Topics inFinancial Mathematics
IVAN G. AVRAMIDINew Mexico Tech
• Financial terminology, options
• Random Variables, stochastic processes, ran-dom walks, stochastic calculus, SDE
• Hedging, no arbitrage principle, Black-Scholesequation
• Stochastic volatility, jump diffusion,
• Heat kernel method
• Methods for calculation of the heat kernel
1
Financial Instruments
Real assets are necessary for producing goodsand services for the survival of society
Financial assets (also called securities) arepieces of paper that entitle its holder to a claimon a fraction of the real assets and to the in-come generated by these real assets
Financial instrument is a specific form of afinancial asset
• equities (stocks, shares) represent a sharein the ownership of a company
• fixed income securities (bonds) promise ei-ther a single fixed payment or a stream offixed payents
• derivative securities are financial asseststhat 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 tem-
porary 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 as-set (called the underlying asset) for an agreedamount (called the strike price E) at a speci-fied time T in the future (called the expirationdate)
The payoff function of the call option is itsvalue at expiry
(S − E)+ = max (S − E,0)
A put option is a contract that gives the rightto sell a particular asset for an agreed amountat a specified time in the future
The payoff function of the put option is
(E − S)+
Other option types: American, Bermudan, bi-nary, spreads, straddles, strangles butterfly spreads,condors, calendar spreads, LEAPS
4
Random Variables
Random variable X
Probability density function fX
Expectation value
E(X) =∫R
dx fX(x)x
Variance
Var(X) = E(X2)− [E(X)]2
Covariance of random variables
Cov(Xi, Xj) = E(XiXj)− E(Xi)E(Xj)
Correlation matrix
ρ(Xi, Xj) =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 pro-
cess 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(dX2) = dt .
For several Wiener processes one defines the
correlation matrix ρij
E(dXi) = 0 E(dXidXj) = ρijdt
7
Ito’s Lemma
Idea
dt ∼ dX2 ∼ ε
For a function F of a stochastic process Xt
dF (X) =dF
dXdX +
1
2
d2F
dX2dt
Rule: replace in Taylor series
dX2 7→ dt
Ito’s Lemma for a function F (t, Xi) of severalvariables
dF =∂F
∂tdt +
n∑i=1
∂F
∂XidXi +
1
2
n∑i,j=1
ρij∂2F
∂Xi∂Xjdt
Rule: replace in Taylor series
dXidXj 7→ ρijdt
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
S(t) = S(0) = exp{[
µ−1
2σ2]t + σ[X(t)−X(0)]
}
Mean-reverting random walk
dS = (ν − µS)dt + σdX
Solution
S(t) = ν + [S(0)− ν]e−µt
+σ
{X(t)− µ
∫ t
0ds 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
Π = V −∂V
∂SS
Portfolio change (Ito’s lemma)
dΠ =∂V
∂tdt+
∂V
∂SdS+
1
2
∂2V
∂S2dS2−
∂V
∂S(dS+DSdS)
=
{∂V
∂t+
1
2σ2S2∂2V
∂S2−DS
∂V
∂S
}dt
10
Black-Scholes Equation
No-arbitrage principle: A completely risk-freechange in the portfolio value must be the sameas the growth one would get if one puts theequivalent amount of cash in a risk-free interest-bearing account (bond)
dΠ = rΠdt
Black-Scholes equation (linear parabolic PDE)(∂
∂t+ L
)V = 0
where (elliptic PDO)
L =1
2σ2S2 ∂2
∂S2+ (r −D)S
∂
∂S− r
Remark: µ is eliminated!
Homogeneous in S. Can be solved by Mellintransform in S (or Fourier transform in x =logS)
11
Domain: [0, T ]× R+
Terminal condition
V (T, S) = f(S)
Boundary conditions
V (t,0) = 0 ,
limS→∞
[V (t, S)− S] = 0
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:
f(S) =1
E2 − E1
[(S − E1)+ − (S − E2)+
]
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
L =1
2
n∑i,j=1
CijSiSj∂2
∂Si∂Sj+
n∑i=1
(r −Di)Si∂
∂Si− r
where
Cij = σiσjρij
Homogeneous in Si. Can be solved by multi-
dimensional Fourier transform in xi = logSi.
