More information

A random variable is a variable whose value is a numerical outcome of a random phenomenon.

A random variable is a variable whose value is a numerical outcome of a random phenomenon. The Practice of Statistics, d ed ates, Moore, ad Stares Itroductio We are ofte more iterested i the umber of times a give outcome ca occur tha i the possible outcomes themselves For example, if we toss

More information

Anomaly Correction by Optimal Trading Frequency

Anomaly Correction by Optimal Trading Frequency Aomaly Correctio by Optimal Tradig Frequecy Yiqiao Yi Columbia Uiversity September 9, 206 Abstract Uder the assumptio that security prices follow radom walk, we look at price versus differet movig averages.

More information

point estimator a random variable (like P or X) whose values are used to estimate a population parameter

point estimator a random variable (like P or X) whose values are used to estimate a population parameter Estimatio We have oted that the pollig problem which attempts to estimate the proportio p of Successes i some populatio ad the measuremet problem which attempts to estimate the mea value µ of some quatity

More information

Introduction to Probability and Statistics Chapter 7

Introduction to Probability and Statistics Chapter 7 Itroductio to Probability ad Statistics Chapter 7 Ammar M. Sarha, asarha@mathstat.dal.ca Departmet of Mathematics ad Statistics, Dalhousie Uiversity Fall Semester 008 Chapter 7 Statistical Itervals Based

More information

Journal of Statistical Software

Journal of Statistical Software JSS Joural of Statistical Software Jue 2007, Volume 19, Issue 6. http://www.jstatsoft.org/ Ratioal Arithmetic Mathematica Fuctios to Evaluate the Oe-sided Oe-sample K-S Cumulative Samplig Distributio J.

More information

Online appendices from The xva Challenge by Jon Gregory. APPENDIX 10A: Exposure and swaption analogy.

Online appendices from The xva Challenge by Jon Gregory. APPENDIX 10A: Exposure and swaption analogy. APPENDIX 10A: Exposure ad swaptio aalogy. Sorese ad Bollier (1994), effectively calculate the CVA of a swap positio ad show this ca be writte as: CVA swap = LGD V swaptio (t; t i, T) PD(t i 1, t i ). i=1

More information

Positivity Preserving Schemes for Black-Scholes Equation

Positivity Preserving Schemes for Black-Scholes Equation Research Joural of Fiace ad Accoutig IN -97 (Paper) IN -7 (Olie) Vol., No.7, 5 Positivity Preservig chemes for Black-choles Equatio Mohammad Mehdizadeh Khalsaraei (Correspodig author) Faculty of Mathematical

More information

Sequences and Series

Sequences and Series Sequeces ad Series Matt Rosezweig Cotets Sequeces ad Series. Sequeces.................................................. Series....................................................3 Rudi Chapter 3 Exercises........................................

More information

Research Article The Probability That a Measurement Falls within a Range of n Standard Deviations from an Estimate of the Mean

Research Article The Probability That a Measurement Falls within a Range of n Standard Deviations from an Estimate of the Mean Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 70806, 8 pages doi:0.540/0/70806 Research Article The Probability That a Measuremet Falls withi a Rage of Stadard Deviatios

More information

A New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions

A New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions A New Costructive Proof of Graham's Theorem ad More New Classes of Fuctioally Complete Fuctios Azhou Yag, Ph.D. Zhu-qi Lu, Ph.D. Abstract A -valued two-variable truth fuctio is called fuctioally complete,

More information

Pricing 50ETF in the Way of American Options Based on Least Squares Monte Carlo Simulation

Pricing 50ETF in the Way of American Options Based on Least Squares Monte Carlo Simulation Pricig 50ETF i the Way of America Optios Based o Least Squares Mote Carlo Simulatio Shuai Gao 1, Ju Zhao 1 Applied Fiace ad Accoutig Vol., No., August 016 ISSN 374-410 E-ISSN 374-49 Published by Redfame

