(Informal) Introduction to Stochastic Calculus

(Informal) Introduction to Stochastic Calculus Paola Mosconi Banca IMI Bocconi University, 19/02/2018 Paola Mosconi 20541 Lecture 2-3 1 / 68

Disclaimer The opinion expressed here are solely those of the author and do not represent in any way those of her employers Paola Mosconi 20541 Lecture 2-3 2 / 68

Main References D. Brigo and F. Mercurio Interest Rate Models Theory and Practice. With Smile, Inflation and Credit Springer (2006) Appendix C S. Shreve Stochastic Calculus for Finance II Springer (2004) Chapters 1-6 Paola Mosconi 20541 Lecture 2-3 3 / 68

From Deterministic to Stochastic Differential Equations Outline 1 From Deterministic to Stochastic Differential Equations 2 Ito s formula 3 Examples 4 Change of Measure 5 No-Arbitrage Pricing 6 Exercises Paola Mosconi 20541 Lecture 2-3 4 / 68

From Deterministic to Stochastic Differential Equations Probability Space Preamble [...] In continuous-time finance, we work within the framework of a probability space (Ω,F, P). We normally have a fixed final time T, and then have a filtration, which is a collection of σ-algebras {F(t);0 t T}, indexed by the time variable t. We interpret F(t) as the information available at time t. For 0 s t T, every set in F(s) is also in F(t). In other words, information increases over time. Within this context, an adapted stochastic process is a collection of random variables {X(t);0 t T}, also indexed by time, such that for every t, X(t) is F(t)-measurable; the information at time t is sufficient to evaluate the random variable X(t). We think of X(t) as the price of some asset at time t and F(t) as the information obtained by watching all the prices in the market up to time t. Two important classes of adapted stochastic processes are martingales and Markov processes. Shreve, Chapter 2 Paola Mosconi 20541 Lecture 2-3 5 / 68

From Deterministic to Stochastic Differential Equations Probability Space Probability Space: Definition A probability space (Ω, F, P) can be interpreted as an experiment, where: ω Ω represents a generic result of the experiment Ω is the set of all possible outcomes of the random experiment a subset A Ω represents an event F is a collection of subsets of Ω which forms a σ-algebra (σ-field) P is a probability measure (See Shreve, Chapter 1) Paola Mosconi 20541 Lecture 2-3 6 / 68

From Deterministic to Stochastic Differential Equations Probability Space Information In order to price a derivative security in the no-arbitrage framework, we need to model mathematically the information on which our future decisions(contingency plans) are based. Given a non empty set Ω and a positive number T, assume that for each t [0,T] there is a σ-algebra F t. F t represents the information up to time t. If t u, every set in F t is also in F u, i.e. F t F u F. The information increases in time, never exceeding the whole set of events The family of σ-fields (F t ) t 0 is called filtration. A filtration tells us the information that we will have at future times, i.e. when we get to time t we will know for each set in F t whether the true ω lies in that set. (See Shreve, Chapter 2) Paola Mosconi 20541 Lecture 2-3 7 / 68

From Deterministic to Stochastic Differential Equations Probability Space Random Variables and Stochastic Processes: Definitions Given a probability space (Ω,F, P), equipped with a filtration (F t) t, 0 t T: A random variable X is defined as a measurable function from the set of possible outcomes Ω to R, i.e. X : Ω R (+ some technical conditions See Shreve, Chapter 1) A stochastic process (X t) t is defined as a collection of random variables, indexed by t [0,T]. For each experiment result ω, the map t X t(ω) is called the path of X associated to ω. A stochastic process is said to be adapted if, for each t, the random variable X t is F t-measurable. Paola Mosconi 20541 Lecture 2-3 8 / 68

From Deterministic to Stochastic Differential Equations Probability Space Expectations: Definitions Let X be a random variable on a probability space (Ω,F, P). The expectation (or expected value) of X is defined as E[X] = Ω X(ω)dP(ω) provided that X is integrable i.e. Ω X(ω) dp(ω) < Let G F be a sub-algebra of F. The conditional expectation of X given G is any random variable which satisfies: 1 Measurability: E[X G] is G- measurable 2 Partial averaging: (See Shreve, Chapter 1,2) E[X G](ω)dP(ω) = X(ω)dP(ω) A G A A Paola Mosconi 20541 Lecture 2-3 9 / 68

