Introduction To Stochastic Calculus Applied To Finance 2nd Edition Damien Lamberton, Bernard Lapeyre - Solutions

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Fourier Transform Consider the Fourier transform F[f(u)] := ∞ −∞ eiuyf(y)dy . For the example f(y) := e−a|y| and complex a show that A −A eiuyf(y)dy converges for A → ∞ and Re(a) > 0.
Project: Implementing a PIDE Set up a computer program to solve Merton’s jump diffusion (7.30) numerically. To this end, concentrate on European-style vanilla options. Set up boundary conditions using (4.18), and use a BDF implicit scheme. Think of how to choose xmin, xmax in relation to the
For a L´evy process Xt adapted to a filtration Ft show E(eXt | Fs) − eXs = E(eXt−s ) − eX0 .
Payoffs of Spreads We consider portfolios of two or more options of the same type with the same underlying stock. K1, K2, K are strikes with K1 < K2.a) A butterfly spread is a portfolio with • one long call with strike K1, • one long call with strike K2, • two short calls with strike K =
Transformation of Nonlinear Black–Scholes Models According Section 7.2, consider the following nonlinear PDE Vt + 1 2 σ2(t, S, VSS)S2VSS + (r − δ)SVS − rV + ˆp max(Ψ − V, 0) = 0 , where σ2(t, S, VSS) depends on the particular model; r is the risk-free interest rate and δ is the
Barles–Soner Model The differential equation of Barles and Soner is: df(x) dx = f(x)+1 2 xf(x) − x with f(0) = 0 .a) By numerical computations, analyze the solution for −2 ≤ x ≤ 2.b) Construct an approximating function ˆf(x) in a piecewise fashion.
Let ΔW be the increment of a Wiener process, see Section 1.6.1. Show E(|ΔW|) = √ Δt 2 π .
Binomial Tree for Two Assets A two-asset binomial tree with (x, y)-coordinates representing the assets, and time-coordinate t, is assumed to develop as follows: Each node with position (x, y) may develop for t → t + Δt with equal probabilities 0.25 to the four positions (xu, yA), (xu, yB), (xd,
TVD of a Model Problem Analyze whether the upwind scheme (6.22), the Lax–Friedrichs scheme (6.25) and the Lax–Wendroff scheme (6.30) applied to the scalar partial differential equation ut + aux = 0 , a> 0, t ≥ 0, x ∈ IR satisfy the TVD property. Hint: Apply Lemma 6.3.
Towards the Black–Scholes Equationa) For the model equation (6.2a) set up the vector a and the matrix b for the general notation (1.41).b) Let LLtr be the Cholesky decomposition of the ρ-matrix, and ˜b := bL. Show trace(˜b˜btrVSS) = n i,j=1 ρijσiσjSiSj ∂2V ∂Si∂Sj .c) Show dV =
Upwind Scheme Apply von Neumann’s stability analysis to ∂u ∂t + a ∂u ∂x = b ∂2u ∂x2 , a> 0,b> 0 using the upwind scheme for the left-hand side and the centered second-order difference quotient for the right-hand side.
Neumann Stability Analysis Assume a difference scheme in the form (6.32) w(ν+1) j = l dl w(ν)and make use of the Fourier transform (6.17) w(ν) j = n −1 k=0 c (ν) k eikηjΔx for η = 2π nΔx .a) What are the coefficients dl for the FTCS method (6.16)?b) Prove linear independence n −1
PDEs for Arithmetic Asian Optionsa) Use the higher-dimensional Itˆo-formula (→ Appendix B2) to show that the value function V (S, A, t) of an Asian option satisfies dV = ∂V ∂t + S ∂V ∂A + μS ∂V ∂S + 1 2 σ2S2 ∂2V ∂S2 dt + σS ∂V ∂S dW , where S is the price of the asset
Project: Monte Carlo Valuation of Exotic Options Perform Monte Carlo valuations of barrier options, basket options, and Asian options, each European style.
