Introduction to risk theory and mathematical finance

ÃL. Stettner∗, Institute of MathematicsPolish Academy of Sciences, Sniadeckich 8, 00-956 Warsaw

1 Introduction

As risk is an inevitable part of all human activity, we naturally seek to be able to bettermeasure it, estimate it, and even reduce it. The notion of risk carries connotations ofchaos, the unexpected and undesired behavior of an observed phenomenon. It is difficultto anticipate how a chaotic model will behave, since as the name itself indicates sucha model does not have predictable dynamics. But while it is hard to say how a chaoticprocess will behave at any specific moment, things are quite different if we take a longer-term perspective, chiefly looking at the mean value of certain functions that hinge onthe process. For example, the investors of a bank or insurance company are much moreinterested in the overall market situation than in individual transactions they might standto lose on. The point is to be sure they come out on top in the appropriate long-termperspective, regardless of short-term fluctuations.

The lecture notes below consists mainly of the extended transparences prepared forthe series of lectures held with SSDNM at the Maria Curie-Sklodowska University onNovember 21, 23 and 30th, 2011.

2 Risk in the history of finance and insurance

2.1 What is risk?

We can say that risk is a failure, an unexpected result. Risk is the probability that ahazard will turn into a disaster.The probability that a disaster will happen.Vulnerability and hazards are not dangerous, taken separately. But if they come together,they become a risk.On the other hand risks can be reduced or managed.

∗SSDNM, Maria Curie-Sklodowska University, November 21, 23 and 30th, 2011

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Therefore the problem is first to model risk, measure it and then to think about itsminimization, its control or management.From financial point of view we can have two approaches to risk: for one group, let sayspeculators it is an opportunity to create extra profit (gain), for the other group the mainproblem is how to eliminate risk in our investments. Consequently there are investorswho are looking for risk and investors who are fighting with risk introducing saver (moresecure) financial instruments.

We start lectures with the historical part which is mainly based on the data taken fromWikipedia.

2.2 History of insurance and trade

Already 2000 B.C. Egyptian stone workers organized a kind of insurance based on commonsharing of funeral costs.The caravan trade of Asia and the Arabian penisula was subject to various risks and therewas a necessity to find a kind of risk sharing guarantee that loosing cargo the trader willbe able to survive.The trade very soon moved from land to sea stimulated construction and development oftrading vessels, oars and sails, which again were subject to various risk.The above factor required suitable codification. In Babylonian times, around 2100 B.C.,the Code of Hammurabi contained the system insurance policies used later on by Mediter-ranean sailing merchants. If a merchant received a loan to fund his shipment he paid alsothe lender an additional sum in exchange for the lender’s guarantee to cancel the loan inthe case of the shipment to be stolen.Greeks and Romans introduced the origin of health and live insurance 600 B.C. creatingguilds which cared for the families of decease members and paying funeral expenses.In Middle Ages the guilds protected their members from loss by fire and shipwreck, paidransoms to pirates, and provided respectable burials as well as support in times of sicknessand povertyFirst actual insurance contract was signed in Genoa in 1347. Policies were signed byindividuals, either alone or in a group. They each wrote their name and the amountof risk they were willing to assume under the insurance proposal. A new profession hasappeared - underwriters, who prepared insurance contracts.In 1693, the astronomer Edmond Halley created a basis for underwriting life insuranceby developing the first mortality table An Estimate of the Degrees of the Mortality ofMankind, drawn from curious Tables of Births and Funerals at the City of Breslau; withan Attempt to ascertain the Price of Annuities upon lives. Edmond Halley English as-tronomer, geophysicist, mathematician, meteorologist, and physicist who is best knownfor computing the orbit of the Halley’s comet.

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In 1756, Joseph Dodson made it possible to scale the premium rate to age.In the meantime the insuring cargo while being shipped was widespread.In London Lloyds Coffee House started to be popular place where merchants, ship-owners,and underwriters met to transact their business. Lloyds grew into one of the first moderninsurance companies, Lloyds of London.

The first American insurance company was founded in the British colony of Charleston,SC. In 1787 and 1794 respectively, the first fire insurance companies were formed in NewYork City and Philadelphia. The first American insurance corporation was sponsored by achurch the Presbyterian Synod of Philadelphia for their ministers and their dependents.Then other needs for insurance were discovered and, in the 1830s, the practice of classifyingrisks was begun.People accepted the fact that they needed to pay premiums to protect themselves andtheir loved ones in case of loss, including major losses like fires. The insurance companieshad a rude awakening to this fact in 1835 when the New York fire struck. The losses wereunexpectedly high and they had no reserves prepared for such a situation. As a resultof this, Massachusetts lead the states in 1837 by passing a law that required insurancecompanies to maintain such reserves. The great Chicago fire in 1871 reiterated the needfor these reserves, especially in large dense cities.Insurance companies had to work together to find a solution to the challenge of largelosses. So they got together and devised a system called reinsurance whereby losses weredistributed among many companies.The Workmens Compensation Act of 1897 in Britain required employers to insure theiremployees against industrial mishaps. This also fostered what we know today as publicliability insurance, which came strongly into play when the automobile arrived on thescene.In the 19th century, many societies were founded to insure the life and health of theirmembersNow insurance was the accepted thing to do. Everybody needed to protect themselvesagainst the many risks in life. Farmers wanted crop insurance. People wanted depositinsurance at their banks. Travelers wanted travel insurance. Everybody turned to insur-ance companies to give them peace of mind. And really, isnt that what insurance is thepaying of a premium to protect against some form of loss.(Gareth Marples)

2.3 History of banking

In Egypt and Mesopotamia gold is deposited in temples for safe-keeping. But it lies idlethere, while others in the trading community or in government have desperate need of it.

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In Babylon at the time of Hammurabi, in the 18th century BC, there are records of loansmade by the priests of the temple. The concept of banking has arrived.Greek and Roman financiers: from the 4th century BC hey take deposits, make loans,change money from one currency to another and test coins for weight and purity; booktransactions. Moneylenders can be found who will accept payment in one Greek city andarrange for credit in another.In Rome by the 2nd century AD a debt can officially be discharged by paying the appro-priate sum into a bank, and public notaries are appointed to register such transactions.During the 13th century bankers from north Italy, collectively known as Lombards, grad-ually replace the Jews in their traditional role as money-lenders to the rich and powerful.The business skills of the Italians are enhanced by their invention of double-entry book-keeping. Creative accountancy enables them to avoid the Christian sin of usury; intereston a loan is presented in the accounts either as a voluntary gift from the borrower or as areward for the risk taken.Florence is well equippped for international finance thanks to its famous gold coin, theflorin. First minted in 1252, the florin is widely recognized and trusted. It is the hardcurrency of its day.By the early 14th century two families in the city, the Bardi and the Peruzzi, have grownimmensely wealthy by offering financial services. They arrange for the collection andtransfer of money due to great feudal powers, in particular the papacy. They facilitatetrade by providing merchants with bills of exchange, by means of which money paid in bya debtor in one town can be paid out to a creditor presenting the bill somewhere else (aprinciple familiar now in the form of a cheque).

2.4 History of stock exchange

A stock exchange is a corporation or mutual organization which provides the facilities forstock brokers to trade company stocks and other securities. Stock exchanges also providefacilities for the issue and redemption of securities, as well as other financial instrumentsand capital events including the payment of income and dividends.In 12th century France the courratiers de change were concerned with managing andregulating the debts of agricultural communities on behalf of the banks. As these menalso traded in debts, they could be called the first brokers.In the late 13th century commodity traders in Bruges gathered inside the house of aman called Van der Bourse, and in 1309 they institutionalized this until now informalmeeting and became the ”Bruges Bourse”. The idea spread quickly around Flanders andneighbouring counties and ”Bourses” soon opened in Ghent and Amsterdam.In the middle of the 13th century Venetian bankers began to trade in government securities.In 1351 the Venetian Government outlawed spreading rumors intended to lower the price

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of government funds. There were people in Pisa, Verona, Genoa and Florence who alsobegan trading in government securities during the 14th century. This was only possiblebecause these were independent city states not ruled by a duke but a council of influentialcitizens.The Dutch later started joint stock companies, which let shareholders invest in businessventures and get a share of their profits - or losses. In 1602, the Dutch East India Companyissued the first shares on the Amsterdam Stock Exchange. It was the first company toissue stocks and bonds.In 2010, a study published by job search website CareerCast ranked actuary as the number1 job in the United States (Needleman 2010). The study used five key criteria to rankjobs: environment, income, employment outlook, physical demands and stress.

3 Modeling in finance and actuary

3.1 Precursors of the mathematical finance and actuary -part 1

We start with two precursors of mathematical actuary:Ernst Filip Oskar Lundberg (31 December 1876 - 2 June 1965), Swedish actuary, founderof mathematical risk theory and managing director of several insurance companie 1903doctoral thesis, Approximations of the Probability Function/Reinsurance of CollectiveRisks, UppsalaHarald Cramer (September 25, 1893 October 5, 1985), Stockholm University,They have introduced a basic model of surplus of the insurance company.

3.2 Modeling of a surplus of insurance company

For simplicity we assume first that we a discrete time modelThe surplus of the company Xn at time n satisfies the following recursive equation

Xn+1 = Xn −An + Yn+1

where Yn represents premia minus payout of claims at time n and it is a sequence of(i.i.d.) random variables with distribution G. The term An above stands for an action ofthe company - possible investments or payout of the dividends.More popular model is a continuous time model introduced by Lundberg and Cramercalled Cramer-Lundberg (C-L) model in their papers (Filip Lundberg 1903, 1926,Harald Cramer 1930, 1955). Then the surplus of the company Xt at time t is of the form

Xt = x + ct−Nt∑

i=1

Yi (1)

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where (Nt) = sup n ≥ 0 :∑n

i=1 Ti ≤ t Poisson process with intensity λ (random variablesTi are i.i.d. exponential law with parameter λ), and 0 ≤ Yi is again i.i.d. with distributionG. We denote µ = EYi, We denote by c - the premium rate and as is written abovethe time between the arrivals of claims (Ti) is exponential with parameter λ. Notice thatthe process

∑Nti=1 Yi is called compound Poisson process. (Xt) is an example of the Levy

process (the process with stationary and independent increments)Let τ = inf n : Xn < 0 be a bankruptcy, ruin time. As a measure of risk in insurancewe frequently consider P τ ≤ T.

3.3 Precursors of the mathematical finance and actuary -part 2

The precursor of mathematics of finance is Louis Bachelier, (born 11 marca 1870 roku,died 26 April 1946 roku), who defended at Sorbonne in 1990 his PhD called ”Theorie de laSpeculation”, later on published in Annales de l’Ecole Normale Superieure (trans. RandomCharacter of Stock Market Prices). Louis Bachelier for the first time used Brownian motion(the process with stationary and independent Gaussian increments) to model fluctuationsof asset prices. The way he modeled prices he could have them both positive nad negative.His work was for a long time forgot.Paul Samuelson 15.05.1915-13.12.2009, 1970 Nobel Prize winner improved Bachelier modelconsidering exponent of Brownian motion and acknowledged the contribution of LouisBachelier, who after his papers started to be cited.