From Deterministic to Stochastic Differential Equations Probability Space Conditional Expectations Let (Ω,F, P) be a probability space, G F and X,Y be (integrable) random variables. Linearity of conditional expectations E[aX +by G] = ae[x G]+bE[Y G] Taking out what is known If Y and XY are integral r.v and X is G-measurable then: Independence If X is integrable and independent of G E[XY G] = X E[Y G] E[X G] = E[X] Iterated conditioning (tower rule) If H G and X is an integrable r.v., then: (See Shreve, Chapter 2) E[E[X G] H] = E[X H] Paola Mosconi 20541 Lecture 2-3 10 / 68

From Deterministic to Stochastic Differential Equations Martingales Martingales I Let (Ω,F, P) be a probability space endowed with a filtration (F t) t, where 0 t T. Consider a process (X t) t satisfying the following conditions: Measurability: F t includes all the information on X t up to time t, i.e. (X t) t is adapted to (F t) t. Integrability: The relevant expected values exist. If: E[X T F t ] = X t for each 0 t T (1) we say the process is a martingale. It has no tendency to rise or fall. Paola Mosconi 20541 Lecture 2-3 11 / 68

From Deterministic to Stochastic Differential Equations Martingales Martingales II In other words... if t is the present time, the expected value at a future time T, given the current information, is equal to the current value a martingale represents a picture of a fair game, where it is not possible to lose or gain on average the martingale property is suited to model the absence of arbitrage, i.e. there is no safe way to make money from nothing (no free lunch) Go to No-Arbitrage Pricing Paola Mosconi 20541 Lecture 2-3 12 / 68

From Deterministic to Stochastic Differential Equations Martingales Submartingales, Supermartingales and Semimartingales A submartingale is a similar process (X t ) t satisfying: E[X T F t ] X t for each t T i.e. the expected value of the process grows in time. A supermartingale satisfies: E[X T F t ] X t for each t T i.e. the expected value of the process decreases in time. A process (X t ) t that is either a submartingale or a supermartingale is termed a semimartingale. Go to Martingales: Exercises Paola Mosconi 20541 Lecture 2-3 13 / 68

From Deterministic to Stochastic Differential Equations Variations/Covariantions Quadratic Variation: Definition Given a stochastic process Y t with continuous paths, its quadratic variation is defined as: Y T = lim Π 0 n [ Yti (ω) Y ti 1 (ω) ] 2 i=1 where 0 = t 0 < t 1 <... < t n = T and Π = {t 0,t 1,...,t n} is a partition of the interval [0,T]. Π represents the maximum step size of the partition. In form of a second order integral: T Y T = [dy s(ω)] 2 0 or in differential form: d Y t = dy t(ω)dy t(ω) Paola Mosconi 20541 Lecture 2-3 14 / 68

From Deterministic to Stochastic Differential Equations Variations/Covariantions Quadratic Variation: Deterministic Process A process whose paths are differentiable for almost all ω satisfies Y t = 0. If Y is the deterministic process Y : t t, then dy t = 0 and dtdt = 0 Paola Mosconi 20541 Lecture 2-3 15 / 68

From Deterministic to Stochastic Differential Equations Variations/Covariantions Quadratic Covariation: Definition The quadratic covariation of two stochastic processes Y t and Z t, with continuous paths, is defined as follows: Y,Z T = lim Π 0 n [ Yti (ω) Y ti 1 (ω) ][ Z ti (ω) Z ti 1 (ω) ] or in form of a second order integral: T Y,Z T = dy s(ω)dz s(ω) 0 i=1 or in differential form: d Y,Z t = dy t(ω)dz t(ω) Paola Mosconi 20541 Lecture 2-3 16 / 68

From Deterministic to Stochastic Differential Equations Deterministic Differential Equations Deterministic Differential Equations (DDE) EXAMPLE: Population Growth Model Let x(t) = x t R,x t 0, denote the population at time t, and assume for simplicity a constant (proportional) population growth rate, so that the change in the population at t is given by the deterministic differential equation: where K is a real constant. dx t = K x t dt, x 0 Assume now that x 0 is a random variable X 0 (ω) and that the population growth is modeled by the following differential equation: dx t (ω) = K X t (ω)dt, X 0 (ω) Paola Mosconi 20541 Lecture 2-3 17 / 68

From Deterministic to Stochastic Differential Equations Deterministic Differential Equations From Deterministic to Stochastic Differential Equations The solution to this equation is: X t (ω) = X 0 (ω)e Kt where all the randomness comes from the initial condition X 0 (ω). As a further step, suppose that even K is not known for certain, but that also our knowledge of K is perturbed by some randomness, which we model as the increment of a stochastic process {W t (ω)}, t 0, so that dx t (ω) = (K dt +dw t (ω))x t (ω), X 0 (ω), K 0 (2) where dw t (ω) represents a noise process that adds randomness to K. Eq. (2) represents an example of stochastic differential equation (SDE). Paola Mosconi 20541 Lecture 2-3 18 / 68