Variable Volatility (Project) For variable volatility σ(S, t) and constant K, T, r, δ , PDEs of the type ∂y ∂τ − 1 2 σˆ2(x, τ) ∂2y ∂x2 − 1 4 y = 0 are to be solved, with τ = T − t and transformations S ↔ x, V ↔ y from the Black–Scholes model given by (A6.2), (A6.3);
Prove Lemma 5.12, and for u ∈ C2 the assertion u − wh1 = O(h).
Assume a function v(ζ) with α ≤ ζ ≤ β and v(α) = 0.a) Show (v(ζ))2 ≤ (ζ − α) ζ α (v (x))2 dx . (Hint: Recall v(ζ) = ζ α v (x) dx, and apply the Schwarzian inequality (C3.7).)b) Usea) to show β α (v(ζ))2 dζ ≤ 1 2 (β − α) 2 β α (v (x))2 dx .
Consider the domain D := {(x, y)| x ≥ 0, y ≥ 0, 1 ≤ x + y ≤ 2} tiled by 12 triangles Dk, where triangles and nodes are numbered as in Figure 5.13.a) Set up the index set I with entries Ik = {ik, jk, lk}, which assigns node numbers to the kth triangle for 1 ≤ k ≤ 12.b) Formulate the
In the three-dimensional (x, y, w)-space let the plane w(x, y) = c1+c2 x+c3 y interpolate the three points (xi, yi, wi), i = 1, 2, 3. Show ⎛ ⎝ 1 x1 y1 1 x2 y2 1 x3 y3 ⎞ ⎠ ⎛ ⎝ c1 c2 c3 ⎞ ⎠ = ⎛ ⎝ w1 w2 w3 ⎞ ⎠ . By inversion, establish a formula for ∇w = (c2, c3)tr.
The boundary ∂D of the trapezoidal domain D in Figure 5.8 consists of four straight lines. What are the four unit outward vectors n orthogonal to ∂D? Give a parameter representation of the boundary.
a) Prove the equivalence of (5.26) and (5.27). Specialize this to the onedimensional case of the Black–Scholes equation.b) Show btr∇u + ru = ∇ · (bu) + γu and determine γ.c) With the transformation x := log( S1 K1 ), y := log( S2 K2 ) and writing u(x, y, t) for V leads to the PDE ut + 1
Suppose the situation of two asset prices S1(t) and S2(t) for t > 0 governed by GBM (3.28), with initial price point (S1(0), S2(0)). Barriers of a barrier option can be aligned such that the probability of (S1(t), S2(t)) reaching the barrier has the same constant value.a) Show that this curve of
Calculating Options with Finite Elements Design an algorithm for the pricing of standard options by means of finite elements. To this end proceed as outlined in Section 5.3. Start with a simple version using an equidistant discretization step Δx. If this is working properly change the algorithm to
Finite-Element Matrices For the hat functions ϕ from Section 5.2 calculate for arbitrary subinterval Dk all nonzero integrals of the form ϕiϕj dx, ϕ iϕj dx, ϕ iϕ j dx and represent them as local 2 × 2 matrices.
Complexity With n underlyings and time t an option problem comprises n+1 independent variables. Assume that we discretize each of the n+1 axes with M grid points, then Mn+1 nodes are involved. Hence the complexity C of the n-factor model is C := O(Mn+1), which amounts to an exponential growth with
Project Implement Kim’s integral equation method (Section 4.8.4).
Consider the functions d1 and d2 of (4.61). For the three cases SSf(τ), calculate the limit for ξ → 0+ of rKe−rξ F(−d2(S, ξ; Sf(τ − ξ))) − δSf(τ) e−δξ F(−d1(S, ξ; Sf(τ − ξ)))
Analytic Method for the American Put (Project) Implement both the Algorithm 4.16 and Algorithm 4.17. For Algorithm 4.17 choose as initial guess the average of the strike and the lower bound (A5.1). A secant method (C1.5) is a good choice for the iteration. Think of how to combine Algorithms 4.16
a) Derive (4.49).b) Derive (4.56).
Extrapolation of Higher Order Similar as in Section 4.7 assume an error model η∗ = η(Δ) − γ1Δ2 − γ2Δ3 and three calculated values η1 := η(Δ) , η2 := η Δ 2 , η3 := η Δ 4 . Show that η∗ = 1 21(η1 − 12η2 + 32η3).
