University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Chapter 3: Black-Scholes Equation and Its Numerical Evaluation
3.1 Ito Integral3.1.1 Convergence in the Mean and Stieltjes Integral
Definition 3.1 (Convergence in the Mean)
A sequence Xnn∈lN of random variables is said to converge in the mean to a random variable X,
if E(X2n) < ∞ , E(X2) < ∞ and if lim
n→∞E((X − Xn)2) = 0. We use the notation l.i.m.n→∞ Xn = X.
Definition 3.2 (Stieltjes Integral)
Let b ∈ C([0,T]) and assume that µ ∈ BV([0,T]) is a function of bounded variation, i.e., for a se-
quence 0 = t(N)0 < t
(N)1 < ... < t
(N)N = T of partitions of [0,T] with max
1≤j≤N|t
(N)j − t
(N)j−1| → 0 as N → ∞
limN→∞
N∑j=1
|µ(tj) − µ(tj−1)| < ∞ .
The Stieltjes integral of b with respect to µ is defined according to
T∫
0
b(t) dµ(t) = limN→∞
N∑j=1
b(tj−1) (µ(tj) − µ(tj−1)) .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.1.2 First and Second Variation of a Wiener Process
Adopting the notation from the previous subsection, for a Wiener process Wt the mapping
t 7−→ Wt is not of bounded variation, i.e., for the first variation of Wt we have
limN→∞
N∑j=1
|Wt(N)j
− Wt(N)j−1
| = ∞ .
However, using E((Wt − Ws)2) = t − s, for the second variation of Wt we find
N∑j=1
E((Wt(N)j
− Wt(N)j−1
)2) =N∑
j=1(t
(N)j − t
(N)j−1) = t
(N)N − t
(N)0 = T .
whence
l.i.m.N→∞
N∑j=1
(Wt(N)j
− Wt(N)j−1
)2 = T .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.1.3 Motivation of the Ito Integral
Assume that the temporal evolution of an asset occurs according to a Wiener process Wt and
denote by the function b = b(t) the number of units of the asset in a portfolio at time t.
(i) Discrete time trading
If trading is only allowed at discrete times tj, given by a partition 0 = t0 < t1 < ... < tN = T of
the time interval [0,T], the trading gain turns out to beN∑
j=1b(tj−1) (Wtj − Wtj−1
) .
Definition 3.3 (Ito Integral over a Step Function)
Assume that b is a step function b(t) = b(tj−1) , tj−1 ≤ t < tj , 1 ≤ j ≤ N. The Ito integral of b is
defined by means ofT∫
0
b(t) dWt =N∑
j=1b(tj−1) (Wtj − Wtj−1
) .
In case the values b(tj−1) are random variables, b is called a simple process.
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
(ii) Continuous time trading
If trading is allowed at all times t ∈ [0,T] and b ∈ C([0,T]), one might be tempted to define the
continuous time trading gain as the limit N → ∞ of its discrete counterpart. However, this limit
does not exist, since the Wiener process Wt has an unbounded first variation.
A remedy is to approximate b by step functions bn,n ∈ lN, in the sense that
limn→∞
E(T∫
0
(b(t) − bn(t))2 dt) = 0 .
In view of the isometry
E(T∫
0
(bn(t) − bm(t)) dWt)2) = E(
T∫
0
(bn(t) − bm(t))2 dt) ,
and the Cauchy convergenceE(
T∫
0
(bn(t) − bm(t))2 dt) → 0 ,
we deduce that the Ito integralsT∫0bn(t) dWt represent a Cauchy sequence with respect to
the convergence in the mean.
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.1.4 Ito Integral
Definition 3.4 (Ito Integral)
Assume that b = b(t) is a stochastically integrable function in the sense that there exists a se-
quence bnn∈lN of simple processes such that
limn→∞
E(T∫
0
(b(t) − bn(t))2 dt) = 0 .
Then, the Ito integral of b is defined according to
T∫
0
b(t) dWt := l.i.m.n→∞
T∫
obn(t) dWt .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.2 Stochastic Differential Equations and Ito Processes
Definition 3.5 (Stochastic Differential Equation)
Let (Ω,F ,P) be a probability space and let Xt, t ∈ lR+ be a stochastic process X : Ω × lRt → lR.