3.4 Asset prices modeling

We consider to approach to modelin of asset prices:discrete time: the price of the asset Si(t), for i = 1, 2 . . . , d at time t has the followingrandom rate of return ζ

Si(t + 1)− Si(t)Si(t)

= ζi(z(t + 1), ξ(t + 1)) (2)

which may depend on economical factors (z(t)), and i.i.d. noise ξ(t).An alternative is a continuous time: model, where Si(t) is the price of the ith asset attime t and is of the form

Si(t) = Si(0)e∫ t

0ai(z(s),S(s))ds+

∫ t

0σi(z(s),S(s))·dL(s) (3)

in which L stands for Levy or Wiener process and (z(s)) is again a process of economicalfactors.

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Both models are in dynamic, the asset prices change in discrete or continuous time, andwe are thinking for a longer time horizon T .

3.5 Precursors of the mathematical finance and actuary -part 3

The notion of risk in financial investments was not recognized among economists for along time (although all practitioner usually took it into account. The major step wascontribution of Harry Markowith in his PHD thesis. Harry Max Markowitz (born 24 Au-gust 1927 in Chicago, Illinois) - an American economist and a recipient of the John vonNeumann Theory Prize and the Nobel Memorial Prize in Economic Sciences. Markowitzis a professor of finance at the Rady School of Management at the University of California,San Diego (UCSD). He is best known for his pioneering work in Modern Portfolio The-ory, studying the effects of asset risk, return, correlation and diversification on probableinvestment portfolio returns. His main result was published in ”Portfolio Selection” TheJournal of Finance, Vol. 7, No. 1. (Mar., 1952), pp. 77-91, and later appeared in his bookPortfolio Selection: Efficient Diversification of Investments (1959)Here is what Harry Markowitz, Nobel prize 1990 told during Nobel price ceremonyThe basic concepts of portfolio theory came to me one afternoon in the library whilereading John Burr Williams’s Theory of Investment Value. Williams proposed that thevalue of a stock should equal the present value of its future dividends. Since futuredividends are uncertain, I interpreted Williams’s proposal to be to value a stock by itsexpected future dividends. But if the investor were only interested in expectedvalues of securities, he or she would only be interested in the expected valueof the portfolio; and to maximize the expected value of a portfolio one needinvest only in a single security. This, I knew, was not the way investors didor should act. Investors diversify because they are concerned with risk as wellas return. Variance came to mind as a measure of risk. The fact that portfoliovariance depended on security covariances added to the plausibility of the approach. Sincethere were two criteria, risk and return, it was natural to assume that investors selectedfrom the set of Pareto optimal risk-return combinations.

4 Markowitz portfolio theory

4.1 Portfolio analysis - basics

Let Si(t) be the price of the ith asset at time t (i = 1, 2, . . . , d). Since we consider so calledstatic approach we let t = 0, 1.

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Random rate of return ith asset is of the form

ζi :=Si(1)− Si(0)

Si(0)=

Si(1)Si(0)

− 1

The expected rate of return of the ith asset is equal to µi := Eζi

We denote by µ = [µ1, . . . , µd]T the vector of the expected rates of return, and by ζ :=

[ζ1, . . . , ζd]T , the vector of the random rates of return.

Let Σ the covariance matrix of ζ defined as follows:Σ = E

(ζ − µ) (ζ − µ)T

= (Σij)

with T standing for transponse of the matrix.Let θ := [θ1, . . . , θd]

T be the vector of investment strategies (at time 0) - portions of capitalinvested in d assets, i.e. θi portion of capital invested in the i-th asset.Then random portfolio rate of return R(θ) is equal to θT ζ.In fact, since: if x is initial capital then xθi

Si(0) is the amount of purchased ith assets at time

0 and xθiSi(0)Si(1) is the value of the capital invested in the i-th asset at time 1 We clearly

have that

R(θ) =

∑di=1

xθiSi(0)Si(1)− x

x=

d∑

i=1

θi

Si(0)Si(1)− 1 =

d∑

i=1

θi

Si(0)(Si(1)− Si(0)) =

d∑

i=1

θiζi

Consequently the expected portfolio rate of return is of the form

E R(θ) = θT µ (4)

and the variance of the portfolio rate of return is equal to

V ar(R(θ)) = θT Σθ. (5)

In fact, we show the last formula - we have

V ar(R(θ)) = E

(d∑

i=1

θi(ζi − µi)

)2 =

E(θT (ζ − µ))((ζ − µ)T θ)

= θT E

(ζ − µ)(ζ − µ)T

θ

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4.2 Markowitz theory - foundations

We maximize expected portfolio rate of return taking into account the risk (minimizingit?)What is risk?, how to measure portfolio risk?We introduce the portfolio risk function Risk(R(θ)), which in the Markowitz theory isequal to Risk(R(θ)) = V ar(R(θ))If fact we have two criterion (minimax) problem: we want ot maximize the expectedportfolio rate of return and minimize its variance.Consequently we introduce so called Markowitz order º.The strategy θ (portfolio rate of return R(θ) is better than θ′ (rate of return of R(θ′)), wewrite as θ º θ′ lub R(θ) º R(θ′), if E R(θ) ≥ E R(θ′), and Risk(R(θ)) ≤ Risk(R(θ′)).The strategy θ is better than θ′ if the rate of return of θ is greater than that of θ′ and therisk corresponding to the strategy θ is not greater than the risk corresponding to θ′.With each strategy one can associate a point on the plane R2, (Risk(R(θ)), E R(θ))The strategy θ is maximal, if there are no strategy θ′ different than θ such that θ′ º θ.The maximal elements (strategies) in this order form on the plane R2 the set calledefficient frontier.

4.3 Classical Markowitz theory - analytic approach

The problem we formulated is a convex analysis problem: to minimize

θT Σθ (6)

under the constraintsθT µ = µp i θTJ = 1,where J = [1, . . . , 1]T , while µp is the fixed portfolio expected rate of return. For fixedportfolio rate of return we minimize risk understood as portfolio variance. Such problemcan be solved using the results of the bookMarkowitz, H., Portfolio Selection Efficient Diversification of Investments, Wiley, 1959.When the matrix Σ is nonsingular, the problem can be is solved using Lagrange multipliers.Consider the functionG(θ, κ) = θT Σθ + κ1(θT µ− µp) + κ2(θTJ − 1)A necessary condition for optimality is of the form∂G∂θi

= 0 ∂G∂κj

= 0 dla j = 1, 2We have

Theorem 1 If µ does not have the same coordinates and Σ is nonsingular then

infθ

V ar(R(θ)) = BT H−1B, (7)

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where H = AT Σ−1A, A = [µ,J ], B =

[µp

1

].

Proof. By necessary conditions with κ := [κ1, κ2]T we have

2Σθ + Aκ = 0 AT θ = B (8)

we solve the first equation with respect to θ:θ = −1

2 Σ−1AκFrom the second equation of (8) we haveAT Σ−1Aκ = −2Bi.e. assuming that H is invertible we haveκ = −2(AT Σ−1A)−1B := −2H−1B.Note that H is symmetric:H = AT Σ−1A therefore HT = AT Σ−1T A = H since Σ and Σ−1 are symmetric.We are now in position to calculate the portfolio variance using the obtained formulae forθ i κθT Σθ = −1

2 θT ΣΣ−1Aκ = θT AH−1B = (AT θ)T H−1B = BT H−1Bwhich completes the proof, assuming that H is invertible.

Let H =

[a bb c

], and υ = detH. H is 2 × 2 matrix and a = µT Σ−1µ, b = µT Σ−1J ,

c = J T Σ−1J .If matrix Σ is nonsingular and all coordinates of µ are nonidentical then υ > 0.In fact, from the definition of H we havea = µT Σ−1µ, b = µT Σ−1J = J T Σ−1µ, c = J T Σ−1Jalso υ = ac− b2.Since Σ is positive definite (yT Σy > 0 when y 6= 0), the matrix Σ−1 is also positive definitei.e. a > 0 whenever µ 6= 0 and c > 0.Furthermore(bµ− aJ )T Σ−1(bµ− aJ ) = bba− abb− abb− aac = a(ac− b2) = aυ > 0whenever µ does not have the same coordinates. The proof of Theorem 1 is completed.

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The formula (7) characterizes the efficient frontier for variance as a measure of risk. Thiswill be the upper part of the parabola in coordinate system (Risk(R(θ)), E R(θ))Using matrix H =

[a bb c

]we have

minθ

V ar(R(θ)) =1υ

(cµ2p − 2bµp + a) (9)

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which gives us formula for the above mentioned parabola. The coordinates of the originof the parabola are of the form:µmv = b

c and V ar(R(θmv)) = 1c .

4.4 Optimal strategy

The formula for optimal strategy obtained in the proof of Theorem 1 is

θ =−12

Σ−1Aκ = Σ−1AH−1B (10)

Since

H−1 =1υ

[c −b−b a

](11)

(note that A = [µ,J ], B =

[µp

1

]). we finally obtain

θopt =1υ

Σ−1A

[cµp − b−bµp + a

]=

1υ

Σ−1 (µ(cµp − b) + J (−bµp + a)) =

1υ

Σ−1 ((aJ − bµ) + (cµ− bJ )µp)

which is the formula for an optimal strategy.

4.5 Minimal variance portfolio

The formula for optimal strategy obtained above is parametrized by expected portfoliorate of return µp i.e. we have

θopt =1υ

Σ−1 ((aJ − bµ) + (cµ− bJ )µp) (12)

If now µp = bc = µmv, which corresponds to the vertex of efficient frontier parabola we

obtain thatθopt = 1

υΣ−1(a− b2

c

)J = Σ−1J 1

c

is the strategy minimizing variance we denote θmv for which we have V ar(R(θmv)) = 1c .

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4.6 Sharpe coefficient - tangent portfolio

We study now the approach considered by another 1990 Nobel prize winner (William F.Sharpe (born in Boston in 1934).Minimal variance strategy usually provides relatively small portfolio rate of return. Henceit is quite natural to look for other portfolios from the efficient frontier.One opportunity is to maximize so called Sharp coefficient

µp√V ar(R(θ))

, (13)

We are looking for the greatest intersection coefficient of the line starting from the originwith efficient frontier (upper part of hiperbola) i.e. for linear part of the line tangent toefficient frontier (upper part of hiperbola) starting from the origin (we tacitly assume herethat µmv > 0)In fact, V ar(R(θopt)) = 1

υ (cµ2p − 2bµp + a). Hence

√V ar(R(θopt)) =

√1υ (cµ2

p − 2bµp + a)To find optimal tangent portfolio we calculate

dµp

d√

V ar(R(θopt))=

d

√V ar(R(θopt))

dµp

−1

=

1

2√

1υ (cµ2

p − 2bµp + a)

1υ

(2cµp − 2b)

−1

=υ√

1υ (cµ2

p − 2bµp + a)

cµp − b

We would like to have dµp

d√

V ar(R(θopt)= µp√

V ar(R(θopt)), i.e.

µp(cµp − b) = υV ar(R(θopt)) = cµ2p − 2bµp + a which gives bµp = a and

µp =a

b= µtg. (14)

If µtg = ab we have

V ar(R(θtg)) = 1υ (c

(ab

)2 − 2bab + a) = 1

υ ( ca2

b2− a) = a

b2.

