Luiss

Quantitative Finance
Luiss Business School
MASTER EMERGES
Mario Dell’Era
External Professor at Pisa University
m.dellera@ec.unipi.it
October 29, 2013
Mario Dell’Era Quantitative Finance

Postulate of yield of money over the time
Every realistic model of Financial market can not regardless of the
cost and the return of money, that is usually quantiﬁed by the
interest rates.
It is easy to see that the cost of money directly affects the strategy of
an agent, that must decide which is the more proﬁtable investment
on market.

In economic and ﬁnancial practice the cost of money has a central
role. Daily experience teaches that those who deposit a euro at a
bank expects its euro grows over time at a rate determined by the
current interest rate, or in other words:
those who give up today to a ﬁnancial availability, shifting over
time, requests that he be paid an appropriate fee called interest;
who today requires the availability of a sum of which can have at
a given future, it must match an appropriate reward called
discount;

to availability of a euro today is to start a process of
capitalization, such as through a bank deposit, and thus
generate a built-in one year as capital employed increased by an
interest. Therefore, the possibility of capitalization makes the
euro value of the euro possessed today’s most realized one year;
the value of money is a function of the time in which is available;
the postulate of time preference (or impatience or postulate
postulate of return on money) according to which any rational
individual would prefer to have an amount immediately rather
than in a subsequent age.

Taking as a basis the postulate of impatience, you can groped to
answer questions such as the following:
What is the value of a euro of today in the future?
What it is value today of a euro which one takes in the future?

Capitalization and Discounting operations
From the beginning, ﬁnancial mathematics (and actuarial) has
focused on operations that allow you to move the money over time
according to two main types of operations:
Capitalization: the value of money is forward shifted over the
time;
Discounting : the value of money is backward shifted over the
time.

Capitalization Operations
T = {t : 0 = t0 < · · · < tn = T}
1 Simple Capitalization
Mtn
= M0(1 + nr)
2 Composed Capitalization
Mtn
= M0(1 + r)n
3 Continuous Capitalization
MT = M0erT

Discounting Operations
T = {t : 0 = t0 < · · · < tn = T}
1 Simple Discounting
Zt0
= Ztn
(1 + nr)−1
2 Composed Discounting
Zt0
= Ztn
(1 + r)−n
3 Continuous Discounting
Z0 = ZT e−rT

Common Economic Sense, and Financial Laws
Intuitively, a market is efficient if it has no possibility of achieving
certain receipts without any risk, this circumstance and also defined
opportunities for arbitrage. We define below the set of financial laws
underlying the mathematical modeling of a market, able to guarantee
the formation of rational prices (arbitrage pricing), which exclude
possibility of arbitrage.

Common Economic Sense, and Financial Laws
Economists identify, driven by common economic sense, the following
laws as necessary and sufﬁcient for the purpose:
Postulate of yield of money over the time;
Law of one price
Law of the linearity of the amounts
Law of the monotonicity of the amounts

Law of one Price
To illustrate the financial laws consider assigned a securities market
which M on the time horizon [0, T] with a schedule T = {0, T},
having assumed for the hypothesis postulated yield money.
For a first definition of efficient market (or rational or coherent)
economists assume the validity of a very important financial law,
known as law of one price, according to which:
if two contracts have the same payoff to the same expiration
date in the future, then the cost of acquisition of the contracts is
the same.

Law of one Price
Let A and B two contracts with the pay table given by:
t = 0 t = T
Long A −p (X) X
Long B −p (Y) Y
where the payoffs X and Y are, in general, random number
generator.

Law of one Price
Suppose that the law of one price is violated: p(X) > p(Y) and,
whatever happens, X = Y. Then we can create a contract C
given by the intersection of sale of A and B of the purchase with
cash ﬂow given by:
t = 0 t = T
Short A p (X) −X
Long B −p (Y) Y
C p (X) − p (Y) > 0 −X + Y = 0

Law of one Price
In summary, a market M is efﬁcient if it is truth that:
all contracts which have certainly the same payoff at maturity
t = T, have also the same price at the time t = 0:
X = Y ⇔ p (X) = p (Y)

Another important Financial law is the following:
If the price of payoff X is p(X) then for each c ∈ R the price of
the payoff cX is cp(X). With more generality:
∀a, b ∈ R p (aX + bY) = a p (X) + b p (Y)

