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Topics in Financial Mathematics IVAN G. AVRAMIDI New Mexico Tech • Financial terminology, options • Random Variables, stochastic processes, random walks, stochastic calculus, SDE • Hedging, no arbitrage principle, Black-Scholes equation • Stochastic volatility, jump diffusion, • Heat kernel method • Methods for calculation of the heat kernel 1 Financial Instruments Real assets are necessary for producing goods and services for the survival of society Financial assets (also called securities) are pieces of paper that entitle its holder to a claim on a fraction of the real assets and to the income generated by these real assets Financial instrument is a specific form of a financial asset • equities (stocks, shares) represent a share in the ownership of a company • fixed income securities (bonds) promise either a single fixed payment or a stream of fixed payents • derivative securities are financial assests that are derived from other financial assets 2 Efficient Market Hypothesis Efficient market concept is the hypothesis that the market is in equilibrium: • the securities have their fair prices • the market prices have only small and temporary deviations from their fair prices • changes in prices of securities are, up to a drift, random The random time evolution of the price of a security is a stochastic process 3 Options A call option is a contract that gives the right (but not the obligation!) to buy a particular asset (called the underlying asset) for an agreed amount (called the strike price E) at a specified time T in the future (called the expiration date) The payoff function of the call option is its value at expiry (S − E)+ = max (S − E, 0) A put option is a contract that gives the right to sell a particular asset for an agreed amount at a specified time in the future The payoff function of the put option is (E − S)+ Other option types: American, Bermudan, binary, spreads, straddles, strangles butterfly spreads, condors, calendar spreads, LEAPS 4 Random Variables Random variable X Probability density function fX Expectation value E(X) = Z R dx fX (x)x Variance Var(X) = E(X 2) − [E(X)]2 Covariance of random variables Cov(Xi, Xj ) = E(XiXj ) − E(Xi)E(Xj ) Correlation matrix ρ(Xi, Xj ) = q Cov(Xi, Xj) Var(Xi)Var(Xj ) 5 Stochastic Processes A stochastic process X(t) is a one-parameter family of random variables, 0 ≤ t ≤ T A Wiener process X(t) (Brownian motion, or Gaussian randow walk) is a stochastic process characterized by the conditions: • X(0) = 0 • X(t) is continuous (almost surely) • the increments dX(t) = X(t + dt) − X(t) are independent random variables with the normal distribution N (0, dt) centered at 0 with variance dt 6 Properties of Wiener Processes The Wiener process has the properties E(dX) = 0, E(dX 2) = dt . For several Wiener processes one defines the correlation matrix ρij E(dXi) = 0 E(dXidXj ) = ρij dt 7 Ito’s Lemma Idea dt ∼ dX 2 ∼ ε For a function F of a stochastic process Xt dF 1 d2F dF (X) = dX + dt 2 dX 2 dX Rule: replace in Taylor series dX 2 7→ dt Ito’s Lemma for a function F (t, Xi) of several variables n n X ∂F ∂F 1 X ∂ 2F dF = dt + dXi + ρij dt ∂t 2 i,j=1 ∂Xi∂Xj i=1 ∂Xi Rule: replace in