Consequently using (9) we obtain

θtg =1υ

Σ−1(

aJ − bµ) + (cµ− bJ )a

b

)= Σ−1 µ

b. (15)

We have looked for the line tangent to√

V ar(R(θopt)). One could could also look for theline tangent to the graph of V ar(R(θopt)), which we shall study in the next subsection.

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4.7 Maximization of the tangency coefficient to the efficient frontierparabola

Now we want to maximizeµp

V ar(R(θ))(16)

i.e. tangency coefficient to the upper part of efficient frontier parabolacµ2

p − 2bµp + a− υV ar(R(θopt)) = 0.

The upper part of the parabola is given by µp = 2b+√

∆2c with ∆ = 4b2−4c(a−υV ar(R(θopt))) =

4υ(cV ar(R(θopt))− 1)The tangency coefficient should satisfy

dµp

dV ar(R(θmin)) = 12c

12√

∆4υc = υ√

∆and therefore the tangent line is of the form µ = υ√

∆V ar + z.

It intersect the origin when z = 0. We look for µ and V ar from the upper parabola parti.e. 2cµ = 2b +

√∆ and µ

√∆ = υV ar, which gives 2cµ2 = 2bµ + υV ar. Therefore

cµ2 + a− 2cµ2 = 0, and µ =√

ac = µst.

HenceV ar(R(θst)) = 1

υ (c(

ac

)2 − 2bac + a) = 1

υ (a2

c − 2 bac + a) = 1

υca(a− 2b + c)andθst = 1

υΣ−1((aJ − bµ) + (cµ− bJ )a

c

)= 1

υΣ−1(a(1− b

c)J + (a− b)µ).

4.8 The role of minimal variance and tangent portfolios

The optimal strategy is of the formθopt = 1

υΣ−1 ((aJ − bµ) + (cµ− bJ )µp)The optimal tangent strategy θtg = Σ−1 µ

b and the optimal minimum variance strategy isθmv = Σ−1J 1

c .Hence

θopt =−b2 + bcµp

υθtg +

ac− bcµp

υθmv (17)

which means that optimal strategy is a linear (affine) combination of tangent andminimal variance strategies.

4.9 Value functionals

Two cost functional problem (minmax) we considered so far leads us to an idea to replacetwo criterions problem by one criterion.We maximize two parameter functional

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F (E R(θ) , Risk(R(θ))) (18)

over all admissible portfolio strategies θ. The choice of the point of efficient frontier isreplaced by the choice of risk parameter λ.The function F should be increasing with respect to the first coordinate and decreasingwith respect to the second. The risk aversion is measured by the parameter λ ≥ 0. Themost natural form of the function F is

F (x, y) = x− 12λy. (19)

We maximize F (E R(θ) , V ar(R(θ))) = E R(θ)− 12λV ar(R(θ)) = θT µ− 1

2λθT Σθ withrespect to such theta θ that θTJ = 1. Again we use Lagrange multipliers method. Weform the functionG(θ, κ) = θT µ− 1

2λθT Σθ + κ(θTJ − 1)Necessary condition of optimality is:∂G∂θi

= 0 ∂G∂κ = 0

We have

Theorem 2 If µ does not have the same coordinates and matrix Σ is nonsingular then

θopt =1λ

Σ−1(

µ + J λ− b

c

). (20)

Proof.From the necessary condition we get µ− λΣθ + J κ = 0 and J T θ = 1. We solve the firstequation with respect to θ and obtainθ = 1

λ

(Σ−1 (µ + J κ)

)

Substituting this to the second equation we obtain1λJ T Σ−1 (µ + J κ) = 1.Hence (using nonsingularity of Σ) we haveκ = λ−J T Σ−1µ

J T Σ−1J = λ−bc .

Recalling that c = J T Σ−1J i b = J T Σ−1J we obtainθopt = 1

λΣ−1(µ + J λ−b

c

)= b

λθtg + (1− bλ)θmv.

Since θtg = Σ−1µ1b and θmv = Σ−1J 1

c we finally have

θopt =b

λθtg + (1− b

λ)θmv (21)

which is an analogy (14) for Markowitz model.When λ = b (b = J T Σ−1µ) we have θopt = θtg and when λ →∞ we have θopt → θmv.

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Furthermore µopt = θToptµ = 1

λµT Σ−1µ+ 1λJ T Σ−1µλ−b

c = aλ + b

λλ−b

c = aλ + b

c− b2

cλ = υcλ +µmv

since a = µT Σ−1µ, b = J T Σ−1µ, and µmv = bc . Furthermore (since c = J T Σ−1J and

V ar(R(θmv)) = 1c ) we have

V ar(R(θopt)) = θToptΣθopt =

1λ

(µT + J T λ− b

c)Σ−1Σ

1λ

Σ−1(µ + J λ− b

c) =

1λ2

(a + 2bλ− b

c+ (

λ− b

c)2c) =

1λ2

(a +2bλ− 2b2 + λ2 − 2bλ + b2

c) =

1λ2

(a +λ2 − b2

c) =

υ

cλ2+

1c

=υ

cλ2+ V ar(R(θmv)).

4.10 Form of the variance corresponding to the optimal strategy

We have µopt = υcλ + µmv, µmv = b

c and V ar(R(θmv)) = 1c .

Moreover

V ar(R(θopt)) =υ

cλ2+ V ar(R(θmv)) =

(µopt − µmv)2cυ

+ V ar(R(θmv)) =

1υ

(cµ2

opt − 2bµopt +b2

c+

υ

c

)=

1υ

(cµ2

opt − 2bµopt + a)

This is an analogy to (9) (minθ V ar(R(θ)) = 1υ (cµ2

p − 2bµp + a)) obtained for Markowitzmodel.We now can summarize the obtained results:Summary:Using optimal strategies for F (E R(θ) , V ar(R(θ))) = E R(θ)−1

2λV ar(R(θ)) =θT µ − 1

2λθT Σθ for different λ we obtain the whole part of the efficient frontier from theorigin of the parabola (minimal risk point) to the Sharpe point (tangency point with theeffective frontier hiperbola).

4.11 Comments and remarks

We complete this part of the lecture with row comments:1. The methodology considered (both Markowitz and one criterion aim functional) doesnot impose any constraints on the portfolio strategies (we admit both short selling andshort borrowing). Vector θ can admit arbitrary values (both positive and negative butθTJ = 1); If we are looking for nonnegative strategies we have to study (in a number ofmodels) further part of efficient frontier (using continuity of the aim functional);2. FunctionFλ (E R(θ) , V ar(R(θ))) = E R(θ) − 1

2λV ar(R(θ))

15

describes a constant risk aversion - the derivative of this function with respect to λ isconstant. An alternative approach leads to study risk sensitive functionals of the formFλ(R(θ)) = −1

λ lnE exp −λR(θ), the idea of which is explained in the next section.

5 Risk sensitive portfolio

5.1 Motivation

Let g(λ) = lnE exp −λX. By Taylor expansion of this function we haveg(λ) = g(0) + λg′(0) + λ2

2 g′′(0) + remainder(λ)

where g′(λ) =E−Xe−λX

Ee−λX and

g′′(λ) =EX2e−λXEe−λX−(E−Xe−λX)2

(Ee−λX)2 .

Hence−1λ g(λ) = E X − 1

2λV ar(X) + remainder(λ), and thereforeFλ(R(θ)) = −1

λ lnE exp −λR(θ) = E R(θ) − 12λV ar(R(θ)) + remainder(λ)

Consequently maximizing Fλ(R(θ)) we maximize E R(θ) and minimize (with a weight12λ) V ar(R(θ)) and in fact also higher moments (but with also higher powers of λ).

5.2 Risk sensitive stationary portfolio

We study further risk sensitive cost functional introduced aboveFλ(R(θ)) = −1

λ lnE exp −λR(θ)We have the following two results:Fact 1. θ 7→ Fλ(R(θ)) is (strictly) concave (Holder inequality). Consequently there is atmost one maximum point of θ 7→ Fλ(R(θ)) θTJ = 1.Fact 2.(Jensen inequality) We have Fλ(R(θ)) ≤ E

θT ζ

.

How to solve the maximization problem of θ 7→ Fλ(R(θ)) under the constraint θTJ = 1.Natural suggestion would be to use Lagrange multiplier’s method introducing the functionG(θ, κ) = −1

λ ln E exp −λR(θ)+ κ(θTJ − 1).Necessary condition of optimality is then :∂G∂θi

= 0 ∂G∂κ = 0,

from which we obtain:−1λ

Eexp−λθT ζ(−λ)ζiEexp−λθT ζ + κ = 0 and J T θ = 1

We would like to find θ(κ) satisfying the first equation. Is it possible? In general it is notpossible. We show in the next subsection a counterexample.

16

5.3 Counterexample

Assume we have two assets i.e. d = 2. Then θ2 = 1−θ1. Assume furthermore that randomrates of return ζi are independent with two values onlyζ1 ∈ a1, b1,ζ2 ∈ a2, b2 where −1 < a1 < 0 < b1, −1 < a2 < 0 < b2, so that we have no arbitrageopportunity (profit without risk - this notion will be studied in details later). Moreoverlet a1 < a2, b1 > b2. Then we have−λθ1(b2 − a1) ¹ Fλ(R(θ)) ¹ λθ1(b1 − a2) where ¹ means an order of approximation. Asa result we have (since there are no restriction on θ1 it may take any positive or negativevalue)Summary: supθ Fλ(R(θ)) = ∞,This means that the Lagrange multipliers method can not be used.However, when we restrict ourselves to θi ≥ 0 everything is fine, although it may bedifficult to find an explicit solution. Therefore we use approximate methods to find anoptimal (unique) portfolio.

6 Markowitz problem with semivariance

6.1 Problem formulation

Variance of the portfolio as a measure of risk can not be considered as the best measure.It penalizes both positive positions of the portfolio (above the mean) as negative position(below the mean). Positive position are good for an investor. In fact risk correspondsonly to the negative positions. This justification leads to considering semivariance (sV ardefined below) as a proper measure of risk. Namely we consider the risk of the form

Risk(R(θ)) = SV ar(R(θ)) = E

(∑di=1 θi(ζi − µi)

)−2

We would like now to minimize E

(∑di=1 θi(ζi − µi)

)−2

under constraints θT µ = µp and θTJ = 1.Denote by ri = ζi − µi. Clearly Eri = 0 and θ1 = 1 − ∑d

i=1 θi, and the problem is tominimizeE

(∑di=1 θiri

)−2

= E

(r1 +

∑di=2 θi(ri − r1)

)−2

under constraints∑d

i=2 θi(µi − µ1) = µp − µ1.In what follows we shall consider two cases:Case 1. µi = µ1, for each i; then µ = µp, and the problem leads to unconstraint mini-mization

17

min(θ2,...,θd) E

(r1 +

∑di=2 θi(ri − r1)

)−2

Case 2. µi 6= µ1 for some i;Before we continue we need to solve an auxiliary problem.

6.2 An auxiliary lemma

Consider the problem Problem (A): minx∈Rm E(A + BT x)−

2

where A and B are random variables and vectors respectively such that B = (B1, . . . , Bm)T ,EB2

i < ∞, EBi = 0 and EA2 < ∞. We have

Lemma 1 The problem (A) admits an optimal solution.