Consider the cash ﬂow of three contracts:
t = 0 t = T
A −ap (X) aX
B −bp (Y) bY
C p (aX + bY) −aX − bY
D = A + B + C p (aX + bY) − ap (X) − bp (Y) 0

The last important Financial law is the following:
If the price of payoff X is p(X) and the price of payoff Y is p(Y)
and, with certainly, at maturity T will be X > Y, then the price
p(X) can not be lower than the p(Y). In other words:
X > Y ⇔ p (X) > p (Y)
We prove that if it is not valid the law of monotony of the amounts,
then there exists an opportunity for arbitrage.

Consider the pay table:
t = 0 t = T
A −p (X) X
B p (Y) −Y
C = A + B p (Y) − p (X) X − Y
Since X > Y, the payoff of the operation C is certainly positive at
maturity date t = T, then it is an opportunity for arbitrage.

Financial Risks
The risk is the central concept of modern finance. As we shall
see, every financial transaction may be viewed as trading risk.
Every economic activity is, as always, influenced by risk factors.
In order to eliminate or at least reduce the Financial Risk have
been invented several types of financial contracts, and they are
continually invented new ones.

Financial Risks
The sources of risk are many, we list a few:
Market Risk: which depends on the risk factors that affect the
overall progress in market prices.
Credit Risk: risk incurred by a party for the eventual inability
(partial or total) of the counterparty to fulﬁll the commitments
assumed in a contract.
Liquidity Risk: risk due to the mismatch between supply and
demand on the market and that makes it impossible, or delayed,
purchase and sale transactions.

Market Risk and Forward contracts
Forward
A type of contract that, since ancient times, has been used to
eliminate the risk of market is the Forward contract in which two
parties are organizing a private market for the exchange deferred to
an activity. More precisely:
A Forward contract is a ﬁnancial contract consisting of the
stipulation between two parties, for the purchase/sale of an
asset/commodity at a predetermined price at the time t0 that it
will be exchanged at a speciﬁed future date T > t0.

Forward
In a Forward contract:
Who decided to purchase the asset is one that takes a long
position or being long side of the contract.
Who decided to sell the asset is one who takes a short position
or which is the short side of the contract. It is said that the sale is
short sale if, at the conclusion of the contract, the seller does not
have the right to deliver.
The price agreed by the parties is called delivery price or
Forward price or Strike price.

Financial Markets
Over the time have been born organized markets, with offices
dedicated to trading regulation, and its respect. In other words we
passed from Private accords, as well as is (OTC) to a regulated and
organized market in which every days are traded several million of
contracts.
Roles of Traders on Markets
Assume that the main figures that give rise to financial contracts are:
Hedgers: It’s who covering a market position, assuming a long
or short position, by opening another.
Speculators: It’s who creates new position on market.
Arbitrageurs: It’s who by crossing several positions on market,
attempts to take a gain without any risk.

Financial Engineering
Derivatives
One of the most fertile of financial areas is the creation of financial
instruments, known as derivatives, whose value is derived from an
asset or index. Derivatives are abstract financial instruments, whose
Payoffs are functions of the underlyings. We can show hereafter the
main kind of derivatives contracts as:
Bonds
Forward
Futures
Options

Monetary Market Model
In the following we will refer to a market multi-period time horizon
[0, T] with calendar:
T = {t0, . . . , tn}
, dove 0 = t0 < · · · < tn = T.In an efﬁcient market model, we assume
that there are default free bonds on the market for each maturity of
T. We’ll ignore the credit risk and liquidity. Assume perfect divisibility
of securities.We will assume:
the possibility of short sales: each agent can take on debt
positions of each title being transferred.
In particular, each agent can borrow by issuing bonds.

Bonds
We will call the bond a contract in which two parties at the time t0
agree to exchange an amount ﬁxed future dates from a schedule:
T = {t0, . . . , tn}
, dove 0 = t0 < · · · < tn = T.For simplicity, we will consider only
ﬁxed-income securities in which the issuer of the bond receives at
time t0 amount P and agrees to pay periodically to the creditor, the
dates of the calendar residual {t1, · · · , tn}, interest on and the
repayment at maturity T = tn of invested capital at the date of signing
of the contract.

