Stochastic Models in Finance

Begleitmaterial zur Vorlesung von

o. Univ.Prof. Dr. Georg Pflug

Hinweise:

Dies ist nur ein Begleitmaterial und ersetzt nicht den Besuch der Vorlesung.

Dieses Werk ist urheberrechtlich geschutzt. Jede Vervielfaltigung ist verboten.

Institut fur Statistik und Decision Support Systeme

Universitatsstraße 5/9, 1010 Wien

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1 A Glossary of Important Terms

• The term Asset (Aktiva) describes the collection of all values a company holds,

such buildings, machinery, but also securities and other financial contracts. Assets

appear on the active side of the company’s balance sheet. Besides the cited tangibles,

sometimes also intangibles, like the company’s reputation are counted as assets.

• A security (Wertpapier) is a certificate proving the the owner has invested in some

organizaions equity or debt. Securities are standardized contracts, allowing the

holder to sell its rights without informing the other party in the marketplace (A

shareholder may sell the share without informing the company, a bondholder may

sell the bond without informing the debtor). The opposite of securities are OTC-

contracts (over the counter contracts), which are nonstandardized and where the

two parties are fixed for the duration of the contract.

• A bond (Anleihe) is a fixed income security, which pays to its holder regularly

the coupon/interest and at maturity time the face value. We distinguish between

(1) government bonds (the government is the issuer) and (2) corporate bonds (a

company is the issuer).

Bonds are subject to interest risk (the market interest rate may change) and the

default risk. If the bond issuer is not able to meets the payment obligations, default

occurs and some legal actions are taken. Typically, on default only a fraction (the

recovery rate) of the face value is paid.

Bonds are priced at the bond market. The pricing depends on modelling the mar-

ket interest rate process and the default risk. The default risk is estimated by

rating agencies (e.g. Moody’s and Standard&Poor) and expressed in rating classes

(AAA,AA,A,BBB,BB,B,CCC,D). D means default.

• A stock (Aktie) is a security, which gives the holder ownership rights of the company.

Stockholders select by voting managers and control the firms activity. Stocks are

priced through the bid-ask mechanism of a stock exchange and are subject to the

value risk. Also called shares.

• equity (Stammkapital) is the shareholder’s stake in the company.

• hybrid instruments, such as convertibles, mandatory convertibles and warrants are

combinations of bonds and stock securities.

• A convertible bond (Wandelanleihe) is a corporate bond, which gives the owner the

option to exchange it at a predetermined date to a predetermined number of stocks1.

1”predetermined” may mean that the number of stocks or the price are numerically fixed at the issue

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• A warrant is is a corporate bond, which gives the owner the option to buy at a

predetermined date a predetermined number of stocks at a predetermined price.

• Derivatives are contracts, which perform in close relation to another basic contract,

the underlying. Examples are: Options (underlying is the stock to be bought or

sold), futures (underlying is the today’s commodity price), swaps (underlying is the

interest rate or exchange rate process), credit derivatives (underlying is the basic

credit portfolio).

• An option gives its holder the right to buy or sell stocks at a predetermined price.

A european option may be exercised only at the predetermined date. An american

option may be exercised at any time before. A (plain vanilla) call option gives

the right to buy a given number stocks at a predetermined price (the strike price).

A (plain vanilla) put option gives the right to sell a given number stocks at a

predetermined price. Other than plain vanilla options are called exotic options

(e.g. a barrier option, which is activated if the price of the underlying crosses some

predetermined level. A barrier option comes in the forms of a trigger option: the

right starts, or a knockout option: the right ends). A Margrabe option also called

exchange option is the right to exchange one type of stock by another type of stock.

• A forward is an agreement to buy or sell a commodity or a financial asset at a

specified future date for a fixed price. It is a completed contract and the commodity

or financial asset will be delivered, unlike an option, which may be exercised or not.

Forwards are OTC contracts.

• A future is the undertaking to buy or sell a standard quantity of a financial asset

or commodity at a future date and at a fixed price. Futures resemble forwards, but

are standardized contracts (i.e. every future contract has standardized terms that

dictate the size, the unit of price quotation, the delivery date and contract months)

and must be traded on a recognized exchange.

• Swaps are exchanges of the cash flows between two counterparties designed to offset

interest rates or currency risks and to match their assets to their liabilities. The

parties to a swap do not exchange principal, or the underlying fixed amount of debt,

but just cash-flow, or the interest payments.

• liquidity of the market is the property that standard transactions are possible (at

least at the trading times)

time of the contract or that at least the exact value may be calculated using a formula, which was agreedon at the issue time of the contract. In this case the number may depend on the actual stock price or onother variables observable in the market

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• completeness of the option market means that contracts with all possible strike

prices and maturities are available and traded.

2 The time value of money: deterministic interest

rates

The basic instrument to deal with is a financial contract. A financial contract, agreed with

to parties A and B obliges these parties to transfer amounts ct of money (the cash-flows

of the contract) at certain times t. At the time of contracting, neither the exact amounts

or the exact times must be determined, however the exact formula how to get to these

data should be fixed (e.g. a reference to the stock market or to some indices such as the

”Sekundarmarktrendite”).

Distinguish financial contracts from commodity contracts, where one part pays money

while the other delivers some commodity at agreed times, quantities and costs, again

either determined in advance or based on some formula involving unknows at the time

of contracting. Examples of commodity contracts are energy contracts such as energy

futures or swing options.

Contracts in this sense are: bank accounts, credits and loans, shares (stocks), bonds

(governmental fixed income securities) corporate bonds (private fixed income securities),

derivatives (futures, forward contracts, options, swaps, etc.)

?

?

year

6 66

6

?

6

?

?

6 6

1 2 3 T − 1 T

c1 c2 c3 c4

A cash-flow structure c = (c1, . . . , cT )

Money today is better than the same amount tomorrow, or in some other future time

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(Why?). If the market determines that K unit today equals K(1 + r) units in one year,

we call r the (one-year) interest rate.

For bank accounts, if a capital is hold for more than one year, typically a capitalization of

the accrued interests occur at the beginning of each calender year. Thus K units on the

first of January 2000 result is a capital of K(1+ r)t units on the first of January 2000+ t.

Notice the Bernoulli inequality:

(1 + r)t > 1 + rt

for r > 0, t > 0.

If the capital K(t) varies over the year, the accrued interest in the year is

r ·∫ 1

0

K(t) dt

(unit is one year) or in discrete time

r · 1

365

365∑

d=1

K(d)

(unit is one day) i.e. the interest is calculated on the basis of the average capital during

the year. Thus, a capital of one unit in January and zero in the other month produces the

same interest as a capital of one unit in December and zero in the other month, although

in the first case the bank had the capital much earlier and should pay more interest (but

does not).

For some contracts, capitalization occurs several times per year (quarterly, monthly).