Proof.Assume first that B1, . . . , Bm are linearly independent P (

∑mi=1 αiBi = 0) = 1 implies

α1 = . . . = αm = 0.Let S =

(k, y) ∈ Rm+1 : k ∈ [0, 1], ‖y‖ = 1

and c = inf(k,y)∈S E[(kA + BT y)−]2.

Clearly the mapping (k, y) → E[(kA+BT y)−]2 is continuous so that there is (k∗, y∗) suchthat c = E[(k∗A + BT y∗)−]2. If c = 0 we have k∗ > 0 since otherwise E[(BT y∗)−]2 = 0and P (BT y∗ ≥ 0) = 1, EBT y∗ = 0 and finally P (BT y∗ = 0) = 1, y∗ = 0 but ‖y∗‖ = 1 acontradiction.Therefore c = 0 ⇒ k∗ > 0 and then y∗

k∗ is an optimal solution to the Problem (A).When c > 0 then for ‖x‖ ≥ 1 we haveE[(A + BT x)−]2 = ‖x‖2E[( A

‖x‖ + BT x‖x‖)

−]2 ≥ ‖x‖2c

so that we have coercivity of x → E(A + BT x)−

2, and since it is a continuous function

it admits a minimizer x∗.Assume now that B1, . . . , Bm are not independent. If P (B = 0) = 1 and x ∈ Rm isa minimizer. If P (B 6= 0) > 0 there is a subset D of B1, . . . , Bm whose elements arelinearly independent and every element in this set is a linear combination of D. Supposethat such subset is B1, . . . , Bk and let B = (B1, . . . , Bk)T . By the proof there is a

minimizer x∗ for x → E(A + BT x)−

2and x∗ = (x∗, 0, . . . , 0)T is a minimizer. This

completes the proof of lemma.2

6.3 Semivariance main result

We are now in position to prove the main result

18

Theorem 3 There is a minimizer toE

(∑di=1 θiri

)−2

= E

(r1 +

∑di=2 θi(ri − r1)

)−2

under constraints∑di=2 θi(µi − µ1) = µp − µ1

Proof. We continue to study the cases considered earlier: Case 1. µi = µ1 (continuation):by Lemma 1 we have a minimizer to

min(θ2,...,θd) E

(r1 +

∑di=2 θi(ri − r1)

)−2

Case 2. Assume that there is i such that µ1 6= µi. For simplicity let i = 2. We then haveθ2 = µp−µ1

µ2−µ1−∑d

i=3 θi(µi−µ1)µ2−µ1

and therefore the problem is reduced to min over (θ3, . . . , θd) of

E

(r1 + µp−µ1

µ2−µ1(r2 − r1) +

∑di=2 θi(ri − r1)− (r2 − r1)

(µi−µ1)µ2−µ1

)−2

and by Lemma 1 again there is a minimizer.

6.4 Semivariance result generalizations

The result on semivariance can extended to the following general downside risk functions(instead of semiviarance): f(x) = 0 for x ≥ 0, and f(x) > 0 for x < 0.As an example one can consider the function f(x) = (x−)p with p > 0.It can be shown that whenever general downside risk function f is lower semicontinuous(lsc) and f(kx) ≥ g(k)f(x) and limx→∞ g(x) = ∞ then the problem

E

f

((∑di=1 θi(ζi − µi)

)−)→ min

with constraints∑d

i=2 θi(µi − µ1) = µp − µ1 has a minimal solution.For details see Jin, H., Markowitz, H., Zhou, XY, A note on semivariance, Math. Fin. 16(2006), 53-61.We complete this part of lecture with the list of most important references.

6.5 References:

De Giorgi, E., – A Note on Portfolio Selection under Various Risk Measures, Workingpaper, University of Zurich 2002.Elton, E. J., Gruber, M. J. – Modern Portfolio Theory and Investment Analysis, Wiley,1981.Jin, H., Markowitz, H., Zhou, XY, A note on semivariance, Math. Fin. 16 (2006), 53-61Markowitz, H., – Portfolio Selection Efficient Diversification of Investments, Wiley, 1959.Sharpe, W. F., – Portfolio Theory and Capital Markets, McGraw-Hill, 1970.

19

Szego, G. P., – Portfolio Theory with Application to Bank Asset Management, AcademicPress, 1980.Whittle, P., – Risk-sensitive Optimal Control, Wiley, 1990.

7 Monetary Measures of Risk

Variance of the portfolio rate of return or semivariance were just examples of of measureof risk. In this section we start to consider a general approach to measures of risk. Ourconsideration will be then continued in the final section 12.

7.1 Definitions

We denote by financial position X : Ω → R net worth position at maturity. and let X bethe class of financial positions.

Definition 1 A function ρ : X → R is monetary measure of risk when:- if X ≤ Y we have ρ(X) ≥ ρ(Y ) (monotonicity)- if m ∈ R then ρ(X + m) = ρ(X)−m (translation invariance; cash additivity)

Directly from the definition we have properties:ρ(X + ρ(X)) = 0|ρ(X)− ρ(Y )| ≤ ‖X − Y ‖where ‖X‖ stands for L∞ norm. We also frequently use the following normalizationρ(0) = 0.

Definition 2 Monetary measure of risk is convex measure wheneverρ(λX + (1− λ)Y ) ≤ λρ(X) + (1− λ)ρ(Y ) for λ ∈ [0, 1]

The meaning of convex risk measure is that diversification does not increase the risk.

Definition 3 Convex measure of risk is called a coherent measure of risk whenif λ ≥ 0 we have ρ(λX) = λρ(X) (positive homogeneity).

We have the following properties of the coherent measures of risk:ρ(0) = 0ρ(X + Y ) ≤ ρ(X) + ρ(Y ) (subadditivity).

20

7.2 Acceptance sets

Given a monetary measure of risk ρ we define so called acceptance set as Aρ :=X ∈ X |ρ(X) ≤ 0.Clearly:ρ convex iff Aρ is convex,ρ is positively homogeneous iff Aρ is a cone.ρ(X) = inf m ∈ R|m + X ∈ Aρ.A particular example of coherent risk measures isworst case measure ρmax(X) = −essinfωX(ω) (Aρ is the the set of all nonnegativebounded random variables).For every monetary measure of risk ρ we have ρ(X) ≤ ρ(essinfωX(ω)) = ρmax(X), sothat ρmax is a maximal monetary measure of risk.Notice furthermore that ρ(X) = −EX is also a coherent measure of risk.Also, ρ(X) = supQ∈Q EQ −X with Q a family of absolutely continuous measures withrespect to P , is a coherent measure of risk.

7.3 Value at Risk

The Basel Committee on Banking Supervision (BCBS) (a committee of banking supervi-sory authorities that was established by the central bank governors of the Group of Tencountries in 1975) recommended as a measure of risk for financial investments so calledvalue at risk (VaR) defined as follows for a given α which is usually less that 0.05 (in factBasel II recommendation is α = 0.01)

V aRα(X) = inf m ∈ R|P m + X < 0 ≤ α .

Therefore V aRα(X) is a minimal capital m that added to our financial position X allowsus to keep the ruin probability (i.e. the probability of negative value of X +m not greaterthan α.In statistics we introduce so called quantiles of random variables. For a given α ∈ (01, )we have upper and lower quantiles defined as follows:α quantiles: upper isq+α (X) := inf x ∈ R : P X ≤ x > α = sup x ∈ R : P X < x ≤ α

lower quantile isq−α (X) := inf x ∈ R : P X ≤ x ≥ α = sup x ∈ R : P X ≤ x < αUsually lower and upper quantiles coincide: this is in the case of continuous random vari-ables (i.e. random variables X with continuous distribution function FX(x) = P X ≤ x.In general q−α (X) ≤ q+

α (X) and we have a sharp inequality whenever P X = α > 0. Inthe last case we have an α quantiles interval [q−α (X), q+

α (X)].

21

Notice thatP −X ≤ m ≥ 1− α and P X < −m ≤ α.Therefore we have the following equivalent formula for value at risk V aRα(X) = q−1−α(−X) =−q+

α (X). It is worth to point out sometimes in the literature we have value at risk of Xdefined as a minus lower quantile of X, i.e. V aRα(X) = −q−α (X) (see papers and booksof H. Follmer and A. Schied, or R. Cont).

7.4 Properties of the value at risk

In this section we consider the properties of the value at risk. Namely we have that valueat risk defined as V aRα(X) = inf m ∈ R|P m + X < 0 ≤ α is such that1. if X ≥ 0 then V aRα(X) ≤ 0,2. if X ≥ Y then V aRα(X) ≤ V aRα(Y )3. V aRα(λX) = λV aRα(X) for λ ≥ 04. V aRα(m + X) = V aRα(X)−mConsequently V aRα is a monetary measure of risk. Unfortunately V aRα in general is nota convex measure of risk as is shown in the example considered below, which was presentedby Paul Embrechts.

7.5 Important example (Embrechts)

Assume that we have 100 i.i.d. loans of 100 units with 2% coupon, that are subject tocredit risk with default probability 1%.We consider two portfolios: X1 consisting of 100 independent loans each of 100 units, andX2 one loan of 100× 100 units.We have X1 =

∑100i=1 Yi i X2 = 100Y1, where P Yi = −100 = 0.01 and P Yi = 2 = 0.99.

One can show that

V aR5%(X1) > V aR5%(X2)

that is V aR5%(∑100

i=0 Yi) >∑100

i=1 V aR5%(Yi) = −200.In fact, q+

0.05(Yi) = 2, and q+0.05(

∑100i=0 Yi) = −1.64 · 101.4887 + 98 < 2 using normal

approximation (by central limit theorem) for sequence Yi.

7.6 Conditional value at risk

Absence of the property that diversification of portfolio minimizes risk is an importantdeficiency of value at risk. This was the reason that scientists started to look for othermeasures of risk. The measure of risk we consider now will be defined for integrable

22

continuous random variables X and is called conditional VaR (denoted by CVaR) orexpected shortfall, average value at risk. It is defined as followsCV aRα(X) = E −X|X + V aRα(X) ≤ 0and it is the expected value if minus X, given X plus V aRα(X) does not exceed 0.We have the following immediate properties of CV aRα

1. CV aRα(λX) = λCV aRα(X) for λ ≥ 02. CV aRα(m + X) = CV aRα(X)−mFor the next properties we shall need the following

Lemma 2 If X ∈ L1, x ∈ R s.t. P X ≤ x > 0 then for any A s.t. P (A) ≥ P X ≤ xwe have

E X|A ≥ E X|X ≤ xProof.