From Deterministic to Stochastic Differential Equations Stochastic Differential Equations Stochastic Differential Equations (SDE) More generally, a SDE is written as dx t (ω) = f t (X t (ω))dt +σ t (X t (ω))dw t (ω), X 0 (ω) (3) The function f corresponds to the deterministic part of the SDE and is called the drift. The function σ t is called the diffusion coefficient. The randomness enters the SDE from two sources: the noise term dw t(ω) the initial condition X 0 (ω) The solution X of the SDE is also called a diffusion process. In general the corresponding paths t X t (ω) are continuous. Paola Mosconi 20541 Lecture 2-3 19 / 68

From Deterministic to Stochastic Differential Equations Stochastic Differential Equations Noise Term Which kind of process is suitable to describe the noise term dw t (ω)? Paola Mosconi 20541 Lecture 2-3 20 / 68

From Deterministic to Stochastic Differential Equations Brownian Motion Brownian Motion The process whose increments dw t (ω) are candidates to represent the noise process in the SDE given by Eq. (3) is the Brownian motion. The most important properties of Brownian motion are that: it is a martingale it accumulates quadratic variation at rate one per unit time. This makes stochastic calculus different from ordinary calculus. Paola Mosconi 20541 Lecture 2-3 21 / 68

From Deterministic to Stochastic Differential Equations Brownian Motion Brownian Motion: Definition Definition Given a probability space (Ω,F,(F t) t, P), for each ω Ω there is a continuous function W t, t 0 such that it depends on ω and W 0 = 0. Then, W t is a Brownian motion if and only if for any 0 < s < t < u and any h > 0 it has: Independent increments: W u(ω) W t(ω) independent of W t(ω) W s(ω) Stationary increments: W t+h (ω) W s+h (ω) W t(ω) W s(ω) Gaussian increments: W t(ω) W s(ω) N(0,t s) Although the paths are continuous, they are (almost surely) nowhere differentiable, i.e. does not exist. Go to Brownian Motion: Exercises W t(ω) = d dt Wt(ω) Paola Mosconi 20541 Lecture 2-3 22 / 68

From Deterministic to Stochastic Differential Equations Brownian Motion Property 1: Martingality Brownian motion is a martingale. Proof Let 0 s t. Then: E[W t F s] = E[W t W s +W s F s] = E[W t W s F s]+e[w s F s] = E[W t W s]+w s = W s Paola Mosconi 20541 Lecture 2-3 23 / 68

From Deterministic to Stochastic Differential Equations Brownian Motion Property 2: Quadratic Variation The quadratic variation of a Brownian motion W is given almost surely by: W T = T for each T or, equivalently: dw t (ω)dw t (ω) = dt Brownian motion accumulates quadratic variation at rate one per unit time This comes from the fact that the Brownian motion moves so quickly that second order effects cannot be neglected. Instead, a process with differentiable trajectories cannot move so quickly and therefore second order effects do not contribute (derivatives are continuous). See Shreve, Chapter 3 for the proof. Paola Mosconi 20541 Lecture 2-3 24 / 68

From Deterministic to Stochastic Differential Equations Brownian Motion Property 2.bis: Quadratic Covariation If W is a Brownian motion and Z a deterministic process t t it follows: W,t T = 0 for each T or, equivalently: dw t (ω)dt = 0 Paola Mosconi 20541 Lecture 2-3 25 / 68

From Deterministic to Stochastic Differential Equations Stochastic Integrals Integral Form of an SDE The integral form of the general SDE, given by Eq. (3), i.e. t t X t (ω) = X 0 (ω)+ f s (X s (ω))ds + σ s (X s (ω))dw s (ω) (4) 0 0 contains two types of integrals: t fs(xs(ω))ds is a Riemann-Stieltjes integral. 0 t 0 σs(xs(ω))dws(ω) is a stochastic generalization of the Riemann-Stieltjes integral, such that the result depends on the chosen points of the sub-partitions used in the limit that defines the integral. Paola Mosconi 20541 Lecture 2-3 26 / 68

From Deterministic to Stochastic Differential Equations Stochastic Integrals Stochastic Integrals A stochastic integral is an integral of the type: T 0 φ t (ω)dw t (ω) where φ t is an adapted process, and W t a Brownian motion. The problem we face when trying to assign a meaning to the above integral, is that the Brownian motion paths cannot be differentiated w.r.t. time. If g(t) is a differentiable function, then we can define: T T φ t(ω)dg(t) = φ t(ω)g (t)dt 0 where the right endside is an ordinary integral w.r.t time. This will not work with Brownian motion. 0 Paola Mosconi 20541 Lecture 2-3 27 / 68