Taylor series dXidXj 7→ ρij dt 8 Stochastic Differential Equations Brownian motion with drift dS = µdt + σdX Solution S(t) = S(0) + µt + σ[X(t) − X(0)] Lognormal random walk dS = µSdt + σSdX Solution 1 µ − σ 2 t + σ[X(t) − X(0)] 2 S(t) = S(0) = exp Mean-reverting random walk dS = (ν − µS)dt + σdX Solution S(t) = ν + [S(0) − ν]e−µt +σ X(t) − µ Z t 0 ds eµ(s−t)X(s) 9 Hedging Stock (log-normal random walk) dS = (µ − D)Sdt + σSdX where µ is the drift, D is the dividend yield, σ is the volatility Let V be a (European call) option Risk-free portfolio ∂V Π=V − S ∂S Portfolio change (Ito’s lemma) ∂V ∂V 1 ∂ 2V 2 − ∂V (dS+DSdS) dΠ = dt+ dS+ dS ∂t ∂S 2 ∂S 2 ∂S ( = ∂ 2V ∂V 1 ∂V + σ 2S 2 2 − DS ∂t 2 ∂S ∂S ) dt 10 Black-Scholes Equation No-arbitrage principle: A completely risk-free change in the portfolio value must be the same as the growth one would get if one puts the equivalent amount of cash in a risk-free interestbearing account (bond) dΠ = rΠdt Black-Scholes equation (linear parabolic PDE) ∂ +L V =0 ∂t where (elliptic PDO) 1 2 2 ∂2 ∂ L= σ S + (r − D)S −r 2 2 ∂S ∂S Remark: µ is eliminated! Homogeneous in S. Can be solved by Mellin transform in S (or Fourier transform in x = log S) 11 Domain: [0, T ] × R+ Terminal condition V (T, S) = f (S) Boundary conditions V (t, 0) = 0 , lim [V (t, S) − S] = 0 S→∞ 12 Payoff Function f (S) is usually a piecewise linear function Call : f (S) = (S − E)+ Put : f (S) = (E − S)+ Binary call : f (S) = θ(S − E) Binary put : f (S) = θ(E − S) Call spread: h i 1 f (S) = (S − E1)+ − (S − E2)+ E2 − E1 13 Basket (Rainbow) Options Multiple assets Si dSi = (µi − Di)Sidt + σiSidXi Multiple correlated Winer processes Xi E(dXi) = 0 E(dXidXj ) = ρij Multi-dimensional Black-Scholes operator n n X 1 X ∂2 ∂ L = Cij SiSj + (r − Di)Si −r 2 i,j=1 ∂Si∂Sj ∂Si i=1 where Cij = σiσj ρij Homogeneous in Si. Can be solved by multidimensional Fourier transform in xi = log Si. 14 Assumptions of Black-Scholes • Hedging is done continuously • There are no transaction costs • Volatility is a known constant • Interest rates and dividends are known constants • Underlying asset path is continuous • Underlying asset is unaffected by trade in the option • Hedging eliminates all risk 15 Stochastic Volatility Assume that the volatility of an asset S is a function of n stochastic factors v i, i = 1, 2, . . . , n. dS = rS dt + σ(t, S, v)S dX dv i = ai(t, v) dt + n X bij (t, v) dWj j=1 where E(dXdWi) = ρ0i, E(dWidWj ) = ρij Then the valuation operator is n X ∂2 1 2 2 ∂2 i L = σ S + A σS 2 2 ∂S ∂S∂v i i=1 n n X ∂2 1 X ∂ ij i ∂ −r B + a + + rS i 2 i,j=1 ∂v i∂v j ∂S i=1 ∂v where Ak (t, v) = B kl (t, v) = n X bkj (t, v)ρj0 j=1 n X bki(t, v)ρij bjl (t, v) i,j=1 16 Two-dimensional Models SDE dS = rS dt + σ(v, S)S dX , dv = a(v) dt + b(v) dW Valuation operator 1 2 σ (S, v)S 2∂S2 + ρb(v)σ(S, v)S∂S ∂v L = 2 1 2 + b (v)∂v2 + rS∂S + a(v)∂v − r 2 Domain: [0, T ] × R+ × R+ Terminal condition at t = T and boundary conditions at S, v → 0 and S, v → ∞. Non-constant coefficients. Cannot be solved exactly. Can be studied by using the methods of asymptotic analysis (theory of singular perturbations) 17 Jump Process Poisson process with intensity λ is a stochastic process such that there is a probability λdt of a