If capitalization takes place n times per year, after one year the capital K is worth

K(1 + r/n)n, which, for n →∞ tends to exp(r).

Example. r = 0.05

1 + r 1.05

(1 + r/2)2 1.050625

(1 + r/10)10 1.05114

(1 + r/360)360 1.051267

exp(r) 1.051271

2.1 Pricing of interest rate dependent contracts

For t = 1, . . . , T , let rt be the yearly interest rates for not default prone zero coupon

bonds maturing at time t. The stock exchange fixes by its bid-ask mechanisms πt, the

today’s price of a default free zero-coupon bond with maturity t and principal value 1.

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From this, we calculate the interest rate for maturity t: r0,t := π−1/tt − 1, i.e.

πt = (1 + r0,t)−t.

The collection of interest rates is the yield curve: (r0,1, . . . , r0,T )

1 2 3 4 5 6 70.03

0.035

0.04

0.045

0.05

0.055Yield curve

Years

inte

rest

rat

e

A typical yield curve

We begin with looking at financial contracts, which pay fixed, predetermined sums at

fixed, predetermined times, for simplicity, the times 1, 2, . . . . Suppose that a contract

consists in exchanging the amount ct at time t, for t = 1, . . . , T . The values ct may be

positive (the contract holder gets money) or negative (the contract holder pays money).

The vector c = (c1, . . . , cT ) is called the cash-flow vector produced by the contract. The

today’s fair price of this contract is denoted by π(c1, . . . , cT ). We will require that the

pricing operator π is a linear mapping from RT to R, however bearing in mind that later

on this requirement will be weakened to certain nonlinear pricing rules.

Pricing rule A: The price equals the discounted cash-flow

Introduce the notation πt := π(0, . . . , 0︸ ︷︷ ︸t−1

, 1, 0, . . . , 0) = (1 + r0,t)−t for a contract paying 1

at time t and nothing at other times. Because of the linearity of the pricing operator

π(c1, . . . , cT ) =T∑

t=1

ct(1 + r0,t)−t =

T∑t=1

ctπt.

2.2 Sensitivity, elasticity and duration

Recall the notions of sensitivity and elasticity of functions: The sensitivity of a function

f is its derivative

f ′(x) =∂

∂xf(x).

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The elasticity is∂ log f(x)

∂ log x=

x · f ′(x)

f(x).

If the yield curve is flat, i.e. r0,t ≡ r, then the price of the contract (c1, . . . , cT ) is a

function of only one parameter r. Introduce

• the (interest rate) sensitivity

S(c1, . . . , cT ; r) = −∂π(c1, . . . , cT )

∂(1 + r)= −∂π(c1, . . . , cT )

∂r

=T∑

t=1

ct∂πt

∂r=

T∑t=1

ctt(1 + r)−t−1

• (interest rate) elasticity

D(c1, . . . , cT ; r) = −∂ log π(c1, . . . , cT )

∂ log(1 + r)= − ∂

∂ log(1 + r)log[

T∑t=1

ct(1 + r)−t]

=

∑Tt=1 ctt(1 + r)−t

∑Tt=1 ct(1 + r)−t

=S(c1, . . . , cT ; r) · (1 + r)

π(c1, . . . , cT ).

Since

D(0, . . . , 0︸ ︷︷ ︸t−1

, 1, 0, . . . , 0; r) = − ∂ log πt

∂ log(1 + r)= t.

this elasticity is also called the ”duration”, or more precisely, the MaCauley dura-

tion. Notice that, in contrast to the sensitivity, the duration is not linear in ct.

Suppose that our portfolio consists of assets and liabilities. We may view assets and

liabilities as just types of general financial contracts. Both produce future cash-flows,

however with different signs. Typically, an asset will produce nonnegative cashflows in

the future, whereas a liability will require nonpositive cashflows. An asset has a positive

and a liability a negative today’s price.

Let the assets be numbered 1, . . . , MA and the liabilities be numbered MA + 1, . . . ,M =

MA + ML. Each of the M contracts produces cash flows c(i)t , i = 1, . . .M . Suppose that

we hold xi, i = 1,M pieces of each contract. Then the cash flows produced at time t are∑Mi=1 c

(i)t . The price of the total AL-portfolio is

π =M∑i=1

xi π(c(i)1 , . . . , c

(i)T ),

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the interest sensitivity of the the portfolio is

S =M∑i=1

xi

T∑t=1

c(i)t t(1 + r)−t−1,

the duration is

D =

∑Mi=1 xi

∑Tt=1 c

(i)t t(1 + r)−t

∑Mi=1 xi

∑Tt=1 c

(i)t (1 + r)−t

= S · (1 + r)/π.

If a portfolio of assets and liabilities is composed in such a manner that the interest

sensitivity is zero, we say that the portfolio is immunized w.r.t interest rate changes.

Notice that S is zero if and only if D is zero. Thus the duration of an immunized

portfolio is zero. This is equivalent to say that a small change in interest rates will not

influence the value of the whole portfolio.

Exercise. Suppose that we may buy two types of assets with cashflows c(1)1 = 5, c

(1)2 = 10

and c(2)1 = 6, c

(2)2 = 6. On the other hand, our liability has cashflows c

(3)1 = −29, c

(3)2 =

−41. The interest rate is 5%. Find an immunized portfolio.

2.3 Forward interest rates and instant interest rates

Suppose that the yield curve r0,t is given. For s < t, the forward interest rates rs,t are

defined through the relation

(1 + r0,s)s · (1 + rs,t)

t−s = (1 + r0,t)t.

We call ρt = limδ→0 rt,t+δ the instant interest rate (overnight rate, spotrate), if it exists.

Let T − t = nδ. By

(1 + rt,T )T−t = (1 + rt,t+δ)δ · (1 + rt+δ,t+2δ)

δ · · · (1 + rt+(n−1)δ,t+nδ)δ

we get by taking logarithms that

log(1 + rt,T ) =1

T − tδ

n−1∑i=0

log(1 + rt+iδ,t+(i+1)δ).

and passing to the limit n →∞

log(1 + rt,T ) =1

T − t

∫ T

t

log(1 + ρu) du.

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and

1 + rt,T = exp

(1

T − t

∫ T

t

log(1 + ρu) du

)=

[exp

(∫ T

t

log(1 + ρu) du

)]1/(T−t)

.

Introducing the product integral∫∏b

a

f(u) du = exp

[∫ b

a

f(u) du

]

one gets that

1 + rt,T =

[∫∏T

t

log(1 + ρu) du

]1/(T−t)

.

Notice that the product integral satisfies for a < b < c

∫∏b

a

f(u) du ·∫∏c

b

f(u) du =

∫∏c

a

f(u) du.

3 Stochastic interest rate models and other price mod-

els

Let Rs,t be a stochastic process to model the forward interest rates rs,t that maintains the

following minimum consistency criteria

(1 + Rs,t)t−s(1 + Rt,u)

u−t = (1 + Rs,u)u−s (1)

The models can be discrete or continuous in time and/or space. The usual way to model

random forward interest rates is to model the logarithmic spotrates X(u) = log(1 + ρu).