From Deterministic to Stochastic Differential Equations Stochastic Integrals Stochastic Integrals: Partitions Take the interval [0, T] and consider the following partitions of this interval: ( ) Ti n i = min T, 2 n i = 0,1,..., where n is an integer. For all i > 2 n T all terms collapse to T, i.e. T n i = T. Foreachnwehavesuchapartition,andwhenn increasesthepartitioncontains more elements, giving a better discrete approximation of the continuous interval [0,T]. Paola Mosconi 20541 Lecture 2-3 28 / 68

From Deterministic to Stochastic Differential Equations Stochastic Integrals Ito and Stratonovich Integrals Then define the integral as: T 0 n 1 [ ] φ s (ω)dw s (ω) = lim φ t n n i (ω) W T n i+1 (ω) W T n i (ω) i=0 where t n i is any point in the interval [ T n i,t n i+1). By choosing: t n i := T n i (initial point), we have the Ito integral; t n i := Tn i +Tn i+1 2 (middle point), we have the Stratonovich integral. Paola Mosconi 20541 Lecture 2-3 29 / 68

From Deterministic to Stochastic Differential Equations Stochastic Integrals Properties of Ito Integral Let I(t) = t φ(u)dw(u) be an Ito integral. I(t) has the following properties: 0 1 Continuity: as a function of the upper limit t, the paths of I(t) are continuous 2 Adaptivity: for each t, I(t) is F t-measurable 3 Linearity: for J(t) = t 0 γ(u)dw(u) then I(t)±J(t) = and ci(t) = t 0 c φ(u)dw(u) 4 Martingality: I(t) is a martingale t 5 Ito isometry: E[I 2 (t)] = E[ t 0 φ2 (u)du] 6 Quadratic variation I t = t 0 φ2 (u)du 0 [φ(u)±γ(u)]dw(u) Paola Mosconi 20541 Lecture 2-3 30 / 68

From Deterministic to Stochastic Differential Equations Stochastic Integrals Ito Integral vs Stratonovich Integral Ito = t 0 Stratonovich = W s (ω)dw s (ω) = W t(ω) 2 1 2 2 t t 0 W s (ω)dw s (ω) = W t(ω) 2 2 Ito Martingale property No standard chain rule Stratonovich No martingale property Standard chain rule Paola Mosconi 20541 Lecture 2-3 31 / 68

From Deterministic to Stochastic Differential Equations Solutions to SDEs Solution to a General SDE Consider the general SDE, given by Eq. (3): dx t (ω) = f t (t,x t (ω))dt +σ t (t,x t (ω))dw t (ω), X 0 (ω) Existence and uniqueness of the solution are guaranteed by: Lipschitz continuity: f(t,x) f(t,y) C x y and σ(t,x) σ(t,y) C x y x,y R d Linear growth bound: f(t,x) D(1+ x ) and σ(t,x) D(1+ x ) x R d (See Øksendal (1992) for the details.) Paola Mosconi 20541 Lecture 2-3 32 / 68

From Deterministic to Stochastic Differential Equations Solutions to SDEs Interpretation of the Coefficients: DDE Case For a deterministic differential equation with f a smooth function, we have: lim h 0 lim h 0 dx t = f(x t )dt x t+h x t h xt=y (x t+h x t ) 2 h xt=y = f(y) = 0 Paola Mosconi 20541 Lecture 2-3 33 / 68

From Deterministic to Stochastic Differential Equations Solutions to SDEs Interpretation of the Coefficients: SDE Case For a stochastic differential equation dx t (ω) = f(x t (ω))dt +σ(x t (ω))dw t (ω) functions f and σ can be interpreted as: { } lim E Xt+h (ω) X t (ω) h 0 h X t = y = f(y) { lim E [Xt+h (ω) X t (ω)] 2 } h 0 h X t = y = σ 2 (y) The second limit is non-zero because of the infinite velocity of the Brownian motion. Moreover, if the drift f is zero, the solution is a martingale. Paola Mosconi 20541 Lecture 2-3 34 / 68

Ito s formula Outline 1 From Deterministic to Stochastic Differential Equations 2 Ito s formula 3 Examples 4 Change of Measure 5 No-Arbitrage Pricing 6 Exercises Paola Mosconi 20541 Lecture 2-3 35 / 68