1P (A)

∫

AufX(u)du ≥ 1

P X ≤ x∫

X≤xufX(u)dy

hint: approximate X by discrete r.v.2

We now have further properties of CVaR:3. X ≥ Y implies that CV aRα(X) ≤ CV aRα(Y )In fact, since P (X + V aRα(X) ≤ 0) = α = P (Y + V aRα(Y ) ≤ 0) we have

CV aRα(Y ) = −E Y |Y + V aRα(Y ) ≤ 0 ≥ −E X|Y + V aRα(Y ) ≤ 0 ≥−E X|X + V aRα(X) ≤ 0 = CV aRα(X)

4. CV aRα(X + Y ) ≤ CV aRα(X) + CV aRα(Y )In fact, we have

CV aRα(X + Y ) = E −X|X + Y + V aRα(X + Y ) ≤ 0+E −Y |X + Y + V aRα(X + Y ) ≤ 0 ≤ E −X|X + V aRα(X) ≤ 0+E −Y |Y + V aRα(Y ) ≤ 0

Consequently CVaR is a coherent measure of risk. Furthermore we have the followingimportant representation (see the book of H. Follmer and A. Schied):5. CV aRα(X) = 1

α

∫ α0 V aRu(X)du = − 1

α

∫ α0 q+

u (X)du.Furthermore CV aRα from 5 we see that CV aRα(X) is a continuous function of the pa-rameter α and6. limα→0 CV aRα(X) = −essinfXlimα→1 CV aRα(X) = −EX.

23

7.7 Measures of dispersion (deviation)

We have already considered two specific measures of risk: variance and semivariance. Nowwe continue the topic in a more general setting. LetX = L2

Definition 4 D : X → [0,∞] is a measure of dispersion iffD(X + c) = D(X) for any c ∈ RD(0) = 0, D(αX) = αD(X) for α > 0D(X + Y ) ≤ D(X) + D(Y )D(X) ≥ 0, and D(X) > 0 for X 6= const

We have the following examples of measure of dispersion: standard deviation, negativesemi standard deviationσ−(X) =

√E(max(EX −X, 0))2

positive semi standard deviationσ+(X) =

√E(max(X −EX, 0))2.

8 Elliptic distributions

In this section we introduce a very important family of elliptic random variables introducedby Douglas Kelker in 1970 from the Department of Statistics and Applied Probability,University of Alberta.

8.1 Definition

Definition 5 We say that random vector X = (X1, . . . , Xd)T has elliptic law, if there isa vector µ, a positive definite symmetric matrix Ω, a nonnegative function gd such that,∫∞0 x

d2−1gd(x)dx < ∞, and a norming constant cd such that the density fX of the vector

X is of the form

fX(x) = cd|Ω|−1/2gd(12(x− µ)T Ω−1(x− µ)),

where |Ω| is the determinant of Ω.

One can show that

cd =Γ(d

2)(2π)d/2

(∫ ∞

0xd/2−1gd(x)dx

)−1

.

where Γ(z) =∫∞0 xz−1e−xdx and for positive integer d we have Γ(d) = d!, while Γ(d+ 1

2) =1·2·3·(2d−1)

2d

√π i Γ(1

2) =√

π.

24

8.2 Properties of elliptic distributions

Characteristic function of the elliptic vector X is given by

φX(t) = E(eitX

)= eitT µψ(

12tT Ωt)

where a function ψ(t) is called characteristic generator.We shall use the following notation: X ∼ Ed(µ,Ω, ψ) or X ∼ Ed(µ,Ω, gd).If

∫∞0 g1(x)dx < ∞ there exists EX and EX = µ. If furthermore

|ψ′(0)| < ∞or equivalently

∫∞0

√xg1(x)dx < ∞ then

Cov(X) := E(X −EX)(X − EX)T

= −ψ′(0)Ω.

The following property is crucial in applicability of elliptic random variables:

Lemma 3 If X ∼ Ed(µ,Ω, gd), A is m× d matrix (m ≤ d) and b - m dimensional vectorthen

AX + b ∼ Em(Aµ + b, AΩAT , gm)

i.e. linear combination of elliptic distributions with the same generator ψ iselliptic with generator ψ.

Therefore

Corollary 1 Marginal law of X ∼ Ed(µ,Σ, gd) is elliptic

Xk ∼ E1(µk, ω2k, g1)

where ω2k is the k-th element of the diagonal of Ω, the density of Xk is of the form

fXk(x) =

c1

ωkg1

(12

(x− µk

ωk

)2)

.

and

Corollary 2 Main propertyIf X ∼ Ed(µ,Ω, gd) then for Y = θ1X1 + θ2X2 + . . . + θdXd = θT X we have

Y ∼ E1(θT µ, θT Ωθ, g1).

25

8.3 Examples

We list below more important examples of elliptic random variables:1. Multidimensional normal X ∼ Nd(µ,Σ)the density is of the form: (we identify Ω = Σ)

fX(x) =cd√|Σ| exp

−1

2(x− µ)T Σ−1(x− µ)

with cd = (2π)−d2 ; the characteristic function

φX(t) = exp

itT µ− 12tT Σt

so that g(u) = e−u and ψ(t) = e−t (since ψ′(0) = −1 Σ = Cov(X)).2. Multidimensional StudentX ∼ td(µ,Ω; p) with p > d

2the density is of the form:

fX(x) =cd√|Ω|

[1 +

(x− µ)T Ω−1(x− µ)2kp

]−p

where cd = Γ(p)

Γ(p− d2)(2πkp)

−d2 , and kp is a constant dependent on p,

we have here gd(u) = (1 + ukp

)−p;

In particular case when p = d+ν2 i kp = ν

2 we have multidimensional t-Student with νdegrees of freedom and then Ω = ν

ν−2Σ.In the case when p = d+m

2 for positive integer m i d and kp = m2 we have

fX(x) =Γ(d+m

2 )

(πm)d2 Γ(m

2 )√|Σ|

[1 +

(x− µ)T Ω−1(x− µ)m

]− d+m2

.

In general case for kp = 2p−32 with p > 3

2 we have Cov(X) = Ω.Then in particular case for p = d+m

2

fX(x) =Γ(d+m

2 )

(π(d + m− 3))d2 Γ(m

2 )√|Ω|

[1 +

(x− µ)T Ω−1(x− µ)d + m− 3

]− d+m2

with Cov(X) = Ω.Moreover, when

26

fX(x) =cd√|Ω|

[1 +

(x− µ)T Ω−1(x− µ)2kp

]−p

and 12 < p ≤ 3

2 there are no variance (we have so called heavy tails)when 1

2 < p ≤ 1 we have that EX does not exist. For m = 1 we have a multidimensionalCauchy distribution

fX(x) =Γ(d+1

2 )π−d+12

√|Ω|[1 + (x− µ)T Ω−1(x− µ)

]− d+12

3. multidimensional logistic hg(u) = e−u

(1+e−u)2,

4. multidimensional exponential when g(u) = e−rus.

9 Portfolio analysis with elliptic rate of return

Now we shall continue the portfolio analysis using elliptic rate of return of asset prices.

9.1 Assumption and preliminary results

Our principal assumption in this section is:Assumption: random rate of return ζ is Ed(µ,Ω, ψ)The portfolio rate of return R(θ)) (for a strategy θ = (θ1, . . . , θd)) is elliptic with the law

E1(θT µ, θT Ωθ, ψ).

Furthermore V ar(R(θ)) = −ψ′(0)ω2, where ω2 = θT Ωθ, and

R(θ)− θT µ

ω∼ E1(0, 1, ψ)

This procedure allows standardization of the elliptic random variables.

9.2 Risk measures for elliptic rate of returns

We assume that ζ is Ed(µ,Ω, ψ) and consequently R(θ) is E1(θT µ, θT Ωθ, ψ).Consider first a measure of risk called probability of the shortfall

Risk(R(θ)) = P R(θ) ≤ q .

Using standardization we obtain

27

Risk(R(θ)) = FY (q − θT µ

ω),

where FY is the distribution of E1(0, 1, ψ).Consequently this leads to the following lower bound for the expected portfolio rate ofreturn

θT µ + καω ≥ q,

where κα is an α quantile of E1(0, 1, ψ).

9.3 Value at Risk - V aRα

We shall now calculate value at risk for the portfolio with elliptic rates of return Valueat Risk (V aRα):

V aRα(R(θ)) = inf x : P R(θ) + x ≤ 0 ≤ αWe recall that it is the minimal value added to the portfolio rate of return which guaranteesnonpositive rate of return with probability at most α. We have

V aRα(R(θ)) = −καω − θT µ

where κα is α quantile of E1(0, 1, ψ).

9.4 Conditional V aRα or CV aRα

For conditional V aRα (CV aRα), called also shortfall (expected shortfall) CV aRα(R(θ)) =E −R(θ)|R(θ) + V aRα ≤ 0 , which is the expected value of −R(θ) given nonpositiveR(θ) + V aRα we use the following representation form (see the book of Follmer Schied)

CV aRα(R(θ)) =1α

∫ α

0V aRβdβ = −ω

1α

∫ α

0κβdβ − θT µ.

Remark 1 We see that if we had no ω, where ω2 = θT Ωθ, for a given α both V aRα(R(θ))and CV aRα(R(θ)) would be a linear function of the investment strategy θ or in other wordsthey would depend on the expected portfolio rate of return θT µ only. Unfortunately this isnot the case.

28

9.5 Risk functions for elliptic rates of return

Since CV aRα is a coherent measure of risk it is reasonable consider the following opti-mization problem: maximize

Fλ(E(R(θ), CV aRα(R(θ)) = E(R(θ))− 12λCV aRα(R(θ)).

Notice that

Fλ(E(R(θ), CV aRα(R(θ))) = (1 +12λ)θT µ +

12λ√

θT Ωθ1α

∫ α

0κβdβ.

The second term is negative since for small α (usually below 0.05) the value of∫ α0 κβdβ is

negative. We are not able to solve the problem explicitly. One can find the maximum of

Fλ(E(R(θ), CV aRα(R(θ)))

using approximate methodsIn fact, consider Lagrange multiplier’s method to the function

Fλ(E(R(θ), CV aRα(R(θ))).

We form

G(θ, κ) = (1 +12λ)θT µ +

12λ√

θT Ωθ1α

∫ α

0κβdβ + κ(θTJ − 1).

Necessary condition for optimality is∂G∂θi

= 0 ∂G∂κ = 0

Hence

(1 +12λ)µ +

12λ

1√θT Ωθ

2Ωθz(α) + J κ = 0

with J T θ = 1.We are not able to solve θ from the first equation, which earlier together with the secondequation gives us κ.The difficulties come because of the existence of the term with square root. Since

√x ≤ x

for x ≥ 1 one can optimize the modified risk function for elliptic rates of return

Fmλ (θ) = (1 +

12λ)θT µ +

12λθT Ωθ

1α

∫ α

0κβdβ,

which is diminished the term with θT Ωθ.

29

We use to this new problem the Lagrange multiplier’s method and have

G(θ, κ) = (1 +12λ)θT µ +

12λθT Ωθ

1α

∫ α

0κβdβ + κ(θTJ − 1)

Necessary condition for optimality is then of the form∂G∂θi

= 0 ∂G∂κ = 0

from which we obtain

(1 +12λ)µ +

12λ2Ωθz(α) + J κ = 0

with J T θ = 1.From the first equation

θ =1

λz(α)Ω−1

(−(1 +

12λ)µ− J κ

),

substituting this to the second equation we obtain

J T 1λz(α)

Ω−1(−(1 +

12λ)µ− J κ

)= 1

or

1λz(α)

(−(1 +

12λ)J T Ω−1µ− J T Ω−1J κ

)= 1

we obtainκ =

((−(1 +

12λ)J T Ω−1µ

)− λz(α)

)1

J T Ω−1J .