Brennan–Schwartz Algorithm Let A be a tridiagonal matrix as in (C1.6), and b and g vectors. The system of equations Aw = b is to be solved such that the side condition w ≥ g is obeyed componentwise. Assume for the case of a put wi = gi for 1 ≤ i ≤ if and wi > gi for if < i ≤ n, where if
Front-Fixing for American Options Apply the transformation ζ := S Sf(t) , y(ζ, t) := V (S, t) to the Black–Scholes equation (4.1).a) Show ∂y ∂t + σ2 2 ζ2 ∂2y ∂ζ2 + $ (r − δ) − 1 Sf dSf dt % ζ ∂y ∂ζ − ry = 0 (4.72)b) Set up the domain for (ζ, t) and formulate the
Semidiscretization, Method of Lines For a semidiscretization of the Black–Scholes equation (1.2) consider the semidiscretized domain 0 ≤ t ≤ T, S = Si := iΔS , ΔS := Smax m , i = 0, 1,...,m for suitable values of Smax > K and m. On this set of lines parallel to the taxis define for τ :=
Smooth Pasting of the American Put Suppose a portfolio consists of an American put and the corresponding underlying. Hence the value of the portfolio is Π := V Am P + S, where S satisfies the SDE (1.33). Sf is the value for which we have high contact, compare (4.22).a) Show that dΠ = ⎧ ⎨ ⎩
Perpetual Put Option For T → ∞ it is sufficient to analyze the ODE σ2 2 S2 d2V dS2 + (r − δ)S dV dS − rV = 0 . Consider an American put with high contact to the payoff V = (K − S)+ at S = Sf . Show:a) Upon substituting the boundary condition for S → ∞ one obtains V (S) = c S K
Implement Algorithms 4.13 and 4.14. Test example: Example 1.6 and others.
Gauß–Seidel as Special Case of SOR Let the n×n matrix A = ((aij )) additively be partitioned into A = D−L−U, with D diagonal matrix, L strict lower triangular matrix, U strict upper triangular matrix, x ∈ IRn, b ∈ IRn. The Gauß–Seidel method is defined by (D − L)x(k) = Ux(k−1) +
Boundary Conditions of American Options Show that the boundary conditions of American options satisfy lim x→±∞ y(x, τ) = lim x→±∞ g(x, τ), where g is defined in Problem 4.7.
Boundary Conditions of a European Call Show that under the transformation (4.3) Se−δ(T −t) − Ke−r(T −t) = exp x 2 (qδ + 1) + τ 4 (qδ + 1)2 − expx 2 (qδ − 1) + τ 4 (qδ − 1)2 holds, and prove (4.19). Hints: Either transform the Black–Scholes equation (4.1) with S := S¯
Crank–Nicolson Order Let the function y(x, τ) solve the equation yτ = yxx and be sufficiently smooth. With the difference quotient δxxwi,ν := wi+1,ν − 2wi,ν + wi−1,ν Δx2 the local discretization error of the Crank–Nicolson method is defined := yi,ν+1 − yi,ν Δτ − 1 2
Stability of the Fully Implicit Method The backward-difference method is defined via the solution of the equation (4.11). Prove the stability. Hint: Use the results of Section 4.2.4 and w(ν) = A−1w(ν−1).