Ito s formula Chain Rule Deterministic Case For a deterministic differential equation such as dx t = f(x t )dt given a smooth transformation φ(t,x), we can write the evolution of φ(t,x t ) via the standard chain rule: dφ(t,x t ) = φ t (t,x t)dt + φ x (t,x t)dx t (5) Paola Mosconi 20541 Lecture 2-3 36 / 68

Ito s formula Chain Rule Stochastic Case: Ito s Formula I Let φ(t,x) be a smooth function and X t (ω) the unique solution to the SDE (3). The chain rule Ito s formula reads as: or, in compact notation: dφ(t,x t (ω)) = φ t (t,x t(ω))dt + φ x (t,x t(ω))dx t (ω) + 1 2 2 φ x 2(t,X t(ω))dx t (ω)dx t (ω) (6) dφ(t,x t ) = φ t (t,x t)dt + φ x (t,x t)dx t + 1 2 φ 2 x 2(t,X t)d X t Paola Mosconi 20541 Lecture 2-3 37 / 68

Ito s formula Chain Rule Stochastic Case: Ito s Formula II The term dx t (ω)dx t (ω) can be developed by recalling the rules on quadratic variation and covariation: thus giving: dw t (ω)dw t (ω) = dt, dw t (ω)dt = 0, dtdt = 0 [ φ dφ(t,x t (ω)) = t (t,x t(ω))+ φ x (t,x t(ω))f(x t (ω)) + 1 2 2 φ x 2(t,X t(ω))σ 2 (X t (ω)) ] dt + φ x (t,x t(ω))σ(x t (ω))dw t (ω) Go to Ito s Formula: Exercises Paola Mosconi 20541 Lecture 2-3 38 / 68

Ito s formula Leibniz Rule Leibniz Rule It applies to differentiation of a product of functions. Deterministic Leibniz rule For deterministic and differentiable functions x and y: d(x t y t ) = x t dy t +y t dx t Stochastic Leibniz rule For two diffusion processes X t (ω) and Y t (ω): d(x t (ω)y t (ω)) = X t (ω)dy t (ω)+y t (ω)dx t (ω)+dx t (ω)dy t (ω) or, in compact notation: d(x t Y t ) = X t dy t +Y t dx t +d X,Y t Paola Mosconi 20541 Lecture 2-3 39 / 68

Examples Outline 1 From Deterministic to Stochastic Differential Equations 2 Ito s formula 3 Examples 4 Change of Measure 5 No-Arbitrage Pricing 6 Exercises Paola Mosconi 20541 Lecture 2-3 40 / 68

Examples Linear SDE Linear SDE with Deterministic Diffusion Coefficient I A SDE is linear if both its drift and diffusion coefficients are first order polynomials in the state variable. Consider the particular case: dx t (ω) = (α t +β t X t (ω))dt +v t dw t (ω), X 0 (ω) = x 0 (7) where α, β, v are deterministic functions of time, regular enough to ensure existence and uniqueness of the solution. The solution is: X t (ω) = e t 0 βsds [x 0 + = x 0 e t 0 βsds + t 0 t 0 t e s 0 βudu α s ds + e t s βudu α s ds + t 0 0 ] e s 0 βudu v s dw s (ω) e t s βudu v s dw s (ω) (8) Paola Mosconi 20541 Lecture 2-3 41 / 68

Examples Linear SDE Linear SDE with Deterministic Diffusion Coefficient II The distribution of the solution X t (ω) is normal at each time t: ( X t N x 0 e t t 0 βsds + e t ) t s βudu α s ds, e 2 t s βudu vs 2 ds 0 0 Major examples: Vasicek SDE (1977) and Hull and White SDE (1990). Paola Mosconi 20541 Lecture 2-3 42 / 68

Examples Linear SDE Vasicek Model (1977) The Vasicek model has been introduced in 1977 to describe the evolution of interest rates. The dynamics of the short rate process r t is given by the following SDE: dr t = k(θ r t )dt +σdw t with k, θ and σ strictly positive constants, and initial condition r s. 1 Solution: r t = θ +(r s θ)e k(t s) +σ t s e k(u t) dw u 2 Distributional properties of the solution: E[r t F s] = r s e k(t s) +θ [1 e k(t s)] var[r t F s] = σ2 2k [1 e 2k(t s)] Paola Mosconi 20541 Lecture 2-3 43 / 68

