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MATHEMATICAL FINANCE CHEAT SHEET
Normal Random Variables
A random variable X is Normal N ( μ , σ^2 ) (aka. Gaussian) under a measure P if and only if EP
e θ^ X^
= e θ μ +^
1 2 θ^ (^2) σ 2 , for all real θ. A standard normal Z ∼ N (0, 1) under a measure P has density
φ (x ) =
p 2 π
e −x 2 / 2
. P [Z ≤ x ] = Φ (x ) :=
∫ (^) x
−∞
φ (z ) d z.
Let X = (X 1 , X 2 ,... , Xn )′^ with Xi ∼ N ( μ i , qi i ) and Cov [Xi , X (^) j ] = qi j for i , j = 1,... , n. We call μ := ( μ 1 ,... , μ n )′^ 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 ) =
p ( 2 π )n^ detQ
exp
Å
(x − μ )′Q −^1 (x − μ )
ã , x ∈ R n^.
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
Å
θ ′ μ +
θ ′ Q θ
ã , for all θ ∈ R n .
If Z ∼ N (0,Q ) and c ∈ R n^ then X = c ′Z ∼ N (0, c ′Q c ). If C ∈ R m×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) / 2 EP [h(Z + c )]. Let X ∼ N (0,Q ), h be a integrable function of x ∈ R n^ , and c ∈ R n^. 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 ( Wf (t ))t ≥ 0 be independent Brownian motions. Given a correla- tion coefficient ρ ∈ [−1, 1], define
W^ c (t ) := ρ W (t ) +
p 1 − ρ^2 Wf (t ),
then ( Wc (t ))t ≥ 0 is a Brownian motion and E [W (t ) Wc (t )] = ρ t.
Identifying Martingales
If Xt = X (t ) is a diffusion process satisfying d X (t ) = μ (t , Xt ) d t + σ (t , Xt )d W (t )
and EP [(
∫ T
0 σ (s^ ,^ Xs^ )
(^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 some F -previsible process ( σ (t ))t ≥ 0 , then we have the simpler condition
EP
ñ
exp
Ç
∫ T
0
σ (s )^2 d s
åô
< ∞ ⇒ X is a martingale.
Itô’s Formula
For Xt = 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 , Xt ), we have
d Y (t ) =
∂ g ∂ t
(t , Xt ) d t +
∂ g ∂ x
(t , Xt ) d Xt +
σ (t )^2
∂^2 g ∂ x 2
(t , Xt )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 ( Wf (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 Wf (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
ï d Q d P
XT
ò , for all claims XT knowable by time T ,
- EQ [Xt |Fs ] = ζ − 1 s EP^ [ ζ t^ Xt^ |Fs^ ], for^ s^ ≤^ t^ ≤^ T^ , where ζ t 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 an F -previsible process satis- fying the boundedness condition EP
î exp
Ä
1 2
∫ T
0 γ (t^ )
(^2) d t
äó < ∞, then there exists a measure Q such that:
d Q d P
= exp
Ç
∫ T
0
γ (t ) d W (t ) −
∫ T
0
γ (t ) 2 d t
å
,
∫ (^) 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 a F -previsible process ( γ (t ))t ≥ 0 such that
W^ f (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
Ç
∫ (^) t
0
γ (t ) d W (t ) −
∫ T
0
γ (t )^2 d t
å
.
