Zhu J. Applications of Fourier Transform to Smile Modeling: Theory and Implementation

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6 1 Option Valuation and the Volatility Smile

We construct a portfolio as follows: a long position of stock in amount of

and a

short position of call in amount of 1. The cost of this portfolio is

G(t)=

S(t) −C(t).

In a short time interval (t,t +dt), the change of the portfolio in value is given by

dG(t)=

dS(t) −dC(t). (1.8)

Inserting the SDEs (1.1) and (1.7) into the above equation yields

dG(t)=−



∂



dt, (1.9)

where the random term dW (t) disappears, and therefore the portfolio is not exposed

to the instantaneous risk linked to dW(t). According to the deﬁnition of a fair price,

this portfolio should have a risk-free return r, namely

G(t)=G(0)e

which corresponds to an ODE

dG(t)=rG(t)dt = r



S(t) −C(t)



dt. (1.10)

By comparing this equation with the equation (1.9), we obtain a PDE

∂

+ rS

∂

−rC = 0. (1.11)

This PDE is called the Black-Scholes PDE with which a call price C(t) can be

solved. Note t is the calendar time. Denoting T as the time to maturity and

∂

−

∂

, we rewrite the Black-Scholes PDE as follows:

∂

−rC =

∂

. (1.12)

Imposing an initial condition for T = 0,

C(S

,T = 0)=max[S

−K,0],

and solving the PDE (1.12) yields the celebrated Black-Scholes formula,

= S

N(d

) −Ke

−rT

N(d

) (1.13)

with

1.2 The Black-Scholes Model 7

ln(S

/K)+(r +

√

ln(S

/K)+(r −

√

The above idea that we construct a dynamic risk-less portfolio at every time is called

the dynamic hedging.

The put option with payoff at maturity

P(T)=max[K −S(T), 0]

can be valued with a similar formula

= Ke

−rT

N(−d

) −S

N(−d

1.2.2 Risk-Neutral Valuation

In the context of the dynamic hedging as shown in (1.9) and (1.10), the drift term

of the stock process S(t) does not play a role any more. In fact, the PDE (1.11)

implies that the drift terms of call and stock processes are equal to the risk-free

interest rate r. Particularly we have a new dynamically hedged stock process and a

call process respectively,

dS(t)=S(t)rdt +

S(t)dW(t), (1.14)

dC(t)=C(t)rdt +

S(t)

∂

dW(t). (1.15)

Taking the expectation on both sides of the above SDE for a call option yields an

ODE

dy(t)=ry(t)dt, y(t)=E[C(t)]. (1.16)

This is equivalent to

y(T)=E[C(T )] = C

In other words, we have a more simple expression for the call price,

= e

−rT

E[C(T)]. (1.17)

This means that the fair price of a call is simply the discounted present value of

its expectation at maturity if we undertake a dynamic hedging strategy. In fact, the

above procedure is a simple example of the Feynman-Kac theorem that may build

up an equivalence relationship between the PDE (1.11) and the expectation solution

(1.17). Instead of solving a PDE, we can calculate the expected value E[C(T )] and

discount it with a risk-free interest rate, then obtain the price of a call. This approach

to pricing a call option illustrated in (1.17) is the so-called risk-neutral valuation.

8 1 Option Valuation and the Volatility Smile

Additionally, the stock price given in (1.14) is also referred to as a risk-neutral stock

process where the drift term is replaced by the risk-free interest rate r.

Since the principle of the risk-neutral valuation is strongly linked to the Feynman-

Kac theorem, here is the best place to state it in more details.

Theorem 1.2.1. The Feynman-Kac theorem: Assume that

dX(t)=a(X,t)dt + b(X,t)dW(t)

is an It

o process starting at time t = 0, and f (x) is a continuous function, f (x,t) ∈

2×1

[R×[0, ∞) → R].Ify(x,T ) satisﬁes the following PDE,

∂

y(x,T)

∂

= a(x,T)

∂

y(x,T)

∂

(x,T)

∂

y(x,T)

∂

−q(x)y(x,T ) (1.18)

with an initial condition

y(x

,T = 0)= f (x

,0),

where q(x) ∈C(R) is lower bounded, then

y(x

,T)=E



exp



−



q(X(t))dt



f (X(T))





. (1.19)

This means that the value of y(x, T ) at time T is equal to the expected value of

f (X(T)) discounted by q. Vise versa, if the expected value of (1.19) exists, then the

PDE (1.18) holds.

The Feynman-Kac theorem builds up a bridge between the deterministic world

represented by the PDE (1.18) and the stochastic world represented by the expected

value (1.19). If we have difﬁculty in calculating the expected value (1.19), we can at

least obtain it by numerically solving the PDE, as the Feynman-Kac theorem states.