Moreover, assume that a(·, ·) : Ω × lR × lR+ → lR and b(·, ·) : Ω × lR × lR+ → lR are stochastically
integrable functions of t ∈ lR+. Then, the equation
(⋆) dXt = a(Xt, t) dt + b(Xt, t) dWt
is called a stochastic differential equation. Note that (⋆) has to be understood as a symbolic no-
tation of the stochastic integral equation
(⋆⋆) Xt = X0 +t∫
0
a(Xs, s) ds +t∫
0
b(Xs, s) dWs .
The functions a and b are referred to as the drift term and the diffusion term, respectively.
Definition 3.6 (Ito Process)
A stochastic process Xt satisfying (⋆) is said to be an Ito process.
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Examples: (i) Wiener process as a special Ito process
Setting a ≡ 0 and b ≡ 1 in (⋆⋆), we obtain
Xt = X0 +t∫
0
dWs = X0 + Wt − W0 = X0 + Wt .
(ii) Stochastic differential equation for a bond with risk-free interest rate r ∈ lR+
In case a(Xt, t) = rXt, r ∈ lR+ and b ≡ 0, we get
Xt = X0 + rt∫
0
Xs ds ⇐⇒ dXt = rXt dt .
This is a stochastic differential equation for a bond Xt with risk-free interest rate r ∈ lR+.
(iii) Ito integral of a Wiener process
Setting Xt = Wt , Wt := Wtj−1, tj−1 ≤ t < tj, and observing l.i.m.N→∞
∑Nj=1(Wtj − Wtj−1
)2 = T, we get
T∫
0
Wt dWt = l.i.m.N→∞
T∫
0
Wt dWt = l.i.m.N→∞
N∑j=1
Wtj−1(Wtj − Wtj−1
) =
=1
2W2
T −1
2l.i.m.N→∞
N∑j=1
(Wtj − Wtj−1)2 =
1
2W2
T −T
2.
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.2 Ito’s LemmaLemma 3.1 (Ito’s Lemma)
Let Xt, t ∈ lR+, be an Ito process X : Ω × lR+ → lR and f : C2(lR × lR+ × lR+). Then, the stochas-
tic process ft := f(Xt, t) is also an Ito process which satisfies
() dft = (∂f
∂t+ a
∂f
∂x+
1
2b2 ∂
2f
∂x2) dt + b
∂f
∂xdWt .
Proof. Taylor expansion of f(Xt+∆t, t + ∆t) around (Xt, t) results in
f(Xt+∆t, t + ∆t) = f(Xt, t) +∂f
∂t(Xt, t) ∆t +
∂f
∂x(Xt, t) (Xt+∆t − Xt) +
+1
2
∂2f
∂t2(Xt, t) (∆t)2 +
1
2
∂2f
∂x2(Xt, t) (Xt+∆t − Xt)
2 +∂
2f
∂x∂t(Xt, t) ∆t (Xt+∆t − Xt) +
+ O((∆t)2) + O((∆t)(Xt+∆t − Xt)2 + O((Xt+∆t − Xt)
3) .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Passing to the limit ∆t → 0 yields
(+) dft =∂f
∂tdt +
∂f
∂xdXt +
1
2
∂2f
∂x2dX2
t + O((dt)2) + O(dt (dXt)2) + O((dXt)
3) .
Taking into account that Xt is an Ito process and dW2t = dt, it follows that
(++) dX2t = (a dt + b dWt)
2 = a2 (dt)2 + 2 a b dt dWt + b2 dW2t = b2 dt + O((dt)3/2) .
Using (++) in (+), we finally obtain
dft =∂f
∂tdt +
∂f
∂x(a dt + b dWt) +
1
2b2 ∂
2f
∂x2dt =
= (∂f
∂t+ a
∂f
∂x+
1
2
∂2f
∂x2) dt + b
∂f
∂xdWt .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Example: Explicit Solution of a Stochastic Differential Equation
The solution of the stochastic differential equation
dXt = µ Xt dt + σ Xt dWt
is given byXt = X0 exp((µ −
1
2σ
2)t + σ Wt) .
Proof. We apply Ito’s Lemma to
Xt = f(Yt, t) := X0 exp((µ −1
2σ
2)t + σ Yt)
with Yt = Wt and a ≡ 0 , b ≡ 1.
