Financial Mathematics - ALL THEORY YOU NEED TO PASS THE EXAM, Dispense di Matematica Finanziaria

Scarica Financial Mathematics - ALL THEORY YOU NEED TO PASS THE EXAM e più Dispense in PDF di Matematica Finanziaria solo su Docsity! 07/02/2024, 16:16 1. Vectors file:///Users/danieleallegri/Downloads/vectors/1 Vectors 3b8db3ad3d2943cd8e53e186be9ad2c8.html 1/6 (a, b) a ∈ A b ∈ B A,B A × B (a, b) a ∈ A b ∈ B A × B = {(a, b) : a ∈ A, b ∈ B} (a, b) A B A × B = B × A A = B A × A = A2 n ≥ 0 (x ,x , ...,x )1 2 n n = 3 (x ,x ,x )1 2 3 R =3 R × R × R n = 4 R =4 {(x ,x ,x ,x ) :1 2 3 4 x ,x ,x ,x ∈1 2 3 4 R} =x x 1 x 2 ... x n x ,x , ...x 1 2 n Rn n x =T (x x ... x )1 2 n 07/02/2024, 16:16 1. Vectors file:///Users/danieleallegri/Downloads/vectors/1 Vectors 3b8db3ad3d2943cd8e53e186be9ad2c8.html 2/6 / x t / =x t (x ,x , ...,x )/(t , t , ..., t )1 2 n 1 2 n x n t n Rn Rn Rn e , ..., e 1 n e i i = 1, ...,n =e 0 ... 0 1 0 ... 0 R3 =e1 1 0 0 =e2 0 1 0 =e3 0 0 1 , ∈x y Rn x =k y ∀ k ∈k {1, 2, ...,n} =x ∀ ∈x x Rn =x ⇒y =y ∀ , ∈x x y Rn =x y =y ⇒z =x ∀ , , ∈z x y z R =x  y , x y >x y x >k y ∀ k ∈k {1, 2, ...,n} ≥x ⇔y x ≥k y ∀ k ∈k {1, 2, ...,n} =0 (0, ..., 0) ∈ Rn x >x 0 x >k 0 ∀ k ∈ {1, ...,n} x ≥x 0 x ≥k 0 ∀ k ∈ {1, 2, ...,n} , ∈x y Rn x y 07/02/2024, 16:16 1. Vectors file:///Users/danieleallegri/Downloads/vectors/1 Vectors 3b8db3ad3d2943cd8e53e186be9ad2c8.html 5/6 ∣∣λ ∣∣ =x ∣λ∣•∣∣ ∣∣   ∀  ∈x x Rn λ ∈ R ∀  , ∈x y R , ∣∣ +n x ∣∣ ≤y ∣∣ ∣∣ +x ∣∣ ∣∣y ∣∣ ∣∣ =x 1 ∣∣ ∣∣ =e2 =0 + 1 + 2 + ... + 02 2 2 2 =1 1 ∈x Rn ∣∣ ∣∣x x =x  0 Rn , ∈x y Rn d( , )x y d( , ) =x y (x − y ) i=1 ∑ n i i 2 d( , )x y −x y ∣∣ −x ∣∣ =y =(x − y ) + (x − y ) + ... + (x − y )1 1 2 2 2 2 n n 2 (x − y ) i=1 ∑ n i i 2 d( , ) ≥x y 0   ∀  , ∈x y R    and   d( , ) =n x y 0 ⇔ =x y d( , ) =x y d( , )   ∀  , ∈y x x y Rn d( , ) ≤x y d( , ) +x z d( , )   ∀  , , ∈z y x y z Rn V Rn V = 0,V ⊆ Rn ∀  , ∈x y V ∀ λ ∈ R +x ∈y V ,λ ∈x V Rn V = 0,V ⊆ Rn Rn V ⊆ Rn λ +x β ∈y V    ∀  , ∈x y V , ∀ λ,β ∈ R k x ,x , ...,x ∈1 2 k Rn e(x ,x , ...,x )1 2 k ∈x Rn =x λ i=1 ∑ k ixi C(x ,x , ...,x ) =1 2 k { ∈x R :n =x λ ,λ ∈∑i=1 ixi i R} Rn C( , ..., )x1 xk , ..., x1 xk , ..., x1 xk C( , , ..., )x1 x2 xk V Rn , ..., x1 xk V = C( , ..., )x1 xk 07/02/2024, 16:16 1. Vectors file:///Users/danieleallegri/Downloads/vectors/1 Vectors 3b8db3ad3d2943cd8e53e186be9ad2c8.html 6/6 V = { }0 Rn 0 Rn (1 0 ) (1 1 ) (1 2 ) R2 R3 R3 C = { , , ..., }x1 x2 xk