More information

REVISIT OF STOCHASTIC MESH METHOD FOR PRICING AMERICAN OPTIONS. Guangwu Liu L. Jeff Hong

REVISIT OF STOCHASTIC MESH METHOD FOR PRICING AMERICAN OPTIONS. Guangwu Liu L. Jeff Hong Proceedigs of the 2008 Witer Simulatio Coferece S. J. Maso, R. R. Hill, L. Möch, O. Rose, T. Jefferso, J. W. Fowler eds. REVISIT OF STOCHASTIC MESH METHOD FOR PRICING AMERICAN OPTIONS Guagwu Liu L. Jeff

More information

Diener and Diener and Walsh follow as special cases. In addition, by making. smooth, as numerically observed by Tian. Moreover, we propose the center

Diener and Diener and Walsh follow as special cases. In addition, by making. smooth, as numerically observed by Tian. Moreover, we propose the center Smooth Covergece i the Biomial Model Lo-Bi Chag ad Ke Palmer Departmet of Mathematics, Natioal Taiwa Uiversity Abstract Various authors have studied the covergece of the biomial optio price to the Black-Scholes

More information

Lecture 4: Parameter Estimation and Confidence Intervals. GENOME 560 Doug Fowler, GS

Lecture 4: Parameter Estimation and Confidence Intervals. GENOME 560 Doug Fowler, GS Lecture 4: Parameter Estimatio ad Cofidece Itervals GENOME 560 Doug Fowler, GS (dfowler@uw.edu) 1 Review: Probability Distributios Discrete: Biomial distributio Hypergeometric distributio Poisso distributio

More information

Analytical Approximate Solutions for Stochastic Volatility. American Options under Barrier Options Models

Analytical Approximate Solutions for Stochastic Volatility. American Options under Barrier Options Models Aalytical Approximate Solutios for Stochastic Volatility America Optios uder Barrier Optios Models Chug-Gee Li Chiao-Hsi Su Soochow Uiversity Abstract This paper exteds the work of Hesto (99) ad itegrates

More information

The pricing of discretely sampled Asian and lookback options: a change of numeraire approach

The pricing of discretely sampled Asian and lookback options: a change of numeraire approach The pricig of discretely sampled Asia ad lookback optios 5 The pricig of discretely sampled Asia ad lookback optios: a chage of umeraire approach Jesper Adrease This paper cosiders the pricig of discretely

More information

5. Best Unbiased Estimators

5. Best Unbiased Estimators Best Ubiased Estimators http://www.math.uah.edu/stat/poit/ubiased.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 5. Best Ubiased Estimators Basic Theory Cosider agai

More information

The Valuation of the Catastrophe Equity Puts with Jump Risks

The Valuation of the Catastrophe Equity Puts with Jump Risks The Valuatio of the Catastrophe Equity Puts with Jump Risks Shih-Kuei Li Natioal Uiversity of Kaohsiug Joit work with Chia-Chie Chag Outlie Catastrophe Isurace Products Literatures ad Motivatios Jump Risk

More information

CAPITAL PROJECT SCREENING AND SELECTION

CAPITAL PROJECT SCREENING AND SELECTION CAPITAL PROJECT SCREEIG AD SELECTIO Before studyig the three measures of ivestmet attractiveess, we will review a simple method that is commoly used to scree capital ivestmets. Oe of the primary cocers

More information

Combining imperfect data, and an introduction to data assimilation Ross Bannister, NCEO, September 2010

Combining imperfect data, and an introduction to data assimilation Ross Bannister, NCEO, September 2010 Combiig imperfect data, ad a itroductio to data assimilatio Ross Baister, NCEO, September 00 rbaister@readigacuk The probability desity fuctio (PDF prob that x lies betwee x ad x + dx p (x restrictio o

More information

Department of Mathematics, S.R.K.R. Engineering College, Bhimavaram, A.P., India 2

Department of Mathematics, S.R.K.R. Engineering College, Bhimavaram, A.P., India 2 Skewess Corrected Cotrol charts for two Iverted Models R. Subba Rao* 1, Pushpa Latha Mamidi 2, M.S. Ravi Kumar 3 1 Departmet of Mathematics, S.R.K.R. Egieerig College, Bhimavaram, A.P., Idia 2 Departmet