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Example: Stochastic Differential Equation for the Value of an Asset
We assume that the value of an asset is described by the random variable St, t ∈ lR, satisfying
the geometric Brownian motion
ln(St) = ln(S0) + (µ −1
2σ
2) t + σ Wt .
with given drift µ and given volatility σ ∈ lR+.
We note that ln(St) is an Ito process, since we may write
d(ln(St)) = (µ −1
2σ
2) dt + σ dWt .
We apply Ito’s Lemma with f(x) = exp(x) and a = µ − 12σ
2, b = σ. Observing ∂f
∂x(ln(St)) = St and
∂2f
∂x2(ln(St)) = St yields
dSt = d(exp(ln(St)) = (µ −1
2σ
2) St dt +1
2σ
2 St dt + σ St dWt = µ St dt + σ St dWt .
Interpretation: The relative change dSt/St of the value consists of a deterministic part µ dt and
a stochastic part σ dWt which represents the volatility.
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.3 Derivation of the Black-Scholes Equation for European Options
3.3.1 Basic Assumptions
We consider a financial market under the following assumptions
• The value of the asset St, t ∈ lR+ satisfies the stochastic differential equation
dSt = µ St dt + σ St dWt , µ ∈ lR , σ ∈ lR+.
• Bonds Bt, t ∈ lR+ are subject to the risk-free interest rate r ∈ lR+, i.e., dBt = r Bt dt.
• There are no dividends, the market is arbitrage-free, liquid and frictionless (i.e., no
transaction costs, no taxes etc.).
• The asset or parts of it an be traded continuously and short-sellings are allowed.
• All stochastic processes are continuous (i.e., a crash of the stock market can not be
modeled).
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.3.2 Self-Financing Portfolio
We consider a portfolio consisting of c1(t) bonds Bt and c2(t) assets St as well as a European
option with value V(St, t). Assuming that the option has been sold at time t ∈ R+, the portfolio
has the valueYt = c1(t) Bt + c2(t) St − V(St, t) .
Definition 3.7 (Self-Financing Portfolio)
A portfolioYt = c(t) · St :=
n∑i=1
ci(t) Si(t) .
consisting of ci(t) investments Si(t),1 ≤ i ≤ n (bonds, stocks, options), is said to be self-financing
if changes in the portfolio are only financed by either buying or selling parts of the portfolio,
i.e., if for sufficiently small ∆t > 0 there holds
Yt = c(t) · St = c(t − ∆t) · St =⇒ dYt = c(t) · dSt .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.3.3 Black-Scholes Equation
Theorem 3.1 (Black-Scholes Equation)
Under the previous assumptions on the financial market, let Y = Yt be a risk-free, self-financing
portfolio consisting of a bond B = Bt, a stock S = St, and a European option with value V = Vt.
Then, the value V of the option satisfies the parabolic partial differential equation
∂V
∂t+
1
2σ
2 S2 ∂2V
∂S2+ r S
∂V
∂S− r V = 0 ,
which is known as the Black-Scholes equation.
The final condition at time t = T (maturity date) is
V(S,T) =
(S − K)+ for a European call
(K − S)+ for a European put.
The boundary conditions at S = 0 and for S → ∞ are given by
V(0, t) =
0 for a European call
K exp(−r(T − t)) for a European put,
V(S, t) = O(S) for a European call
limS→∞
V(S, t) = 0 for a European put.
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Proof. According to Ito’s Lemma, V satisfies the stochastic differential equation
(⋆) dV = (∂V
∂t+ µ S
∂V
∂S+
1
2σ
2 S2 ∂2V
∂S2) dt + σ S
∂V
∂SdW .
Inserting the stochastic differential equations for V,B and S
dV = c1 dB + c2 dS − dY , dB = r B dt , dS = µ S dt + σ S dW
into (⋆), we obtain
() dY = [c1 r B + c2 µ S − (∂V
∂t+ µ S
∂V
∂S+
1
2σ
2 S2 ∂2V
∂S2)] dt + (c2 σ S − σ S
∂V
∂S) dW .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
The assumption of a risk-free portfolio implies the non-existence of stochastic fluctuations, i.e.,
the coefficient in front of dW has to be set to zero: c2 = ∂V/∂S.