Discrete Dividend Payment Assume that a stock pays a dividend D at ex-dividend date tD, with 0 < tD < T .a) Assume that a known dividend is paid once per year. Calculate a corresponding continuous dividend rate δ under the assumptions S˙ = (μ − δ)S, μ = 0, S(1) = S(0) − D > 0 . Generalize
Binary Random Variate Let α, β, p with 0
SDE in Standard Form Let us denote (1.41) as “standard form” of a system of SDEs, with uncorrelated Wiener processes W(1) t ,...,W(m) t . What is the vector a and the matrix b fora) the example of equation (3.28),b) the Heston model of equation (1.43). For the Heston model, first transform the
Error of Biased Monte Carlo Assume MSE = ζ(h, N) := α2 1h2β + α2 N as error model of a Monte Carlo simulation with sample size N, based on a discretization of an SDE with stepsize h, where α1, α2 are two constants.a) Argue why for some constant α3 C(h, N) := α3 N h is a reasonable model for
Project: Monte Carlo Experiment Construct as hitting curve a parabola with horizontal tangent at (S, t) = (K, T ), similar as in Figure 3.10. The parabola is defined by the intersection with the S-axis, (S, t)=(β, 0). Choose K = 10, r = 0.006, σ = 0.3, and S0 = 9 and simulate for several values
Monte Carlo for European Options Implement a Monte Carlo method for single-asset European options, based on the Black–Scholes model. Perform experiments with various values of N and a random number generator of your choice. Compare results obtained by using the analytic solution formula for St
Bias of the Euler Approximation Given is the SDE dSt = St(μ dt + σ dWt) with constant μ, σ. Let Sˆ denote an Euler approximation at t2 := 2Δt, calculated with two steps of length Δt, starting at t0 := 0 with the value S0.a) Calculate E(Sˆ).b) Calculate the bias E(Sˆ) − S0 exp[μt2] .
Monte Carlo and European Option For a European put with time to maturity τ := T − t prove that V (St, t)=e−rτ ∞ 0 (K − ST ) + 1 ST σ √2πτ exp −[ln(ST /St) − (r − σ2 2 )τ] 2 2σ2τ * dST = e−rτKF(−d2) − StF(−d1), where d1 and d2 are defined in (A4.10). Hints: The
Error of the Milstein Scheme To which formula does the Milstein scheme reduce for linear SDEs? Perform the experiment outlined in Example 3.2 using the Milstein scheme of Algorithm 3.5. Set up a table similar as in Table 3.1 to show ε(h) ≈ h for Example 3.2.
Brownian Bridge For a Wiener process Wt consider Xt := Wt − t T WT for 0 ≤ t ≤ T . Calculate Var(Xt) and show that ) t 1 − t T Z with Z ∼ N (0, 1) is a realization of Xt
In addition to (3.14) further moments are E(ΔW) = E(ΔW3) = E(ΔW5)=0, E(ΔW2) = Δt, E(ΔW4)=3Δt2. Assume a new random variable ΔW satisfying P ΔW = ± √ 3Δt = 1 6 , P ΔW = 0 = 2 3 and the additional random variable ΔY := 1 2 ΔW Δt . Show that the random variables ΔW
By transformation of two independent standard normally distributed random variables Zi ∼ N (0, 1), i = 1, 2, two new random variables are obtained by ΔW := Z1 √ Δt, ΔY := 1 2 (Δt) 3/2 Z1 + 1 √3 Z2 . Show that ΔW and ΔY have the moments of (3.14).
Moments of Itˆo Integrals for Weak Solutionsa) Use the Itˆo isometry E ⎡ ⎣ ! b a f(t, ω) dWt "2 ⎤ ⎦ = b a E % f 2(t, ω) & dt to show its generalization E [I(f)I(g)] = b a E[fg] dt , where I(f) = b a f(t, ω) dWt . Hint: 4fg = (f + g)2 − (f − g)2.b) For ΔY := t t0 s t0
Integration by Parts for Itˆo Integralsa) Show t t0 s dWs = tWt − t0Wt0 − t t0 Ws dsb) Denote ΔY := t t0 s t0 dWz ds. Show by usinga) that t t0 s t0 dz dWs = ΔW Δt − ΔY .
Itˆo Integral in Equation (3.9) Let the interval 0 ≤ s ≤ t be partitioned into n subintervals, 0 = t1 < t2 < ... < tn+1 = t. For a Wiener process Wt assume Wt1 = 0.a) Show n j=1 Wtj Wtj+1 − Wtj = 1 2 W2 t − 1 2 n j=1 Wtj+1 − Wtj 2b) Use Lemma 1.9 to deduce Equation (3.9).