jump 1 in Q in the time step dt, that is, dQ = 0 with probability (1 − λdt) 1 with probability λdt so that, E(dQ) = λdt Define the jump process by J dN = −λm dt + e − 1 dQ where J is a random variable with the probability density function ω(J) and m= Z∞ dJ ω(J) eJ − 1 −∞ such that E(dN ) = 0 18 Jump Diffusion with Stochastic Volatility SDE dS = rS dt + σ(v, S)S dX + SdN dv = a(v) dt + b(v) dW Assume that there is no correlation between the Wiener processes X and W and the Poisson process Q, that is, E(dXdQ) = E(dW dQ) = 0 19 Hedging No arbitrage principle E(dΠ) = rE(Π) dt Valuation operator L = L̄ + λLJ , where L̄ is the PDO defined above and LJ is an integral operator defined by h (LJ V )(t; S, v) = E V (t; eJ S, v) − V (t, S, v) −E(eJ − 1)S∂S V (t, S, v) = Z∞ h dJ ω(J) V (t; eJ S, v) − V (t; S, v) i −∞ −mS∂S V (t; S, v) Notice that LJ = ω̂ (−iS∂S ) − mS∂S − 1 where ω̂ is the characteristic function ω̂(z) = Z ∞ −∞ dJ ω(J)eizJ 20 i Double-exponential Distribution Double-exponential distribution ! ! p p− J J +θ(−J) exp ω(J) = θ(J) + exp − δ+ δ+ δ− δ− where p± ≥ 0 are the probabilities of positive and negative jumps, and δ± > 0 are the means of positive and negative jumps Characteristic function p+ p− + ω̂(z) = 1 − izδ+ 1 − izδ− Average jump amplitude p+ p− m= + − 1. 1 − δ+ 1 + δ− As δ+, δ− → 0 the probability density degenerates and the average jump amplitude vanishes, that is, ω(J) → δ(J) , m → 0. 21 Heston Model SDE dS = µSdt + √ v S dX , √ dv = κ (θ − v) + η v dW where κ is the mean reverting rate, θ is the long-term volatility, and η is the volatility of volatility Valuation operator " L = η2 1 v S 2∂S2 + 2ρηS∂S ∂v + ∂v2 2 2 # +rS∂S + κ (θ − v)∂v − r Homogeneous in S and linear in v Can be solved by Mellin transform in S (or Fourier transform in x = log S) and Laplace transform in v 22 SABR Model SDE dS = vS 1−α dX , dv = ηv dW where 0 ≤ α ≤ 1 Valuation operator i 1 2 h 2−2α 2 1−α 2 2 L = ∂S + 2ρηS ∂S ∂v + η ∂v v S 2 Defines a Riemannian metric on the hyperbolic plane H 2 with constant negative curvature −η 2/2. Can be solved by the tools of geometric analysis 23 SABR Model with Mean-Reverting Volatility SDE dS = vS 1−α dX , dv = κ (θ − v)dt + ηv dW Valuation operator i 1 2 h 2−2α 2 1−α 2 2 ∂S + 2ρηS ∂S ∂v + η ∂v L = v S 2 +κ (θ − v)∂v Can be solved in perturbation theory in parameter κ 24 Change of Variables Change of variables τ = T − t, x = log S, Valuation equation (∂τ − L)V = 0 where L = L̄ + λLJ , i 1h 2 2 2 2 σ (x, v)∂x + 2ρb(v)σ(x, v)∂x∂v + b (v)∂v L̄ = 2 1 2 + r − σ (x, v) ∂x + a(v)∂v − r 2 LJ = ω̂ (−i∂x) − m∂x − 1 The operator LJ acts as follows (LJ V )(τ ; x, v) = Z∞ dJ ω(J)V (τ ; x + J, v) −∞ −V (τ ; x, v) − m∂xV (τ ; x, v) 25 Heat Kernel Heat equation (∂τ − L)U (τ ; x, v, x0, v 0) = 0 Initial condition (and boundary conditions at v = 0 and at infinity) U (0; x, v, x0, v 0) = δ(x − x0)δ(v − v 0) Heat semigroup representation U (τ ; x, v, x0, v 0) = exp(τ L)δ(x − x0)δ(v − v 0) Option price V (τ ; x, v) = Z ∞ −∞ dx0 Z ∞ 0 dv 0 U (τ ; x, v, x0, v 0)f (x0) where f (x) is the payoff function; for call option f (x) = (ex − E)+ Thus, the knowledge of the heat kernel gives the value of all options with any payoff function. 