SeminarQF

FinancialProducts and

Structures

Mario Dell’Era

FinancialTheoryPostulate of yield ofmoney over the time

CapitalizationOperations

DiscountingOperations

Economic Laws:Arbitrage Principle

Financial Risks

FinancialEngineering

Efficient Marketmodel

Complete MarketModel

Asset PricingTheorems

ModelingBlack&Scholes

Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial Products and StructuresFinancial Theory and Modeling

Mario Dell’EraHead of Quant front Office at ENOI spa and External Professor at Pisa University

[email protected]

Head of Quant front Office

Quantitative Finance at Pavia University 2015

Mario Dell’Era (Head of Quant front Office ) Financial Products and Structures Seminar of Quantitative Finance 1 / 40


Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

1 Financial TheoryPostulate of yield of money over the time

Capitalization OperationsDiscounting OperationsEconomic Laws: Arbitrage Principle

Financial RisksFinancial EngineeringEfficient Market modelComplete Market ModelAsset Pricing Theorems

2 ModelingBlack&ScholesHeston

3 Greeks and HedgingAnalytical methodMonteCarlo method



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryPostulate of yield of money over the time

Postulate of yield of money over the time

Every realistic model of Financial market can not regardless ofthe cost and the return of money, that is usually quantified bythe interest rates.

It is easy to see that the cost of money directly affects thestrategy of an agent, that must decide which is the moreprofitable investment on market.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method


In economic and financial practice the cost of money has acentral role. Daily experience teaches that those who deposit aeuro at a bank expects its euro grows over time at a ratedetermined by the current interest rate, or in other words:

those who give up today to a financial availability, shiftingover time, requests that he be paid an appropriate fee calledinterest;who today requires the availability of a sum of which canhave at a given future, it must match an appropriate rewardcalled discount;



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method


The value of money is a function of the time that is available.From the beginning, financial mathematics (and actuarial) hasfocused on operations that allow you to move the money overtime according to two main types of operations:

Capitalization: the value of money is forward shifted over thetime;Discounting : the value of money is backward shifted overthe time.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryCapitalization Operations

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



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryDiscounting Operations

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



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryEconomic Laws

Arbitrage Principle

Intuitively, a market is efficient if it has no possibility of achievingcertain receipts without any risk, this circumstance and alsodefined opportunities for arbitrage.

We define below the set of financial laws underlying themathematical modeling of a market, able to guarantee theformation of rational prices (arbitrage pricing), which excludepossibility of arbitrage.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryEconomic Laws

Economic LawsEconomists identify, driven by common economic sense, thefollowing laws as necessary and sufficient for the purpose:

Postulate of yield of money over the time;Law of one priceLaw of the linearity of the amountsLaw of the monotonicity of the amounts



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryLaw of one price

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 numbergenerator.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryLaw 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 givenby the intersection of sale of A and B of the purchase with cashflow 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



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryLaw of the linearity of the amounts

Law of the linearity of the amounts

If the price of payoff X is p(X ) then for each c ∈ R the price ofthe payoff cX is cp(X ). With more generality:

∀a,b ∈ R p (aX + bY ) = a p (X ) + b p (Y )



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryLaw of the linearity of the amounts

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



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryLaw of the monotonicity of the amounts

Law of the monotonicity of the amounts

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 pricep(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.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryLaw of the monotonicity of the amounts

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 atmaturity date t = T , then it is an opportunity for arbitrage.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryFinancial Risks

The sources of risk are many, we list a few:Market Risk: which depends on the risk factors that affectthe 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 commitmentsassumed in a contract.Liquidity Risk: risk due to the mismatch between supplyand demand on the market and that makes it impossible, ordelayed, purchase and sale transactions.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial TheoryFinancial Engineering

DerivativesOne of the most fertile of financial areas is the creation offinancial instruments, known as derivatives, whose value isderived from an asset or index. Derivatives are abstract financialinstruments, whose Payoffs are functions of the underlyings. Wecan show hereafter the main kind of derivatives contracts as:

BondsForwardFuturesOptions



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Market Model HypothesisEfficient Market model

Hypothesis for an Abstract or Ideal market model

The theoretical goal that we are aiming, is the definition of amarket model which, starting from a limited set of simplehypotheses:

Market Frictionless,Competitive MarketMarkov Processes,Absence of Arbitrage opportunities,

that allow to modeling a mathematical structure in which toevaluate the price of any financial contract.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Complete Market Model

Replication Strategies

An efficient market model is also defined complete, if the value ofevery traded contracts can be replaced by strategies on market.Following this methodology, we are able to obtain theno-arbitrage price for every Derivatives contract by a suitablestrategies, where the fair price is the actualized expected futurecash flow.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

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 oneRisk-Neutral probability measure are equivalent to each other.

