Cheat sheet for math finance, Study notes of Mathematics

Download Cheat sheet for math finance and more Study notes Mathematics in PDF only on Docsity! MATHEMATICAL FINANCE CHEAT SHEET Normal Random Variables A random variable X is Normal N(µ,σ2) (aka. Gaussian) under a measure P if and only if EP  e θX  = e θµ+ 1 2 θ 2σ2 , for all real θ . A standard normal Z ∼N(0, 1) under a measure P has density φ(x ) = 1 p 2π e −x 2/2. P[Z ≤ x ] =Φ(x ) := ∫ x −∞ φ(z )d z . Let X = (X1, X2, . . . , Xn )′ with X i ∼N(µi , qi i ) and Cov[X i , X j ] = qi j for i , j = 1, . . . , n . We call µ := (µ1, . . . ,µn )′ the mean and Q := (qi j )ni , j=1 the covariance matrix of X . Assume detQ > 0, then X has a multivariate normal distribution if it has the den- sity φ(x ) = 1 p (2π)n detQ exp  − 1 2 (x −µ)′Q−1(x −µ) ‹ , x ∈Rn . We write X ∼ N(µ,Q ) if this is the case. Alternatively, X ∼ N(µ,Q ) under P if and only if EP[e θ ′X ] = exp  θ ′µ+ 1 2 θ ′Qθ ‹ , for all θ ∈Rn . If Z ∼N(0,Q ) and c ∈Rn then X = c ′Z ∼N(0, c ′Q c ). If C ∈Rm×n (i.e., m×n matrix) then X =C Z ∼N(0, C Q C ′) and C Q C ′ is a m ×m covariance matrix. Gaussian Shifts If Z ∼ N(0, 1) under a measure P, h is an integrable function, and c is a constant then EP[e c Z h (Z )] = e c 2/2EP[h (Z + c )]. Let X ∼N(0,Q ), h be a integrable function of x ∈Rn , and c ∈Rn . Then EP[e c ′X h (X )] = e 1 2 c ′Q c EP[h (X + c )]. Correlating Brownian Motions Let (W (t ))t≥0 and (fW (t ))t≥0 be independent Brownian motions. Given a correla- tion coefficient ρ ∈ [−1, 1], define cW (t ) :=ρW (t ) + p 1−ρ2 fW (t ), then (cW (t ))t≥0 is a Brownian motion and E[W (t )cW (t )] =ρt . Identifying Martingales If X t = X (t ) is a diffusion process satisfying d X (t ) =µ(t , X t )d t +σ(t , X t )d W (t ) and EP[( ∫ T 0 σ(s , X s )2 d s )1/2]<∞ (or,σ(t , x )≤ c |x | as |x | →∞), then X is a martingale ⇐⇒ X is driftless (i.e., µ(t )≡ 0 with P-prob. 1). Novikov’s Condition In the case d X (t ) =σ(t )X (t )d W (t ) for someF -previsible process (σ(t ))t≥0, then we have the simpler condition EP – exp ‚ 1 2 ∫ T 0 σ(s )2 d s Œ™ <∞⇒ X is a martingale. Itô’s Formula For X t = X (t ) given by d X (t ) =µ(t )d t +σ(t )d W (t ) and a function g (t , x ) that is twice differentiable in x and once in t . Then for Y (t ) = g (t , X t ), we have d Y (t ) = ∂ g ∂ t (t , X t )d t + ∂ g ∂ x (t , X t )d X t + 1 2 σ(t )2 ∂ 2g ∂ x 2 (t , X t )d t . The Product Rule Given X (t ) and Y (t ) adapted to the same Brownian motion (W (t ))t≥0, d X (t ) =µ(t )d t +σ(t )d W (t ), d Y (t ) = ν(t )d t +ρ(t )d W (t ). Then d (X (t )Y (t )) = X (t )d Y (t ) +Y (t )d X (t ) +d 〈X , Y 〉(t ) ︸ ︷︷ ︸ σ(t )ρ(t )d t . In the other case, if X (t ) and Y (t ) are adapted to two different and independent Brownian motions (W (t ))t≥0 and (fW (t ))t≥0, d X (t ) =µ(t )d t +σ(t )d W (t ), d Y (t ) = ν(t )d t +ρ(t )d fW (t ). Then d (X (t )Y (t )) = X (t )d Y (t ) +Y (t )d X (t ) as d 〈X , Y 〉(t ) = 0. Radon-Nikodým Derivative Given P and Q equivalent measures and a time horizon T , we can define a random variable d Q d P defined on P-possible paths, taking positive real values, such that • EQ[XT ] = EP • d Q d P XT ˜ , for all claims XT knowable by time T , • EQ[X t |Fs ] = ζ −1 s EP [ζt X t |Fs ], for s ≤ t ≤ T , where ζt is the process EP[ d Q d P |Ft ]. Cameron-Martin-Girsanov Theorem If (W (t ))t≥0 is a P-Brownian motion and (γ(t ))t≥0 is anF -previsible process satis- fying the