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 con-
stants
• 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 afunction of n stochastic factors vi, i = 1,2, . . . , n.
dS = rS dt + σ(t, S, v)S dX
dvi = ai(t, v) dt +n∑
j=1
bij(t, v) dWj
where E(dXdWi) = ρ0i, E(dWidWj) = ρij
Then the valuation operator is
L =1
2σ2S2 ∂2
∂S2+
n∑i=1
AiσS∂2
∂S∂vi
+1
2
n∑i,j=1
Bij ∂2
∂vi∂vj+ rS
∂
∂S+
n∑i=1
ai ∂
∂vi− r
where
Ak(t, v) =n∑
j=1
bkj(t, v)ρj0
Bkl(t, v) =n∑
i,j=1
bki(t, v)ρijbjl(t, v)
16
Two-dimensional Models
SDE
dS = rS dt + σ(v, S)S dX ,
dv = a(v) dt + b(v) dW
Valuation operator
L =1
2σ2(S, v)S2∂2
S + ρb(v)σ(S, v)S∂S∂v
+1
2b2(v)∂2
v + rS∂S + a(v)∂v − r
Domain: [0, T ]× R+ × R+
Terminal condition at t = T and boundary con-
ditions at S, v → 0 and S, v →∞.
Non-constant coefficients. Cannot be solved
exactly. Can be studied by using the meth-
ods of asymptotic analysis (theory of singular
perturbations)
17
Jump Process
Poisson process with intensity λ is a stochas-
tic 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
dN = −λm dt +(eJ − 1
)dQ
where J is a random variable with the proba-
bility density function ω(J) and
m =
∞∫−∞
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(dWdQ) = 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 isan integral operator defined by
(LJV )(t;S, v) = E[V (t; eJS, v)− V (t, S, v)
]−E(eJ − 1)S∂SV (t, S, v)
=
∞∫−∞
dJ ω(J)[V (t; eJS, v)− V (t;S, v)
]
−mS∂SV (t;S, v)
Notice that
LJ = ω̂ (−iS∂S)−mS∂S − 1
where ω̂ is the characteristic function
ω̂(z) =∫ ∞−∞
dJ ω(J)eizJ
20
Double-exponential Distribution
Double-exponential distribution
ω(J) = θ(J)p+
δ+exp
(−
J
δ+
)+θ(−J)
p−δ−
exp
(J
δ−
)where p± ≥ 0 are the probabilities of positiveand negative jumps, and δ± > 0 are the meansof positive and negative jumps
Characteristic function
ω̂(z) =p+
1− izδ++
p−1− izδ−
Average jump amplitude
m =p+
1− δ++
p−1 + δ−
− 1 .
As δ+, δ− → 0 the probability density degener-ates 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 =1
2v
[S2∂2
S + 2ρηS∂S∂v +η2
2∂2
v
]
+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 = logS) and Laplace
transform in v
22
SABR Model
SDE
dS = vS1−α dX ,
dv = ηv dW
where 0 ≤ α ≤ 1
Valuation operator
L =1
2v2[S2−2α∂2
S + 2ρηS1−α∂S∂v + η2∂2v
]
Defines a Riemannian metric on the hyper-
bolic plane H2 with constant negative curva-
ture −η2/2.
Can be solved by the tools of geometric anal-
ysis
23
SABR Model withMean-Reverting Volatility
SDE
dS = vS1−α dX ,
dv = κ(θ − v)dt + ηv dW
Valuation operator
L =1
2v2[S2−2α∂2
S + 2ρηS1−α∂S∂v + η2∂2v
]+κ(θ − v)∂v
Can be solved in perturbation theory in param-
eter κ
24
Change of Variables
Change of variables
τ = T − t, x = logS,
Valuation equation
(∂τ − L)V = 0
where
L = L̄ + λLJ ,
L̄ =1
2
[σ2(x, v)∂2
x + 2ρb(v)σ(x, v)∂x∂v + b2(v)∂2v
]+[r −
1
2σ2(x, v)
]∂x + a(v)∂v − r
LJ = ω̂ (−i∂x)−m∂x − 1
The operator LJ acts as follows
(LJV )(τ ;x, v) =
∞∫−∞
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, x′, v′) = 0
Initial condition (and boundary conditions atv = 0 and at infinity)
U(0;x, v, x′, v′) = δ(x− x′)δ(v − v′)
Heat semigroup representation
U(τ ;x, v, x′, v′) = exp(τL)δ(x− x′)δ(v − v′)
Option price
V (τ ;x, v) =∫ ∞−∞
dx′∫ ∞0
dv′U(τ ;x, v, x′, v′)f(x′)
where f(x) is the payoff function; for call op-tion
f(x) = (ex − E)+
Thus, the knowledge of the heat kernel givesthe value of all options with any payoff func-tion.