Martingale Representation Theorem
Suppose (M (t ))t ≥ 0 is a Q -martingale process whose volatility
EQ [M (t )^2 ] = σ (t ) satisfies σ (t ) 6 = 0 for all t (with Q -probability one). Then if (N (t ))t ≥ 0 is any other Q - martingale, there exists an F -previsible process ( φ (t ))t ≥ 0 such that
∫ T
0 φ (t^ )
(^2) σ (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 := (X 1 , X 2 ,... , 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 ) ∈ R n×m^ and W is a m-dimensional Brownian motion. The quadration covariation of the components Xi and X (^) j is
〈Xi , X (^) j 〉(t ) =
∫ (^) t
0
Σ i (s ) ′ Σ j (s ) d s ,
or in differential form d 〈Xi , X (^) j 〉(t ) = Σ i (t )′ Σ j (t ) d t , where Σ i (t ) is the i th^ column of Σ (t ). The quadratic variation of Xi (t ) is 〈Xi 〉(t ) =
∫ (^) t 0 Σ i^ (s^ )
i (s^ )^ d s^. The multi-dimensional Itô formula for Y (t ) = f (t , X 1 (t ),... , Xn (t )) is
d Y (t ) =
∂ f ∂ t
(t , X 1 (t ),... , Xn (t ))d t +
∑^ n
i = 1
∂ f ∂ xi
(t , X 1 (t ),... , Xn (t ))d Xi (t )
∑^ n
i , j = 1
∂^2 f ∂ xi ∂ x (^) j
(t , X 1 (t ),... , Xn (t ))d 〈Xi , X (^) j 〉(t ).
The (vector-valued) multi-dimensional Itô formula for Y (t ) = f (t , X (t )) = (f 1 (t , X (t )),... , fn (t , X (t )))′ where fk (t , X ) = fk (t , X 1 ,... , Xn ) and Y (t ) = (Y 1 (t ), Y 2 (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 Xi (t )
∑^ n
i , j = 1
∂^2 fk (t , X (t )) ∂ xi ∂ x (^) j
d 〈Xi , X (^) j 〉(t ).
Stochastic Exponential
The stochastic exponential of X is Et (X ) = exp(X (t ) − 12 〈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 ) = ζε t +
∫ (^) t
a
ε t ε − u^1 m(u) d u, ε t := exp
∫ (^) t
a
μ (u) d u
= ζ exp
Ä
∫ (^) t
a
μ (u) d u
ä
∫ (^) t
a
m(u) exp
Ä
∫ (^) t
u
μ (r ) d r
ä 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 ) = ζ Et +
∫ (^) t
a
Et E (^) u− 1 [m(u) − q (u) σ (u)] d u +
∫ (^) t
a
Et E (^) u− 1 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) −
∫ (^) t
a
σ (u) 2 d u
Fundamental Theorem of Asset Pricing
Let X be some FT -measurable claim, payable at time T. The arbitrage-free price V of X at time t is
V (t ) = EQ
ñ
exp
Ä
∫ T
t
r (s ) d s
ä X Ft
ô
,
where Q is the risk-neutral measure.
Market Price Of Risk
Let Xt = X (t ) be the price of a non-tradable asset with dynamics d X (t ) = μ (t ) d t + σ (t )d W (t ) where ( σ (t ))t ≥ 0 and ( μ (t ))t ≥ 0 are previsible processes and (W (t ))t ≥ 0 is a P -Brownian motion. Let Y (t ) := f (Xt ) be the price of a tradable asset where f : R → R is a deterministic function. Then the market price of risk is
γ (t ) :=
μ t f ′(Xt ) + 12 σ^2 t f ′′(Xt ) − r f (Xt ) σ t f ′(Xt )
and the behaviour of Xt under the risk-neutral measure Q is given by
d X (t ) = σ (t )d Wf (t ) +
r f (Xt ) − 1 2 σ
2 t f^
′′(X
t ) f ′(Xt )
d t.
Black’s Model
Consider a European option with strike price K on a asset with value VT at ma- turity time T. Let FT be the forward price of VT , F 0 the current forward price. If log VT ∼ N (F 0 , σ^2 T ) then the Call and Put prices are given by
C = P (0, T )(F 0 Φ (d 1 ) − K φ (d 2 )), P = P (0, T )(K Φ (−d 2 ) − F 0 Φ (−d 1 )),
where d 1 =
log( EQ (VT ) / K ) + σ^2 T / 2 σ
p T
and d 2 = d 1 − σ
p T.