In most cases, however, we know via the Feynman-Kac theorem that simulating a

stochastic process and then computing the expected value is equivalent to solving a

corresponding PDE. Mathematically, the Black-Scholes PDE (1.11) coincides with

the risk-neutral valuation completely.

In most ﬁnancial literature, it becomes a standard and convenient way to work

directly with the risk-neutral process. Nevertheless, we should never forget that one

of the important driving reasons for a risk-neutral process is the dynamic hedging.

The other way to construct a risk-neutral process is to introduce the so-called market

price of risk to compensate for the excess return to the risk-free return in incomplete

markets.

1.2.3 Self-ﬁnancing and No Arbitrage

In this section we generalize the concept of dynamic hedging, and brieﬂy address

how to construct an arbitrage-free portfolio within which options, or more precisely

1.2 The Black-Scholes Model 9

contingent claims, can be valued fairly. To this end, we assume a portfolio compris-

ing n assets A

that are governed by the following SDEs,

(t)

dt +

(t), i = 1,2,··· ,n.

Next we denote a vector process

(t)=(

(t),

(t), ··· ,

(t)) as a trading strategy,

the initial investment G(0) of the desired portfolio with such trading strategy is

simply

(

∑

i=1

(0)A

(0).

To ensure a fair measurement, we require that this portfolio should behave like a

closed fund without any cash in-ﬂows and out-ﬂows. More mathematically, if a

trading strategy

satisﬁes the following condition,

G(t;

)=G

(

∑

i=1



(u)dA

(u), (1.20)

we regard this trading strategy as a self-ﬁnancing trading strategy .

On the other hand, a total integral over dA

(u) and d

(u) yields

G(t;

)=G

(

∑

i=1



(u)dA

(u)+

∑

i=1



(u)d

(u). (1.21)

By a comparison of (1.20) and (1.21), we could arrive at

∑

i=1



(u)d

(u)=0, (1.22)

which is the necessary condition for a self-ﬁnancing strategy.

If for any contingent claim H(T ) at time T exists a self-ﬁnancing strategy

(t),

so that

H(T )=H

∑

i=1



(u)dA

(u),

then the market spanned by A

is a complete market. Market completeness means

that any contingent claim could be duplicated with the existing traded assets by

using a self-ﬁnancing strategy

(t). In the given formulation of the market, the risk

sources of market comes from n random variables W

(t), each of which controls the

dynamics of the asset A

, therefore each risk could be traded and well priced. Any

contingent claim that is linked to one or more risks in this market can be duplicated

perfectly via the traded assets and a given self-ﬁnancing strategy. However, if the

number of risk sources W is smaller than the number of assets A, some assets may

share a common risk, and therefore some of these assets become redundant to the

market. If the number of risk sources W is larger than the number of assets A,itis

10 1 Option Valuation and the Volatility Smile

clear that some risks can not be adequately traded, the contingent claim linked to this

non-tradable risk is then unattainable to the market. This market is then incomplete.

A typical incomplete market is the market described by a stochastic volatility

model where the stochastic volatility is exposed to an additional risk sources. There-

fore the number of the risk sources in a stochastic volatility model is larger than the

number of the underlying asset. Consequently, stochastic volatility models are gen-

erally incomplete. To make the risk-neutral valuation applicable, we need introduce

a market price of the volatility risk to obtain a risk-neutral process for stochastic

volatility. We will return to this topic in Chapter 3.

Now we are at a stage to rigorously deﬁne an arbitrage-free strategy.

Deﬁnition 1.2.2. No Arbitrage: An arbitrage-free strategy

must be self-ﬁnancing,

and satisfy the following conditions,

 0 =⇒ G(t;

)  0, ∀ t > 0 (1.23)

and

= 0 =⇒ G(t;

)=0, ∀ t > 0. (1.24)

The ﬁrst condition says that any negative initial investment can not generate a

positive future value. In other words, one can not make money by borrowing money.

The second condition gives a clear restriction: zero investment leads to zero future

value. Both conditions exclude any form of “free lunch”.

To show that the valuation of options in the Black-Scholes model is arbitrage-

free, we assume a market consisting of three securities: stock S, bond B and call C.

Namely, we have A

= B, A

= S, A

= C, and then construct a portfolio with an

initial value of zero,

(

= 0. (1.25)

Additionally it follows

dG(t)=

dB(t)+

dS(t)+

dC(t)

rB(t)+

S(t)

)dt +(

S(t)

)dW(t).