Implementing Euler’s Method Implement Algorithm 1.11. Start with a test version for one scalar SDE, then develop a version for a system of SDEs. Test examples:a) Perform the experiment of Figure 1.17.b) Integrate the system of Example 1.15 for α = 0.3, β = 10 and the initial values S0 = 50, σ0
Testing the Distribution Let X be a random variate with density f and let a1 < a2 < ... < al define a partition of the support of f into subintervals, including the unbounded intervals xal. Recall from (B1.1), (B1.2) that the probability of a realization of X falling into ak ≤ xb) Let τj be the
Algorithm for the Radical-Inverse Function Use the idea i = dkbk−1 + ... + d1 b + d0 to formulate an algorithm that obtains d0, d1, ..., dk by repeated division byb. Reformulate φb(i) from Definition 2.17 into the form φb(i) = z/bj+1 such that the result is represented as rational number.
Bounds on the Discrepancy (Compare Definition 2.15) Showa) 0 ≤ DN ≤ 1,b) D∗ N ≤ DN ≤ 2mD∗ N (show this at least for m ≤ 2),c) D∗ N ≥ 1 2N for m = 1.
Experiment on Monte Carlo Integration To approximate the integral 1 0 f(x) dx calculate a Monte Carlo sum 1 N N i=1 f(xi) for f(x)=5x4 and, for example, N = 100000 random numbers xi ∼ U[0, 1]. The absolute error behaves like cN −1/2. Compare the approximation with the exact integral for
Error of the Monte Carlo Integration The domain for integration is Q = [0, 1]m. For ΘN := 1 N N i=1 f(xi) , E(f) := f dx, g := f − E(f) and σ2(f) from (2.14b) showa) E(g)=0b) σ2(g) = σ2(f)c) σ2(δN ) = E(δ2 N ) = 1 N2 ( "g(xi))2 dx = 1 N σ2(f) Hint on (c): When the random points xi
Correlated Distributions Suppose we need a two-dimensional random variable (X1, X2) that must be normally distributed with mean 0, and given variances σ2 1, σ2 2 and prespecified correlation ρ. How is X1, X2 obtained out of Z1, Z2 ∼ N (0, 1)?
Programming Assignment: Normal Deviatesa) Write a computer program that implements the Fibonacci generator Ui :=Ui−17 − Ui−5 Ui :=Ui + 1 in case Ui < 0 in the form of Algorithm 2.7. Tests: Visual inspection of 10000 points in the unit square.b) Write a computer program that implements
Uniform Distribution For the uniformly distributed random variables (V1, V2) on [−1, 1]2 consider the transformation X1 X2 = V 2 1 + V 2 2 1 2π arg((V1, V2)) where arg((V1, V2)) denotes the corresponding angle. Show that (X1, X2) is distributed uniformly.
Inverting the Normal Distribution Suppose F(x) is the standard normal distribution function. Construct a rough approximation G(u) to F −1(u) for 0.5 ≤ u < 1 as follows:a) Construct a rational function G(u) (−→ Appendix C1) with correct asymptotic behavior, point symmetry with respect to (u,
Cauchy-Distributed Random Numbers A Cauchy-distributed random variable has the density function fc(x) := c π 1 c2 + x2 . Show that its distribution function Fc and its inverse F −1 c are Fc(x) = 1 π arctan x c + 1 2 , F −1 c (y) = c tan(π(y − 1 2 )). How can this be used to generate
Coarse Approximation of Normal Deviates Let U1, U2, ... be independent random numbers ∼ U[0, 1], and Xk := k +11 i=k Ui − 6 . Calculate mean and variance of the Xk.
Lattice of the Linear Congruential Generatora) Show by induction over j Ni+j − Nj = aj(Ni − N0) mod Mb) Show for integer z0, z1, ..., zm−1 ⎛ ⎜⎜⎝ Ni Ni+1 . . . Ni+m−1 ⎞ ⎟⎟⎠ − ⎛ ⎜⎜⎝ N0 N1 . . . Nm−1 ⎞ ⎟⎟⎠ = (Ni − N0) ⎛ ⎜⎜⎝ 1 a . . . am−1 ⎞
Deficient Random Number Generator For some time the generator Ni = aNi−1 mod M, with a = 216 + 3, M = 231 was in wide use. Show for the sequence Ui := Ni/M Ui+2 − 6Ui+1 + 9Ui is integer! What does this imply for the distribution of the triples (Ui, Ui+1, Ui+2) in the unit cube?