26 Solution Methods • Analytic: integral transforms (Fourier, Laplace, Mellin) • Geometric: Riemannian geometry, negative curvature, diffusion on hyperbolic plane • Singularly perturbed pde: asymptotic expansion of the heat kernel as τ → 0 • Perturbative: semi-groups, Volterra series, perturbation theory • Numeric: finite differences, Monte-Carlo, binomial trees • Functional: path integrals, Feynman-Kac formula 27 Example: Black-Scholes Heat Kernel Heat Kernel U (τ ; x, x0) = exp(τ L)δ(x − x0) Valuation operator L = a∂x2 + b∂x − r 2 where a = σ2 , 2 b = r − D − σ2 We have −rτ 2 exp (b∂x) exp a∂x exp(τ L) = e By using x−x 2 0 −1/2 exp − exp a∂x δ(x − x ) = (4πa) 2 0 4a exp (b∂x) f (x) = f (x + b) we get U (τ ; x, x0) = (4πa)−1/2 exp 2 x − x0 + b −rτ − 4a 28 Elliptic Operators Second-order PDO L= n X g ij (x) i,j=1 n X ∂2 i(x) ∂ + P (x) A + i ∂xi∂xj ∂x i=1 Symbol σ(x, p) = n X g jk (x)pj pk − i n X Aj (x)pj − P (x) . j=1 j,k=1 Leading symbol σL(x, p) = n X g jk (x)pj pk . j,k=1 Operator L is elliptic if for any point x in M and for any real p 6= 0 the leading symbol σL(x, p) is positive The matrix (gij ) = (g ij )−1 is positive definite and plays the role of a Riemannian metric 29 Heat Kernel of Operators with Constant Coefficients Fourier transform U (τ ; x, x0) = Z Rn X dp 0 )j exp −τ σ(p) + i p (x − x j (2π)n j For a differential operator (Gaussian integral) U (τ ; x,x0) = (4πτ )−n/2g 1/2 X 1 i j × exp τ P − gij A A 4 i,j 1X × exp − gij (x − x0)iAj 2 i,j 1 X × exp − gij (x − x0)i(x − x0)j 4τ i,j where g = det gij 30 Indegro-Differential Operators Pseudo-differential operator L = L̄ + Σ(−i∂) where Σ(p) 6= 0 for any p 6= 0 and Σ(p) ∼ |p|−m as p → ∞ Heat kernel U (τ ; x, x0) = Z Rn dp exp {−τ [σ(p) + Σ(p)]} n (2π) X × exp i j pj (x − x0)j 31 Heat Kernel Asymptotic Expansion Asymptotic ansatz as τ → 0 U (τ ; x, x0) = (4πτ )−n/2 exp − × ∞ X d2(x, x0) ! 4τ τ k ak (x, x0) k=0 where d(x, x0) is the geodesic distance Recursion differential equations (transport equations along geodesics) for coefficients ak Can be solved in form of a covariant Taylor series in x close to x0 32 Perturbation Theory for Heat Semigroup Let L = L0 + εL1 be a negative operator The heat semigroup is a one-parameter family of operators (for τ ≥ 0) U (τ ) = exp(τ L) Volterra series U (τ ) = U0(τ ) + ∞ X k=1 εk Zτ 0 dτk Zτk dτk−1 · · · 0 Zτ2 dτ1 0 × U0(τ − τk )L1U0(τk − τk−1) · · · · · · U0(τ2 − τ1)L1U0(τ1) Thus the heat kernel U (τ, x, x0) = τ2 2 2 ε L1 + ε[L0, L1] 1 + ετ L1 + 2 n o 3 +O(τ ) U0(τ ; x, x0) 33 Discretization Let τk = kτ /N , k = 0, 1, . . . , N . Then U (τ ) = lim U (τN − τN −1)U (τN −1 − τN −2) N →∞ · · · U (τ2 − τ1)U (τ1) Then the heat kernel is U (τ ; x, x0) = lim Z N →∞ RN n dx1 . . . dxN U (τN − τN −1, x, xN −1) ×U (τN −1 − τN −2, xN −1, xN −2) · · · U (τ2 − τ1, x2, x1)U (τ1, x1, x0) 34 Path Integrals As τ → 0 n o 0 −n/2 1/2 0 U (τ ; x, x ) = (4πτ ) g exp −S(τ, x, x ) where Zτ i dxj dx S(τ, x, x0) = ds gij 4 i,j ds ds 0 n1 X o 1X 1 X dxi j i j + gij A + gij A A − P 2 i,j ds 4 i,j is the action functional Let M be the space of all all continuous paths x(s) starting at x0 at τ = 0 and ending at x at s = τ , that is, x(0) = x0 , x(τ ) = x , Then the heat kernel is represented as a path integral U (τ, x, x0) = Z Dx(s) exp[−S(τ, x, x0)] M 35