Since we require that G(t) is an arbitrage-free portfolio, it means G(t)=0,t  0,

or dG(t)=0,t  0, this is equivalent to

rB(t)+

S(t)

= 0, (1.26)

S(t)

= 0. (1.27)

We form a linear equation system via the above three equations and obtain

⎛

⎝

rB(t) S(t)

μμ

0 S(t)

σσ

B(t) S(t) C(t)

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

1.2 The Black-Scholes Model 11

Given the processes of S and C givenin(1.1) and (1.7), it is easy to solve this

equation system with the following solution,

= −

∂

S +C

= −

∂

, (1.28)

= 1.

Inserting this solution into the ﬁrst equation immediately yields the Black-Scholes

PDE,

∂

+ rS

∂

−rC = 0.

This replication exercise illustrates that there is no arbitrage in the Black-Scholes

model. So far, we have derived the option pricing formula within the Black-Scholes

model with three different but linked concepts: dynamic hedging, risk-neutral pric-

ing and no-arbitrage. While dynamic hedging and no-arbitrage have strong eco-

nomic and ﬁnancial implications, risk-neutral pricing is a simpliﬁed technical for-

mulation of the former two concepts.

1.2.4 Equivalent Martingale Measures

As we have seen above, the principle of risk-neutral valuation just replaces the drift

of the historical (real) asset process with the risk-free interest rate. For a better com-

parison, we recall both processes.

Historical asset process:

dS(t)

S(t)

dt +

dW(t).

Risk-neutral asset process:

dS(t)

S(t)

= rdt +

dW(t).

An implication of the risk-neutral process is that the discounted asset price is a

martingale. Roughly speaking, martingale is a process with an identical conditional

expected value at each time point, and therefore naturally linked to fair play. Here

we will brieﬂy show two results: (1) There exists an equivalent martingale measure

between the historical asset process and its risk-neutral counterpart. In other words,

the risk-neutral valuation implies an equivalent martingale measure. (2) For any self-

ﬁnancing portfolio, there exists also a martingale measure that ensures the absence

12 1 Option Valuation and the Volatility Smile

of arbitrage, hence also guarantees the risk-neutral valuation. To this end, we ﬁrst

deﬁne a martingale and an equivalent martingale measure in more details.

Deﬁnition 1.2.3. Martingale: A stochastic process X(t) is a martingale based on a

ﬁltration F =(F

)

t0

, if it satisﬁes the following three conditions

1. X(t) is F

-measurable at any time t;

2. E[|X(t)||F

] is well-deﬁned, i.e., E[|X(t)||F

] < ∞;

3. E[X(s)|F

]=X(t), s ≥ t.

The ﬁrst two conditions are purely technical and ensure the conditional expecta-

tion E[X(s)|F

] exists. The last condition is the essence of a martingale and states

that any conditional expected value of X(s) based on F

is just X(t), and hence

keeps unchanged for any s,t  s. For example, a standard Brownian motion is a

simple martingale.

A stochastic process is always associated with a measure that characterizes the

distribution law of increments. We denote P and Q as the measures for the histor-

ical and the risk-neutral processes respectively. In fact, the measure Q of a risk-

neutral process with respect to the measure P for any event is always continuous,

this relation establishes an equivalence between two measures. Generally, given a

measurable space (

), two measures P and Q are equivalent, denoted by P ∼ Q,

P(A) > 0 =⇒ Q(A) > 0, ∀A ∈

and

P(A)=0 =⇒ Q(A)=0.

Using two equivalent measures, we could deﬁne a Radon-Nikodym derivative,

M(t)=

(t),

which enables us to change a measure to another. It follows immediately

[XM]=



)MdP(



)dQ(

)=E

[X].

This interchangeability of the expected values under two different measures con-

ﬁrms the important role of a Radon-Nikodym derivative as intermediary between

two measures.

In most cases, we want to know more, and namely how to change a measure to

another. The Girsanov theorem gives us some concrete instructions to change the

measures for an It

o process.

Theorem 1.2.4. The Girsanov theorem: Given a measurable space (

,F,P), and

an It

o process X(t),

dX(t)=a(X,t)dt + b(X,t)dW (t).

Denote M(t) as a (an exponential) martingale under the measure P,

1.2 The Black-Scholes Model 13

M(t)=exp



−



(u)du+



(u)dW

(u)



with

[M(t)] = 1.

Additionally, W and W

are correlated with dWdW

dt.

Then we have the following results:

1. M(t) deﬁnes a Radon-Nikodym derivative

M(t)=

∗

(t).