The process Rs,t depends on the process X(t) by

Rt,T = exp[

∫ T

t

X(u) du]1/(T−t) − 1.

The process Bt,T , which serves as stochastic discount,

Bt,T = (1 + Rt,T )−(T−t) = exp[−∫ T

t

X(u) du]

is called a stochastic deflator.

Denote by πT (t) the price of a zero coupon bond with face value 1 at time t with maturity

T . There are several pricing rules, i.e. rules how to get a price from the stochastic spotrate

process

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• The local expectation rule

πT (t) = E[exp(−∫ T

t

X(u) du)] = E[Bt,T ]

• The unbiased expectation rule

πT (t) = exp[−∫ T

t

E(X(u)) du]

• The return-to-maturity rule

πT (t) =

[E[exp(

∫ T

t

X(u) du)]

]−1

• The yield-to-maturity rule

πT (t) =

[E[exp(− 1

T − t

∫ T

t

X(u) du)]

]T−t

Typically, the prices πT (0) are observed and these prices are used for calibrating the

process X(t).

3.1 Discrete time processes

In discrete time, the spot rate process is (Xs); s = 0, 1, . . . and

(1 + Rs,t)t−s = (1 + Xs) · (1 + Xs+1) · · · (1 + Xt−1).

We typically consider processes, which are driven by a mean zero driving process Zt. First

order processes are of the form

Xt+1 = F (Xt, Zt).

3.1.1 The additive model (Random walk model)

The simplest model is the Bernoulli random walk model. Here, the driving process is

Zt = 2Yt − 1, where Yt ∼ B(1, 1/2). The process is

Xt+1 = Xt + u · (2Yt − 1)

with some starting value X0 = x0. It is easy to see that E(Xt) = x0 and Var(Xt) = u2 · t.The process is not stationary.

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3.1.2 The multiplicative model (Black-Derman-Toy or Lattice model)

The additive model has the disadvantage that the process may fall negative. The multi-

plicative model avoids this.

Xt+1 = Xt · v2Yt−1

Notice that log Xt follows the recursion

log Xt+1 = log Xt + (log v) · (2Yt − 1).

3.1.3 The autoregressive process with mean reversion

Xt+1 = Xt + a(µ−Xt) + Zt

where (Zt) is a zero mean i.i.d. process.

3.2 Continuous time processes

3.2.1 The Wiener process - Brownian motion

Let Xt be a symmetric random walk, i.e.

Xt+1 = Xt + (2 · Yt − 1),

where Yt is a sequence of independent Bernoulli B(1, 1/2) random variables. Let Wn(t) =1√nX[nt], where [nt] is the largest integer not exceeding nt. Notice that because of con-

struction, the process Wn has the following properties

• E(Wn(t)) = 0, Var(Wn(t)) = [nt]n

• Wn is a martingale: E(Wn(t)|Wn(s)) = Wn(s) s < t

• Wn has independent increments

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• Wn has stationary increments

As n → ∞, the process Wn converges to a limiting process, called the Wiener process

W (t).

The Wiener process W (t) has following properties:

• W (t) ∼ N(0, t), in particular E(W (t)) = 0, Var(W (t)) = t

• Cov(W (s),W (t)) = min(s, t)

• W is a martingale: E(W (t)|W (s)) = W (s) s < t

• W has independent increments

• W has stationary increments

Let us prove the last two assertions. By the martingale property E(W (t)|W (s)) = W (s))

for s < t we have that

Cov(W (t)−W (s),W (s)) = E[(W (t)−W (s)) ·W (s)] = EE[(W (t)−W (s)) ·W (s)|W (s)]= E[W (s)E(W (t)−W (s)|W (s))] = 0.

Hence

Cov(W (t),W (s)) = Cov(W (t)−W (s),W (s)) + Var(W (s)) = s

and

Var(W (t)−W (s)) = Var(W (t)) + Var(W (s))− 2Cov(W (t),W (s))

= t + s− 2s (since s < t)

= t− s

Assuming s < t < u < w it holds that

Cov(W (v)−W (u),W (t)−W (s)) = Cov(W (v),W (t))− Cov(W (v),W (s))

−Cov(W (u),W (t)) + Cov(W (u),W (s))

= t− s− t + s = 0

i.e. the Wiener process has uncorrelated, hence independent increments.

The Wiener process is the basis for stochastic calculus: The notation dW (t) denotes an

infinitesimal increment of W . The infinitesimal increment dW (t) is a stochastic version

of the the deterministic increment dt. Here are some formulas for dt:

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• dt2 = 2t dt

• (dt)2 = 0

However, for dW we have

• [dW (t)]2 = dt

Proof: Let t1, . . . , tn a partition of [0, T ].

E[n∑

i=1

(W (ti)−W (ti−1))2] =

n∑i=1

Var(W (ti)−W (ti−1))

=n∑

i=1

ti − ti−1 = T

Var

(n∑

i=1

(W (ti)−W (ti−1))2

)=

n∑i=1

Var(W (ti)−W (ti−1))2 = 2

n∑i=1

(ti − ti−1)2 → 0

as the partition gets finer and finer. Here we used the fact that for a normal variable

V ∼ N(0, σ2), E(V 2) = σ2 and Var(V 2) = E(V 4)− σ4 = 3σ4 − σ4 = 2σ4.

Thereforen∑

i=1

[W (ti)−W (ti−1)]2 →

∫ T

0

(dW (t))2

n∑i=1

[W (ti)−W (ti−1)]2 →

∫ T

0

dt = T,

hence (dW (t))2 = dt.

3.3 Stochastic differential equations (SDEs)

An autonomous stochastic differential equation is defined by a drift function f and a

diffusion function σ. It reads

dX(t) = f(X(t)) dt + σ(X(t)) dW (t) (2)

These processes are generated as follows. Divide the interval [0,1] in n parts (equally

sized), the equation (2) can be considered as a limit of the following equation for n

tending to ∞Xn(ti) = Xn(ti−1) + f(X(ti−1))(ti − ti−1) + σ(Xn(ti−1)) (W (ti)−W (ti−1))︸ ︷︷ ︸

∼N(0,ti−ti−1)

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The random increment

∆Xn(ti) = Xn(ti)−Xn(ti−1)

has the following property

E[∆Xn(ti)|Xn(ti)] = f(Xn(ti−1))(ti − ti−1) + o(ti − ti−1)

Var[∆Xn(ti)|Xn(ti)] = σ2(Xn(ti−1))(ti − ti−1) + o(ti − ti−1).

Proposition. The process (2) allows a stationary solution iff there are constants c1 and

c2 such that g(x) := m(x)[c1S(x) + c2] is a density, where

S(x) =

∫ x

0

s(u) du

s(x) = exp(−2

∫ x

0

f(u)

σ2(u)du)

m(x) =1

s(x)σ2(x).