More information

Lecture 5: Sampling Distribution

Lecture 5: Sampling Distribution Lecture 5: Samplig Distributio Readigs: Sectios 5.5, 5.6 Itroductio Parameter: describes populatio Statistic: describes the sample; samplig variability Samplig distributio of a statistic: A probability

More information

An Empirical Study of the Behaviour of the Sample Kurtosis in Samples from Symmetric Stable Distributions

An Empirical Study of the Behaviour of the Sample Kurtosis in Samples from Symmetric Stable Distributions A Empirical Study of the Behaviour of the Sample Kurtosis i Samples from Symmetric Stable Distributios J. Marti va Zyl Departmet of Actuarial Sciece ad Mathematical Statistics, Uiversity of the Free State,

More information

Dr. Maddah ENMG 624 Financial Eng g I 03/22/06. Chapter 6 Mean-Variance Portfolio Theory

Dr. Maddah ENMG 624 Financial Eng g I 03/22/06. Chapter 6 Mean-Variance Portfolio Theory Dr Maddah ENMG 64 Fiacial Eg g I 03//06 Chapter 6 Mea-Variace Portfolio Theory Sigle Period Ivestmets Typically, i a ivestmet the iitial outlay of capital is kow but the retur is ucertai A sigle-period

More information

18.S096 Problem Set 5 Fall 2013 Volatility Modeling Due Date: 10/29/2013

18.S096 Problem Set 5 Fall 2013 Volatility Modeling Due Date: 10/29/2013 18.S096 Problem Set 5 Fall 2013 Volatility Modelig Due Date: 10/29/2013 1. Sample Estimators of Diffusio Process Volatility ad Drift Let {X t } be the price of a fiacial security that follows a geometric

More information

Overlapping Generations

Overlapping Generations Eco. 53a all 996 C. Sims. troductio Overlappig Geeratios We wat to study how asset markets allow idividuals, motivated by the eed to provide icome for their retiremet years, to fiace capital accumulatio

More information

AN IMPROVED BINOMIAL METHOD FOR PRICING ASIAN OPTIONS

AN IMPROVED BINOMIAL METHOD FOR PRICING ASIAN OPTIONS Commun. Korean Math. Soc. 28 (2013), No. 2, pp. 397 406 http://dx.doi.org/10.4134/ckms.2013.28.2.397 AN IMPROVED BINOMIAL METHOD FOR PRICING ASIAN OPTIONS Kyoung-Sook Moon and Hongjoong Kim Abstract. We

More information

arxiv: v5 [cs.ce] 3 Dec 2008

arxiv: v5 [cs.ce] 3 Dec 2008 PRICING AMERICAN OPTIONS FOR JUMP DIFFUSIONS BY ITERATING OPTIMAL STOPPING PROBLEMS FOR DIFFUSIONS ERHAN BAYRAKTAR AND HAO XING arxiv:0706.2331v5 [cs.ce] 3 Dec 2008 Abstract. We approximate the price of

More information

The Limit of a Sequence (Brief Summary) 1

The Limit of a Sequence (Brief Summary) 1 The Limit of a Sequece (Brief Summary). Defiitio. A real umber L is a it of a sequece of real umbers if every ope iterval cotaiig L cotais all but a fiite umber of terms of the sequece. 2. Claim. A sequece

More information

Limits of sequences. Contents 1. Introduction 2 2. Some notation for sequences The behaviour of infinite sequences 3

Limits of sequences. Contents 1. Introduction 2 2. Some notation for sequences The behaviour of infinite sequences 3 Limits of sequeces I this uit, we recall what is meat by a simple sequece, ad itroduce ifiite sequeces. We explai what it meas for two sequeces to be the same, ad what is meat by the -th term of a sequece.