Bonds
CBB
A typical title of the fixed-interest or coupon bearing bonds CBB
with investment horizon [0, T] generates, for the buyer, a
deterministic flow amounts given by:
X = −P I{t0} +
n
i=1
cF I{ti } + F I{tn} (1)
or cash flow
O =
t0 t1 · · · tn−1 tn
−P cF · · · cF (c + 1) F
(2)

Bonds
ZCB
One ZCB can be considered one CBB with zero coupon rate and
therefore cash ﬂow for the buyer is given by:
X = −Zn
0 I{t0} + I{tn} (3)
or cash ﬂow
O =
t0 t1 · · · tn−1 tn
−Z (t0, tn) 0 · · · 0 1
(4)

Term Structure of ZCB.
Given a market in which are negotiable ZCBs, if at the time t0 the
prices recorded for maturities {t1, t2, · · · , tn}, are respectively:
Z(t0, t1), Z(t0, t2), · · · , Z(t0, tn)
this sequence is the term structure of prices at time t0, that
satisﬁes the property of monotonicity:
1 = Z (t0, t0) > Z (t0, t1) > · · · > Z (t0, tn) > 0

Efficient Market model
Hypothesis for an Abstract or Ideal market model
The theoretical goal that we are aiming, is the definition of a market
model which, starting from a limited set of simple hypotheses:
Market Frictionless,
Competitive Market,
Markov Processes,
Absence of Arbitrage opportunities,
that allow to modeling a mathematical structure in which to evaluate
the price of any financial contract.

Complete Market Model
Replication Strategies
An efficient market model is also defined complete, if the value
of every traded contracts can be replaced by strategies on
market. Following this methodology, we are able to obtain the
no-arbitrage price for every Derivatives contract by a suitable
strategies, where the fair price is the actualized expected future
cash flow.

Asset Pricing Theorems
1st
Asset Pricing Theorem
Given a Markov’s market model, the following conditions:
absence of arbitrage and the existence at least of one Risk-Neutral
probability measure are equivalent to each other.
2nd
Asset Pricing Theorem
Given a complete Markov’s market model, the following conditions:
absence of arbitrage and the existence of a unique Risk-Neutral
probability measure are equivalent to each other.

Pricing for Arbitrage: Forward price
Considering a contract Forward a game with zero-sum, we have:
Replication Strategy
t T
Short Forward 0 ˆFT
t − ST
Buy S −St ST
Sell ˆFT
t zcb ZT
t
ˆFT
t −ˆFT
t
Net ZT
t
ˆFT
t − St 0

no-arbitrage Forward price
It is intuitively clear that a transaction with no ﬁnal amount
certainly must have a rational cost of acquisition null and
therefore the equilibrium price of the Forward contract is given
by:
ˆFT
t =
St
ZT
t

Pricing for Arbitrage: Futures
Futures
Futures are standard contracts, whose Payoff is given by the
difference between the spot price St of the underlying
asset/commodity, and the futures price fT
t at time t and maturity T as:
(St − fT
t ).
Futures can be traded only on regulated markets, in which there is
the cash compensation that adjusts the margins at the end of
trading days.

Cox-Ross-Rubinstein
no-arbitrage Futures price
To determine the relationship between spot prices and Futures prices
we can follow the following trading strategy suggested by
Cox-Ross-Rubinstein:
t0 t1 t2 t3
−f3
0
M1
0 f3
0
+ M1
0
“
f3
1
− f3
0
”
0 0
0 −M1
0 f3
1
M1
0 M2
1 f3
1
+ M1
0 M2
1
“
f3
2
− f3
1
”
0
0 0 −M1
0 M2
1 f3
2
M1
0 M2
1 M3
2
f3
2
+ M1
0 M2
1 M3
2
“
f3
3
− f3
2
”
−f3
0
0 0
“
M1
0 M2
1 M3
2
”
∗ X
thus
f3
0 = M1
0 M2
1 M3
2 ∗ X

Example
Forward and Futures
As result of climate changes, a major Company that produces Fuel by
wheat, worried about the price of wheat in a year, in order to hedge
oneself against the risk of appreciation of the latter, it decides to buy
the grain necessary for the production, to meet the needs of an entire
year, using the Forward and Futures contracts.
Be given a schedule It = {0, T1, T2, T3, T4}, where T1 = 3 − month,
T2 = 6-month, T3 = 9-months and T4 = 12-months. Let S0 = 5.00$ the
spot price of wheat in kg. Supposing to enter into a contract Forward
and in a contract Futures, for delivering of 100.000 kg of wheat in
one year and in six months respectively. The interest rates for bonds
with spot stipulation in T = 0 and maturity T1, T2, T3 and T4 are
respectively: r1 = 3%, r2 = 4%, r3 = 5% and r4 = 6%.

