Stochastic Volatility Model and Option Pricing Methodology

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Contents
Question 1...............................................................................................................................................1
Question 2...............................................................................................................................................5
Question 3...............................................................................................................................................7
Question 4...............................................................................................................................................8
Question 5...............................................................................................................................................9
Question 6.............................................................................................................................................11

Question 1
Let St
(0):=1 denote the underlying asset process and Stein model of the variance process. The
general form of stochastic volatility model is characterized by σ t is being the expected rate of
return process, where Xt is denoting the dividend yield βSS being the volatility of variance.
KSS is the speed of mean reversion BQ and W Qare two independent Q-Brownian motions
generating the filtration Ft, dθSS, which is the mean reversion level. Ft , t∈[0,T] with T>0
possibly has correlated with EQ[(ST −k )+¿∨Ft ¿]. The general form of a stochastic volatility
model is,
dSt:=σ t St dBt
Q, S0>0,
dσ t:= KSS(θSS−σ t)dt+ βSS( ρSS dBt
Q+√ 1−ρSS
2 dW t
Q), σ 0>0,
The equation is obtained by forming a risk-free portfolio, where this time it involves a
position in an extra derivative as compared to the constant volatility case, due to the second
source of randomness, or namely the variance stochastic dynamics. The price of a derivative
contract V should be satisfied.
dσ t:= KSS (θSS−σ t)dt+βSS(ρSS dBt
Q+√1−ρSS
2 dW t
Q), σ 0>0
Let us apply the values of numerical parameters as follows,
S0:= 100, σ 0:=θSS ≔0.1, KSS ≔2 , ρSS:=0.5 βSS:=0.05, T:=1 ρS:=0.5
dσ t: = 2 (0.1−σt ) dt+0.05 (0.5 dBt
Q+√1−(0.5)2 dW t
Q), σ 0>0
∂ σ
∂ t =2 ( 0.1−σt ) ∂ σ
∂ t +0.05 (0.5 ∂ B
∂ t +
√ 1−(0.5)2 ∂W
∂t ), σ 0>0
∂ σ
∂ t =2 ( 0.1−σt ) ∂ σ
∂ t +0.05 (0.5 ∂ B
∂ t + 1
2(1-(0.5)2) ∂W
∂t
Where, βSS( ρSS dBt
Q+√ 1−ρSS
2 dW t
Q), is the market price of volatility risk and r is the risk free
rate. For bearing the additional volatility risk, investors require extra return of amount ψSS(u;t,
σ t , xt )
The general definition will be presented. The option pricing methodology, which will be
discussed in the Fourier transform of a real function is ψSS(u;t,σ , x,x.
ψSS(u;t,σ t , xt ): = EQ[eiu X T
∨Ft]

With i being the imaginary unit, and u being a real number (u ∈ R). In case F( xt)=f XT ( x ) is a
probability density function of a random variable X, the transform above is the characteristic
function of X to be denoted by φX(u).
Random variables are fully described by their characteristic functions, that is, if φX(u) is
known then the distribution of X is completely defined. Also, by knowing the characteristic
function,
CSS(t, S, σ):= EQ[(ST −k )+¿∨Ft ¿], t∈[0,T]
C= EQ[eiu X T
∨Ft]
f XT ( x )=F−1[φX(u)]du= 1
√ 2 π ∫
0
T
eiu X T
σ X(u)du
CT(k)= EQ[eiu X T
∨Ft]
Thus, we can consider the call Price's model of the ODE for C, D and E such as for all
u∈R and all t∈ [0, T] the equation is,
ψSS(u;t,σ t , xt )= exp(c(u, T-t)+ D(u, T-t) σ + 1
2 E ( u ,T −t ) σ2 +iux)
Let us consider the value of C (u,0) where u∈R
C (u,T-t)=EQ[(ST −k )+¿¿| 1
2 σ ∫
0
T
eiu X T
σ X(u)du
X=log ST k=log k σ (X)are the risk natural values of the call price,
C (u,T-t)=EQ[(ST −k )+¿¿| 1
2 σ ∫
0
T
ex−ek q(x)dx
lim
k → ∞
CT (k )=s0
Let us consider the limit values as, T>0
ψSS = σt (S−(T +1) xt)
T 2+T −s2+ i(2T +1) s