More information

Maximum Empirical Likelihood Estimation (MELE)

Maximum Empirical Likelihood Estimation (MELE) Maximum Empirical Likelihood Estimatio (MELE Natha Smooha Abstract Estimatio of Stadard Liear Model - Maximum Empirical Likelihood Estimator: Combiatio of the idea of imum likelihood method of momets,

More information

43. A 000 par value 5-year bod with 8.0% semiaual coupos was bought to yield 7.5% covertible semiaually. Determie the amout of premium amortized i the 6 th coupo paymet. (A).00 (B).08 (C).5 (D).5 (E).34

More information

Topic-7. Large Sample Estimation

Topic-7. Large Sample Estimation Topic-7 Large Sample Estimatio TYPES OF INFERENCE Ò Estimatio: É Estimatig or predictig the value of the parameter É What is (are) the most likely values of m or p? Ò Hypothesis Testig: É Decidig about

More information

Neighboring Optimal Solution for Fuzzy Travelling Salesman Problem

Neighboring Optimal Solution for Fuzzy Travelling Salesman Problem Iteratioal Joural of Egieerig Research ad Geeral Sciece Volume 2, Issue 4, Jue-July, 2014 Neighborig Optimal Solutio for Fuzzy Travellig Salesma Problem D. Stephe Digar 1, K. Thiripura Sudari 2 1 Research

More information

A New Approach to Obtain an Optimal Solution for the Assignment Problem

A New Approach to Obtain an Optimal Solution for the Assignment Problem Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): 231-7064 Idex Copericus Value (2013): 6.14 Impact Factor (2015): 6.31 A New Approach to Obtai a Optimal Solutio for the Assigmet Problem A. Seethalakshmy

More information

1 The Black-Scholes model

1 The Black-Scholes model The Blac-Scholes model. The model setup I the simplest versio of the Blac-Scholes model the are two assets: a ris-less asset ba accout or bod)withpriceprocessbt) at timet, adarisyasset stoc) withpriceprocess

More information

Institute of Actuaries of India Subject CT5 General Insurance, Life and Health Contingencies

Institute of Actuaries of India Subject CT5 General Insurance, Life and Health Contingencies Istitute of Actuaries of Idia Subject CT5 Geeral Isurace, Life ad Health Cotigecies For 2017 Examiatios Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which

More information

Subject CT5 Contingencies Core Technical. Syllabus. for the 2011 Examinations. The Faculty of Actuaries and Institute of Actuaries.

Subject CT5 Contingencies Core Technical. Syllabus. for the 2011 Examinations. The Faculty of Actuaries and Institute of Actuaries. Subject CT5 Cotigecies Core Techical Syllabus for the 2011 Examiatios 1 Jue 2010 The Faculty of Actuaries ad Istitute of Actuaries Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical

More information

Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1

Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1 Chapter 8 Cofidece Iterval Estimatio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 1 Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for

More information

Mixed and Implicit Schemes Implicit Schemes. Exercise: Verify that ρ is unimodular: ρ = 1.

Mixed and Implicit Schemes Implicit Schemes. Exercise: Verify that ρ is unimodular: ρ = 1. Mixed ad Implicit Schemes 3..4 The leapfrog scheme is stable for the oscillatio equatio ad ustable for the frictio equatio. The Euler forward scheme is stable for the frictio equatio but ustable for the

More information

Productivity depending risk minimization of production activities

Productivity depending risk minimization of production activities Productivity depedig risk miimizatio of productio activities GEORGETTE KANARACHOU, VRASIDAS LEOPOULOS Productio Egieerig Sectio Natioal Techical Uiversity of Athes, Polytechioupolis Zografou, 15780 Athes

More information

Simulation Efficiency and an Introduction to Variance Reduction Methods

Simulation Efficiency and an Introduction to Variance Reduction Methods Mote Carlo Simulatio: IEOR E4703 Columbia Uiversity c 2017 by Marti Haugh Simulatio Efficiecy ad a Itroductio to Variace Reductio Methods I these otes we discuss the efficiecy of a Mote-Carlo estimator.

More information

Faculdade de Economia da Universidade de Coimbra

Faculdade de Economia da Universidade de Coimbra Faculdade de Ecoomia da Uiversidade de Coimbra Grupo de Estudos Moetários e Fiaceiros (GEMF) Av. Dias da Silva, 65 300-5 COIMBRA, PORTUGAL gemf@fe.uc.pt http://www.uc.pt/feuc/gemf PEDRO GODINHO Estimatig

More information

Unbiased estimators Estimators

Unbiased estimators Estimators 19 Ubiased estimators I Chapter 17 we saw that a dataset ca be modeled as a realizatio of a radom sample from a probability distributio ad that quatities of iterest correspod to features of the model distributio.