Compute:
(1) the Forward price for delivering of 100.000 kg of wheat in one
year;
(2) the Futures price for delivering of 100.000 kg of wheat in six
months;
Suppose lending rate and borrowing rate equal to each other.

(1) The Forward price is given by the following equation based on
arbitrage principle: FT4
0 = S0er4(T4−0)
× 100.000 $.
(2) For the Futures price we have to consider the forward rate rf
among T2 and T1; which can be obtained by: rf = r2T2−r1T1
T2−T1
. Thus
using the Cox-Ross-Rubinstein law, one has the Futures price as
required: fT2
0 = er1(T1−0)+rf (T2−T1)
S0 × 100.000 $.

Derivatives
Options can be of different style:
European style;
American style;
Bermuda style.
Options of European Style
An Option is a contract that gives to the holder the right and do not
the obligation, to buy or to sell at maturity T, the underlying asset, at
the strike price established at the signing time.
This right has a price, that is the price of the option which has to be
evaluated with great careful from the issuer.

Vanilla Options
Call Option
A Call option is a financial contract between two parties, the buyer
and the seller of this type of option. The buyer of the call option has
the right, but not the obligation to buy an agreed quantity of a
particular commodity or financial instrument (the underlying) from the
seller of the option at a certain time (the expiration date) for a certain
price (the strike price). The seller (or ”writer”) is obligated to sell the
commodity or financial instrument should the buyer so decide. The
buyer pays a fee (called a premium) for this right. Indicate its payoff
as follows:
ψCall = (ST − K)
+
= max (0, ST − K)

Vanilla Options
Call Option

Vanilla Options
Put Option
A Put option is a contract between two parties to exchange an asset
(the underlying), at a speciﬁed price (the strike), by a predetermined
date (the expiry or maturity). One party, the buyer of the put, has the
right, but not an obligation, to re-sell the asset at the strike price by
the future date, while the other party, the seller of the Put, has the
obligation to repurchase the asset at the strike price if the buyer
exercises the option. Indicate its payoff as follows:
ψPut = (K − ST )
+
= max (0, K − ST )

Vanilla Options
Put Option

Complete Market Model in discrete time
Binomial Model
Consider a single time step δt. We know the asset price S0 at the
beginning of the time step; the price S1 at the end of the period is
a random variable. We start with price S0, at the next instant we
assume that the price may take either value S0u or S0d, where
d < u, with probabilities pu and pd respectively.
S0u
S0
S0d
t = 0 δt

Binomial Model
its future value, depending on the realized state, will be either
Π1 = ∆S0u + βeδt
or Π1 = ∆S0d + βeδt
Let us try to ﬁnd a portfolio which will exactly replicate the option
payoff:
∆S0u + βeδt
= fu
∆S0d + βeδt
= fd
Solving this system of two linear equations in two unknown
variables ∆, β.

Binomial Model
we get
∆ =
fu − fd
S0(u − d)
β = e−rδ ufu − dfd
u − d
In order to avoid the arbitrage, the initial value of this portfolio
must be exactly to f0:
f0 = ∆S0 + β
=
fu − fd
u − d
+ e−rδ ufu − dfd
u − d
= e−rδt erδt
− d
u − d
fu +
u − erδt
u − d
fd

Risk Neutral Measure
Binomial Model
It is important to note that this relationship is independent on the
objective probabilities pu and pd ; we can nevertheless interpret the
above equation as an expected value if we set
qu =
erδt
− d
u − d
, qd =
u − erδt
u − d
where d < u. We may notice that
qu + qd = 1;
qu and qd are positive if d < erδt
< u, which must be the case if
there is no arbitrage strategy involving the riskless and the risky
asset; hence we may interpret qu and qd as probabilities.