V V V C C V C = { , , ..., }x1 x2 xk V S C span(C) = {λ +x1 λ +x2 ... + λ ∣λ , ...,λ ∈kxk 1 k R} span(C) = V V { , , ..., }x1 x2 xk C V k , , ..., ∈x1 x2 xk Rn k , , ..., ∈x1 x2 xk Rn , , ..., x1 x2 xk ∃ k λ , ...,λ 1 k λ = i=1 ∑ k ixi 0 , , ..., x1 x2 xk =xk λ →∑i=1 k−1 ixi =0 λ −∑i=1 k−1 ixi xk 0 , , ..., x1 x2 xk λ =k −1 = 0 , , ..., ∈x1 x2 xk Rn λ =∑i=1 k ixi 0 λ =1 λ =2 ... = λ =k 0 Rn λ =∑i=1 n iei ⇔0 λ =i 0 ∀ i = 1, ...,n =0 λ +∑i=1 k−1 ixi λ • k 0 λ =1 ... = λ =k−1 0 λ =k  0 λ +1x1 λ =2x2 ⇔0 λ =1x1 −λ 2x2 λ =1  0 → =x1 − λ 1 λ 2x2 V ⊆ Rn V { , ..., }x1 xk V V V dim(V ) 07/02/2024, 16:17 2. Matrices file:///Users/danieleallegri/Downloads/matrices/2 Matrices f76c1bec063e401793c18ba6c22e34de.html 1/5 m,n ∈ N) A = a 11 a 21 ... a m1 a 12 a 22 ... a m2 ... ... ... ... a 1n a 2n ... a mn A = [a ], i =ij 1, ...,m ;  j = 1, ...,n a ij A i j M (R)m,n a ∈ R A = [a ]ij A A A AT A A = AT λ , ...,λ 1 n λ =1 λ =2 ... = λ =n 1 I n 07/02/2024, 16:18 2. Matrices file:///Users/danieleallegri/Downloads/matrices/2 Matrices f76c1bec063e401793c18ba6c22e34de.html 4/5 A = [a ]ij n n > 1 r det(A) = a A ∑s=1 n rs rs s det(A) = a A ∑r=1 n rs rs A AT det(A) = det(A )T det(A) = a •a •a 11 22 m,n A = a•I n det(A) = an det(I ) =n 1 det(A) = 0 A det(A) = 0 A det(A) = 0. AB det(AB) = det(A)• det(B) det(A) = 0 ⟹ { , , ..., }x1 x2 xk det = 0 A n A ij A : A =ij (−1) M ; i =i+j ij 1, ...,m; j = 1, ...,n A =∗ [A ]ij A A∗ (A )∗ T A ⟺ det = 0 A =−1 •(A ) det(A) 1 ∗ T ⟺ det = 0 det(A) 07/02/2024, 16:18 2. Matrices file:///Users/danieleallegri/Downloads/matrices/2 Matrices f76c1bec063e401793c18ba6c22e34de.html 5/5 k A ∈ M (R)m,n k A k A k mxn k ≤ min{m,n} A ∈ M (R)m,n r ≥ 0 A r r A r(A) min{m,n} r ≤ {m,n} min{m,n} =( m n) m!(n−m)! n! min{m,n} r(A) ≤ min{m,n} − 1 min{m,n} − 1 = 0 r(A) V dim(V ) = r(A) A V f : A ⊆ R →n R n ∈x A f( ) ∈x R f x A ⊆ Rn f R f(A) = {y ∈ R, ∃(x ,x , ...,x ) ∈1 2 n A : y = f(x ,x , ...,x ) =1 2 n f( )}x f : R →n R f( +x ) =y f( ) +x f( )     ∀  , ∈y x y Rn f(α ) =x αf( )     ∀  ∈x x R , ∀ α ∈n R ⟺ ∃ ∈a R :n ∀  ∈x R →n f( ) =x • =aT x (a , ..., a ) =1 n x 1 ⋮ x n a x i=1 ∑ n i i f f : A ⊆ R →2 R (x, y) → z f : A ⊆ R →2 R (f) = {(x, y, z) ∈ R :3 z = f(x, y)} f R3 n > 2 (f) f : A ⊆ R →2 R c ∈ R c c A A c f A =c f {(x, y) ∈ R :2 f(x, y) = c} f (0, 0, c) c f(x, y) = c c ∈ R f : A ⊆ R →2 R ∃  lim f(x, y)(x,y)→(x ,y )0 0 f (x , y )0 0 (x , y )0 0 f (x , y )0 0 f : A ⊆ R →2 R (x , y )0 0 A (x , y ) ∈0 0 A f (x , y ) ⇒0 0 lim f(x, y) =(x,y)→(x ,y )0 0 f(x , y )0 0 lim f(x, y)(x,y)→(x ,y )0 0 (x , y )0 0 f (x , y )0 0 ∄ f lim f(x, y) =(x,y)→(x ,y )0 0 f(x , y )0 0 e•∞, , (e =∞ C 0 e  0), , 0 ∞ ∞ 0 R f : A ⊆ R →2 R A A (x , y )0 0 (x , y )0 0 x y f(x, y ) →0 x f(x , y) →0 y x y f(x, y) x y (x , y )0 0 f : A ⊆ R →2 R (x , y )0 0 A (x , y ) ∈0 0 A A (x , y ) = ∂x ∂f 0 0 f (x , y ) =x ′ 0 0 = x→x 0 lim x − x 0 f(x, y ) − f(x , y )0 0 0 h→0 lim h f(x + h, y ) − f(x , y )0 0 0 0 f x (x , y )0 0 (x , y ) = ∂y ∂f 0 0 f (x , y ) =y ′ 0 0 = y→y 0 lim y − y 0 f(x , y) − f(x , y )0 0 0 h→0 lim h f(x , y + h) − f(x , y )0 0 0 0 f y (x , y )0 0 f : A ⊆ R →2 R (x , y )0 0 A A (x , y )∂x ∂f 0 0 (x , y )∂y ∂f 0 0 f (x , y )0 0 ∇f(x , y ) =o 0 [ (x , y )      (x , y )] ∂x ∂f 0 0 ∂y ∂f 0 0 T f (x , y )0 0 f (x , y )0 0 f (x , y )x ′ 0 0 x → f(x, y )0 (x , y , z )0 0 0 f (x , y )y ′ 0 0 y → f(x , y)0 (x , y , z )0 0 0 f : A ⊆ R →2 R :f A ⊆ R →2 R f (x , y )0 0 (x, y) =f ax + by + c (x , y ) =f 0 0 f(x , y )0 0 lim =(x,y)→(x ,y )0 0 d((x,y),(x ,y ))0 0 f(x,y)− (x,y)f 0 (x, y)f (x , y ) =f 0 0 f(x , y )0 0 lim =(x,y)→(x ,y )0 0 d((x,y),(x ,y ))0 0 f(x,y)− (x,y)f 0 f (x , y )0 0 (x, y) =f f(x , y ) +0 0 f (x , y )(x −x ′ 0 0 x ) +0 f (x , y )(y −y ′ 0 0 y )0 f z = f(x, y) f f (x , y )0 0 f (x , y )(x −x ′ 0 0 x ) +0 f (x , y )(y −y ′ 0 0 y )0 df(x , y )0 0 f x ′ f y ′ f : A ⊆ R →2 R (x , y ) ∈0 0 A f x ′ f y ′ U x ,y 0 0 (x , y )0 0 f (x , y )0 0 e1 A f : A ⊆ R →2 R e1 A (x, y) ∈ A f : A ⊆ R →2 R A (x , y ) ∈0 0 A f (x , y )0 0 (x , y )0 0 f : A ⊆ R →2 R e1 A x f x ′ f y ′ y f x ′ f y ′ ∀ (x, y) ∈ A ( (x, y)) = ∂x ∂f ∂x ∂f (x, y) = ∂ x2 ∂ f2 f (x, y)xx ′′ ( (x, y)) = ∂x ∂f ∂y ∂f (x, y) = ∂x∂y ∂ f2 f (x, y)yx ′′ ( (x, y)) = ∂y ∂f ∂x ∂f (x, y) = ∂y∂x ∂ f2 f (x, y)xy ′′ ( (x, y)) = ∂y ∂f ∂y ∂f (x, y) = ∂ y2 ∂ f2 f (x, y)yy ′′ (x, y) ∈ A f A f ∈ e (A)2 f(x, y) H (x, y)f H (x, y) =f [f (x, y)xx ′′ f (x, y)yx ′′ f (x, y)xy ′′ f (x, y)yy ′′ ] f ∈ e (A)2 (x, y) ∈ A (x, y) = ∂y∂x ∂ f2 f (x, y) =xy ′′ f (x, y) =yx ′′ (x, y) ∂x∂y ∂ f2 H (x, y)f ∀ (x, y) ∈ A f : A ⊆ R →2 R A f ∈ e (A)2 (x , y ) ∈0 0 A f ∇f(x , y ) =0 0 )0 (x , y )0 0 H (x , y )f 0 0 det (H (x , y )) >f 0 0 0 f (x , y ) <xx ′′ 0 0 0 H (x , y )f 0 0 det (H (x , y )) >f 0 0 0 f (x , y ) >xx ′′ 0 0 0 H (x , y )f 0 0 det (H (x , y )) <f 0 0 0 H (x , y )f 0 0 det (H (x , y )) =f 0 0 0 f (x , y ) <xx ′′ 0 0 0 det (H (x , y )) =f 0 0 0 f (x , y ) <yy ′′ 0 0 0 H (x , y )f 0 0 det (H (x , y )) =f 0 0 0 f (x , y ) >xx ′′ 0 0 0 det (H (x , y )) =f 0 0 0 f (x , y ) >yy ′′ 0 0 0 (x , y )0 0 (x , y ) ∈0 0 A f B (x , y )r 0 0 (x , y ) ∈1 1 A : f(x , y ) <1 1 f(x , y )0 0 (x , y ) :2 2 f(x , y ) >2 2 f(x , y )0 0 Financial Mathematics 1 Financial Mathematics Definition: Financial Contract A financial contract is an agreement between two or more parties for regulating the exchange of monetary amount changeable at different dates. Through the financial contract, the parties establish the trading rules, and, in particular, the due dates, the level of amounts or the interest rate. As a consequence, the financial contract generates a financial operation. Definition: Financial Operation A financial operation is an operation of exchange of monetary amounts, each characterized by its maturity (due date) and its currency. By the point of view of a party involved in the contract, we can formally define a financial operation as a couple of vectors ( ) of the same dimension, where is the vector of the monetary amounts, each of them is due on the corresponding date of the ordered vector , with , where is the contract date. A financial operation can be denoted by or . The monetary amounts of are if they are income, if they are outlays. Note that in the definition of financial operation, the first date involved, , is . In the case that , the contract date is not included in the definition of the operation, however it is possible to include it associating a null amount if in it is not planned the exchange of money. In particular, if we talk about spot operation; if we talk about forward operation. In a spot contract/operation you pay or receive something at contract date ; while in a forward contract/operation the contract terms are agreed at but the delivery and payment will occur at a future date. Given a financial operation we talk about Investment if the sign of the first monetary amount is negative and the subsequent are positive, i.e.: Financing if the first monetary amount is positive and the subsequent are negative: Remark: The exchange of monetary amounts of a financial operation occurs through few rules or schemes that go under the name of financial regimes. Then we present the ,x t =x (x , x , ..., x )1 2 n =t (t , t , ..., t )1 2 n t ≤0 t <1 t <2 ... < tn t0 /x t (x , ..., x )/(t , ..., t )1 n 1 n x > 0 < 0 t1 ≥ t0 t >1 t0 x =0  0 x =0 0 t0 t0 /x t =x (−,+,+, ..., +) =x (+,−,−, ..., −) Financial Mathematics 2 classical financial regimes of linear interest, the commercial discount and the compound interest. F (n) = (1 + i)n F (t) = C(1 + i) =t Cut C = F (t) =(1+i)t 1 F (t)vt I(t) I(t) = F (t) − C = C((1 + i) −t 1) = C(u −t 1) δ i δ = log(1 + i) δ F (t) = Ceδt δ C = F (t)e−δt 0 < i < 1 <1−dt 1 (1 + i) <t 1 + it 0 < t < 1 =1−dt 1 (1 + i) =t 1 + it t = 0 t = 1 >1−dt 1 (1 + i) >t 1 + it t > 1 t = 1 t = 1 i[2] i[3] i[4] k 1 + i = (1 + i )[k] k k i i[k] i =[k] (1 + i) − k 1 1 i = (1 + i ) −[k] k 1 i(m) m m m m m i(m) m i =(m) mi[m] m δ i(m) δ i(m) δ = i m→+∞ lim (m) δ δ s t 0 ≤ s ≤ t F (s,C; t) t C > 0 s F (s,C; t) = C + I(s,C; t) C λ C > 0 s t 0 ≤ s ≤ t F (s,λC; t) = λF (s,C; t) C F (s,C; t) s t t − s s t C s s 1 t 0 ≤ s ≤ s ≤1 t F (s ,F (s,C; s ); t) =1 1 F (s,C; t) F (s,C; t) < F (s,C; t ) ∀ C1 ∀ s t t 1 0 ≤ s ≤ t ≤ t 1 F (s,C; s) = C C s ≥ 0 δ F (s,C; t) = Ceδ(t−s) k + m h 0 ≤ k ≤ n − 1 1 ≤ h ≤ m n m m 1 m 1 nm m 1 nm m 1 a = in (m) = m 1 i[m] 1 − (1 + i )[m] −nm = m 1 i[m] 1 − vn = mi[m] i i 1 − vn a mi[m] i in i[m] (1 + i ) =[m] −m v a = in (m) a mi[m] i in lim i =m→+∞ (m) lim mi =m→+∞ [m] δ a = in (∞) a = m→+∞ lim in (m) a = m→+∞ lim mi[m] i in a δ i in a i∞ (∞) a = i∞ (∞) a = n→∞ lim i∞ (∞) a = n→∞ lim δ i in δ 1 m 1 p W (0) = a = in = i 1−(1+i)−n i 1−vn R W (0) = R i[m] 1 − (1 + i )[m] −p S t = 0 n i S = Ra in R n S n α = in a in 1 α in C n n C = Rs in R n C n σ = in s in 1 C = 1 t = n σ in α = in i + σ in 1 = ia + in vn C R h C h C I h D h h D =h C +h+1 C +h+2 ... + C =n C k=h+1 ∑ n k R =h C +h I           h =h 1, ...,n D =0 C +1 C +2 ... + C =n C = h=1 ∑ n h C t 0 ≤ t ≤ n t V (0) C = V (0) V (0) < C V (0) > C s = s(t) t s ≤ t ≤ s + 1 t V (t) V (t) = R (1 + h=1 ∑ n−s s+h i )1 −(s+h−t) U(t) P (t) V (t) = U(t) + P (t) U(t) = I (1 + h=1 ∑ n−s s+h i )1 −(s+h−t) P (t) = C (1 + h=1 ∑ n−s s+h i )1 −(s+h−t) h ∈ {1, ...,n} C =h n C h D =h = k=h+1 ∑ n n C C n n − h h(h = 1, ...,n) I =h iD =h−1 iC n n − h + 1 R R = = a in C Cα in 1 + i R = C (1 +n i) = C (1 +n−1 i) =2 ... = C (1 +1 i)n C =h R(1 + i) =−(n−h+1) Cα (1 + in i)−(n−h+1) I h I =h R − C =h Cα [1 − in (1 + i) ]−(n−h+1) U h h = 1, ...,n P h D h U =h [ C − i 1 i k=1 ∑ n−h h+k C (1 + k=1 ∑ n−h h+k i ) ] =1 −k [D − i 1 i h P ]h P h V h V =h P +h [D − i 1 i h P ]h