The stationary density is g(x).

3.3.1 The geometric brownian motion (GBM)

The geometric brownian motion is

Y (t) = exp(σW (t)− tσ2/2)

where W (t) is a Wiener process.

Using the fact that E(eZ)=eµ+σ2/2 for Z ∼ N(µ, σ2) we get that

E[Y (t)] = E[exp(σW (t)− tσ2/2)]

= exp(tσ2/2)) · exp(−tσ2/2))

= 1

and Var(Y (t)) = exp(tσ2/2)− 1.

The geometric brownian motion is not stationary, its variance increases.

Proposition. The Ito formula.

Let dX(t) = f1(X(t)) dt + σ1(X(t)) dW (t) and let Y (t) = h(X(t), t). Then Y (t) satisfies

the following stochastic equation

dY (t) = [ht(X(t), t)+hx(X(t), t)f1(X(t))+1

2hxx(X(t), t)σ2

1(X(t))] dt+hx(X(t))σ1(X(t)) dW (t).

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Here ht = ∂∂t

h(x, t), hx = ∂∂x

h(x, t), hxx = ∂2

∂x2 h(x, t).

Proposition. Using Ito’s formula we may show that the GBM satisfies

dY (t)

Y (t)= σ dW (t).

Proof. Since Y (t) = exp (σW (t)− tσ2/2) we use Ito’s formula for f1 = 0, σ1 = 1,

h(x, t) = exp(σx− tσ2/2), ht = −h · σ2/2, hx = σ · h, hxx = σ2 · h, and get

dY (t) = dh(W (t), t)

=(−Y (t)σ2/2 + Y (t)σ2/2

)dt + σY (t) dW (t)

= σY (t) dW (t)

Since there is no drift, the GBM is a martingale.

In contrast, the GBM with drift is

Y (t) = exp(σW (t)− tσ2/2 + µt

).

It fulfills the SDE

dY (t) = µY (t) dt + Y (t)σ dW (t)

or - equivalently -dY (t)

Y (t)= µ dt + σ dW (t).

3.3.2 The Vasicek model

dX(t) = a(µ−X(t)) dt + σ dW (t)

where a is the mean reversion force, µ is the long term mean and σ2 is the constant volatil-

ity. This model allows negative interest rates. This process has stationary distribution

N(µ, σ2

2a).

3.3.3 The Cox-Ingersoll-Ross (CIR) model

dX(t) = a(µ−X(t)) dt + σ√

X(t) dW (t)

where a is the mean reverting force, µ is the long term mean and σ is the volatility

parameter. Such a model disallows negative values for interest rates. This process has

stationary distribution Gamma(2aµσ2 − 1, σ2

2a).

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3.3.4 The mean reverting GBM

dX(t) = a(µ−X(t)) dt + σX(t) dW (t).

This process has a stationary distribution, which is of Pareto type.

3.3.5 Pricing of zero coupon bonds according to the local expectation rule

Suppose that the spot interest rate follows the SDE

dX(t) = f(X(t)) dt + σ(X(t)) dW (t)

Let π(r, t, T ) be the price of a zero coupon bond at time t, which matures at time T ,

given that at time t the spot rate is r, i.e. π(r, t, T ) = E[exp(− ∫ T

tX(u) du)|X(t) = r].

Then π(r, t, T ) follows the following partial differential equation

π(r, t, T ) · r = πt(r, t, T ) + f(r)πr(r, t, T ) +1

2σ2(r)πrr(r, t, T )

with boundary condition

π(r, T, T ) = 1.

Proof.

π(r, t, T ) = E[exp(−∫ t+h

t

X(u)du) · exp(−∫ T

t+h

X(u) du)|X(t) = r]

= exp(−X(t) · h + o(h)) · E[exp(−∫ T

t+h

X(u) du|X(t) = r]

Since X(·) is Markovian, we have

π(r, t, T )[exp(X(t) · h + o(h))− 1] = E[π(X(t + h), t + h, T )|X(t) = r]− π(r, t, T )

and therefore noticing that X(t) = r,

π(r, t, T )(hr + o(h))

= E[hπt(r, t, T ) + (X(t + h)− r)πr(r, t, T ) +1

2(X(t + h)− r)2 πrr(r, t, T )|X(t) = r]

= hπt(r, t, T ) + πr(r, t, T )E(X(t + h)− r) +1

2πrr(r, t, T )E[X(t + h)− r)2]

= hπt(r, t, T ) + πr(r, t, T )hf(r) +1

2πrr(r, t, T )hσ2(r)

16

Therefore, dividing by h and taking the limit w.r.t h → 0, we get the final result

π(r, t, T ) · r = πt(r, t, T ) + f(r)πr(r, t, T ) +1

2σ2(r)πrr(r, t, T ).

Examples.

1. If the spot interest rate follows the SDE

dX(t) = c dt + b dW (t)

then

π(r, t, T ) = exp(−rτ − c

2τ 2 +

b2

6τ 3)

with τ = T − t.

2. If the spot interest rate is a Vasicek process

dX(t) = a(µ−X(t)) dt + σ dW (t)

then

π(r, t, T ) = exp−τy +1

a(e−at − 1)(r − y)− σ2

4a3(e−at − 1)2)

with

y = µ− σ2

2a2.

Proposition. The Girsanov Theorem: Change of measure for diffusion pro-

cesses.

Assume that under a probability measure P , the process X is a diffusion process with

drift

dXt = µt dt + σt dW t

Then one may construct a probability measure Q on the same probability space such that

under Q, the process X has the same diffusion, but no drift, i.e.

dXt = σt dWt

with Wt being a Wiener process under Q. The density of Q with respect to P is given by

the Girsanov formula

dQ

dP= exp

(−

∫ (µt

σt

)dW t − 1

2

∫ (µt

σt

)2

dt

).

17

where W is a Wiener process under P .

Proof. We start with a normal density with mean µ and covariance matrix Σ:

f(x, µ, Σ) =1

(2π)n2 det(Σ)

exp

(∼ 1

2(x− µ)T Σ−1(x− µ)

).

If P has density f(x, µ, Σ) and Q has density f(x, 0, Σ), then

dP

dQ=

f(x, µ, Σ)

f(x, 0, Σ)= exp

(−1

2(x− µ)T Σ−1(x− µ) +

1

2xT Σ−1x

)

= exp

(xT Σ−1µ− 1

2µT Σ−1µ

)

We show the formula only for deterministic µt and σt. For the given processes

dXt = µt dt + σt dW t under P

dXt = σt dWt under Q

we choose a partition (t1, . . . , tn) and set

Dti = Xti+1−Xti = σti · (Wti+1

−Wti).

