[...]... option can choose to 1 The price K as well as other prices are meant as the price of one unit of an asset, say, in $ R.U Seydel, Tools for Computational Finance, Universitext, DOI: 10.1007/97 8-3 -5 4 0-9 292 9-1 1, c Springer-Verlag Berlin Heidelberg 2009 1 2 Chapter 1 Modeling Tools for Financial Options • • • • sell the option at its current market price on some options exchange (at t < T ), retain the... double-index notation the recursion Vji = e−rΔt (pVj+1,i+1 + (1 − p)Vj,i+1 ) (1.13) So far, this recursion for Vji is merely an analogy, which might be seen as a further assumption But the following Section 1.5 will give a justification for (1.13), which turns out to be a consequence of the no-arbitrage principle and the risk-neutral valuation 20 Chapter 1 Modeling Tools for Financial Options For European... Modeling Tools for Financial Options 0.45 T 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 S 0 0 50 100 150 200 250 Fig 1.10 Tree in the (S, t)-plane for M = 32 (data of Example 1.6) A consequence of this approach is that up to terms of higher order the relation √ u = eσ Δt holds (−→ Exercise 1.6) Therefore the extension of the tree in S-direction matches the volatility of the asset So the tree is well-scaled... the (S, t)-plane Figure 1.4 illustrates the character of such a surface for the case of an American put For the illustration assume T = 1 The figure depicts six curves obtained by cutting the option surface with the planes t = 0, 0.2, , 1.0 For t = T the payoff function (K − S)+ of Figure 1.2 is clearly visible 8 Chapter 1 Modeling Tools for Financial Options Shifting this payoff parallel for all 0... 1.2 The resulting profit diagram shows a negative profit for some range of S-values, which of course means a loss (see Figure 1.3) V K K S Fig 1.3 Profit diagram of a put The payoff function for an American call is (St −K)+ and for an American put (K − St )+ for any t ≤ T The Figures 1.1 and 1.2 as well as the equations (1.1C), (1.1P) remain valid for American type options The payoff diagrams of Figures... exercise the put, selling the underlying for the strike price K The profit of this arbitrage strategy is K − S − V > 0 This is in conflict with the no-arbitrage principle Hence the assumption that the value of an American put is below the payoff must be wrong We conclude for the put Am VP (S, t) ≥ (K − S)+ for all S, t Similarly, for the call Am VC (S, t) ≥ (S − K)+ for all S, t Am Am Eur Eur (The meaning... (Chapter 5) abbreviations: BDF CIR CFL Dow FE FFT FTBS FTCS GBM LCP Backward Difference Formula, see Section 4.2.1 Cox Ingersoll Ross model, see Section 1.7.4 Courant-Friedrichs-Lewy, see Section 6.5.1 Dow Jones Industrial Average Finite Element Fast Fourier Transformation Forward Time Backward Space, see Section 6.5.1 Forward Time Centered Space, see Section 6.4.2 Geometric Brownian Motion, see (1.33)... important role of numerical algorithms is not noticed For example, an analytical formula at hand [such as the Black–Scholes formula (A4.10)] might suggest that no numerical procedure is needed But closed-form solutions may include evaluating the logarithm or the computation of the distribution function of the normal distribution Such elementary tasks are performed using sophisticated numerical algorithms... Derivatives A3 Forwards and the No-Arbitrage Principle A4 The Black-Scholes Equation A5 Early-Exercise Curve B Stochastic Tools B1 Essentials of Stochastics B2 Advanced Topics B3 State-Price Process ... So the tree is well-scaled and will cover a relevant range of S-values Forward Phase: Initializing the Tree Now the factors u and d can be considered as known, and the discrete values of S for each ti until tM = T can be calculated The current spot price S = S0 for t0 = 0 is the root of the tree (To adapt the matrix-like notation to the two-dimensional grid of the tree, this initial price will be also . Köln Mathematisch-Naturwiss. Fakultät Mathematisches Institut Weyertal 8 6-9 0 50931 Köln Germany seydel@ math.uni-koeln.de ISBN: 97 8-3 -5 4 0-9 292 8-4 e-ISBN: 97 8-3 -5 4 0-9 292 9-1 DOI: 10.1007/97 8-3 -5 4 0-9 292 9-1 Library of Congress Control Number: 2008943076 This. say, in $. R.U. Seydel, Tools for Computational Finance, Universitext, 1 DOI: 10.1007/97 8-3 -5 4 0-9 292 9-1 1, c  Springer-Verlag Berlin Heidelberg 2009 2 Chapter 1 Modeling Tools for Financial Options •. R¨udiger U. Seydel Tools for Computational Finance Fourth Edition ABC Prof. Dr. Rüdiger U. Seydel Universität Köln Mathematisch-Naturwiss. Fakultät Mathematisches Institut Weyertal 8 6-9 0 50931