Binomial Model
the option price can be interpreted as the discounted expected
value of payoff under those probabilities:
f0 = e−rδt
EQ[f1|F0 = S0] = e−rδt
(qufu + qd fd )
It’s worth noting that the expected value of S1 under probabilities
qu and qd is
EQ[S1|S0] = quS0u + qd S0d = S0erδt
The last observation explains why the “artiﬁcial probabilities” qu
and qd are called risk-neutral.

Multi-periodic Binomial Model
Cox-Ross-Rubinstein
Consider to repeat the above scheme on a trading calendar
T = {t0, t1, t2, t3}.
Suppose S is a risk asset, that follows the Markov process
Sn (ξ1, . . . , ξn) = Sn−1 (ξ1, . . . , ξn−1) u
1 + ξn
2
+ d
1 − ξn
2
=
= Sn−1 (ξ1, . . . , ξn−1) u(1+ξn)/2
d(1−ξn)/2
(5)
and Bn = ertn
, a riskless bond, where n=0,1,2,3.

where 0 < d < ertn
< u
Q {ξn = +1} = qu =
ertn
− d
u − d
Q {ξn = −1} = qd =
u − ertn
u − d
(6)
for which we have our process is a martingale
u qu + d qd = 1
EQ (Sn|Fn−1) = Sn−1 (uqu + dqd ) = Sn−1

S0 u u u
S0 u u
S0 u S0 u u d
S0 S0 u d
S0 d S0 u d d
S0 d d
S0 d d d
t0 t1 t2 t3

Option Pricing
Consider a Binomial Market model, thus the price of an option,
whose payoff is a function f(Sn) of the underlying asset S, is
given by:
f0 = f(S0) = e−rtn
EQ[f(Sn)|F0]
= e−rtn
n
k=0
n
k
qk
u qn−k
d f S0uk
dn−k

Option Pricing
Vanilla Options
C0 = e−rT
n
k=0
n
k
max 0, S0uk
dn−k
− K qk
u qn−k
d
P0 = e−rT
n
k=0
n
k
max 0, K − S0uk
dn−k
qk
u qn−k
d

Complete Market Model in continuous time
Black-Scholes market model
Suppose that on markets there exist only risk securities as St ,
riskless or ﬁxed income bonds as Bt and derivatives, whose
value is a function f(ST , T) also said Payoff .Considering the
dynamic of St follows a Geometric Brownian motion, whose
analytical form is given by the following SDE (stochastic
differential equations):
dSt = µSt dt + σSt dWt
and the dynamic of the ﬁxed income as:
dBt = rBt dt

Black-Scholes market model
Deﬁne Black-Scholes market model, the set of equations:



dSt = µSt dt + σSt dWt →asset
dBt = rBt dt →ﬁxed income
f = f(ST , T) →Payoff
in which µSt is the drift term, σSt is the diffusion term and r is the
interest rate.

Risk Neutral Measure
By Asset Pricing theorems we know that in any market models of
Markov, in which there is no arbitrage opportunities, a risky asset
can not gain in mean more than a ﬁxed income security.Thus we
need to change the probability measure of Geometrical Brownian
motion following Girsanov’s theorem:
dWt = γt dt + d ˜Wt
such that for γt = r−µ
σ , one has:
dSt = rSt dt + σSt d ˜Wt

Risk Neutral Black-Scholes market model
Deﬁne risk neutral Black-Scholes market model, the set of
equations:



dSt = rSt dt + σSt d ˜Wt →asset
dBt = rBt dt →ﬁxed income
f = f(ST , T) →Payoff
Since we have supposed that the underlying asset follows a
Geometrical Brownian motion, and the value over the time of any
derivative is built as function of the underlying asset f(t, St ), then to
describe its dynamic we need to use the Itˇo’s lemma as follows:

Itˇo’s Lemma
Let be given a function f = f(t, S) ∈ C1,2
, i.e. so that, f is a
function continuos one time with respect to the variable t and two
times with respect to the variable S. Computing the Taylor
expansion at ﬁrst order for t and at second order for S of f(t, S)
one has:
df =
∂f
∂t
dt +
∂f
∂S
dS +
1
2
∂2
f
∂S2
dS2
Therefore substituting in the latter dS = rSdt + σSd ˜Wt , we have:
df(t, S) =
∂f
∂t
dt +
∂f
∂S
“
rSdt + σSd ˜Wt
”
+
1
2
∂2
f
∂S2
“
rSdt + σSd ˜Wt
”2
=
∂f
∂t
dt +
∂f
∂S
“
rSdt + σSd ˜Wt
”
+
1
2
∂2
f
∂S2
“
(rSdt)2
+ 2rSdt × σSd ˜Wt + (σSd ˜Wt )2
”