2nd Asset Pricing Theorem

Given a complete Markov’s market model, the followingconditions:absence of arbitrage and the existence of a uniqueRisk-Neutral probability measure are equivalent to each other.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial ModelingComplete Market Model in continuous time

Complete Market Model in continuous time

Suppose that on markets there exist only risk securities as St ,riskless or fixed income bonds as Bt and derivatives, whose valueis a function f (ST ,T ) also said Payoff .

Considering the dynamic of St follows a Geometric Brownianmotion, whose analytical form is given by the following SDE(stochastic differential equations):

dSt = µStdt + σStdWt , (1)

and the dynamic of the fixed income as:

dBt = rBtdt . (2)



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial ModelingBlack&Scholes

Black&ScholesDefine Black-Scholes market model, the set of equations:

dSt = µStdt + σStdWt →asset

dBt = rBtdt →fixed income

f = f (ST ,T )→Payoff

(3)

in which µSt is the drift term, σSt is the diffusion term and r is theinterest rate.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial ModelingRisk Neutral Measure

Risk Neutral MeasureBy Asset Pricing theorems we know that in any market models ofMarkov, in which there is no arbitrage opportunities, a risky assetcan not gain in mean more than a fixed income security.

Thus we need to change the probability measure of GeometricalBrownian motion following Girsanov’s theorem:

dWt = γtdt + dWt (4)

such that for γt = r−µσ , one has:

dSt = rStdt + σStdWt (5)



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial Market ModelsRisk Neutral Black&Scholes model

Risk Neutral Black&Scholes modelDefine risk neutral Black-Scholes market model, the followingset of stochastic differential equations (SDE):

dSt = rStdt + σStdWt →asset

dBt = rBtdt →fixed income

f = f (ST ,T )→Payoff

(6)



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial Market ModelsBlack&Scholes PDE

By Ito’s lemma and Asset Pricing theorems, the expected value ofthe stochastic process df (t ,St ) has to be equal to the fixedincome return:

EQ[df (t ,St )] = rf (t ,St )

namely

EQ

[∂f (t ,S)

∂t+ rS

∂f (t ,S)

∂S+σ2S2

2∂2f (t ,S)

∂S2 + σS∂f (t ,S)

∂SdWt

]= rf (t ,St ),



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial Market ModelsBlack&Scholes PDE

Black&Scholes PDEsolving the above expected value, one obtains the famous PDE ofBlack&Scholes:

∂f (t ,S)

∂t+ rS

∂f (t ,S)

∂S+σ2S2

2∂2f (t ,S)

∂S2 = rf (t ,St ) (7)

where f (ST ,T ) = Φ(ST ) is the payoff of any derivative contract.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial ModelingStochastic Volatility market models

Stochastic Volatility market models

The assumption of constant volatility is not reasonable, since werequire different values for the volatility parameter for differentstrikes and different expiries to match market prices. The volatilityparameter that is required in the Black-Scholes formula toreproduce market prices is called the implied volatility. This is acritical internal inconsistency, since the implied volatility of theunderlying should not be dependent on the specifications of thecontract. Thus to obtain market prices of options maturing at acertain date, volatility needs to be a function of the strike. Thisfunction is the so called volatility skew or smile.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial ModelingHeston

HestonDefine risk neutral Heston market model, the following set ofstochastic differential equations (SDE):

dSt = rStdt +√νtStdW (1)

t ,

dνt = κ(θ − νt )dt + α√νtdW (2)

t ,

dBt = rBtdt ,f (T ,ST , νT ) = Φ(ST ),

(8)

where EQ[dW (1)t dW (2)

t ] = ρdt for some constant ρ ∈ [−1,+1],and suppose also that both processes St and νt , are nonnegative:St ∈ [0,+∞), ν ∈ (0,+∞), t ∈ [0,T ] and α ∈ R+.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Financial ModelingHeston’s PDE

Heston’s PDE

By Ito’s lemma we have the following parabolic kind PDE ofHeston:

∂f∂t

+12ν

(S2 ∂

2f∂S2 + 2ραS

∂2f∂S∂ν

+ α2 ∂2f

∂ν2

)+ rS

∂f∂S

+ κ(θ − ν)∂f∂ν

= rf , (9)

where f (T ,ST , νT ) = Φ(ST ) is the payoff of any derivativecontract.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Greeks and Hedging

Greeks and Hedging

The Greeks are the risk factor sensitivities of the option price.They are partial derivatives of the model price of an option withrespect to one or more of its risk factors. The popular names anddefinition of the most common option price sensitivities are:

∆ =∂f∂S

, Γ =∂2f∂S2 , ν =

∂f∂σ

, Θ =∂f∂t, ρ =

∂f∂r

;

sometimes one uses also: νσ = ∂2f∂σ2 , νSσ = ∂2f

∂σS .



