Download Topics in Financial Mathematics

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