Notice that

E[Dti ] = µti(ti+1 − ti)

Var[Dti ] = σ2ti(ti+1 − ti)

Then, under Q

fµ(D1....Dn)

f0(D1....Dn)= exp

(∑ µti(ti+1 − ti)

σ2ti(ti+1 − ti)

(Xti+1−Xti)−

1

2

∑ µ2ti(ti+1 − ti)

2

σ2ti(ti+1 − ti)

)

which converges as n →∞ to

exp

(∫µt

σ2t

dXt − 1

2

∫µ2

t

σ2t

dt

)= exp

(∫ T

0

µt

σt

dWt − 1

2

∫ T

0

(µt

σt

)2 dt

)

We need however dQ/dP under P , which is:

dQ

dP= exp

(−

∫µt

σt

dWt +1

2

∫ (µt

σt

)2

dt

)

Since

dXt = σt dWt under Q

dXt = µt dt + σt dW t under P

18

We have that

σt dWt = µt dt + σt dW t

where W is a Wiener process under P . Therefore, under P ,

dQ

dP= exp

(−

∫µt

σt

dWt +1

2

∫ (µt

σt

)2

dt

)

= exp

(−

∫ (µt

σ2t

)σt dWt +

1

2

∫ (µt

σt

)2

dt

)=

= exp

(−

∫µt

σt

dWt −∫

µt

σ2t

· µt dt +1

2

∫ (µt

σt

)2

dt

)=

= exp

(−

∫ (µt

σt

)dWt − 1

2

∫ (µt

σt

)2

dt

).

4 Pricing Rules for financial contracts

To find prices for financial contracts in the primary market (the market of goods) as well as

in the secondary market (the market of financial contracts) is done by a tatonnement (ask

and bid matching) procedure on stock exchange or options and future exchange markets

or by a bargaining procedure for over-the-counter contracts. What role can mathematics

and statistics play in pricing?

By statistical methods, one may analyze the historical path of prices and fit some models

for prediction. This is however retrospective and gives only a little help to understand

future bargaining results.

There are however situations, in which one may use mathematical methods to determine

prices. This is the situation, where a contract A has to be priced, but this contract can be

brought into close relation to contracts B1, . . . , Bk, for which prices are already known.

In such a situation, there is no freedom in pricing A, but its price has to be consistent

with the prices B1, . . . , Bk. This is completely analogous to linear algebra: if a point A

in in the linear span of the points B1, . . . , Bk, then the value of any linear operator on A

is determined by its values on B1, . . . , Bk.

It may be evident (though also sometimes debatable) that a contract for selling 5 kilos

of apples and 3 kilos of pears can be priced knowing the price for the contracts of 1 kilo

apples and 1 kilo pears respectively. The first contract is in the span of the second two.

However, in stochastic price models, the situation is not as simple as that.

19

Pricing rules in stochastic markets depend on models. Thus, given a model M, we may

infer from the prices of B1, . . . Bk the price of A, if A is in the span of B1, . . . , Bk. Now

suppose that the parties, irrespective of whether they have done the mathematical calcu-

lations or not, agree on a different price for A than our model M would predict. This does

typically not embarrass the financial modeler. He would simply modify the model M to

a new model M′ in such a way that it would predict the observed prices of A correctly.

Thus the role of contract A, priced by the market, was to improve the model. The new

model can now be used to price a further new contract C, again the parties may agree on

the price or bargain a different price – an endless game.

From this description one may see that mathematical pricing is a to be seen in an endless

correction loop between calculation of prices and estimation of models.

We review here the state-of-the-art pricing rules:

4.1 Deterministic pricing

Rule A: price = discounted cash-flow

π(c1, . . . , cT ) =T∑

t=1

ctπ(0, . . . , 0︸ ︷︷ ︸t−1

, 1, 0, . . . , 0) =T∑

t=1

ctπt.

(This is a linear pricing rule)

4.2 Stochastic pricing

If either the cash-flows Ct or the interest rates R0,t or both are stochastic, then we may

use the simple expectation rule

Rule B: price = expected, discounted cash-flow

π(C1, . . . , CT ) = E[T∑

t=1

Ct(1 + R0,t)−t].

This again a linear pricing rule.

20

4.3 Pricing through stochastic optimization

1. The price of an underlying contract is determined by stock exchange ask-bid pricing

mechanism.

2. What is the correct price of a derivative contract (the payment of the derivative

contract is a function of the value of the underlying contract)

Rule C: The price of a derivative contract is the minimal initial capital needed to

replicated (or superreplicate) the cash-flow of the derivative contract by

implementing an appropriate trading strategy.

Let St be the stochastic price of the underlying at time t and CT the cash-flows from the

derivative contract at maturity T (CT = f(ST )), where f is typically nonlinear.

Example: European call option with maturity T and exercise value K:

Ct =

0 for 0 < t < T

max(ST −K, 0) for t = T

The price π of this option is the minmal solution of the following optimization problem:

∥∥∥∥∥∥∥∥∥∥∥∥∥

Minimize (in xt, yt) : w

subject to

x0 + y0S0 ≤ w the initial capital condition

xt(1 + Rt) + ytSt+1 ≥ xt+1 + yt+1St+1 t = 0, . . . , T − 1

the self-financing condition

xT + yT ST ≥ CT the (super)replication condition

(3)

An implicit constraint is that the processes xt and yt are measurable w.r.t. the σ-algebra

Ft generated by (R1, . . . , Rt; S1, . . . , St).

xt is the amount to be invested in the bond (random interest rate Rt) and yt is the number

of shares hold of the underlying.

Remarks:

• The problem (3) does not contain any probabilities. Only the possible values of the

processes (Rt, St) and the generated σ-algebras enter the calculations.

21

• The pricing rule is homogeneous, i.e.

π(λCT ) = λπ(CT )

and subadditive, i.e.

π(C(1)T + C

(2)T ) ≤ π(C

(1)T ) + π(C

(2)T ).

The pricing rule is linear, iff the dual feasible set is a singleton (unique martingale

measure).

4.3.1 Tree models

A stochastic process St is called a tree process, if the conditional distribution of S1, S2, . . . , St−1

given St is degenerated.

A tree process taking only finitely many values may be represented by a finite tree.

N = 0, 1, 2, . . . N is the node set

0 is the root

T is set of terminal nodes (level T )

n− is the predecessor of the node n

n+ is the set of successors of node n

0

1

4

6

8

5

7

9

10

4

2

3

p 4

v 0

p 7

p 5

p 6

p 8

p 9

p 10

v 1

v 4

v 5

v 6

v 2 v

7

v 8

v 9

v 10

v 3

22

The following processes ”sit” on the nodes of the tree:

the interest rate process Rn, n ∈ N \ Tthe process of underlying’s value Sn, n ∈ NFor European options, the cash-flows Cn are only defined for the terminal nodes (n ∈ T ).

But the approach allows for much more general derivative contracts, which make payments

at all times.