More information

Math 312, Intro. to Real Analysis: Homework #4 Solutions

Math 312, Intro. to Real Analysis: Homework #4 Solutions Math 3, Itro. to Real Aalysis: Homework #4 Solutios Stephe G. Simpso Moday, March, 009 The assigmet cosists of Exercises 0.6, 0.8, 0.0,.,.3,.6,.0,.,. i the Ross textbook. Each problem couts 0 poits. 0.6.

More information

We analyze the computational problem of estimating financial risk in a nested simulation. In this approach,

We analyze the computational problem of estimating financial risk in a nested simulation. In this approach, MANAGEMENT SCIENCE Vol. 57, No. 6, Jue 2011, pp. 1172 1194 iss 0025-1909 eiss 1526-5501 11 5706 1172 doi 10.1287/msc.1110.1330 2011 INFORMS Efficiet Risk Estimatio via Nested Sequetial Simulatio Mark Broadie

More information

Today: Finish Chapter 9 (Sections 9.6 to 9.8 and 9.9 Lesson 3)

Today: Finish Chapter 9 (Sections 9.6 to 9.8 and 9.9 Lesson 3) Today: Fiish Chapter 9 (Sectios 9.6 to 9.8 ad 9.9 Lesso 3) ANNOUNCEMENTS: Quiz #7 begis after class today, eds Moday at 3pm. Quiz #8 will begi ext Friday ad ed at 10am Moday (day of fial). There will be

More information

Research Paper Number From Discrete to Continuous Time Finance: Weak Convergence of the Financial Gain Process

Research Paper Number From Discrete to Continuous Time Finance: Weak Convergence of the Financial Gain Process Research Paper Number 197 From Discrete to Cotiuous Time Fiace: Weak Covergece of the Fiacial Gai Process Darrell Duffie ad Philip Protter November, 1988 Revised: September, 1991 Forthcomig: Mathematical

More information

1 + r. k=1. (1 + r) k = A r 1

1 + r. k=1. (1 + r) k = A r 1 Perpetual auity pays a fixed sum periodically forever. Suppose a amout A is paid at the ed of each period, ad suppose the per-period iterest rate is r. The the preset value of the perpetual auity is A

More information

The Time Value of Money in Financial Management

The Time Value of Money in Financial Management The Time Value of Moey i Fiacial Maagemet Muteau Irea Ovidius Uiversity of Costata irea.muteau@yahoo.com Bacula Mariaa Traia Theoretical High School, Costata baculamariaa@yahoo.com Abstract The Time Value

More information

ad covexity Defie Macaulay duratio D Mod = r 1 = ( CF i i k (1 + r k) i ) (1.) (1 + r k) C = ( r ) = 1 ( CF i i(i + 1) (1 + r k) i+ k ) ( ( i k ) CF i

ad covexity Defie Macaulay duratio D Mod = r 1 = ( CF i i k (1 + r k) i ) (1.) (1 + r k) C = ( r ) = 1 ( CF i i(i + 1) (1 + r k) i+ k ) ( ( i k ) CF i Fixed Icome Basics Cotets Duratio ad Covexity Bod Duratios ar Rate, Spot Rate, ad Forward Rate Flat Forward Iterpolatio Forward rice/yield, Carry, Roll-Dow Example Duratio ad Covexity For a series of cash

More information

Online appendices from Counterparty Risk and Credit Value Adjustment a continuing challenge for global financial markets by Jon Gregory

Online appendices from Counterparty Risk and Credit Value Adjustment a continuing challenge for global financial markets by Jon Gregory Olie appedices from Couterparty Risk ad Credit Value Adjustmet a APPENDIX 8A: Formulas for EE, PFE ad EPE for a ormal distributio Cosider a ormal distributio with mea (expected future value) ad stadard

More information

FINM6900 Finance Theory How Is Asymmetric Information Reflected in Asset Prices?

