An Introduction to Financial Option Valuation 1 potx

An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_13 pot

... derivatives in (24 .10 ) leads to the ﬁnite difference formulaWi +1 j− Wijk+ 1 2σ2Wi +1 j +1 − 2Wi +1 j+ Wi +1 j 1 h2+ (r − 1 2σ2)Wi +1 j +1 − Wi +1 j 1 2h= 0. (24 .11 )Setting h2= ... eliminating the Wi +1 jterms in (24 .11 ), and theformula then reduces to Wij= pWi +1 j +1 + (1 − p)Wi +1 j 1 , (24 .12 )where p= 1 2 1 +√k(r/σ − σ/2).Transforming back to V we ﬁnd thatVij= ... generalization of BTCS isVi +1 j− Vijk− 1 2σ2( jh)2Vi +1 j +1 − 2Vi +1 j+ Vi +1 j 1 h2−rjhVi +1 j +1 − Vi +1 j 1 2h+rVi +1 j= 0. (24.7)The matrix–vector representation of FTCS...

An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_14 pot

... neutrality, 11 5, 11 8 12 0, 14 4, 14 6, 15 1, 15 4, 16 3, 16 7, 18 0, 18 1, 19 4, 232cash-or-nothing, 16 7 16 8sample mean, 34, 48, 64, 14 1, 14 6,204, 215 sample variance, 34, 48SDE, see stochastic differential ... Carlo method, 14 1 14 8, 215 –224,229–232for American put, 18 0 18 2for exotics, 19 4 19 6for Greeks, 14 5 14 8Index 273New York Stock Exchange, 6, 50Newton’s method, 12 4 12 8, 13 1 13 3normal distribution, ... 14 6, 15 2EWMA, see exponentially weighted movingaverageexercise price, 1 exercise strategy, 18 0, 18 1, 18 3exotic option, 7, 18 7 19 6, 222expected payoff, 11 5 11 6, 11 8 12 0expected value, 21, ...

An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_1 pot

... formulas 10 5 11 .1 Motivation 10 5 11 .2 Where is µ? 10 5 11 .3 Time dependency 10 6 11 .4 The big picture 10 6 11 .5 Change of variables 10 8 11 .6 Notes and references 11 1 11 .7 Program of Chapter 11 and walkthrough ... walkthrough 11 1 12 Risk neutrality 11 5 12 .1 Motivation 11 5 12 .2 Expected payoff 11 5 12 .3 Risk neutrality 11 6 12 .4 Notes and references 11 8 12 .5 Program of Chapter 12 and walkthrough 12 0 13 Solving ... method 15 1 16 .1 Motivation 15 1 16 .2 Method 15 1 16 .3 Deriving the parameters 15 3 16 .4 Binomial method in practice 15 4 16 .5 Notes and references 15 6 16 .6 Program of Chapter 16 and walkthrough 15 9 17 ...

An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_3 pptx

... DENNEY AND STEVEN GAINES (Denney and Gaines, 2000)36 Computer simulation0 0.5 1 00.5 1 1.5 10 00 samples0 0.5 1 00.5 1 1.5 10 000 samples0 0.5 1 00.5 1 1.5 10 0000 samples0 0.5 1 00.5 1 1.5 10 00000 ... samples and N(0 ,1) quantiles 1 0 1 2 1 −0.500.5 1 1.52U(0 ,1) samples and U(0 ,1) quantiles Fig. 4.4. Quantile–quantile plots using M = 10 0 samples. Ordered samplesξ 1 ,ξ2, ,ξMon ... theU(0, 1) sample means and variances approach the true values 1 2and 1 12≈ 0.0833(recall Exercise 3.5) and the N(0, 1) sample means and variances approach the truevalues 0 and 1. A more...

An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_4 ppt

... small,thene 1. 96σ√t+(µ− 1 2σ2)t≈ e 1. 96σ√t≈ 1 − 1. 96σ√tande 1. 96σ√t+(µ− 1 2σ2)t≈ e 1. 96σ√t≈ 1 + 1. 96σ√t.So the conﬁdence interval is approximately[S0 (1 − 1. 96σ√t), S0 (1 ... Part II0 0.5 1 1.5 2 2.5 30.9 1 1 .1 1.2 1. 3 1. 4 1. 5Discrete asset pathtiSiFig. 7 .1. Discrete asset path of the form (7 .1) . Discrete points are joined bystraight lines to give the impression ... can be found in (Lowenstein, 20 01, page 71) .If the population of price changes is strictly normal, on the average for any stock anobservation more than ﬁve standard deviations from the mean...