The value of the derivative contract is:

∥∥∥∥∥∥∥∥∥

min w

x0 + y0S0 ≤ w

xn−(1 + Rn−) + yn−Sn ≥ xn + ynSn for all n ∈ N \ T except the root

xn−(1 + Rn−) + yn−Sn ≥ Cn for all terminal nodes n ∈ T

This is a linear program (LP). Recall that to ever linear program of the form

∥∥∥∥min cT x

A · x ≥ b

there corresponds a dual program

∥∥∥∥∥∥

min bT λ

AT · λ = c

λ ≥ 0

with the same optimal value.

In order to dualise (4) we have to introduce for every node a nonnegative dual variable

λn. The dual program is ∥∥∥∥∥∥∥∥∥∥∥

max∑

n∈T λnCn

λ0 = 1

λn = (1 + Rn)∑

m∈n+ λm

λnSn =∑

m∈n+ λmSm

λn ≥ 0

Let

γn = λn · (1 + Rn−) · (1 + Rn−−) · · · (1 + R0)

and

Zn = Sn · (1 + Rn−)−1 · (1 + Rn−−)−1 · · · (1 + R0)−1.

The dual program expressed in the new variables is

23

∥∥∥∥∥∥∥∥∥∥∥

max∑

n∈T γn · (1 + Rn−)−1 · (1 + Rn−−)−1 · · · (1 + R0)−1Cn

γ0 = 1

1 =∑

m∈n+ γm

Zn =∑

m∈n+ γmZm

γn ≥ 0

(4)

The γn may be interpreted as conditional probabilities of the node n given its predecessor.

If one sets the total node probabilities as pn = γn · γn− · γn−− · · · 1, then (Zn) will be a

martingale w.r.t (pn).

Definition. A stochastic process Zt is a martingale w.r.t. the filtration Ft, if for all t we

have that

E(Zt|Ft−1) = Zt−1.

Martingales are sometimes also referred to as fair games. For the tree model, the mar-

tingale condition reads:

Zn =

∑m∈n+ pmZm∑

m∈n+ pm

.

Pricing rule D: (dual of rule C):

Rule D (dual pricing rule): If the price of a derivative contract is finite, its value

equals the maximum of the expected, discounted cash-flow (rule B), where the

maximum is taken over all equivalent probability distributions, which make the

discounted underlying process a martingale.

Consequence. The pricing operator is homogeneous and subadditive. It is additive iff

there is just one equivalent martingale measure (the dual feasibility set contains only one

point). Economists call this the complete market situation.

Examples for stock price models which lead to the complete market situation:

• Binary trees

• The geometric Brownian motion. This process is the basis for the Black-Scholes

formula (see later)

The call-put parity.

Let π(S0, T, CT ) be the today’s price of a derivative contract with maturity T and cash-

flow CT .

24

Recall that the payment function of a European call option is [ST−K]+ = max(ST−K, 0)

and of a European put option is [ST −K]− = −min(ST −K, 0). Since [ST −K]+− [ST −K]− = ST − K, we have - if the dual problem is unique and hence the pricing

rule is linear -

π(S0, T, [ST −K]+)− π(S0, T, [ST −K]−) = π(S0, T, ST )− π(S0, T, K) = S0 −Ke−rT .

i.e.

price of the call option with maturity T and strike price K

− price of the put option with maturity T and strike price K = S0 −Ke−rT .

This relation is called the call-put parity.

4.4 Option pricing for GBM: the Black-Scholes formula

If the binomial lattice model is refined in such way that both the time intervals and the

logarithmic changes tend to zero, then the model tends to the geometric Brownian motion.

This continuous time model is the basis of many calculations in financial mathematics,

mostly because of its simplicity and the fact that it allows a unique martingale measure.

Assume therefore that the stock prices St follow a GBM with drift

St = S0 exp(σWt − tσ2/2 + µt)

where Wt is the Wiener process. The constant σ2 is called the volatility and µ is called

the drift of St.

The riskless bond process is modeled by

Bt = exp(rt); B0 = 1.

The process Bt is considered as numeraire and discounting St with Bt gives the discounted

stock process Zt:

Zt = exp(−rt)St.

The process Zt satisfies a stochastic differential equation (SDE)

dZt = Ztσ dWt + Zt(µ− r +1

2σ2) dt. (5)

A European call option has payment function [ST − K]+. We may however consider

a general payment function f (ST ). What is the correct today’s price π(S0, 0) of this

contract?

25

Let Π(S, t) be the price of this contract at time t, given that the price of the underlying

at time t is S. We know that at time of maturity

Π(ST , T ) = f(S(T )).

To calculate the today’s price Π(S0, 0), we may use the pricing rule C. This however

requires the solution of a stochastic optimization problem in continuous time and space.

Alternatively, one may also use the method of differential equations and the martingale

method (pricing rule D).

1. The method of differential equations. We consider the delta hedge, i.e. a port-

folio consisting of one unit of the derivative contract and −∆ units of the underlying

(the stocks) and call its value V (S, t). The amount of ∆ has to be fixed later.

V (S, t) = Π(S, t)−∆ · S,

i.e.

dV = dΠ−∆ dS

Since

dS = µS dt + σS dW

one gets that

dV =∂Π

∂tdt +

∂Π

∂sdS +

1

2σ2S2∂2Π

∂S2dt−∆dS.

We choose now ∆ = ∂Π∂S

with the effect that only the deterministic part remains, i.e.

dV =

(∂Π

∂t+

1

2σ2S2∂2Π

∂S2

)dt

Since there is only one riskless interest rate on the market, one must have

dV = rV dt

i.e. that (∂Π

∂t+

σ2

2S2∂2Π

∂S2

)dt = r

(Π− ∂Π

∂SS

)dt

leading to the Black-Scholes differential equation

∂Π

∂t+

σ2

2S2∂2Π

∂S2+ rS

∂Π

∂S− rΠ = 0.

Together with the boundary condition

Π(S, T ) = f(S)

26

the solution of the pricing problem may be found.

Let us indicate the solution for the special case of an European option

f(S) = [S −K]+.

It took the two authors Black and Scholes several years to find the analytical solu-

tion, but once the solution is found it is easy to check its correctness:

Π(S, t)

= SΦ

log S

K+

(r + σ2

2

)(T − t)

σ√

T − t

− Ke−r(T−t)Φ

log S

K+

(r − σ2

2

)(T − t)

σ√

T − t

Here

Φ(u) =1√2π

∫ u

−∞e−v2/2 dv.

Since

limb→∞

Φ

(a + b√

b

)=

1 a > 0

0 a < 0

one sees that for t = T

Π(S, T ) = S · 1[S>K] − k · 1[S>K] = [S −K]+

i.e. this solution fulfills the boundary condition. The price of the option at time 0

is

π(S0, T, [ST −K]+) = Π(S0, 0) = (6)

S0Φ

(1

σ√

T

[log

(S0

K

)+ (r + σ2/2)T

])−Ke−rT Φ

(1

σ√

T

[log

(S0

K

)+ (r − σ2/2)T

]).