Itˇo’s Lemma
=
∂f
∂t
dt +
∂f
∂S
“
rSdt + σSd ˜Wt
”
+
1
2
∂2
f
∂S2
“
(rSdt)2
+ 2rSdt × σSd ˜Wt + (σSd ˜Wt )2
”
=
„
∂f
∂t
+
∂f
∂S
rS
«
dt +
∂f
∂S
σSd ˜Wt +
1
2
∂2
f
∂S2
(σSd ˜Wt )2
+
1
2
∂2
f
∂S2
“
(rSdt)2
+ 2rSdt × σSd ˜Wt
”
=
„
∂f
∂t
+
∂f
∂S
rS +
1
2
∂2
f
∂S2
σ2
S2
«
dt
+
1
2
∂2
f
∂S2
“
(rSdt)2
+ 2rσS2
dt
3
2
”
+
∂f
∂S
σSd ˜Wt

Itˇo’s Lemma
=
»„
∂f
∂t
+
∂f
∂S
rS +
1
2
∂2
f
∂S2
σ2
S2
«
dt
+
∂2
f
∂S2
(rσS2
)dt
3
2 +
1
2
∂2
f
∂S2
(rS)2
dt2
+
∂f
∂S
σSd ˜Wt .
in which we have used the famous relation by which is possible
to build the Wiener process Wt beginning to the Random walk
process εt :
dWt = εt
√
dt

Itˇo’s Lemma
In other words, Itˇo considers worthless the terms with grade in dt
greater than one of the Taylor expansion of the function f(t, S); thus
one has the following relation whose name in literature is known like
Itˇo’s lemma:
df(t, S) =
„
∂f(t, S)
∂t
+
∂f(t, S)
∂S
rS +
1
2
∂2
f(t, S)
∂S2
σ2
S2
«
dt
+
∂f(t, S)
∂S
σSd ˜Wt

Itˇo’s Lemma
We can conclude that given a function f(t, S) ∈ C1,2
such that S
follows a Geometric Brownian motion, thus also f(t, S) follows a
Geometric Brownian motion with drift coefﬁcient:
∂f(t, S)
∂t
+
∂f(t, S)
∂S
rS +
1
2
∂2
f(t, S)
∂S2
σ2
S2
.
and diffusion coefﬁcient:
∂f(t, S)
∂S
σS.

Black-Scholes PDE
Therefore as we have seen before, by Asset Pricing theorems,
the expected value of the stochastic process df(t, St ) has to be
equal to the ﬁxed income return:
EQ[df(t, St )] = rf(t, St )
namely
EQ
∂f(t, S)
∂t
+ rS
∂f(t, S)
∂S
+
σ2
S2
2
∂2
f(t, S)
∂S2
+ σS
∂f(t, S)
∂S
d ˜Wt
= rf(t, St )
Black − ScholesPDE:
∂f(t, S)
∂t
+ rS
∂f(t, S)
∂S
+
σ2
S2
2
∂2
f(t, S)
∂S2
= rf(t, St )

Vanilla Options
Black-Scholes Approach: Vanilla Options Pricing
Let be given the Black-Scholes PDE, for which we suppose that
a generic underlying asset S follows a geometric Brownian
motion, and the price of the money creases with an interest rate
equal to r, that we suppose to be a constant. In order to price a
European Call (Put) option we have to solve the following
Cauchy problem, for which the initial condition is our payoff
f(T, S) = (S − K)
+
= max(S − K, 0):
∂f(t, S)
∂t
+
σ2
S2
2
∂2
f(t, S)
∂S2
+ rS
∂f(t, S)
∂S
= rf(t, S)
f(T, S) = (S − K)+
S ∈ [0, +∞) t ∈ [0, T]
where K is the strike price.