On the other hand, the assumption of a risk-free portfolio implies
() dY = r Y dt = r (c1 B + c2 S − V) dt .
Now, substituting () into () yields
r (c1 B +∂V
∂SS − V) dt = (c1 r B + c2 µ S −
∂V
∂t− µ S
∂V
∂S−
1
2σ
2 S2 ∂2V
∂S2) dt =
= (c1 r B −∂V
∂t−
1
2σ
2 S2 ∂2V
∂S2) dt .
The coefficients in front of dt must be equal, and hence, we obtain the Black-Scholes equation
∂V
∂t+
1
2σ
2 S2 ∂2V
∂S2+ r S
∂V
∂S− r V = 0 .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
We note that the final condition at t = T follows from Chapter 1.
In order to derive the boundary conditions at S = 0 and for S → ∞, we also have to distin-
guish between a call and a put:
Case 1: European call
For S = 0, i.e., a stock with zero value, the option to buy such a stock is of value zero as well.
On the other hand, for S ≫ 1, i.e., for a high value of the stock, it is almost sure that the op-
tion will be exercised, and we have V ≈ S − K exp(−r(T − t)). Since the strike K can be neglec-
ted for large S, we thus obtain V = O(S) (S → ∞).
Case 2: European put
For S ≫ 1 it is unlikely that the put V = P will be exercised, i.e., P(S, t) → 0 as S → ∞.
On the other hand, at S = 0, the put-call parity (Theorem 1.1) implies
P(0, t) = (C(S, t) + K exp(−r(T − t)) − S)|S=0 = K exp(−r(T − t)) .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Theorem 3.2 (Black-Scholes Formula for European Calls)
For a European call, the Black-Scholes equation with boundary data and final condition as in
Theorem 3.1 has the explicit solution
(∗) V(S, t) = S Φ(d1) − K exp(−r(T − t)) Φ(d2) ,
where Φ denotes the distribution function of the standard normal distribution
Φ(x) =1
√2π
x∫
−∞
exp(−s2/2) ds , x ∈ lR
and dν ,1 ≤ ν ≤ 2, are given by
d1/2 =ln(S/K) + (r ± σ
2/2)(T − t)
σ√
T − t.
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Proof. The idea of proof is to transform the Black-Scholes equation to ∂u/∂τ = ∂2u/∂x2 for
which an analytical solution is available and then to transform back to the original variables.
Step 1: Transformation
We perform the transformation of variables
x = ln(S/K) , τ =1
2σ
2 (T − t) , v(x, τ ) = V(S, t)/K .
Obviously, x ∈ lR (since S > 0), 0 ≤ τ ≤ T0 := σ2T/2 (since 0 ≤ t ≤ T), and v(x, τ ) ≥ 0
(since V(S, t) ≥ 0). Moreover, the chain rule implies
∂V
∂t= K
∂v
∂t= K
∂v
∂τ
dτ
dt= −
1
2σ
2 K∂v
∂τ,
∂V
∂S= K
∂v
∂x
dx
dS=
K
S
∂v
∂x,
∂2V
∂S2=
∂
∂S(K
S
∂v
∂x) = −
K
S2
∂v
∂x+
K
S
∂2v
∂x2
1
S=
K
S2(−
∂v
∂x+
∂2v
∂x2) .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Hence, the transformed Black-Scholes equation takes the form
−σ
2
2K
∂v
∂τ+
σ2
2S2 K
S2(−
∂v
∂x+
∂2v
∂x2) + r S
K
S
∂v
∂x− r K v = 0 .
Setting κ := 2r/σ2 and T0 := σ2T/2, it follows that
(+)∂v
∂τ−
∂2v
∂x2+ (1 − κ)
∂v
∂x+ κ v = 0 , x ∈ lR , τ ∈ (0,T0]
with initial condition (observe (S − K)+ = K(exp(x) − 1)+)
v(x,0) = (exp(x) − 1)+ , x ∈ lR .
For the elimination of ∂v/∂x and v, we use the ansatz
v(x, τ ) = exp(αx + βτ ) u(x, τ ) , α, β ∈ lR .