26
Solution Methods
• Analytic: integral transforms (Fourier, Laplace,Mellin)
• Geometric: Riemannian geometry, nega-tive curvature, diffusion on hyperbolic plane
• Singularly perturbed pde: asymptotic ex-pansion 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-Kacformula
27
Example: Black-Scholes Heat Kernel
Heat Kernel
U(τ ;x, x′) = exp(τL)δ(x− x′)
Valuation operator
L = a∂2x + b∂x − r
where a = σ2
2 , b = r −D − σ2
2
We have
exp(τL) = e−rτ exp (b∂x) exp(a∂2
x
)By using
exp(a∂2
x
)δ(x− x′) = (4πa)−1/2 exp
−(x− x′
)24a
exp (b∂x) f(x) = f(x + b)
we get
U(τ ;x, x′) = (4πa)−1/2 exp
−rτ −(x− x′ + b
)24a
28
Elliptic Operators
Second-order PDO
L =n∑
i,j=1
gij(x)∂2
∂xi∂xj+
n∑i=1
Ai(x)∂
∂xi+ P (x)
Symbol
σ(x, p) =n∑
j,k=1
gjk(x)pjpk−in∑
j=1
Aj(x)pj−P (x) .
Leading symbol
σL(x, p) =n∑
j,k=1
gjk(x)pjpk .
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) = (gij)−1 is positive definite
and plays the role of a Riemannian metric
29
Heat Kernel of Operatorswith Constant Coefficients
Fourier transform
U(τ ;x, x′) =∫
Rn
dp
(2π)nexp
−τσ(p) + i∑j
pj(x− x′)j
For a differential operator (Gaussian integral)
U(τ ;x, x′) = (4πτ)−n/2g1/2
× exp
τ
P −1
4
∑i,j
gijAiAj
× exp
−1
2
∑i,j
gij(x− x′)iAj
× exp
− 1
4τ
∑i,j
gij(x− x′)i(x− x′)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, x′) =∫
Rn
dp
(2π)nexp {−τ [σ(p) + Σ(p)]}
× exp
i∑j
pj(x− x′)j
31
Heat Kernel Asymptotic Expansion
Asymptotic ansatz as τ → 0
U(τ ;x, x′) = (4πτ)−n/2 exp
(−
d2(x, x′)
4τ
)
×∞∑
k=0
τkak(x, x′)
where d(x, x′) is the geodesic distance
Recursion differential equations (transport equa-
tions along geodesics) for coefficients ak
Can be solved in form of a covariant Taylor
series in x close to x′
32
Perturbation Theoryfor Heat Semigroup
Let L = L0 + εL1 be a negative operator
The heat semigroup is a one-parameter fam-ily of operators (for τ ≥ 0)
U(τ) = exp(τL)
Volterra series
U(τ) = U0(τ) +∞∑
k=1
εkτ∫
0
dτk
τk∫0
dτk−1 · · ·τ2∫0
dτ1
×U0(τ − τk)L1U0(τk − τk−1) · · ·
· · ·U0(τ2 − τ1)L1U0(τ1)
Thus the heat kernel
U(τ, x, x′) ={1 + ετL1 +
τ2
2
(ε2L2
1 + ε[L0, L1])
+O(τ3)}U0(τ ;x, x′)
33
Discretization
Let τk = kτ/N , k = 0,1, . . . , N . Then
U(τ) = limN→∞
U(τN − τN−1)U(τN−1 − τN−2)
· · ·U(τ2 − τ1)U(τ1)
Then the heat kernel is
U(τ ;x, x′) = limN→∞
∫RNn
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, x′)
34
Path Integrals
As τ → 0
U(τ ;x, x′) = (4πτ)−n/2g1/2 exp{−S(τ, x, x′)
}where
S(τ, x, x′) =
τ∫0
ds{14
∑i,j
gijdxi
ds
dxj
ds
+1
2
∑i,j
gijdxi
dsAj +
1
4
∑i,j
gijAiAj − P
}is the action functional
Let M be the space of all all continuous pathsx(s) starting at x′ at τ = 0 and ending at x ats = τ , that is,
x(0) = x′ , x(τ) = x ,
Then the heat kernel is represented as a pathintegral
U(τ, x, x′) =∫M
Dx(s) exp[−S(τ, x, x′)]
35
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