FINM6900 Finance Theory How Is Asymmetric Information Reflected in Asset Prices? FINM6900 Fiace Theory How Is Asymmetric Iformatio Reflected i Asset Prices? February 3, 2012 Referece S. Grossma, O the Efficiecy of Competitive Stock Markets where Traders Have Diverse iformatio, Joural

More information

NPTEL DEPARTMENT OF INDUSTRIAL AND MANAGEMENT ENGINEERING IIT KANPUR QUANTITATIVE FINANCE END-TERM EXAMINATION (2015 JULY-AUG ONLINE COURSE)

NPTEL DEPARTMENT OF INDUSTRIAL AND MANAGEMENT ENGINEERING IIT KANPUR QUANTITATIVE FINANCE END-TERM EXAMINATION (2015 JULY-AUG ONLINE COURSE) NPTEL DEPARTMENT OF INDUSTRIAL AND MANAGEMENT ENGINEERING IIT KANPUR QUANTITATIVE FINANCE END-TERM EXAMINATION (2015 JULY-AUG ONLINE COURSE) READ THE INSTRUCTIONS VERY CAREFULLY 1) Time duratio is 2 hours

More information

x satisfying all regularity conditions. Then

x satisfying all regularity conditions. Then AMS570.01 Practice Midterm Exam Sprig, 018 Name: ID: Sigature: Istructio: This is a close book exam. You are allowed oe-page 8x11 formula sheet (-sided). No cellphoe or calculator or computer is allowed.

More information

Monetary Economics: Problem Set #5 Solutions

Monetary Economics: Problem Set #5 Solutions Moetary Ecoomics oblem Set #5 Moetary Ecoomics: oblem Set #5 Solutios This problem set is marked out of 1 poits. The weight give to each part is idicated below. Please cotact me asap if you have ay questios.

More information

Threshold Function for the Optimal Stopping of Arithmetic Ornstein-Uhlenbeck Process

Threshold Function for the Optimal Stopping of Arithmetic Ornstein-Uhlenbeck Process Proceedigs of the 2015 Iteratioal Coferece o Operatios Excellece ad Service Egieerig Orlado, Florida, USA, September 10-11, 2015 Threshold Fuctio for the Optimal Stoppig of Arithmetic Orstei-Uhlebeck Process

More information

Chapter 5: Sequences and Series

Chapter 5: Sequences and Series Chapter 5: Sequeces ad Series 1. Sequeces 2. Arithmetic ad Geometric Sequeces 3. Summatio Notatio 4. Arithmetic Series 5. Geometric Series 6. Mortgage Paymets LESSON 1 SEQUENCES I Commo Core Algebra I,

More information

Models of Asset Pricing

Models of Asset Pricing APPENDIX 1 TO CHAPTER 4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see

More information

Models of Asset Pricing

Models of Asset Pricing APPENDIX 1 TO CHAPTER4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see

More information

Inferential Statistics and Probability a Holistic Approach. Inference Process. Inference Process. Chapter 8 Slides. Maurice Geraghty,

Inferential Statistics and Probability a Holistic Approach. Inference Process. Inference Process. Chapter 8 Slides. Maurice Geraghty, Iferetial Statistics ad Probability a Holistic Approach Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike 4.0

More information

EXERCISE - BINOMIAL THEOREM

EXERCISE - BINOMIAL THEOREM BINOMIAL THOEREM / EXERCISE - BINOMIAL THEOREM LEVEL I SUBJECTIVE QUESTIONS. Expad the followig expressios ad fid the umber of term i the expasio of the expressios. (a) (x + y) 99 (b) ( + a) 9 + ( a) 9

More information

Hannan and Blackwell meet Black and Scholes: Approachability and Robust Option Pricing

Hannan and Blackwell meet Black and Scholes: Approachability and Robust Option Pricing Haa ad Blackwell meet Black ad Scholes: Approachability ad Robust Optio Pricig Peter M. DeMarzo, Ila Kremer, ad Yishay Masour October, 2005 This Revisio: 4/20/09 ABSTRACT. We study the lik betwee the game

More information

CHAPTER 2 PRICING OF BONDS

CHAPTER 2 PRICING OF BONDS CHAPTER 2 PRICING OF BONDS CHAPTER SUARY This chapter will focus o the time value of moey ad how to calculate the price of a bod. Whe pricig a bod it is ecessary to estimate the expected cash flows ad