boundedness condition EP ” exp € 1 2 ∫ T 0 γ(t )2 d t Š— <∞, then there exists a measure Q such that: • Q is equivalent to P, • d Q d P = exp ‚ − ∫ T 0 γ(t )d W (t )− 1 2 ∫ T 0 γ(t )2 d t Œ , • fW (t ) :=W (t ) + ∫ t 0 γ(s )d s is a Q-Brownian motion. In other words, W (t ) is a drifting Q-Brownian motion with drift −γ(t ) at time t . Cameron-Martin-Girsanov Converse If (W (t ))t≥0 is a P-Brownian motion, and Q is a measure equivalent to P, then there exists aF -previsible process (γ(t ))t≥0 such that fW (t ) :=W (t ) + ∫ t 0 γ(s )d s is a Q-Brownian motion. That is, W (t ) plus drift γ(t ) is a Q-Brownian motion. Ad- ditionally, d Q d P = exp ‚ − ∫ t 0 γ(t )d W (t )− 1 2 ∫ T 0 γ(t )2 d t Œ . Martingale Representation Theorem Suppose (M (t ))t≥0 is a Q-martingale process whose volatility Æ EQ[M (t )2] = σ(t ) satisfiesσ(t ) 6= 0 for all t (with Q-probability one). Then if (N (t ))t≥0 is any other Q- martingale, there exists anF -previsible process (φ(t ))t≥0 such that ∫ T 0 φ(t )2σ(t )2 d t < ∞ (with Q-prob. one), and N can be written as N (t ) =N (0) + ∫ t 0 φ(s )d M (s ), or in differential form, d N (t ) =φ(t )d M (s ). Further,φ is (essentially) unique. Multidimensional Diffusions, Quadratic Covariation, and Itô’s Formula If X := (X1, X2, . . . , Xn )′ is a n-dimensional diffusion process with form X (t ) = X (0)+ ∫ t 0 µ(s )d s + ∫ t 0 Σ(s )d W (s ), where Σ(t ) ∈Rn×m and W is a m-dimensional Brownian motion. The quadration covariation of the components X i and X j is 〈X i , X j 〉(t ) = ∫ t 0 Σi (s ) ′Σ j (s )d s , or in differential form d 〈X i , X j 〉(t ) = Σi (t )′Σ j (t )d t , where Σi (t ) is the i th column of Σ(t ). The quadratic variation of X i (t ) is 〈X i 〉(t ) = ∫ t 0 Σi (s )′Σi (s )d s . The multi-dimensional Itô formula for Y (t ) = f (t , X1(t ), . . . , Xn (t )) is d Y (t ) = ∂ f ∂ t (t , X1(t ), . . . , Xn (t ))d t + n ∑ i=1 ∂ f ∂ xi (t , X1(t ), . . . , Xn (t ))d X i (t ) + 1 2 n ∑ i , j =1 ∂ 2 f ∂ xi ∂ x j (t , X1(t ), . . . , Xn (t ))d 〈X i , X j 〉(t ). The (vector-valued) multi-dimensional Itô formula for Y (t ) = f (t , X (t )) = ( f1(t , X (t )), . . . , fn (t , X (t )))′ where fk (t , X ) = fk (t , X1, . . . , Xn )and Y (t ) = (Y1(t ), Y2(t ), . . . , Yn (t ))′ is given component- wise (for k = 1, . . . , n) as d Yk (t ) = ∂ fk (t , X (t )) ∂ t d t + n ∑ i=1 ∂ fk (t , X (t )) ∂ xi d X i (t ) + 1 2 n ∑ i , j =1 ∂ 2 fk (t , X (t )) ∂ xi ∂ x j d 〈X i , X j 〉(t ). Stochastic Exponential The stochastic exponential of X is Et (X ) = exp(X (t )− 1 2 〈X 〉(t )). It satisfies E (0) = 1, E (X )E (Y ) = E (X +Y )e 〈X ,Y 〉, E (X )−1 = E (−X )e 〈X ,X 〉. The process Z = E (X ) is a positive process and solves the SDE d Z = Z d X , Z (0) = e X (0). Solving Linear ODEs The linear ordinary differential equation d z (t ) d t =m (t ) +µ(t )z (t ), z (a ) = ζ, for a ≤ t ≤ b has solution given by z (t ) = ζεt + ∫ t a εt ε −1 u m (u )d u , εt := exp ∫ t a µ(u )d u  , = ζexp € ∫ t a µ(u )d u Š + ∫ t a m (u )exp € ∫ t u µ(r )d r Š d u . Solving Linear SDEs The linear stochastic differential equation d Z (t ) = [m (t ) +µ(t )Z (t )]d t + [q (t ) +σ(t )Z (t )]d W (t ), Z (a ) = ζ, for a ≤ t ≤ b has solution given by Z (t ) = ζEt + ∫ t a Et E−1 u [m (u )−q (u )σ(u )]d u + ∫ t a Et E−1 u q (u )d W (u ), where Et := Et (X ) and X (t ) = ∫ t a µ(u )d u + ∫ t a σ(u )d W (u ). In other words, Et = exp ∫ t a µ(u )d u + ∫ t a σ(u )d W (u )− 1 2 ∫ t a σ(u )2 d u  .