This price may be obtained also by solving the optimal replication problem. One

may therefore ask about the portfolio strategy, which leads to the optimal value (6).

It turns out, that the number of units of the stock to be held at time t is given by

∂Π(s, t)

∂s|s=St= Φ

(1

σ√

T − t

[log

(St

K

)+ (r + σ2/2)(T − t)

])(7)

and the value kept in stock is

Ke−r(T−t)Φ

(1

σ√

T − t

[log

(St

K

)+ (r − σ2/2)(T − t)

]). (8)

27

Since the stock units equal the derivative of the option price w.r.t the actual stock

value and since this derivative is called the δ, the strategy (7)-(8) is known under

the name of the delta-hedge.

2. Martingale method. We know by Girsanov’s formula that we may find the

(unique) probability Q such that under Q the discounted stock process

dZt = (µ− r)Zt dt + Ztσ dWt

is a martingale

dZt = ZtσdWt

From the pricing rule D, we know that the correct price at time t is

e−r(T−t)EQ(f(ST ) | St)

and in particular for t = 0

π(S0, T, f(ST )) = e−rT EQ[f(ST )].

Since

St = ertZt

with

dZt = σZt dWt

under Q one gets that

dSt = ertZtσ dWt + rertZt dt

= Stσ dWt + r dt

The solution of this SDE is

St = S0 · [exp(σWt − 1

2σ2t) · exp(rt)].

Therefore the distribution of ST is

ST ∼ S0 · exp

[N((r − σ2

2)T, σ2T )

].

Again, we may specialize the result to the European option

f(ST ) = [ST −K]+.

28

and get by integration the Black-Scholes formula, the same result as in (6)

π(S0, T, [ST −K]+) = e−rTEQ([ST −K]+)

= S0Φ

log

(S0

K

)+

(r + σ2

2

)T

σ√

T

− K−rTe Φ

log

(S0

K

)+

(r − σ2

2

)T

σ√

T

.

The moneyness is defined as the Q-probability that the option pays money, i.e.

QST > K = PS0 · exp

(σ√

T · Z + (r − σ2

2)

)> K =

= Pσ√

T · Z +

(r − σ2

2

)> log

K

S0

=

= PZ >log K

S0− (r − σ2

2)

σ√

T =

= PZ <log S0

K− (r − σ2

2)

σ√

T =

= Φ

(log S0

K+ (r − σ2

2)

σ√

T

)

Here Z is a standard normal variable.

Exercise. Explain why the drift µ does not appear in the Black-Scholes formula.

4.4.1 Sensitivity analysis and the greeks

Let π(S, T ) be the today’s (t = 0) price of a derivative, which matures at time T , if the

today’s price of the underlying is S.

Define the following quantities, called the greeks:

δ =∂

∂Sπ(S, T ),

γ =∂2

∂S2π(S, T ),

θ =∂

∂Tπ(S, T ).

29

For instance, for a European call option the Black-Scholes formula gives

δ = Φ

(1

σ√

t

[log

(S

K

)+ (r + σ2/2)t

]).

γ = exp(−1

2

(1

σ√

t

[log

(S

K

)+ (r + σ2/2)t

])2

) · 1

Sσ√

2πt.

θ is more complicated.

4.5 The fundamental theorem of asset pricing

The fundamental theorem of asset pricing states that the absence of arbitrage opportu-

nities is equivalent to the existence of a (not necessarily unique) equivalent martingale

measure. We will derive and comment this theorem.

Suppose that a M -dimensional integrable stochastic price (column) vector process St, t =

1, . . . , T defined on some probability space (Ω,A, P ) is given. Each component of St

describes the price of one specific asset (stock, bonds etc.). It is assumed that each asset

may be traded without transaction costs and with negative holdings (short selling). We

assume that St is adapted to the filtration (Ft). S0 is the today’s price vector and is

deterministic and F0 is the trivial σ-algebra.

An arbitragist starts with no or negative initial capital and gets nonnegative, not iden-

tically zero capital at time T . Suppose that yt is the (row) vector of holdings he has at

time t. Then yt an arbitrage strategy yt must be Ft measurable and must a.s. satisfy

(i) yT0 S0 ≤ 0 start with no capital

(ii) yTt−1St ≥ ytSt self financing condition, t=1, . . . , T-1

(iii) yTT−1ST ≥ 0 nonnegative wealth at maturity

Here is the fundamental theorem:

In order that a stochastic model for asset prices be reasonable, i.e. excludes absurd

argitrage possibilites, it is necessary and sufficient that there exists at least one new

measure Q, which makes the renormalized process Zt = St/S(k)t a martingale. Here

S(k)t may be any positive component of the process St, which serves as numeraire.

30

Let us prove the theorem for tree structures. Suppose that Sn is a vector of M processes

on a tree and yn are the M -vectors of holdings.

For every node n, the self financing condition is that yTn Sn ≤ yn−Sn for all nonterminal

nodes. The program to detect arbitrage is

∥∥∥∥∥∥∥∥∥

max∑

n∈T y>n Sn

yT0 S0 ≤ 0

yTn−Sn ≥ y>n Sn n 6= 0

yT Sn ≥ 0 n ∈ T(9)

Notice that this program has either maximal value of 0 or is unbounded, since every

solution with∑

n∈T yTn Sn > 0 can be multiplied by an arbitrary large positive constant

to give another, better solution. The dual of (9) is

max 0

λnSn =∑

m∈n+

λmSm n ∈ N \ T

λn ≥ 0

Notice that the dual has objective function 0. Therefore it may either be feasible (in this

case the optimal value is 0) or infeasible.

If there are no arbitrage possibilities, then the primal is bounded and the dual is feasible.

Hence the nonexistence of arbitrage is equivalent to the existence of constants λn ≥ 0

satisfying

λnSn =∑

m∈n+

λmSm n ∈ N \ T .

Notice that the constants λ are only determined up to a constant.

Choose now any strictly positive component S(k)n of Sn. Let Zn = Sn/S

(k)n and pn = λn·S(k)

n

λ0S(k)0

.

Then in the no-arbitrage case we have that for the k-th component

pn =∑

m∈n+

pm (10)

and for the other components

Zn =1

pn

∑m∈n+

Zm · pm (11)

Let us check quickly that the existence of a martingale measure is sufficient to prevent

arbitrage. Suppose that there is a trading strategy, which leads to ynSn ≥ 0; n ∈ T and∑n∈T λnyT

n Sn > 0. Because of the martingale property however, 0 <∑

n∈T λnyTn Sn =∑

n∈T pnyTn Zn = yT

0 Z0 = yT0 S0/S

(k)0 . Thus only a positive initial capital may lead to a

nonnegative, not identically zero final wealth.