Consider the random number generator Ni = 2Ni−1 mod 11. For (Ni−1, Ni) ∈ {0, 1, ..., 10}2 and integer tuples with z0 + 2z1 = 0 mod 11 the equation z0Ni−1 + z1Ni = 0 mod 11 defines families of parallel straight lines, on which all points (Ni−1, Ni) lie. These straight lines are to be
Negative Prices Assume Z ∼ N (0, 1), S > 0, σ > 0, and a step (t, S) → (t + Δt, S + ΔS) of the discretized GBM ΔS S = μΔt + σZ√ Δt . What is the probability that the resulting price S + ΔS is negative? Discuss the result and think about remedy.
Improved Binomial Tree The Algorithm 1.4 is to be improved as follows:a) Apply the anchoring of Exercise 1.14.b) Extend the tree by starting at −2Δt as discussed in Section 1.4.6, and calculate approximations for the Greeks delta and gamma by using difference quotients. Use Example 1.5 to
Dividend Payment and the Binomial Method A dividend yield δ can be calculated by annualizing a known dividend payment D per year by setting δ = D/S. For a binomial tree, the effects of paying eithera) a fixed amount D orb) a proportional amount δS are different. Assume a dividend payment at
Binomial Method with p = 0.5 Use the equations (1.5), (1.9) and p = 1/2 to show u = erΔt (1 + eσ2Δt − 1) d = erΔt (1 − eσ2Δt − 1).
Ornstein–Uhlenbeck Process An Ornstein–Uhlenbeck process is defined as solution of the SDE dXt = −αXt dt + γ dWt , α> 0 for a Wiener process W.a) Show Xt = e−αt X0 + γ t 0 eαsdWs b) Suppose the volatility σt is an Ornstein–Uhlenbeck process. Show that the variance vt := σ2 t
General Black–Scholes Equation Assume a portfolio Πt = αtSt + βtBt consisting of αt units of a stock St and βt units of a bond Bt, which obey dSt = μ(St, t) dt + σ(St, t) dWt dBt = r(t)Bt dt The functions μ, σ, and r are assumed to be known, and σ > 0. Further assume the portfolio is
Positive Itˆo Process Let Xt be a positive one-dimensional Itˆo process for t ≥ 0. Show that there exist functions α and β such that dXt = Xt(αt dt + βt dWt) and Xt = X0 exp t 0 (αs − 1 2 β2 s ) ds + t 0 βs dWs *
Bounds and Arbitrage Using arbitrage arguments, show the following bounds for the values VC of vanilla call options:a) 0 ≤ VCb) (S − K)+ ≤ V Am C ≤ S
Portfolios Figure 1.24 sketches some payoffs over S: (a) bull spread, (b) bear spread, (c) strangle, (d) butterfly spread. For each of these payoffs, construct portfolios out of two or three vanilla options such that the portfolio meets the payoff. (a) K1 (c) (b) KI (d) K2 K1 K2 K1 K K2
Extrapolation Let η∗ ∈ IR denote the exact solution of an equation, Δ denotes the grid size of a numerical approximation scheme, and η(Δ) the approximating solution. Further assume an error model η(Δ) − η∗ = c Δq, withc, q ∈ IR. q is the order of the approximation scheme. Suppose
Anchoring the Binomial Grid at K The equation (1.10) has established a kind of symmetry for the grid. As an alternative, one may anchor the grid by requiring (for even M) S0uM/2dM/2 = K .a) Give a geometrical interpretation.b) Derive from equations (1.5), (1.9) and ud = γ for some constant γ (not
Return of the Underlying Let a time series S1, ..., SM of a stock price be given (for example data in the domain www.compfin.de). The simple return Rˆi,j := Si − Sj Sj ,an index number of the success of the underlying, lacks the desirable property of additivity RM,1 = M i=2 Ri,i−1 . (∗) The
Moments of the Lognormal Distribution For the density function f(S;t − t0, S0) from (1.48) showa) ∞ 0 Sf(S;t − t0, S0) dS = S0eμ(t−t0)b) ∞ 0 S2f(S;t − t0, S0) dS = S2 0 e(σ2+2μ)(t−t0) Hint: Set y = log(S/S0) and transform the argument of the exponential function to a squared
Analytical Solution of Special SDEs Apply Itˆo’s lemma to showa) Xt = exp λWt − 1 2λ2t solves dXt = λXt dWtb) Xt = exp (2Wt − t) solves dXt = Xt dt + 2Xt dWt Hint: Use suitable functions g with Yt = g(Xt, t). In (a) start with Xt = Wt and g(x, t) = exp(λx − 1 2λ2t).