Examples Linear SDE Vasicek Model (1977): Zero Coupon Bonds The price at time t of a zero coupon bond with maturity T is: where: P(t,T) = A(t,T)e B(t,T)rt [ 2hexp{(k +h)(t t)/2} A(t,T) = 2h +(k +h)(exp{(t t)h} 1) B(t,T) = 2(exp{(T t)h} 1) 2h+(k +h)(exp{(t t)h} 1) h = k 2 +2σ 2 ] 2kθ/σ 2 Paola Mosconi 20541 Lecture 2-3 44 / 68

Examples Linear SDE Hull-White Model (1990-1994) SDE: dr t = (ϑ(t) k r t )dt +σdw t where k, σ are positive constants, and ϑ is chosen so as to exactly fit the term structure of interest rates currently observed in the market. The solution, r t, can be expressed in terms of the Vasicek solution, x t, and a deterministic shift extension ϕ(t; α), which captures the initial term structure of interest rates: r t = x t +ϕ(t;α) with α = (k,θ,σ) the parameters of the Vasicek model. Paola Mosconi 20541 Lecture 2-3 45 / 68

Examples Linear SDE Hull-White Model (1990-1994): Market Instruments ZCB: P(t,T) = E [e ] T r t sds = e T t ϕ(s;α) E [e ] T x t sds = Φ(t,T;α)P Vasicek (t,t) Swaptions and caps/floors admit a closed-form expression, as a function of the parameters α and of the initial term structure. The parameters of the model, evolving under the risk neutral measure, are derived by calibrating the theoretical prices of market instruments to their corresponding market quotes: α = argmin α N i=1 ( Π Th i (t,t;α) Π Mkt i (t,t) ) 2 Paola Mosconi 20541 Lecture 2-3 46 / 68

Examples Lognormal Linear SDE Lognormal Linear SDE The lognormal SDE can be obtained as an exponential of a linear equation with deterministic diffusion coefficient. Let us take Y t = exp(x t ), where X t evolves according to (7), i.e.: d lny t (ω) = (α t +β t lny t (ω))dt +v t dw t (ω), Y 0 (ω) = exp(x 0 ) Equivalently, by Ito s formula we can write: dy t (ω) = de Xt(ω) = e Xt(ω) dx t (ω)+ 1 2 ext(ω) dx t (ω)dx t (ω) [ = α t +β t lny t (ω)+ 1 ] 2 v2 t Y t dt +v t Y t (ω)dw t (ω) The process Y has a lognormal marginal density. Major examples: Black Karasinski model (1991) and Geometric Brownian Motion. Paola Mosconi 20541 Lecture 2-3 47 / 68

Examples Lognormal Linear SDE Geometric Brownian Motion I The GBM is a particular case of lognormal linear process. Its evolution is defined by: dx t (ω) = µx t (ω)dt +σx t (ω)dw t (ω), X 0 (ω) = X 0 where µ and σ are positive constants. By Ito s formula, one can solve the SDE, by computing d lnx t : X t (ω) = X 0 exp {(µ 12 ) } σ2 t +σw t (ω) FromtheworkofBlackandScholes(1973)on,processesofthistypearefrequently used in option pricing theory to model the asset price dynamics. Paola Mosconi 20541 Lecture 2-3 48 / 68

Examples Lognormal Linear SDE Geometric Brownian Motion II The GBM process is a submartingale: E[X T F t ] = e µ(t t) X t X t The process Y t (ω) = e µt X t (ω) is a martingale, since we obtain: i.e. the drift of the process is zero. dy t (ω) = σy t (ω)dw t (ω) Go to Geometric Brownian Motion: Excercise Paola Mosconi 20541 Lecture 2-3 49 / 68

Examples Square Root Process Square Root Process It is characterized by a non-linear SDE: dx t (ω) = (α t +β t X t (ω))dt +v t Xt (ω)dw t (ω), X 0 (ω) = X 0 Square root processes are naturally linked to non-central ξ-square distributions. Major examples: the Cox Ingersoll and Ross (CIR) model (1985) and a particular case of the constant-elasticity variance (CEV) model for stock prices: dx t (ω) = µx t (ω)dt +σ X t (ω)dw t (ω), X 0 (ω) = X 0 Go to Cox Ingersoll Ross Model Go to SDE: Excercise Paola Mosconi 20541 Lecture 2-3 50 / 68

Change of Measure Outline 1 From Deterministic to Stochastic Differential Equations 2 Ito s formula 3 Examples 4 Change of Measure 5 No-Arbitrage Pricing 6 Exercises Paola Mosconi 20541 Lecture 2-3 51 / 68