Vanilla Options
Changing the variable as follows:
x = ln S + r −
1
2
σ2
(T − t) x ∈ (−∞, +∞)
τ =
σ2
2
(T − t) τ ∈ 0,
σ2
2
T
f(t, S) = e−r(T−t)˜f(τ, x)

Vanilla Options
we have the one-dimension heat equation:
∂˜f(τ, x)
∂τ
=
∂2˜f(τ, x)
∂x2
˜f(0, x) = (ex
− K)
+
x ∈ (−∞, +∞) τ ∈ 0,
σ2
2
T
its solution is known in literature (see Handbook of Linear Partial
Differential Equation by A.Polyanin) and is given by:
˜f(τ, x) =
1
√
4πτ
+∞
−∞
dx ˜f(0, x )e−
(x −x)2
4τ

Vanilla Options
˜f(τ, x) =
1
√
2πτ
+∞
−∞
dx ex
− K
+
e−
(x −x)2
4τ
=
1
√
2πτ
+∞
ln K
dx ex
− K e−
(x −x)2
4τ
=
1
√
2πτ
+∞
ln K
dx ex
e−
(x −x)2
4τ −
1
√
2πτ
K
∞
ln K
dx e−
(x −x)2
4τ
=
1
√
4πτ
+∞
ln K
dx e−
(x −x)2−4τ(x −x)+4τ2
4τ ex+τ
−
1
√
4πτ
K
∞
ln K
dx e−
(x −x)2
4τ

Vanilla Options
=
1
√
4πτ
ex+τ
+∞
ln K
dx e−
[(x −x)−2τ]2
4τ −
1
√
4πτ
K
∞
ln K
dx e−
(x −x)2
4τ
Let us impose: ξ = [(x −x)−2τ]
√
2τ
and η = (x −x)
√
2τ
, thus we have:
dξ =
dx
√
2τ
, ξ =
[(ln K − x) − 2τ]
√
2τ
=
[(ln K − ln S − r − 1
2 σ2
(T − t) − σ2
(T − t)]
σ2(T − t)
,

Vanilla Options
dη =
dx
√
2τ
, η =
[ln K − x]
√
2τ
=
[ln K − ln S − r − 1
2 σ2
(T − t)]
σ2(T − t)
.
Therefore substituting the new variable in the previous integral,
we have:
˜f(τ, x) =
1
√
4πτ
ex+τ
+∞
ξ
√
2τdξe− ξ2
2
−
1
√
4πτ
K
∞
η
√
2τdηe− η2
2

Vanilla Options
=
1
√
2π
ex+τ
+∞
ξ
dξe− ξ2
2 −
1
√
2π
K
∞
η
dηe− η2
2
Remembering that f(t, s) = e−r(T−t)˜f(τ, x), and
x = ln S + r −
1
2
σ2
(T − t), τ =
1
2
σ2
(T − t),

Vanilla Options
we have:
f(t, s)
= e−r(T−t) 1
√
2π
ex+τ
+∞
ξ
dξe− ξ2
2 −
1
√
2π
K
∞
η
dηe− η2
2
= e−r(T−t)
eln S+(r− 1
2 σ2
)(T−t)+ 1
2 σ2
(T−t) 1
√
2π
+∞
ξ
dξe− ξ2
2
− e−r(T−t) 1
√
2π
K
∞
η
dηe− η2
2
= S
1
√
2π
+∞
ξ
dξe− ξ2
2 − K
1
√
2π
∞
η
dηe− η2
2

Vanilla Options
where
N(−ξ) =
1
√
2π
∞
ξ
dξe
−ξ2
2 , N(ξ) =
1
√
2π
ξ
−∞
dξe
−ξ2
2 ,
and N(ξ) + N(−ξ) = 1.

Vanilla Options
Thus we can write the price of European Call option as follows:
f(t, S) = S ∗ N(d1) − e−r(T−t)
K ∗ N(d2)
where d1 = −ξ =
[(ln S/K+(r+ 1
2 σ2
)(T−t)]
√
σ2(T−t)
,
d2 = −η =
[ln S/K+(r− 1
2 σ2
)(T−t)]
√
σ2(T−t)
and d2 = d1 − σ2(T − t).

Vanilla Options
For a European Put option we are able to repeat the same
calculus, for which our payoff is f(T, S) = (K − S)
+
, and one has
the following result:
f(t, S) = e−r(T−t)
K ∗ N(−d2) − S ∗ N(−d1) .

Luiss

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