More information

STRAND: FINANCE. Unit 3 Loans and Mortgages TEXT. Contents. Section. 3.1 Annual Percentage Rate (APR) 3.2 APR for Repayment of Loans

STRAND: FINANCE. Unit 3 Loans and Mortgages TEXT. Contents. Section. 3.1 Annual Percentage Rate (APR) 3.2 APR for Repayment of Loans CMM Subject Support Strad: FINANCE Uit 3 Loas ad Mortgages: Text m e p STRAND: FINANCE Uit 3 Loas ad Mortgages TEXT Cotets Sectio 3.1 Aual Percetage Rate (APR) 3.2 APR for Repaymet of Loas 3.3 Credit Purchases

More information

Chapter Four Learning Objectives Valuing Monetary Payments Now and in the Future

Chapter Four Learning Objectives Valuing Monetary Payments Now and in the Future Chapter Four Future Value, Preset Value, ad Iterest Rates Chapter 4 Learig Objectives Develop a uderstadig of 1. Time ad the value of paymets 2. Preset value versus future value 3. Nomial versus real iterest

More information

Portfolio Optimization for Options

Portfolio Optimization for Options Portfolio Optimizatio for Optios Yaxiog Zeg 1, Diego Klabja 2 Abstract Optio portfolio optimizatio for Europea optios has already bee studied, but more challegig America optios have ot We propose approximate

More information

Annual compounding, revisited

Annual compounding, revisited Sectio 1.: No-aual compouded iterest MATH 105: Cotemporary Mathematics Uiversity of Louisville August 2, 2017 Compoudig geeralized 2 / 15 Aual compoudig, revisited The idea behid aual compoudig is that

More information

Monopoly vs. Competition in Light of Extraction Norms. Abstract

Monopoly vs. Competition in Light of Extraction Norms. Abstract Moopoly vs. Competitio i Light of Extractio Norms By Arkadi Koziashvili, Shmuel Nitza ad Yossef Tobol Abstract This ote demostrates that whether the market is competitive or moopolistic eed ot be the result

More information

A STOCHASTIC GROWTH PRICE MODEL USING A BIRTH AND DEATH DIFFUSION GROWTH RATE PROCESS WITH EXTERNAL JUMP PROCESS *

A STOCHASTIC GROWTH PRICE MODEL USING A BIRTH AND DEATH DIFFUSION GROWTH RATE PROCESS WITH EXTERNAL JUMP PROCESS * Page345 ISBN: 978 0 9943656 75; ISSN: 05-6033 Year: 017, Volume: 3, Issue: 1 A STOCHASTIC GROWTH PRICE MODEL USING A BIRTH AND DEATH DIFFUSION GROWTH RATE PROCESS WITH EXTERNAL JUMP PROCESS * Basel M.

More information

. (The calculated sample mean is symbolized by x.)

. (The calculated sample mean is symbolized by x.) Stat 40, sectio 5.4 The Cetral Limit Theorem otes by Tim Pilachowski If you have t doe it yet, go to the Stat 40 page ad dowload the hadout 5.4 supplemet Cetral Limit Theorem. The homework (both practice

More information

Chapter 11 Appendices: Review of Topics from Foundations in Finance and Tables

Chapter 11 Appendices: Review of Topics from Foundations in Finance and Tables Chapter 11 Appedices: Review of Topics from Foudatios i Fiace ad Tables A: INTRODUCTION The expressio Time is moey certaily applies i fiace. People ad istitutios are impatiet; they wat moey ow ad are geerally

More information

INTERVAL GAMES. and player 2 selects 1, then player 2 would give player 1 a payoff of, 1) = 0.