Therefore

κ =1

J T Ω−1J(−(1 +

12λ)J T Ω−1µ

)− λz(α)J T Ω−1J

and finally the optimal strategy is of the form

θ =−(1 + 1

2λ)λz(α)

Ω−1µ +(1 + 1

2λ)µT Ω−1Jλz(α)J T Ω−1J Ω−1J +

1J T Ω−1J Ω−1J

with z(α) = 1α

∫ α0 κβdβ.

30

9.6 Further alternatives

One can consider also a function

Risk(R(θ)) = (CV aRα(R(θ)) + ER(θ))2.

which corresponds to the square of the CVaR (an analogy to the variance considered as asquare of the standard deviation).Then

Fλ = θT µ− 12λθT Ωθ(z(α))2

the only difference is that z(α) = 1α

∫ α0 κβdβ in the aim function has been replaced by

(z(α))2. The optimal strategy is of the form

θ =−(1 + 1

2λ)λ(z(α))2

Ω−1µ +(1 + 1

2λ)λ(z(α))2

µT Ω−1JJ T Ω−1J Ω−1J +

1J T Ω−1J Ω−1J .

We complete this section with references.

9.7 Further references

De Giorgi, E., – A Note on Portfolio Selection under Various Risk Measures, Workingpaper, University of Zurich 2002.Fang, K.T., Kotz, S., Ng, K.W., Symmetric Multivariate and Related Distributions, Lon-don, Chapman & Hall 1987.Follmer, H., Schied, A. – Stochastic Finance. An Introduction in Discrete Time, W. deGruyter, 2002.Kelker, D., – Distribution Theory of Spherical Distributions and Location-Scale ParameterGeneralization, Sankhya 32 (1970), 419-430Landsman, Z., Valdez, E. A. – Tail Conditional Expectations for Elliptical Distributions,North American Actuarial Journal 7 (2003), 55-71,

10 Risk of the insurance company

This section is devoted to aspect of risk in insurance companies. Probability of ruin(negative value of surplus) was considered as a measure of risk. We shall continue tostudy this measure considering also two typical problems of actuary sciences: optimaldividend and reinsurance problems.

31

10.1 Definition of risk

We consider first ruin probability for Cramer Lundberg model

ψ(x) = P Xt < 0 for some t ≥ 0 ,

with Xt = x + ct−∑Nti=1 Yi.

One can show that when c ≤ λEY1 we have ψ(x) = 1. Therefore our typical assumptionwould be so called ”safety loading”: which means that c > λEY1.We calculate ψ and obtain

ψ(x) =∫ ∞

0λe−λt

(∫ x+ct

0ψ(x + ct− y)G(dy) + (1−G(x + ct))

)dt.

Letting z = x + ct we have

ψ(x) =1ceλ x

c

∫ ∞

xλe−λ z

c

(∫ z

0ψ(z − y)G(dy) + (1−G(z))

)dz

and

ψ′(x) = (λ

c)2eλ x

c

∫ ∞

xe−λ z

c

(∫ z

0ψ(z − y)G(dy) + (1−G(z))

)dz

+λ

ceλ x

c (−1)e−λ xc

(∫ x

0ψ(x− y)G(dy) + (1−G(x))

).

Consequently

cψ′(x) = λψ(x)− λ

(∫ x

0ψ(x− y)G(dy) + (1−G(x))

)

and the inifnitesimal generator of the function ψ defined as

Aψ(x) = limt→0

E

ψ(Xt∧τ )− ψ(x)

t

exists and is of the form

Aψ(x) = λ (E ψ(x− Y1) − ψ(x)) + cψ′(x) = 0.

32

10.2 Optimal dividend problem

We start first with the main problem of insurance company which is optimal distributionof dividends. The surplus is of the form

Xn+1 = Xn − Un + Yn+1,

where Xn is the surplus before dividend at time n,Un is the dividend at time n, which should not exceed the value of surplus i.e. 0 ≤ Un ≤Xn, andYn is a value of premia minus payout of claims at time n, which is a sequence of (i.i.d.)random variables with distribution G.Our strategy is a sequence U = (U0, U1, . . . , Un, . . .) of dividends paid at times 0, 1, . . . , n.Denote by τ = inf n : Xn < 0 the ruin moment.Our optimization problem is to maximize

V U (x) = Ex

τ−1∑

n=0

e−δnUn

the discounted (δ is a discount rate) dividend till the ruin problem. Optimal solution tosuch problem is of the form threshold strategies (with one or more thresholds dependingon the form of distribution G). In the case of one threshold we pay dividend to reach withsurplus a certain threshold and below we pay nothing.

10.3 Optimal reinsurance problem

For an insurance company one possibility to minimize risk is to insure part of claims inanother company. Then the surplus is of the form

Xn+1 = Xn + c(bn)− s(Yn+1, bn)

with Yn ≥ 0 standing for claims (i.i.d. with distribution G), c(b) is premium left ifreinsurance b is chosen, and 0 ≤ s(y, b) ≤ y is the expense of the first insurer (cedent)(self-insurance function), under claim y and reinsurance b. Thereinsurance function is fixedhowever we have a possibility to choose the reinsurance level b The reinsurance functionhas the properties:s(y, b) ≥ s(y, b′) for y ≥ 0 implies that c(b) ≥ c(b′); and also c(bmax) < 0 (s(y, bmax) = y)We also assume that y 7→ c(y, b) is increasing and continuous, b 7→ c(b) ≤ c contin-uous, where c corresponds to the absence of reinsurance and P s(Y, b) > c(b) > 0 iE c(b)− s(Y, b) > 0 for certain b.Our purpose is to choose the strategy B = (b0, b1, . . . , bn, . . .) so as to minimize functional

V B(x) = −Px τ < ∞ .

33

10.4 Forms of self-insurance function s

We can consider various form of self-insurance function s:- proportional reinsurance: s(y, b) = yb, where b ∈ (0, 1) (retention level); reinsurer pays(1− b)y, insurer by- excess of loss reinsurance: s(y, b) = min y, b; reinsurer pays (y− b)+, insurer min y, b- first risk deductible: s(y, b) = (y − b)+, reinsurer pays min y, b,- proportional reinsurance in a layer:s(y) = min y, a+ (y − a− γ)+ + b min (y − a)+, γ, dla a, γ > 0, b ∈ (0, 1)We complete this section with references concerning the insurance problems we formulatedabove.

10.5 References

A. Martin - Lof, ”Lectures on the use of control theory in insurance”, Scand. Acturial J.1994, 1-25,C. Hipp, ”Stochastic control with Application in Insurance”, Stochastic Methods in Fi-nance, LNM 1856, Springer 2004,H. Schmidli, ”Stochastic Control in Insurance”, Springer 2008,T. Rolski, H. Schmidli, V. Schmidt, J. Teugels, ”Stochastic Processes for Insurance andFinance”, Wiley 1998,P. Brockett, X. Xia, ”Operations research in insurance: a review”, Trans. Act. Soc.XLVII (1995), 7-80,

11 Pricing of financial derivatives

11.1 Options (contingent claims)

We consider first two financial instruments used frequently to get protection against riskof the changes of asset prices. They depend on the fact whether we intend to buy an assetat time T called maturity and have a certain guarantee concerning the price we shall payfor it at maturity or we want to sell assets at time T and again want to have a guaranteeconcerning the price for which we can sell them. We have therefore two kinds of options:European call option:The buyer of this contract has the option to buy (exercise the option), at time T oneshare of the stock at the specified (at time t = 0) price K called exercise (striking) price.Possible profit of the buyer at time T is of the form

(S1(T )−K)+

34

European put optionThe buyer of this contract has the option to sell (exercise the option), at time T oneshare of the stock at the specified (at time t = 0) price K called exercise (striking) price.Possible profit of the buyer at time T is of the form

(K − S1(T ))+

In this option maturity T is fixed. An alternative is so called an American option winwhich the buyer is allowed to choose the time of exercise the option. We haveAmerican call option - situation is similar as in the case of European call option with theonly difference that the buyer can choose the exercise time τ . The profit then is of theform

(S1(τ)−K)+

American put option - similar situation to the European put option again with possibilityto choose the exercise time τ

(K − S1(τ))+ .

The exercise time τ is in fact a stopping time: τ ≤ t ∈ Ft for t ≥ 0 i.e. a random timewhich is chosen basing on available information.We have therefore the following time horizon0 T (τ)In options the contract terminates at time T or τ and then we get a possible payoff. Thequestion (and our problem) is: how much the option should cost at time 0? It appearsthat there is a certain ideology behind pricing of financial instruments which is explainedin the next section.

11.2 Wealth process

Given an initial capital v we consider the wealth process V defined as followsV (0) = v, V (t)Our investment strategy consists of the number Ni of i-th assets held in our portfolio,or portions πi of our capital V invested in the i-th assets(N0(t), N1(t), . . . , Nd(t)) or (π0(t), π1(t), . . . , πd(t))Then we have

V (t) =d∑

j=0

Nj(t)Sj(t) = V (t− 1) +d∑

j=0

Nj(t− 1)(Sj(t)− Sj(t− 1)) =

35

V (t− 1) + N(t− 1) ·∆S(t) = V (t− 1)d∑

j=0

πj(t− 1)ζj(z(t), ξ(t))

with ∆Sj(t) = Sj(t)− Sj(t− 1).Such property usually is called selffinancing property of the portfolio or wealth processV v,π(t)With the use of selffinancing portfolios we define option prices. For a given contingentclaim H we define the seler price ps(H) as follows:seller:

ps(H) = inf v : ∃π, V v,π(T ) ≥ H ,

which means that it is a minimal initial capital that allows us to hedge the claim H attime T (to get V v,π(T ) ≥ H with probability 1).We also have the buyer price pb(H) of the contingent claim Hbuyer:

pb(H) = supv : ∃π, V −v,π(T ) + H ≥ 0

which means that it is a maximal value of the initial capital v such that with a debt vwe are able to construct a portfolio which together with income H gives us at time Tnonnegative position with probability 1 (we shall have no debts at time T )We clearly have that

pb(H) ≤ ps(H).

Definition 6 Absence of arbitrage (AA) We say that we have an absence of arbitrage(AA) if there are no portfolio π, such that V π,0(t) ≥ 0 for t ∈ [0, T ] and P

V 0,π(T ) > 0

>

0

The interval [pb(H), ps(H)] is called (AA) interval.

11.3 Martingale measures

On a given (Ω, F, P ) probability space we define so called martingale measure. Denoteby B(t) the value at time t of the banking account consisting of an investment at time 0equal to B(0) = 1. We call it discount factor.

Definition 7 A probability measure Q is called a martingale measure, if Q ∼ P (equiva-lent) and (Si(t)

Bt) is a Q martingale i = 1, 2, . . . , d that is it is Q integrable (EQ

Si(t+1)Bt+1

<

∞) and

EQ

Si(t + 1)Bt+1

|Ft

=

Si(t)Bt

36

The notion of martingale measures has been introduced in the papers M. Harrison, D.Kreps (1979), M. Harrison - S. Pliska (1981), in which the absence of arbitrage, whichis a fundamental economical requirement for any economical consideration was explainedin terms of martingales measures (methodology coming from stochastic processes theory!). These results were first formulated for finite markets (the case when Ω was finite) andtheir generalization for a long time (till 1990) was an open problem.