An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_5 ppt

... = 1, if x > 0,0, if x = 0, 1, if x < 0.7.4 Notes and references 690 0 .1 0.2 0.3 0.4 0.50.60.8 1 1.2 1. 4 1. 6dt = 5 × 10 −3dt = 5 × 10 −4Asset paths0 0 .1 0.2 0.3 0.4 ... (Bj¨ork, 19 98; Dufﬁe, 20 01; Karatzas and Shreve, 19 98; Nielsen, 19 99;Øksendal, 19 98).It is possible to weaken the boundary conditions (8 .17 ) and (8 .18 ) in the Black–Scholes PDE (8 .15 ) without ... Show that (8. 21) can be replaced by (8.22).8.3. Conﬁrm that C(S, t) in (8 .19 ) satisﬁes (8 .16 ), (8 .17 ) and (8 .18 ). [Hint: to deal with (8 .16 ), take the limit t → T−,todeal with (8 .17 ) take the...

An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_7 pdf

... zlabel(’C(S,t)’)Fig. 11 .6. Program of Chapter 11 : ch 11. m.PROGRAMMING EXERCISESP 11. 1. Edit ch 11. m so that it applies to a European put option, as in Figure 11 .4.P 11. 2. Editch 11. m so that it applies to the ... this model, you had these numbers, 11 .7 Program of Chapter 11 and walkthrough 11 1 11 .6 Notes and referencesColour versions of Figures 11 .3, 11 .4 and 11 .5 can be downloaded from this book’swebsite, ... 3 − 2 − 1 0 1 2 3 4 5− 0.8− 0.6− 0.4− 0.200.20.40.6xF(x)Fig. 13 .1. The function F(x) := N(x) −23.0 5 10 15 20 10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 1 100 10 1 BisectionErrorIteration1...

An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_8 pptx

... EXERCISESP15 .1. Adapt ch15 to produce a picture like that in Figure 15 .2.P15.2. Adaptch15 to produce an estimate of the delta. 15 .2 Monte Carlo 14 3 10 1 102 10 3 10 4 10 5 10 6 10 0 .1 100.2 10 0.3 10 0.4Num ... Black–Scholesformula. 14 .5 Implied volatility with real data 13 50 0.5 1 1.522.22.42.62.8σC(σ)starting valueiterates0 2 4 6 8 10 12 14 16 18 10 15 10 10 10 −5 10 0ErrorIterationFig. 14 .1. Newton’s ... 14 .1 illustrates the performance of Newton’smethod in the case where S0= 3, E = 1, r = 0.05, T = 3 and t = 0. We used 15 .4 Monte Carlo for Greeks 14 5 10 1 102 10 3 10 4 10 5 10 6 10 0 .1 100.2 10 0.3Num...

An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_9 pot

... Ae−r(T−t). (17 .10 ) 17 .2.  Show that Ccash(S, t) in (17 .4) satisﬁes (17 .1) , (17 .2) and (17 .3). 17 .3. Differentiate (17 .4) to establish (17 .5), (17 .6) and (17 .7). 17 .4. Using (17 .4) and (17 .10 ), ... practice 15 5Table 16 .1. European put valueapproximations from binomial method Option valueM = 10 0 1. 4 716 M = 200 1. 4762M = 400 1. 4726Black–Scholes 1. 47280 50 10 0 15 0 200 250 1. 46 1. 48 1. 5 1. 52MEuropean ... Explain why the ratio (V 1 1− V 1 0)/(S 1 1− S 1 0) can be regarded as an ap-proximation to the time-zero delta. 16 .6 Program of Chapter 16 and walkthroughThe program ch16 implements the binomial...

An Introduction to Financial Option Valuation Mathematics Stochastics and Computation_1 doc

... Puts Option Oct Nov Dec Oct Nov DecRoyal Bk Scot. 16 00 67.092.5 10 9.529.049.062.5 (16 34.0) 17 00 19 .543.559.082.0 10 0.0 11 2.5The number 16 34.0isthe closing price of The Royal Bank of Scotland’s ... gained an amount S(t 1 ) at time t = t 1 from the short sale,(b) paid out an amount S(t2) at time t = t2from the buy back. 1. 6 Notes and references 730 40 50 60 70 80 90 10 0 11 0 12 005 10 15 20253035404550StrikeCall ... introduce American and other moreexotic options. 1. 6 Notes and referencesThere are many introductory texts that explain how stock markets operate; see, forexample, Dalton (20 01) ; Walker (19 91) . Chapter...

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