Forward Rates, Short Rates, Yields, and Bond Prices
The forward rate at time t that applies between times T and S is defined as
F (t , T,S ) =
S − T
log
P (t , T ) P (t ,S )
The instantaneous forward rate at time t is f (t , T ) = limS →T F (t , T,S ). The instan- taneous risk-free rate or short rate is r (t ) = limT →t f (t , T ). The cash account is given by
B (t ) = exp
∫ (^) t
0
r (s ) d s
and satisfies d B (t ) = r (t )B (t ) d t with B ( 0 ) = 1. The instantaneous forward rates and the yield can be written in terms of the bond prices as
f (t , T ) = −
∂ T
log P (t , T ), R (t , T ) = −
log P (t , T ) T − t
Conversely,
P (t , T ) = exp
Ç
∫ T
t
f (t , u) d u
å
and P (t , T ) = exp(−(T − t )R (t , T )).
Affine Jump Diffusion (AJD) Models
The state vector Xt follows a Markov process solving the SDE
d Xt = μ (Xt )d t + σ (Xt )d Wt + d Zt
where W is an adapted Brownian, and Z is a pure jump process with intensity λ. The moment generating function of the jump sizes is θ (c ) = EQ (exp(c J )). Impose
an affine structure on μ , σσ T^ , λ and the discount rate R , possibly time dependent:
μ (x ) = K 0 +K 1 x ( σ (x ) σ (x )T^ )i j = (H 0 )i j +(H 1 )i j x λ (x ) = L 0 +L 1 x R (x ) = R 0 +R 1 x
Given X 0 , the risk neutral coefficients (K , H , L , θ , R ) completely determine the dis- counted risk neutral distribution of X. Introduce the transform function
ψ (u, X 0 , T ) = EQ
ñ
exp
Ç
∫ T
0
R (Xs )d s
å
e u
T (^) XT F 0
ô
= e α (0,u)+ β^ (0,u)
T (^) x 0
where α and β solve the Ricatti ODEs subject to α (T, u) = 0, β (T, u) = u:
− β ˙ (t , u) =K T 1 β^ (t^ ,^ u) +^
1 2 β^ (t^ ,^ u)
T H 1 β (t , u) + L 1 ( θ ( β (t , u)) − 1 ) − R 1
− α ˙(t , u) =K T 0 β^ (t^ ,^ u) +^
1 2 β^ (t^ ,^ u)
T H 0 β (t , u) + L 0 ( θ ( β (t , u)) − 1 ) − R 0
AJD bond pricing
In ψ , set Li = R 0 = u = 0, R 1 = 1 to obtain the zero coupon bond with maturity T − t via the Ricatti ODEs:
Short rate model K 0 K 1 H 0 H 1 P?–MR? Merton μ σ^2 N–N Dothan μ σ^2 Y–N Vasicek αμ − α σ^2 N–Y CIR αμ − α σ^2 Y–Y Pearson-Sun αμ − α − σ^2 β σ^2 Y–Y Ho & Lee θ (t ) σ^2 N–N Hull & White αμ (t ) − α σ^2 N–Y Extended Vasicek α (t ) μ (t ) − α (t ) σ (t )^2 N–Y Black-Karasinski† α (t ) μ ¯(t ) − α (t ) σ (t )^2 Y–Y
P means the process stays positive, MR means rt is mean-reverting. Closed form so- lutions for bond prices and European options exist for all models except for †, which describes the evolution of d log(rt ) instead of d rt.
AJD option pricing
Define the Fourier transform inversion of the conditional expectation
G (a , b , y ) = EQ
ñ
exp
Ç
∫ T
0
R (Xs )d s
å
e a^
T (^) XT
1 b XT ≤y
ô
ψ (a , X 0 , T ) 2
π
0
ℑ( ψ (a + i v b , X 0 , T )e −i v y^ ) v
d v
The i th entry in X is the log asset price and k = l o g (K ), the log strike. d is a vector whose i th element is 1, else zero. The corresponding call option price is C = G (d , −d , −k ) − K G (0, −d , −k )
The Heath-Jarrow-Morton Framework
Given a initial forward curve T 7 → f (0, T ) then, for every maturity T and under the real-world probability measure P , the forward rate process t 7 → f (t , T ) follows
f (t , T ) = f (0, T ) +
∫ (^) t
0
α (s , T ) d s +
∫ (^) t
0
σ (s , T )′^ d W (s ), t ≤ T,
where α (t , T ) ∈ R and σ (t , T ) := ( σ 1 (t , T ),... , σ n (t , T )) satisfy the technical condi- tions: (1) α and σ are previsible and adapted to Ft ; (2)
∫ T
0
∫ T
0 | α (s^ ,^ t^ )|^ d s d t^ <^ ∞ for all T ; (3) sups ,t ≤T ‖ σ (s , t )‖ < ∞ for all T. The short-rate process is given by
r (t ) = f (t , t ) = f (0, t ) +
∫ (^) t
0
α (s , t ) d s +
∫ (^) t
0
σ (s , t ) d W (s ),
so the cash account and zero coupon T -bond prices are well-defined and obtained through
B (t ) = exp
∫ (^) t
0
r (s ) d s
, P (t , T ) = exp
Ç
∫ T
t
f (t , u) d u
å
.
