Chapter 3: Black-Scholes Equation and Its Numerical …rohop/spring_13/Chapter3.pdf · Numerical Methods for Option Pricing in Finance Chapter 3: Black-Scholes Equation and Its Numerical

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)) .



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 .



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.



(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.



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 .



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.



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.



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) .



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 .



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.



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.



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).



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 .



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.



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 .



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 .



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)) .



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.



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) .



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 .



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 .



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) .



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) .



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)+ .



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).



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])



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)) .



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 .



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).



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.



Approximation of the Error Function erf by Cubic Hermite Interpolation

Approximation of the error function erf by

cubic Hermite interpolation (xmax = 3 , n = 8).



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) .



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])



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

.



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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