Inserting this ansatz into (+) and dividing by exp(αx + βτ ) yields
β u +∂u
∂τ− α
2 u − 2 α∂u
∂x−
∂2u
∂x2+ (1 − κ)(α u +
∂u
∂x) + κ u = 0 .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Choosing α, β ∈ lR such that
β − α2 + (1 − κ) α + κ = 0
− 2 α + (1 − κ) = 0=⇒
α = − 12 (κ − 1)
β = − 14 (κ + 1)2
,
we find that the function
u(x, τ ) = exp(1
2(κ − 1) x +
1
4(κ + 1)2 τ ) v(x, τ )
satisfies the linear diffusion equation
∂u
∂τ−
∂2u
∂x2= 0 , x ∈ lR , τ ∈ (0,T0] ,
with the initial condition
u(x,0) = u0(x) := exp((κ − 1) x/2) (exp(x) − 1)+ = (exp((κ + 1)x/2) − exp((κ − 1)x/2))+ , x ∈ lR .
Step 2: Analytical solution of the linear diffusion equation
The analytical solution of the linear diffusion equation is given by
u(x, τ ) =1
√4πτ
+∞∫
−∞
u0(s) exp(−(x − s)2/4τ ) ds .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
For the evaluation of the integral we use the transformation of variables y = (s − x)/√
2τ .
Together with the expression for the initial condition u0 = u(x,0) we obtain
u(x, τ ) =1
√2π
+∞∫
−∞
u0 (√
2τ y + x) exp(−y2/2) dy =
=1
√2π
[+∞∫
−x/√
2τ
exp(1
2(κ + 1)(x + y
√2τ ))exp(−y2
/2)dy −+∞∫
−x/√
2τ
exp(1
2(κ − 1)(x + y
√2τ ))exp(−y2
/2)dy] .
Using the representation
1√
2π
+∞∫
−x/√
2τ
exp(1
2(κ ± 1)(x + y
√2τ ))exp(−y2
/2)dy = exp(1
2(κ ± 1)x +
1
4(κ ± 1)2τ ) Φ(d1/2) ,
it follows that
u(x, τ ) = exp(1
2(κ + 1)x +
1
4(κ + 1)2τ ) Φ(d1) − exp(
1
2(κ − 1)x +
1
4(κ − 1)2τ ) Φ(d2) .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Step 3: Back-Transformation
In view of v(x, τ ) = V(S, t)/K and v(x, τ ) = exp(−12(κ − 1)x − 1
4(κ + 1)2τ )u(x, τ ) we obtain
V(S, t) = K v(x, τ ) = K exp(−1
2(κ − 1)x −
1
4(κ + 1)2τ ) u(x, τ ) .
Now, inserting
u(x, τ ) = exp(1
2(κ + 1)x +
1
4(κ + 1)2τ ) Φ(d1) − exp(
1
2(κ − 1)x +
1
4(κ − 1)2τ ) Φ(d2) ,
it follows that
V(S, t) = K exp(x) Φ(d1) − K exp(−1
4(κ + 1)2τ +
1
4(κ − 1)2τ ) Φ(d2) .
Finally, observing x = ln(S/K) and τ = σ2(T − t)/2, we arrive at
V(S, t) = S Φ(d1) − K exp(−r(T − t)) Φ(d2) .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Step 4: Checking the Boundary Conditions and the Final Condition
(i) Boundary condition at S = 0
In view of d1/2 → −∞ for S → 0, we have Φ(d1/2) → 0 for S → 0 whence
V(S, t) = S Φ(d1) − K exp(−r(T − t)) Φ(d2) → 0 as S → 0 .
(ii) Boundary condition for S → +∞
Observing Φ(d1) → 1 and Φ(d2)/S → 0 for S → +∞, we get
V(S, t)
S= Φ(d1) − K exp(−r(T − t))
Φ(d2)
S→ 1 as S → +∞ .
(iii) Final condition at t = T
For t → T we get
ln(S/K)
σ√
T − t→
+∞ , S > K
0 , S = K
−∞ , S < K
=⇒ Φ(d1/2) →
1 , S > K
1/2 , S = K
0 , S < K
=⇒
V(S,T) →
S − K , S > K
0 , S ≤ K
= (S − K)+ .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Theorem 3.3 (Black-Scholes Formula for European Puts)
For a European put, the Black-Scholes equation with boundary data and final condition as in
Theorem 3.1 has the explicit solution
(∗∗) V(S, t) = K exp(−r(T − t)) Φ(−d2) − S Φ(−d1) ,
where Φ and d1/2 are given as in Theorem 3.2.