11.4 Fundamental theorem of mathematical finance - discrete time

The above mentioned relationship between absence of arbitrage and martingale measuresin a general discrete setting can be formulated as follows

Theorem 4 (Dalang Morton Willinger 1990) (AA) ≡ existence of a martingale measureQ (Q 6= ∅)

We denote by Q - family of all martingale measuresIn what folows we shall need a notion of generalized martingale and generalized super-matingale.

Definition 8 We say that (Zt) is a generalized (Ft) martingale with respect to measureP whenever

EP[Z+

t+1|Ft

]< ∞, EP

[Z−t+1|Ft

]< ∞

and

EP [Zt+1|Ft] := EP[Z+

t+1|Ft

]− EP

[Z−t+1|Ft

]= Zt

If instead of the last condition we have EP[Z+

t+1|Ft

]− EP

[Z−t+1|Ft

]≤ Zt, then (Zt) is a

generalized (Ft) supermartingale with respect to measure P .

Proof. We prove only sufficiency of the existence of martingale measure.We start with the following

Lemma 4 (Jacod Shiryaev 1998) Generalized martingale (Xt)t=0,1,...,T , such that EP X+T <

∞ or EP X−T < ∞ is a martingale. Whenever sequence (ξn) is Fn-adapted, and (Xn) is a

generalized martingale then martingale transformation

t∑

n=1

ξn−1(Xn −Xn−1)

is also a generalized martingale.

37

Under investment strategy π = (N0(t), N1(t), . . . , Nd(t)) we have

V 0,πt =

t−1∑

n=0

Nn ·∆Sn

where ∆Sn = Sn+1−Sn, is a generalized martingale with respect to measure Q. Thereforeif V 0,π

T ≥ 0 then by Lemma 4, (V 0,πt ) is a martingale with respect to the measure Q, that

isEQ[V 0,π

T ] = V0 = 0,

which means that V 0,πT = 0, so that there are no arbitrage.

2

Next theorem is fundamental in pricing of European contingent claims (claims with fixedmaturity T ). We have

Theorem 5 Under (AA) if EQH− < ∞ then ps(H) = supQ∈QEQH and if EQH+ < ∞then pb(H) = infQ∈QEQH.

Proof. Steps:1. The family

EQ(H|Ft), Q ∈ Q

is directed upward i.e. for Q1, Q2 ∈ Q there is Q3 ∈ Q,

such thatEQ3 [H|Ft] ≥ max

EQ1 [H|Ft], EQ2 [H|Ft]

.

Hence there is a process Yt := esssupQ∈QEQ[H|Ft] is attained by an increasing sequenceof random variables.2. If Q1, Q2 ∈ Q, then there is Q3 ∈ Q, for whichEQ3 [H|Ft] = EQ1 [EQ2 [H|Ft+1]|Ft]3. We prove here the following

Lemma 5 The process Yt := esssupQ∈QEQ[H|Ft] is a generalized Q-supermartingale foreach Q ∈ Q, i.e. EQ[Yt+1|Ft] ≤ Yt, dla t = 0, 1, . . . , T − 1.

Proof. From the step 1 we know that there is a sequence of measures Qn ∈ Q, such thatEQn [H|Ft+1] ↑ Yt+1. By step 2 for a given measure Q ∈ Q there is a sequence Q′

n ∈ Qsuch that

EQn [H|Ft] ≥ EQ[EQn [H|Ft+1]|Ft].

HenceYn ≥ EQ[EQn [H|Ft+1]|Ft]

and by Lebesgue monotonical convergence theorem we obtain Yn ≥ EQ[Yt+1|Ft], whichcompletes the proof.

38

2

To complete the proof of theorem we need another important result so called optionaldecomposition, which we formulate in the next section.

11.5 Optional decomposition theorem and its consequences

4. We continue the proof recalling first the following result:

Theorem 6 (Follmer i Kabanov (1998)) If Q 6= ∅ and Yt is a generalized Q-supermartingalefor each measure martingale measure Q, then there is a strategy π = (Nt)t≥0 and increas-ing Ft adapted process dt with d0 = 0 such thatY0 +

∑T−1t=0 Nt ·∆St − dT = YT

Proof of pricing theorem cont. Since YT = H , by the definition of ps(H) and theabove theorem we have Y0 = supQ∈QEQH ≥ ps(H). It remains to show the inverseinequality. Since

x +T−1∑

t=0

Nt ·∆St ≥ H

from Jacod Shiryaev’s Lemma 4 it follows that (∑n−1

t=0 Nt ·∆St) is a martingale with respectto any Q ∈ Q. Hence we obtain x ≥ EQH for any Q ∈ Q, which completes the proof.

2

11.6 Remarks on the seller price

In the case when there are not transaction costs we have the following simple relationbetween the buyer and seller prices

pb(H) := −ps(−H)

or equivalentlypb(H) = sup

x : ∃π V −x,π

T + H ≥ 0, Pa.e.

,

which is the way we defined buyer price.Since V −x,π = −V x,−π we have

pb(H) = sup x : ∃πV x,πT ≤ H, Pa.e. .

Furthermore

ps(H)− pb(H) = inf x : ∃πV x,πT ≥ H+ inf

x : ∃πV x,π

T ≥ −H

≥ infx + x : ∃π∃πV x,π

T ≥ H,X x,πT ≥ −H

≥ inf x : ∃πV x,π

T ≥ 0 ≥ 0

39

which means that ps(H) ≥ pb(H). Such result is intuitively clear and follows immediatelyfrom Theorem 5.

11.7 Complete markets

Financial market is complete if for any FT measurable bounded contingent claim H thereis a strategy π and initial capital v such that V v,π

T = H, P a.e.. Such strategy is calledreplicating strategy. We have

Theorem 7 (Jacod Shiryaev 1998) Market is complete iff Q is a singleton.

There are not too many markets which are complete. In general markets are incomplete.However we have the following exampleExample of a complete market: Cox Ross Rubinstein model (1979) We consider the cased = 1 with the random rate of return ζt = St−St−1

St−1and (ζt) being a sequence of i.i.d.

random variables. Furthermore we assume thatP ζt = a > 0 and P ζt = b = 1− P ζt = a > 0 where a i b, b > 0 > a > −1, so thatthere are only two possible values of ζ(t): the random rate of return is either below zero(is equal to a and the price of the asset diminishes) or is above zero (is equal to b and theprice of the asset is increasing).For such evolution of asset prices we can characterize martingale measures Q. By the verydefinition we should have

EQ ζt+1|Ft = aQ ζt+1 = a|Ft+ b(1−Q ζt+1 = a|Ft) = 0,

Therefore Q ζt+1 = a|Ft = bb−a , and Q ζt+1 = b|Ft = −a

b−a . It can be further shown thatunder Q, (ζt) are also of i.i.d. random variables.

11.8 Pricing under Cox Ross Rubinstein model

We consider the case when the contingent claim is of the European form H = h(ST ).

Theorem 8 We have

ps(H) = pb(H) =T∑

i=0

T !(T − i)!i!

(b

b− a

)i ( −a

b− a

)T−i

h(S0(1 + a)T−i(1 + b)i),

Proof. It Follows form Bernoulli scheme under measure Q or from direct replicationargument.

2

40

11.9 Continuous time models

In continuous time case the modeling and results are more difficult. Our standard assump-tion is that the asset prices (S(t)) are in general semimartingales (roughly speaking theyare the sums of local martingales and processes with locally finite variation - for simplicitywe shall have in mind everywhere here (S(t)) of the form (3) defined in section 3.4). Thenthe wealth value of our investments (π(t)), where we assume that we are allowed to changethe portfolio instantaneously is of the form

V v,π(t) = v +∫ t

0π(u)V v,π(u)dS(u)

with an integral with respect to the semimartingale (S(t)) considered as a limit of integralwith simple (piecewise constant) integrands. For the purpose of this lecture we skip heremathematical precision which would lead us far away to the stochastic integration theory.In general continuous time setting the only admissible strategies are so called tame strate-gies i.e. such (predictable) strategies π for which there is a deterministic constant K < 0such that for each t we have that V v,π(t) ≥ K. P a.e.. The use of other strategies maylead to arbitrage.The absence of arbitrage is defined now by analogy to discrete time case

Definition 9 Absence of arbitrage (AA) We say that we have an absence of arbitrage(AA) if there are no admissible strategy π, such that V π,0(t) ≥ 0 for t ∈ [0, T ] andP

V 0,π(T ) > 0

> 0.

11.10 Two important examples

We consider now two examples which show that without restricting to tame strategy evenusing simple strategies we are able to create an arbitrage.Example 1.Let (Bt)t≥0 be a Brownian motion starting from 1, i.e. B(0) = 1. Denote by τ =inf t > 0 : Bt = 0 first entry time to 0. We define the asset price St as follows

St = Btan( tπ2

)∧τ

for t ≤ 1, and S1 = Bτ for t = 1.Consider the fixed strategy N(t) = −1Then we have that

(N · S)1 =∫ 1

0N(t)dSt = 1

41

P a.s. and (N · S)0 = 0, so that we have an arbitrage for a simple strategy based onshortselling of one asset.This strategy is not tame since before reaching 0 Brownian motion can take arbitrarilylarge values.Example 2.Assume now that we are given the discrete time process (x(n)) defined as follows x(0) = 1and P x(1) = 1 = 1

2 , P x(1) = −1 = 12 , . . .,

Px(n) = 2n−1

= 1

2 , Px(n) = −2n−1

= 1

2 .We construct the continuous time process (z(t))z(t) =

∑kni=0 x(i) for kn ≤ t < (k + 1)n

and letSt = z(tan( tπ

2 ) ∧ τ) for t ≤ 1, and S1 = z(τ) for t = 1, with τ = inf t > 0 : zt = 0Then for N(t) = −1, (N · S)1 =

∫ 10 N(t)dSt = 1 a.s. and (N · S)0 = 0

so that again we have an arbitrage.

11.11 No free lunch with vanishing risk (NFLVR)

In the case of continuous time we don’t have a version of Theorem 4. We have to introducenew notion of so called no free lunch vanishing risk condition defined as follows(NFLVR): there are no sequence of strategies (πk(t)) such that:∃αk

, PV πk,0(t) ≥ αk, t ∈ [0, T ]

= 1

∀k,∃δ1,δ2>0, PV πk,0(T ) > δ1

> δ2, and

V πk,0(T ) ≥ − 1k

We have then the following theorem

Theorem 9 (F. Delbaen-W. Schachermayer 1994)

(AA) ⇐= (NFLV R) ⇐⇒ Q 6= ∅

where Q the family of equivalent measures, such that (S(t)Bt

) is a local martingale i.e. there

is a sequence of stopping times (τn) such that ∀n

(S(t∧τn)Bt∧τn

)is a Q - martingale.

As we see in continuous time case we don’t have equivalence we have only implication.