Change of Measure Change of Measure The way a change in the underlying probability measure affects a SDE is defined by the Girsanov theorem The theorem is based on the following facts: the SDE drift depends on the particular probability measure P if we change the probability measure in a regular way, the drift of the equation changes while the diffusion coefficient remains the same. The Girsanov theorem can be useful when we want to modify the drift coefficient of a SDE. Paola Mosconi 20541 Lecture 2-3 52 / 68

Change of Measure Radon-Nikodym Derivative Two measures P and P on the space (Ω,F,(F t ) t ) are said to be equivalent, i.e. P P, if they share the same sets of null probability. When two measures are equivalent, it is possible to express the first in terms of the second through the Radon-Nikodym derivative. There exists a martingale ρ t on (Ω,F,(F t) t, P) such that P = ρ t(ω)dp(ω), A F t which can be written as: A dp dp = ρ t Ft The process ρ t is called the Radon-Nikodym derivative of P with respect to P restricted to F t. Paola Mosconi 20541 Lecture 2-3 53 / 68

Change of Measure Expected Values When in need of computing the expected value of an integrable random variable X, it may be useful to switch from one measure to another equivalent one. Expectations E [X] = Ω X(ω)dP (ω) = Ω ] X(ω) [X dp dp (ω)dp(ω) = E dp dp Conditional expectations E [X F t ] = [ E X dp dp ρ t ] F t Paola Mosconi 20541 Lecture 2-3 54 / 68

Change of Measure Girsanov Theorem Consider SDE, with Lipschitz coefficients, under dp: dx t(ω) = f(x t(ω))dt +σ(x t(ω))dw t(ω), x 0 Let be given a new drift f (x) and assume (f (x) f(x))/σ(x) to be bounded. Define the measure P through the Radon-Nikodym derivative: dp { dp (ω) = exp 1 t ( f ) (X 2 s(ω)) f(x s(ω)) t ds + Ft 2 0 σ(x s(ω)) 0 f (X s(ω)) f(x s(ω)) σ(x s(ω)) dw s(ω) } Then P is equivalent to P and the process W defined by: [ ] dwt f (X t(ω)) f(x t(ω)) (ω) = dt +dw t(ω) σ(x t(ω)) is a Brownian motion under P and dx t(ω) = f (X t(ω))dt +σ(x t(ω))dw t (ω), x 0 Paola Mosconi 20541 Lecture 2-3 55 / 68

Change of Measure Example: from P to Q I A classical example involves moving from the real world asset price dynamics P to the risk neutral one, Q, i.e. from ds t = µ S t dt +σs t dw P t under P (9) to ds t = r S t dt +σs t dw Q t under Q (10) The risk neutral measure Q is used in pricing problems while the real-world (or historical) measure P is used in risk management. Paola Mosconi 20541 Lecture 2-3 56 / 68

Change of Measure Example: from P to Q II Start from the asset dynamics under the real-world measure P, eq. (9): ds t = µs t dt +σs t dw P t Consider the discounted asset price process S t = S t e rt. This process satisfies the following SDE: d S t = (µ r) S t dt +σ S t dw P t (11) The goal is to find a measure Q, equivalent to P, such that the discounted asset price process is a martingale under the new measure, i.e. d S t = σ S t dw Q t (12) Paola Mosconi 20541 Lecture 2-3 57 / 68

Change of Measure Example: from P to Q III To this purpose, rewrite eq. (11) as follows: d S t = [ [ (µ r)dt +σdwt P ] µ r St = σ ] dt + dwt P σ S t and define, according to Girsanov theorem, a new Brownian process: Therefore, dwt µ r dt + dwt P. σ d S t = σ S t dw t (13) i.e. S t is a martingale under the equivalent measure P. Comparing (12) with (13), we obtain P Q. Going back to the asset price process S t = S t e rt, we finally get eq. (10): ds t = r S t dt +σs t dw Q t Paola Mosconi 20541 Lecture 2-3 58 / 68

No-Arbitrage Pricing Outline 1 From Deterministic to Stochastic Differential Equations 2 Ito s formula 3 Examples 4 Change of Measure 5 No-Arbitrage Pricing 6 Exercises Paola Mosconi 20541 Lecture 2-3 59 / 68

No-Arbitrage Pricing No-Arbitrage Pricing We refer to Brigo and Mercurio, Chapter 2. As already mentioned, absence of arbitrage is equivalent to the impossibility to invest zero today and receive tomorrow a non-negative amount that is positive with positive probability. In other words, two portfolios having the same payoff at a given future date must have the same price today. Historically, Black and Scholes (1973) showed that, by constructing a suitable portfolio having the same instantaneous return as that of a risk-less investment, the portfolio instantaneous return was indeed equal to the instantaneous risk-free rate, which led to their celebrated option-pricing formula. Paola Mosconi 20541 Lecture 2-3 60 / 68