Proof. The proof follows readily from Theorem 3.2 and the put-call parity (cf. Theorem 1.1).
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Black-Scholes Formula for a European Call and a European Put
Values of a European call (left) and a European put (right) in dependence of S
at times 0 ≤ t ≤ 1 for K = 100 , T = 1 , r = 0.1 , σ = 0.4 (courtesy of [1])
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.3.4 Interpretation of the Option Price as a Discounted Expectation
Consider a European call option V = V(S, t) with Λ(S) = (S − K)+ and the proof of the Black-
Scholes formula (Theorem 3.2): In Step 3, the back transformation can be performed based on
the representationu(x, τ ) =
1√
2π
+∞∫
−∞
u0 (√
2τ y + x) exp(−y2/2) dy .
Using further S := exp(√
2τy), we find
V(S, t) =K
√2π
exp(−(κ − 1)x/2 − (κ + 1)2τ/4)∫
lRu0(
√2τy + x) exp(−y2
/2) dy =
=1
√2π
∫
lRexp(−(κ + 1)2τ/4 + (κ − 1)
√2τy/2 − y2
/2) (exp(√
2τy) S − K)+ dy =
= exp(−r(T − t)) E(Λ(S)) , E(Λ(S)) =∞∫
0
S f(S;S, t) Λ(S) dS ,
where E(Λ(S)) is the expectation of Λ(S) w.r.t. the density function of the log-normal distribution
f(S;S, t) =1
Sσ
√2π(T − t)
exp(−(ln(S/S) − (r − σ
2/2)(T − t))2
2σ2(T − t)) .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.4 Numerical Evaluation of the Black-Scholes FormulaThe evaluation of the Black-Scholes formula requires a numerically efficient approximation of
the error function erf(x) = (2/√
π)∫x0exp(−t2)dt. This can be done by either rational best ap-
proximation in the nonlinear least-squares sense or piecewise polynomial approximation (see
Chapters 3.5 and 4.2, Handout Numerical Analysis I, Fall 2005, on my webpage).
3.4.1 Rational Best Approximation of the error function erf
Recalling the asymptotic properties
limx→∞
erf(x) = 1 , limx→∞
1 − erf(x)
erfx(x)= lim
x→∞
∞∫
xexp(x2 − t2) dt = 0 ,
we look for a rational function erfR
1 − erfR
erfx(x)= a1 η + a2 η
2 + a3 η3
, η :=1
1 + p x
such that erfF(0) = 0 resulting in, e.g., a3 =√
π/2 − a1 − a2. We determine erf∗ as the best
L2-approximation of erf within the class of functions erfR
(⋄) ‖erf∗ − erf‖2L2(0,∞) = inf
a1,a2,p‖erfR − erf‖2
L2(0,∞) = infa1,a2,p
∞∫
0
|erfR(x) − erf(x)|2 dx .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Obviously, (⋄) represents an infinite dimensional nonlinear least-squares problem. We reduce it
to a computationally more accessible finite dimensional nonlinear least-squares problem by ap-
proximating the integral with respect to a partition ∆ = 0 = x0 < x1 < ... < xn < ∞ of the do-
main of integration
(+) minz=(a1,a2,p)
n−1 ‖F(z)‖22 , F(z) :=
erfR(x1) − erf(x1)
·
erfR(xn) − erf(xn)
,
where ‖·‖2 stands for the Euclidean norm in lRn.
Given a start iterate z(0) ∈ lR3, the Gauss-Newton method for the solution of (+) is given by
F′(z(k))T F′(z(k)) ∆z(k) = − F′(z(k))T F(z(k)) ,
z(k+1) = z(k) + ∆z(k).
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.4.2 Piecewise Polynomial Interpolation
Another way to evaluate the error function is to approximate erf on [0,xmax] by a piecewise
polynomial interpolation using either cubic Hermite interpolation or the complete cubic
spline interpoland with respect to a partition ∆ = 0 = x0 < x1 < ... < xn = xmax.