11.12 Black and Scholes formula

The most known (and applicable) result in continuous time mathematical finance is theso called Black Scholes formula established in the paper Black, Fischer; Myron Scholes(1973). ”The Pricing of Options and Corporate Liabilities”. Journal of Political Economy

42

81 (3) 637-654, by Fischer Black, 11.01.1938 30.08.1995, PhD Applied Math. Harward1964, and Myron Scholes, 1.07.1941, PhD Economics, University of Chicago 1969. M.Scholes received Noble price in 1997 for this formula (Black died two years earlier). Theformula concern pricing of European call option with maturity T and striking price K onthe market consisting of the banking account with constant interest rates rt = r, whichmeans that B(t) = ert and one asset with price given by the formula

S(t) = S(0)e∫ t

0ads+

∫ t

0σdW (s) (22)

Consequently we want to price H = (S(T ) − K)+. This market is another (continuoustime) example of complete market which means that the seller and buyer prices are thesame. We have

ps(H) = pb(H) = S(0)N(d1(S(0), T ))−KerT N(d2(S(0), T ))

where d1(s, t) = ln sK

+(r+0.5σ2)t

σ√

t, d2(s, t) = d1(s, t) − σ

√t. Notice that we have an explicit

formula for the price, which is rather simple (can be easily computed) and does not dependon a. The explicit form of the price was the main factor that stimulated frequent use ofthe formula although the market considered here is too simple to model real market.

11.13 References

R.N. Dana, M. Jeanblanc, Financial Markets in Continuous Time, Springer 2003H. Follmer, A. Schied, Stochastic Finance. An Introduction in Discrete Time, W. deGruyter, 2002,J. Jakubowski, A. Palczewski, M. Rutkowski, ÃL. Stettner, Matematyka Finansowa, Instru-menty pochodne, WNT 2003 (in Polish)I. Karatzas, Lecture on the Mathematics of Finance, AMS 1997,A. Shiryaev, Essentiels of Stochastic Finance, World Scientific 1999,S. Shreve, Stochastic Calculus for Finance I. The Binomial Asset Pricing Model. Springer2003,S. Shreve, Stochastic Calculus for Finace II. Continuous-Time Models. Springer 2003,A. Weron, R. Weron, Inzynieria Finansowa, WNT 1998,

12 Coherent risk measures

In the last part of the lectures we come back to coherent measures of risk. We presenthere some most recent result stimulated by the necessity to measure risk in the real worldproblems, partly stimulated by the works of Basel II and III arrangements.

43

12.1 Representation results

(Ω, F, P ) a given probability space we consider here the class of financial positions X = L∞.We have the following fundamental result

Theorem 10 (Artzner Ph., Delbaen F., Eber J.M, Heath D. (1999)) Let ρ be a coherentrisk measure. The following conditions are equivalent:1. there is a closed convex set of probability measures Q such that any q ∈ Q is absolutelycontinuous with respect to P and for X ∈ L∞

ρ(X) = supEQ [−X] ; Q ∈ Q

2. ρ satisfies the Fatou property, i.e. if (Xn) ⊂ L∞ be a sequence of uniformly boundedrandom variables converging in probability to X, thenρ(X) ≤ lim infn→∞ρ(Xn).3. If (Xn) is a uniformly bounded sequence that decreases to X then ρ(Xn) converges toρ(X).

2

Let

ρm(X) = −∫ 1

0q+u (X)m(du)

where m is a probability measure on (0, 1). Such a measure is monotone, cash additiveand positive homogeneous.We have the following examples of such measures:1. V aRα(X) = −q+

α (X), for m = δα,2. expected shortfall (conditional VaR): ESα(X) = 1

α

∫ α0 V aRu(X)du, for m uniform

distribution over (0, α),3. spectral risk measures (weighted VaRs): ρφ(X) =

∫ 10 V aRu(X)φ(u)du, with m(du) =

φ(u)du, where φ : [0, 1] → [0,∞) is a density on [0, 1] and u 7→ φ(u) is decreasing,Furthermore we have

Theorem 11 ρm is subadditive iff it is a spectral risk measure.

which means that the set of coherent measures is equivalent to the set of spectral riskmeasures

12.2 Law invariant measures of risk

In majority of applications the risk measure depends on the distribution (law) of randomfinancial position only.

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Definition 10 We say that a map ρ : L∞ → R is law invariant if ρ(X) = ρ(Y ), wheneverX, Y have the same probability law.

In this section we shall use the following assumptionAssumption:(A) (Ω, F, P ) a fixed probability space with measure P non-atomic; L∞(Ω, F, P )We have

Theorem 12 (Kusuoka 2001) Under (A) the following conditions are equivalent:1. there is a (compact, convex) set M0 of probability measures on [0, 1] such thatρ(X) = sup

∫ 10 ESαm(dα) : m ∈M0

, for X ∈ L∞

2. ρ is a law invariant coherent risk measure with the Fatou property

12.3 Comonotone measures of risk

We start with the following definition

Definition 11 We say that a pair X and Y of random variables is comonotone if(X(ω)−X(ω′))(Y (ω)− Y (ω′)) ≥ 0, P (dω)× P (dω′) a.s.We say that a map ρ : L∞ → R is comonotone if ρ(X + Y ) = ρ(X) + ρ(Y ) for anycomonotone pair X,Y ∈ L∞.

As an example of a comonotone measure of risk may serve expected shortfall ESα.In fact we have the following identification result for the class of comonotone measures ofrisk

Theorem 13 (Kusuoka 2001) Under (A) the following conditions are equivalent1. there is a probability measure m on [0, 1] such that for X ∈ L∞

ρ(X) =∫ 10 ESα(X)m(dα)

2. ρ is a law invariant and comonotone coherent risk measure with the Fatou property.

12.4 Further properties of coherent measures of risk

We present here further results on measures of risk. We have

Theorem 14 (Kusuoka 2001) Under (A) if ρ is law invariant coherent measure of risksuch thatρ(X) ≥ V aRα(X), for X ∈ L∞

then we have ρ(X) ≥ ESα(X) for X ∈ L∞.

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12.5 Historical risk estimators

Since law invariant measures of risk depend on the distribution of random variable onlyinstead of ρ(X) we may write ρ(FX).Then we can estimate risk measures using empirical distributions:

F empX (x) =

1n

n∑

i=1

1x≥xi

where X = (x1, x2, . . . , xn) is a random sample, we denote such estimation as

ρh(X) = ρ(F empX ).

Whenever ρm(X) = − ∫ 10 q+

u (X)m(du) we haveρh(X) = −∑n

i=1 x(i)m(( i−1n , i

n ]) where x(i) is the i-th element of X = (x1, x2, . . . , xn).Consider now the examples of estimators:1. ˆV aR

hα(X) = −x([nα]+1)

2. EShα(X) = − 1

nα

(∑[nα]i=1 x(i) + x([nα]+1)(nα− [nα]

)

3. ρhφ(X) = −∑n

i=1 x(i)

∫ in(i−1)

n

φ(u)du

12.6 Consistency of risk estimators

Risk estimator is consistent for distribution F whenever

ρ(X1, X2, . . . , Xn) → ρ(F ),

as n →∞ a.s. for and i.i.d. sequence Xi with distribution FIn particular,

ˆV aRhα(F ) is consistent for F such that q+

α (F ) = q−α (F ),ES

hα(F ) and ρh

φ(X) are consistent as linear functions of order statistics (van Zwet 1980)For further consideration it will be important for us to introduce so called Levy distancebetween distribution functions:

d(F, G) = inf ε > 0 : F (x− ε)− ε ≤ G(x) ≤ F (x + ε) + ε, for x ∈ RIt is clear that it is a metric which is consistent with the weak topology of laws (generatedby random variables with distributions F and G.Denote byLn(ρ, G) the distribution of the estimator of ρ based on the sample (X1, X2 . . . , Xn) ofi.i.d. random variables with distribution G.

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12.7 Robustness of risk estimators

Consider a family C of distribution functions.

Definition 12 A risk estimator ρ is C robust at F if for any ε > 0 there exists δ > 0 nadN0 ≥ 1 such that for all G ∈ C we haved(F,G) ≤ δ implies d(Ln(ρ, G),Ln(ρ, F )) ≤ ε for n ≥ n0.

The notion of robustness is one of principal requirements in estimation theory. We have

Theorem 15 (Cont, Deguest, Scandolo 2010) Let ρ be a risk measure and F ∈ C. If ρh

- the historical estimator of ρ is consistent with ρ at every G ∈ C, then the following areequivalent1. the restriction of ρ to C is continuous at F (with respect to Levy distance)2. ρh is C robust at F .

Corollary 3 If ρ is continuous in C then ρh is C robust at any F ∈ C.

Proof. (We use Glivienko-Cantelli theorem which says that empirical distributions con-verge uniformly to approximated distribution with probability 1 and in the case of con-tinuous distributions such convergence does not depend on the form of (continuous) dis-tribution)

12.8 Continuity of risk measures

We consider again risk measure ρm(X) = − ∫ 10 q+

u (X)m(du), and let Dρ be the space ofdistributions for which ρm is defined, and

Am = α ∈ [0, 1] : m α > 0We have the following result

Theorem 16 (Huber 1981). If the support of m does not contain 0 nor 1 then ρm iscontinuous for any F ∈ Dρ such that q+

α (F ) = q−α (F ) for any α ∈ Am. Otherwise ρm isnot continuous at any F ∈ Dρ.

Directly from this theorem we obtain

Corollary 4 ˆV aRhα(F ) is Cα robust at any F ∈ Cα where

Cα = F ∈ D : q+α (F ) = q−α (F )

Let m(du) = φ(u)du and φ ∈ Lq. Then ρφ is defined in Dp - the set of distributions withfinite left tail p moment (1

p + 1q = 1). We have

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Corollary 5 For any F ∈ Dp the historical estimator of ρφ is Dp robust at F if and onlyif for some ε > 0 φ(u) = 0 for all u ∈ (0, ε) ∪ (1− ε, 1) i.e. φ vanishes in a neighborhoodof 0 and 1.

Therefore

Corollary 6 The historical estimator of any spectral risk measure ρφ defined on Dp isnot Dp - robust at any F ∈ Dp. In particular, the historical estimator ESα is not D1 -robust at any F ∈ D1.

This is an important deficiency of spectral risk measures and consequently of coherentmeasures of risk. On the other hand V aR is not coherent but its historical estimator isrobust. There is a long discussion among risk specialist which measure is good in practice.It seems that it should be a measure which is between coherent risk measures and valueat risk. In particular the historical estimator of the risk measure

1α2 − α1

∫ α2

α1

V aRu(F )du

is a D1 robust and such measure is not far away from coherent measures of risk.

12.9 References

Acerbi, C., Spectral measures of risk: a coherent representation of subjective risk aversion.J. Bank. Finance, 2002, 26, 1505-1518.Artzner, P., Delbaen, F., Eber, J. and Heath, D., Coherent measures of risk. Math.Finance, 1999, 9, 203-228.Cont, R., Deguest, R., and Scandolo, G., Robustness and sensitivity analysis of riskmeausurement procedures. Quantitative Finance, 10(6):593–606, 2010.Follmer, H. and Schied, A., Convex measures of risk and trading constraints. FinanceStochast., 2002, 6, 429-447.Follmer, H. and Schied, A., Stochastic Finance: An Introduction in Discrete Time, 2004(Walter de Gruyter: Berlin).Huber, P., Robust Statistics, 1981 (Wiley: New York).Kusuoka, S., On law invariant coherent risk measures. Adv. Math. Econ., 2001, 3, 83-95.Van Zwet, W., A strong law for linear functions of order statistics. Ann. Probab., 1980,8, 986-990.

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