a) Suppose that a random variable Xt satisfies Xt ∼ N (0, σ2). Use (B1.4) to show E(X4 t )=3σ4 .b) Applya) to show the assertion in Lemma 1.9, E ⎛ ⎝ j ((ΔWj ) 2 − Δtj ) ⎞ ⎠ 2 = 2 j (Δtj ) 2
In Definition 1.7 the requirement (a) W0 = 0 is dispensable. Then the requirement (b) reads E(Wt − W0)=0 , E((Wt − W0) 2) = t . Use these relations to deduce (1.21). Hint: (Wt − Ws)2 = (Wt − W0)2 + (Ws − W0)2 − 2(Wt − W0)(Ws − W0)
Limiting Case of the Binomial Model Consider a European Call in the binomial model of Section 1.4. Suppose the calculated value is V (M) 0 . In the limit M → ∞ the sequence V (M) 0 converges to the value VC(S0, 0) of the continuous Black–Scholes model given by (A4.10) (−→ Appendix A4). To
Implementing the Binomial Method Design and implement an algorithm for calculating the value V (M) of a European or American option. Use the basic version of Algorithm 1.4. INPUT: r (interest rate), σ (volatility), T (time to expiration in years), K (strike price), S (price of asset), and the
Price Evolution for the Binomial Method For β from (1.11) and u = β + β2 − 1 show u = exp " σ √ Δt# + O " (Δt)3 # .
Implied Volatility For European options we take the valuation formula of Black and Scholes of the type V = v(S, τ, K, r, σ), where τ denotes the time to maturity, τ := T −t. For the definition of the function v see Appendix A4, equation (A4.10). If actual market data V mar of the price are
Calculating the Sample Variance An estimate of the variance of M numbers x1, ..., xM is s2 M := 1 M − 1 M i=1 (xi − x¯) 2, with ¯x := 1 M M i=1 xi The alternative formula s2 M = 1 M − 1 ⎛ ⎝ M i=1 x2 i − 1 M ( M i=1 xi )2⎞ ⎠ () can be evaluated with only one loop i = 1, ..., M,
Standard Normal Distribution Function Establish an algorithm to calculate F(x) = 1 √2π x −∞ exp(−t2 2 ) dt . Hint: Construct an algorithm to calculate the error function erf(x) := 2 √π x 0 exp(−t 2) dt and use erf(x) to calculate F(x). Use quadrature methods (−→ Appendix C1).
Transforming the Black–Scholes Equation Show that the Black–Scholes equation (1.2) ∂V ∂t + σ2 2 S2 ∂2V ∂S2 + rS ∂V ∂S − rV = 0 for V (S, t) with constant σ and r is equivalent to the equation ∂y ∂τ = ∂2y ∂x2 for y(x, τ). For proving this, you may proceed as follows:a)
Put-Call Parity Consider a portfolio consisting of three positions related to the same asset, namely, one share (price S), one European put (value VP), plus a short position of one European call (value VC). Put and call have the same expiration date T , and no dividends are paid.a) Assume a
(b) Prove that the fair price at time n of a European contingent claim h is given by WnEW (h/WN |Fn) .
(a) Prove that the probability measure PW introduced in Question 4 is unique, and satises dPW/dP¤ =WN/S0N.
6. Assume that the market is viable and complete, and denote by P¤ the unique equivalent probability measure with respect to which ˜ S is a martingale.
5. Prove that, in a viable market, there is at most one deterministic numéraire.
4. Prove that the market is viable if and only if there exists a probability PW, equivalent to P, such that SW is a PW-martingale.

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Introduction To Stochastic Calculus Applied To Finance 2nd Edition Damien Lamberton, Bernard Lapeyre - Solutions

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