2. If we deﬁne

∗

(t)=dW (t) −

(t) < dW(t),dW

(t) >= dW(t) −

ργ

(t)dt,

it is then a new Brownian motion in (

,F,P

∗

3. The It

o process X(t) may take a new form under P

∗

X(t)=a(X,t)dt + b(X,t)dW

∗

(t)

= a(X,t)dt + b(X,t)[dW(t) −

ργ

(t)dt]

=[a(X,t) −

ργ

(t)b(X,t)]dt + b(X,t)dW(t). (1.29)

Now we apply the Girsanov theorem to verify the equivalent measures between

the historical stock process and the risk-neutral stock process. To this end, we con-

struct a Radon-Nikodym derivative M(t) as follows

M(t)=

(t)=exp



−



du+



dW(u)



with

−r

Thus we have W(t)=W

(t).Theterm

may be interpreted as an excess return

measured in volatility. Therefore under the measure Q the Brownian motion W

∗

(t)

is equal to

∗

(t)=dW (t) −

dt = dW(t) −

−r

dt,

and we obtain

dS(t)

S(t)

dt +

[dW(t) −

−r

dt]

= rdt +

dW(t),

that is identical to the risk-neutral process. In this sense, the measure Q is called the

risk-neutral measure, and is equivalent to the historical statistical measure P.

14 1 Option Valuation and the Volatility Smile

Now we are able to show that for any self-ﬁnancing portfolio G(t), there ex-

ists also a martingale measure ensuring absence of arbitrage. At ﬁrst we deﬁne a

numeraire N(t) that is any strictly positive and not-dividend paying process. More-

over, we assume

G(t)

N(t)

is a martingale. According to the property of a martingale, it follows

= E



G(t)

N(t)



, ∀t, (1.30)

where P

represents the measure associated with numeraire N(t), and is equivalent

to the original measure of G(t). Based on (1.30), we can check the conditions for

absence of arbitrage. Firstly, given N(t) > 0, if G

 0, then it follows immediately

G(t)  0. Secondly, if G

= 0, it is also easy to see G(t)=0. Hence it is proven that

(1.30) implies no arbitrage. Furthermore, the equation (1.30) implies the existence

of the measure P

associated with N(t).

Generally we have different choices for N(t). As a consequence, there exist dif-

ferent equivalent measures to the original measure. In other words, equivalent mar-

tingale measures are not unique. However, a natural numeraire asset may be a money

market account and is deﬁned by

H(t)=exp





r(u)du



, (1.31)

where r(t) denotes the instantaneous short rate of interest. Obviously, H(t) is

strongly linked to a zero-coupon bond,

H(t)=

B(t)

= 1/exp



−



r(u)du



It can be veriﬁed that the measure under numeraire H(t) is exactly the risk-neutral

measure Q. To see this, we deﬁne

H(t)N

N(t)H

H(t)N

N(t)

and obtain



G(t)

N(t)



= E



G(t)

H(t)



, (1.32)

= E



G(t)

H(t)



= E



B(t)G(t)|F



. (1.33)

This means that, if the money market account is used as a numeraire, the fair price

of the arbitrage-free asset process G(t) is just its present value. If r(t) is a constant,

we have an usual discounting formula,

1.3 Volatility Quotations in Markets 15

= e

−rt

[G(t)|F

Since the money market account is unique, the risk-neutral measure Q is also unique.

Finally we could conclude:

Theorem 1.2.5. Equivalent martingale measure and no arbitrage: Denote G(t) as

the process of a self-ﬁnancing portfolio, P is its historical measure. For any nu-

meraire N(t), there exists an equivalent P

, such that

= E



G(t)

N(t)



, ∀ t,

and G(t) is arbitrage-free under P

.IfN(t) is the money market account, then P

the risk-neutral measure.

Summing up our discussions on no arbitrage (dynamic hedging in Black-Scholes

model), risk-neutral valuation and equivalent martingale measure, we could estab-

lish a cycle relation as follows:

no arbitrage ⇒ risk-neutral valuation ⇒ equivalent martingale measure ⇒ no

arbitrage.

By this relation, we can also conclude that risk-neutral valuation implies no ar-

bitrage, and no arbitrage implies equivalent martingale measure. The latter result is,

however, difﬁcult to be proven directly, see Harrison and Kreps (1979), Harrison

and Pliska (1981), Geman, El Karoui and Rochet (1995), Musiela and Rutkowski

(2005).

1.3 Volatility Quotations in Markets

1.3.1 Implied Volatilities

In the Black-Scholes formula there are ﬁve parameters determining the option price,

they are the spot price S

, the constant interest rate r,thestrikeK, the constant

volatility

and the maturity T. Except for the volatility

, other four parameters are

either speciﬁed by option contract or can be observed directly in markets. Therefore

between the only unknown volatility parameter

and option price there is a one-

to-one correspondence. As long as the market price of a call or a put is given, we

can calculate a volatility matching the option price perfectly. The volatility obtained

from a given option market price is called the implied volatility. Mathematically, an

implied volatility

impl

is a quantity satisfying the following relation,

(

impl

;K,T,S

,r)=C

Market