INTERVAL GAMES. and player 2 selects 1, then player 2 would give player 1 a payoff of, 1) = 0. INTERVAL GAMES ANTHONY MENDES Let I ad I 2 be itervals of real umbers. A iterval game is played i this way: player secretly selects x I ad player 2 secretly ad idepedetly selects y I 2. After x ad y are

More information

r i = a i + b i f b i = Cov[r i, f] The only parameters to be estimated for this model are a i 's, b i 's, σe 2 i

r i = a i + b i f b i = Cov[r i, f] The only parameters to be estimated for this model are a i 's, b i 's, σe 2 i The iformatio required by the mea-variace approach is substatial whe the umber of assets is large; there are mea values, variaces, ad )/2 covariaces - a total of 2 + )/2 parameters. Sigle-factor model:

More information

On Regret and Options - A Game Theoretic Approach for Option Pricing

On Regret and Options - A Game Theoretic Approach for Option Pricing O Regret ad Optios - A Game Theoretic Approach for Optio Pricig Peter M. DeMarzo, Ila Kremer ad Yishay Masour Staford Graduate School of Busiess ad Tel Aviv Uiversity October, 005 This Revisio: 9/7/05

More information

Twitter: @Owe134866 www.mathsfreeresourcelibrary.com Prior Kowledge Check 1) State whether each variable is qualitative or quatitative: a) Car colour Qualitative b) Miles travelled by a cyclist c) Favourite

More information

SUPPLEMENTAL MATERIAL

SUPPLEMENTAL MATERIAL A SULEMENTAL MATERIAL Theorem (Expert pseudo-regret upper boud. Let us cosider a istace of the I-SG problem ad apply the FL algorithm, where each possible profile A is a expert ad receives, at roud, a

More information

Non-Inferiority Logrank Tests

Non-Inferiority Logrank Tests Chapter 706 No-Iferiority Lograk Tests Itroductio This module computes the sample size ad power for o-iferiority tests uder the assumptio of proportioal hazards. Accrual time ad follow-up time are icluded

More information

Lecture 9: The law of large numbers and central limit theorem

Lecture 9: The law of large numbers and central limit theorem Lecture 9: The law of large umbers ad cetral limit theorem Theorem.4 Let X,X 2,... be idepedet radom variables with fiite expectatios. (i) (The SLLN). If there is a costat p [,2] such that E X i p i i=

More information

First determine the payments under the payment system

First determine the payments under the payment system Corporate Fiace February 5, 2008 Problem Set # -- ANSWERS Klick. You wi a judgmet agaist a defedat worth $20,000,000. Uder state law, the defedat has the right to pay such a judgmet out over a 20 year

More information

Further Pure 1 Revision Topic 5: Sums of Series

Further Pure 1 Revision Topic 5: Sums of Series The OCR syllabus says that cadidates should: Further Pure Revisio Topic 5: Sums of Series Cadidates should be able to: (a) use the stadard results for Σr, Σr, Σr to fid related sums; (b) use the method

More information

Bayes Estimator for Coefficient of Variation and Inverse Coefficient of Variation for the Normal Distribution

Bayes Estimator for Coefficient of Variation and Inverse Coefficient of Variation for the Normal Distribution Iteratioal Joural of Statistics ad Systems ISSN 0973-675 Volume, Number 4 (07, pp. 7-73 Research Idia Publicatios http://www.ripublicatio.com Bayes Estimator for Coefficiet of Variatio ad Iverse Coefficiet

More information

This article is part of a series providing

This article is part of a series providing feature Bryce Millard ad Adrew Machi Characteristics of public sector workers SUMMARY This article presets aalysis of public sector employmet, ad makes comparisos with the private sector, usig data from

More information

Chapter Four 1/15/2018. Learning Objectives. The Meaning of Interest Rates Future Value, Present Value, and Interest Rates Chapter 4, Part 1.

Chapter Four 1/15/2018. Learning Objectives. The Meaning of Interest Rates Future Value, Present Value, and Interest Rates Chapter 4, Part 1. Chapter Four The Meaig of Iterest Rates Future Value, Preset Value, ad Iterest Rates Chapter 4, Part 1 Preview Develop uderstadig of exactly what the phrase iterest rates meas. I this chapter, we see that

More information

AY Term 2 Mock Examination

AY Term 2 Mock Examination AY 206-7 Term 2 Mock Examiatio Date / Start Time Course Group Istructor 24 March 207 / 2 PM to 3:00 PM QF302 Ivestmet ad Fiacial Data Aalysis G Christopher Tig INSTRUCTIONS TO STUDENTS. This mock examiatio

More information