The discounted asset price Z (t , T ) = P (t , T ) / B (t ) satisfies
d Z (t , T ) = Z (t , T )
îÄ (^1)
2
S
2 (t , T ) −
∫ T
t
α (t , u) d u ︸ ︷︷ ︸ b (t ,T )
ä d t + S (t , T ) ′ d W (t )
ó ,
where S (s , T ) := −
∫ T
s σ (s^ ,^ u)^ d u. The^ HJM drift condition^ states that Q is EMM (i.e., no arbitrage for bonds) ⇐⇒ b (t , T ) = −S (t , T ) γ (t )′,
where Wf (t ) := W (t )−
∫ (^) t 0 γ (s^ )^ d s^ is a^ Q -Brownian motion. If this holds, then under Q , the forward rate process follows
f (t , T ) = f (0, T ) +
∫ (^) t
0
Ä
σ (s , T )
∫ T
s
σ (s , u)′^ d u ︸ ︷︷ ︸ HJM drift
ä d s +
∫ (^) t
0
σ (s , T ) d Wf (s ),
and the discounted asset Z (t , T ) satisfies d Z (t , T ) = Z (0, T )Et (X ) with
X (t ) =
∫ (^) t
0
S (s , T ) ′ d Wf (s ).
The LIBOR Market Model
For a tenor δ > 0, the LIBOR rate L (T, T, T + δ ) is the rate such that an investment of 1 at time T will grow to 1+ δ L (T, T, T + δ ) at time T + δ. The forward LIBOR rate (i.e., a contract made at time t under which we pay 1 at time T and receive back 1 + δ L (t , T, T + δ ) at time T + δ ) is defined as
L (t , T ) := L (t , T, T + δ ) =
δ
Å
P (t , T ) P (t , T + δ )
ã ,
and satisfies L (T, T ) = L (T, T, T + δ ). Under the real-world probability measure P , The LMM assumes that each LIBOR process (L (t , Tm )) 0 ≤t ≤Tm satisfies
d L (t , Tm ) = L (t , Tm )
μ (t , L (t , Tm )) d t + λ m (t , L (t , Tm ))′d W (t )
where W = (W 1 ,... , W d^ ) is a d -dimensional Brownian motion with instantaneous correlations d 〈W i , W j 〉(t ) = ρ i , j (t ) d t , i , j = 1, 2,... , d. The function λ (t , L ) : [0, Tj ] × R → R N^ ×d^ is the volatility, and μ (t ) : [0, Tj ] → R is the drift. Let 0 ≤ m, n ≤ N − 1. Then the dynamics of L (t , Tm ) under the forward measure P Tn+ 1 is for m < n given by
d L (t , Tm ) = L (t , Tm )
ñ
− λ (t , Tm )
∑^ n
r =m+ 1
σ Tr ,Tr + 1 (t ) ′ d t + λ (t , Tm )d W m (t )
ô
For m = n, d L (t , Tm ) = L (t , Tm ) λ (t , Tm )d W (^) tm and for m > n we have
d L (t , Tm ) = L (t , Tm )
ñ
λ (t , Tm )
∑^ m
r =n+ 1
σ Tr ,Tr + 1 (t )′^ d t + λ (t , Tm )d Wt m
ô
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