The cubic Hermite interpoland h∆ ∈ C1([0,xmax]) with h∆|[xi,xi+1] ∈ P3([xi,xi+1]),0 ≤ i ≤ n − 1, is
given byh∆(xj) = erf(xj) , h′
∆(xj) = erfx(xj) , j ∈ i, i + 1
and has the representation
h∆(x) = erf(xi) ϕ1(t) + erf(xi+1) ϕ2(t) + hi erfx(xi) ϕ3(t) + hi erfx(xi+1) ϕ4(t) , t :=x − xi
hi, hi := xi+1 − xi ,
ϕ1(t) := 1 − 3t2 + 2t3, ϕ2(t) := 3t2 − 2t3
, ϕ3(t) := t − 2t2 + t3, ϕ4(t) := −t2 + t3
.
Having determined h∆, we define erf∗ by erf∗(x) = h∆(x),x ∈ [0,xmax], and erf∗(x) = h∆(xmax),x > xmax.
For the complete cubic spline interpoland we refer to Chapt. 3.5, Handout Numer. Anal. II.
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Approximation of the Error Function erf by Cubic Hermite Interpolation
Approximation of the error function erf by
cubic Hermite interpolation (xmax = 3 , n = 8).
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.5 Greeks and Volatility
3.5.1 GreeksLet V be the value of a call option or a put option. Greeks are derivatives of V and hence can
be used for a hedging of the portfolio.
Definition 3.8 (Greeks)
The Greeks Delta ∆, Gamma Γ, Vega (resp. Kappa) κ, Theta Θ, and Rho ρ are defined by
∆ :=∂V
∂S, Γ :=
∂2V
∂S2, κ :=
∂V
∂σ, Θ :=
∂V
∂t, ρ :=
∂V
∂r.
Theorem 3.4 (Computation of Greeks)
The Greeks have the following representations
∆ = Φ(d1) , Γ = Φ′(d1)/S σ
√T − t , κ = S
√T − t Φ′(d1) ,
Θ = − S σ Φ′(d1)/2√
T − t − r K exp(−r(T − t)) Φ(d2) , ρ = (T − t) K exp(−r(T − t)) Φ(d2) .
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Greeks at t = 0 (dotted line), t = 0.4 (straight line) and t = 0.8 (bold line)
for K = 100 , T = 1 , r = 0.1 , σ = 0.4 (courtesy of [1])
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.5.2 Volatility
The Black-Scholes formulas reveal that the price of options depends on the volatility σ of the
basic asset, but not on the drift µ. Since σ is only known in the past, we need an efficient pre-
diction of σ for the future. The most commonly used predictors are the historical volatility and
the implicit volatility.
3.5.2.1 Historical Volatility
The historical volatility is the annualized standard deviation of the logarithmic changes in the
value of the asset.
Definition 3.8 (Historical Volatility)
Let Si,1 ≤ i ≤ n, be the value of the asset on the day ti and denote by N the average number of
trading days at the stock market. Then, the historical volatility σhist is defined as
σhist =√
N (1
n − 1
n−1∑i=1
[(lnSi+1 − lnSi) −1
n − 1
n−1∑i=1
(lnSi+1 − lnSi)]2)1/2
.
University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
3.5.2.2 Implicit Volatility
Consider a European call option C = C(t) and assume that the prize at some time t0 < T is
known and given by C0 = C(t0). The Black-Scholes formula for European call options (Theo-
rem 3.2) shows that C depends on the volatility σ according to
C(σ) = S Φ(d1(σ)) − K exp(−r(T − t)) Φ(d2(σ)) .
Definition 3.9 (Implicit Volatility)
Assume that C0 satisfies the arbitrage estimate (S − Kexp(−r(T − t))+ ≤ C0 ≤ S. Then, the equa-
tion C(σimp) = C0 admits a unique solution σimp > 0 which is called the implicit volatility.
C(σ) = S Φ(d1(σ)) − K exp(−r(T − t)) Φ(d2(σ)) .
Given a start iterate σ(0)
> 0, the implicit volatility can be computed by Newton’s method
σ(k+1) = σ
(k) −C(σ(k)) − C0
C′(σ(k))= σ
(k) −C(σ(k)) − C0
κ(σ(k)), κ(σ(k)) := S
√T − t Φ′(d1(σ
(k))) .
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