No-Arbitrage Pricing Harrison and Pliska Result (1983) A financial market is arbitrage free and complete if and only if there exists a unique equivalent (risk-neutral or risk-adjusted) martingale measure. Stylized characterization of no-arbitrage theory via martingales: The market is free of arbitrage if (and only if) there exists a martingale measure The market is complete if and only if the martingale measure is unique In an arbitrage-free market, not necessarily complete, the price π t of any attainable claim is uniquely given, either by the value of the associated replicating strategy, or by the risk neutral expectation of the discounted claim payoff under any of the equivalent (risk-neutral) martingale measures: π t = E[D(t,T)Π T F t] Go Back Paola Mosconi 20541 Lecture 2-3 61 / 68

Exercises Outline 1 From Deterministic to Stochastic Differential Equations 2 Ito s formula 3 Examples 4 Change of Measure 5 No-Arbitrage Pricing 6 Exercises Paola Mosconi 20541 Lecture 2-3 62 / 68

Exercises Brownian Motion: Exercises 1 Time reversal Prove that the continuous time stochastic process defined by: B t = W T W T t, t [0,T] is a standard Brownian motion. 2 Brownian scaling Let W t be a standard Brownian motion. Given a constant c > 0, show that the stochastic process X t defined by: X t = 1 c W ct, t > 0 is a standard Brownian motion. Go Back Paola Mosconi 20541 Lecture 2-3 63 / 68

Exercises Martingales: Exercises 1 Let X 1,X 2,... be a sequence of independent random variables in L 1 such that E[X n] = 0 for all n. If we set: S 0 = 0, S n = X 1 +X 2 +... +X n, for n 1 F 0 = {,Ω} and F n = σ(x 1,X 2,...,X n), for n 1 prove that the process (S n) n is an (F n) n-martingale. 2 Consider a filtration (F n) n and an F n-adapted stochastic process (X n) n such that X 0 = 0 and E[ X n ] for all n 0. Also, let (c n) n be a sequence of constants. Define M 0 = 0 and M n = c nx n n c j E[X j X j 1 F j 1] j=1 Prove that (M n) n is an F n-martingale. n (c j c j 1)X j 1, for n 1 j=1 Go Back Paola Mosconi 20541 Lecture 2-3 64 / 68

Exercises Ito s formula: Exercises 1 Consider a standard one-dimensional Brownian motion W t. Use Ito s formula to calculate: and W 2 t = t +2 t t Wt 27 = 351 Ws 25 ds +27 0 0 W sdw s t 0 W 26 s dw s 2 Consider a standard one-dimensional Brownian motion W t. Given k 2 and t 0, use Ito s formula to prove that: E[W k t ] = 1 t 2 k(k 1) 0 E[W k 2 u ]du Use this expression to calculate E[W 4 t ] and E[W 6 t ]. Hint: stochastic integrals are martingales, so their expectation is zero. Go Back Paola Mosconi 20541 Lecture 2-3 65 / 68

Exercises Geometric Brownian Motion: Exercise Consider the Geometric Brownian motion SDE: ds t = µ t S t dt +σ t S t dw t with deterministic time-dependent coefficients, µ t and σ t, and initial condition S 0. Prove that its solution is given by: { t ( ) t } S t = S 0 exp µ u σ2 u du + σ u dw u 2 0 0 Go Back Paola Mosconi 20541 Lecture 2-3 66 / 68

Exercises Cox Ingersoll Ross Model In the Cox Ingersoll Ross model (1985) for interest rates, the dynamics of the short rate process r t is given by the following SDE: dr t = k(θ r t)dt +σ r tdw t with k, θ and σ strictly positive constants, and initial condition r 0. 1 Show that the solution to the above SDE is given by: t r t = θ +(r 0 θ)e kt +σ e k(s t) rsdw s 0 Hint: Consider the Ito processes X t and Y t defined by X t = e kt and Y t = r t and integrate by parts. 2 Calculate the mean E[r t] and the variance var(r t) of the random variable r t. Hint: Use Ito s isometry and assume that all stochastic integrals are martingales, so they have zero expectation. Go Back Paola Mosconi 20541 Lecture 2-3 67 / 68

Exercises SDE: Exercise Consider the following integral SDE: t t Z t = Z u du + e u dw u 0 0 Prove that: Z t = e t W t Go Back Paola Mosconi 20541 Lecture 2-3 68 / 68