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Greeks and HedgingBlack&Scholes model

European Call Option

The price of a European Call Option according to theBlack&Scholes model is:

C (S,K ,T , r , σ) = S ∗ N (d+)− e−rT K ∗ N (d−)

where

d± =1

σ√

Tln

SKe−rT ±

σ√

T2



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Greeks and Hedging

Example: Black&Scholes and Heston model

Compute the Vanilla Option prices using the Black&Scholesmodel and Heston model.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

GreeksBlack&Scholes model

Black&Scholes

∆ (S,K ,T , r , σ) =∂C∂S

= N (d+) ∈ [0,1]

Γ (S,K ,T , r , σ) =∂2C∂S2 =

∂∆

∂S=

1S√

2πσ2Te

„− d2

+2

«> 0

Θ (S,K ,T , r , σ) = −∂C∂T

= −re−rT KN (d−)− Sσ2√

2πTe

„− d2

+2

«< 0

V (S,K ,T , r , σ) =∂C∂σ

= σTS2Γ = S

√T2π

e

„− d2

+2

«> 0

ρ (S,K ,T , r , σ) =∂C∂r

= T (S∆− C) = Te−rT KN (d−) ≥ 0



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Greeks and HedgingBlack&Scholes model

Example: Sensitivities with Black&Scholes model

Compute Greeks using the Black&Scholes model.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Greeks and HedgingMonteCarlo method

Greeks and MonteCarlo methodThe Monte Carlo simulation method can be used also to evaluatethe sensitivities of the derivatives price at varying of theparameters value of the used model.

For example we can consider the ∆, that is the sensitivity of theoption price at varying of the underlying asset value x over thetime:

∆ = e−r∗T ∂EP[F (ST (x))]

∂x, (10)

where F (ST ) is the payoff, and it is a function of x.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Greeks and HedgingGreeks and MonteCarlo method

The most simple way to compute numerically ∆ is to approximatethe derivative with an incremental difference ratio as:

∆ = e−r∗T EP

[F (ST (x + h))− F (ST )(x)

h

]. (11)

The expected value is assessed by MonteCarlo simulationmethod, choosing carefully the size of increment h. This methodis a general approach but it is not efficient from computationalviewpoint, it seems to be useful only when the payoff is a functionroughly smooth either it is know the analytical formula for price ofoption.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method


For example we can image to be in the case of a Vanilla option, inwhich there exists the closed formula for pricing a Call or a Put. Inthis case we pass the derivative within the expected value andone computes the derivative of integrand function as follows:

∆ = e−r∗T ∂EP[F (ST (x))]

∂x= e−r∗T

∫ +∞

0dyF (ey )∂xG(x , y)

= e−r∗T∫ +∞

0dyF (ey )G(x , y)

y − ln x −(

r − σ2

2

)T

σ2Tx,

(12)



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method


where we have taken ST (x) = ey and

G(x , y) =1√

2πσ2Texp

−(

y − ln x −(

r − σ2

2

)T)2

2σ2T

. (13)

Rewriting the above equation with the own variables, we have:

∆ = e−r∗T 1σ2Tx

EP

[F (ST (x))

(ln ST (x)− ln x −

(r − σ2

2

)T)]

;

(14)which is easier to compute than the incremental difference ratioseen before. However this procedure it is not always applicable,and the best way to solve this problem is to use theMalliavin Calculus.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method


Example: Greeks with MonteCarlo method

Compute Greeks using the MonteCarlo method.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Greeks HedgingDelta-Hedging

Delta-Hedging

Consider a portfolio V composed by a European Call optionf (t ,S) and a quantity ∆ of its underlying asset S:

V = f (t ,S) + ∆ ∗ S,

in order to neutralize the changing in value of the portfolio, withrespect to the underlying variation δS; we chose a particular ∆,as well as:

δV =∂f (t ,S)

∂SδS + ∆ ∗ δS = 0

if and only if:

∆ = −∂f (t ,S)

∂S.



Structures

Mario Dell’Era





Financial Risks






Heston

Greeks andHedging

Analytical method

MonteCarlo method

Greeks and HedgingDelta-Theta-Gamma-Hedging

Delta-Theta-Gamma-Hedging

Consider a portfolio V composed by a European Call optionf (t ,S) and a quantity ∆ of its underlying asset S:

V = f (t ,S) + ∆ ∗ S,

such that our portfolio is ∆-hedged. In order to improve theaccuracy we consider also the Greeks Γ and Θ in the portfoliovalue expansion, as well as:

δV =∂f (t ,S)

∂t(δt) +

12∂2f (t ,S)

∂S2 (δS)2 = Θδt +12

ΓδS2,

we obtain Delta-Theta-Gamma-Hedging strategy, imposing:

Θδt +12

ΓδS2 = 0.


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SeminarQF