31

5 Portfolio optimization

Financial decisions have two dimensions: A value dimension, which is measured by a

location parameter of the profit distribution and a risk dimension, which is measured

by a dispersion parameter. Value is expressed by expectation (or some other location

parameter). The dispersion is measured by a translation-invariant functional D.

The optimal portfolio problem consists in finding the composition of a portfolio, which

leads to a compromise between high expected return and low risk. The price of the

portfolio is limited by the available budget.

Let ξ = (ξ(1), . . . , ξ(M))> be the vector of possible returns per unit of price of each of M

assets for one holding period and x = (x1, . . . , xM)> is the vector of asset holdings (again

in unit of price). If a total budget of B is invested today, i.e. B = x1 + · · · + xM , then

the random value at the end of the holding period is Yx =∑M

m=1 xmξ(m).

The standard decision problem is to minimize the risk of the outcome among all feasible

decisions x ∈ X under a the constraint that the the expected return must be larger than

µ. ∥∥∥∥∥∥

Minimize D[Yx]

E[Yx] ≥ µ

x ∈ X(12)

If the risk functional is the variance or the standard deviation, the pertaining model is

the Markowitz model.

Denote by r the expected return vector r = Eξ and by C the covariance matrix C =

E[(ξ − r) · (ξ − r)>] of the asset data. Since the standard deviation is the square root

of the variance, it does not matter, whether the variance or the standard deviation is

considered as the objective function. In the following, the variance is minimized, but the

standard deviation is shown in the risk/return diagrams.

The model is ∥∥∥∥∥∥∥∥∥∥∥

Minimize x>Cx

subject to

r> · x ≥ µ minimal expected return µ

1l>M · x ≤ 1 budget constraint

x ≥ 0 nonnegativity

(13)

This is a quadratic program with linear constraints. The number of variables is M (all

nonnegative), the number of constraints is 2. The Markowitz model has become very

popular, mostly due to the fact that is is simple and its complexity does not increase with

the sample size. In fact, both for theoretical models and for discrete or sampled models,

32

all one has to do is to calculate the covariance matrix and the mean returns first and use

these parameters in the optimization model.

The relation between the lower bound return µ and the minimal variance is called the

efficient frontier. Figure 5 shows the efficient frontier in the upper part of the picture.

Risk (in this case the variance) is shown in the x-axis and return in the y-axis. The

risk/return values of the 6 assets are indicated as numbers. The lower part shows the

composition of the optimal portfolios in the same risk scale as the efficient frontier above.

0.03 0.04 0.05 0.06 0.07 0.08 0.091.003

1.004

1.005

1.006

1.007

1.008

1.009

1.01

12

3

4

5

6

0.03 0.04 0.05 0.06 0.07 0.08 0.090

0.2

0.4

0.6

0.8

1

1

6

Figure 1: A variance efficient frontier. Portfolio weights must be nonnegative.

A variant of this model is the CAPM model (capital asset pricing model), which drops

(unrealistically) the nonnegativity constraints and sets all inequalities to equalities in (??)

∥∥∥∥∥∥∥∥∥

Minimize x>Cx

subject to

r> · x = µ expected return µ

1l>M · x = 1 budget constraint

(14)

For this model, we can find the explicit solution.

Proposition. Assume that C is invertible. Then the solution of (14) is affine-linear in µ

and given by

x∗µ = µ[c

ac− b2C−1r − b

ac− b2C−11lM ] + [

a

ac− b2C−11lM − b

ac− b2C−1r].

33

where

a = r>C−1r

b = r>C−11lM

c = 1l>C−11lM . (15)

The moments of the random return Yµ = x∗>µ · ξ are

E(Yµ) = µ by construction

Var(Yµ) =µ2c− 2µb + a

ac− b2.

i.e. Var(Yµ) is a quadratic function in µ.

Proof. Introducing the Lagrange multipliers λ and γ the Lagrange function is

1

2x>Cx− λ[x>r − µ]− γ[x>1l− 1]

and the necessary conditions are given by the following equations

Cx− λr − γ1lM = 0

r>x = µ

1l>Mx = 1

Thus

x = λC−1r + γC−11lM

and one may calculate λ and γ from the equations

x>r = λr>C−1r + γr>C−11lM = µ

1l>Mr = λ1l>MC−1r + γ1l>MC−11lM = 1

One gets

λ =µc− b

ac− b2

γ =a− µb

ac− b2

with a, b, c given by (15) which leads to the asserted equation for the optimal portfolio

x∗µ. 2

34

0.03 0.04 0.05 0.06 0.07 0.08 0.091.003

1.004

1.005

1.006

1.007

1.008

1.009

1.01

12

3

4

5

6

0.03 0.04 0.05 0.06 0.07 0.08 0.09

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1

6

Figure 2: A variance efficient frontier. Portfolio weights may fall negative. The efficient

frontier function is parabola shaped

5.0.1 The linear relation between Cov(Yµ, ξm) and rm

The return of the optimal portfolio is Yµ = x∗>µ · ξ. The covariance with the return of

the m-th asset is Cov(x∗>ξ, ξm) =∑

j cmjxj = e>mCx∗, where em is the m-th unit vector.

Thus for all m

Cov(x∗>ξ, ξm) = emCx∗(µ) = µ[c

ac− b2rm − b

ac− b2] + [

a

ac− b2− b

ac− b2rm]

= rm[µc− b

ac− b2] +

−µb + a

ac− b2

i.e. all points (rm,Cov(Yµ, ξm) for m = 1, . . . , M lie all on one straight line.

5.0.2 Introducing a risk-free asset

We add now asset 0 to the set of possible assets, which has return r0 and is risk free (i.e.

has zero variance). Suppose we form a new portfolio (including the risk-free asset) out of

an old x (which does not include the risk free asset) in such a way that we take δ of risk

free and a portion (1− δ) of x. This new portfolio has expected return δr0 + (1− δ)x>r

and standard deviation (1− δ)√

x>Cx. Geometrically, in the return-risk plane, it lies on

a straight line segment connecting the points (0, r0) and (√

x>Cx, x>r). Thus by adding

all these line segments to the feasible set, we get the new feasible set of risk/return

combinations.

Denote the efficient frontier function without risk-free asset by µ 7→ σ(µ), where σ(µ) =√x∗>µ Cx∗µ (where C is the covariance matrix of the risk-prone assets).

35

We extend the optimization problem in the following way: The decision vector, the return

vector and the covariance matrix are augmented for the risk free return

x =

(x0

x

), r =

(r0

r

), C =

(0 0

0 C

), ξ =

(r0

ξ

).

The augmented decision problem is∥∥∥∥∥∥∥∥∥

Minimize 12x>Cx

subject to

r>x + r0x0 = µ

1l>M x + x0 = 1

(16)

Here 1lM is the vector of ones of the same length as the number of risky assets.

Proposition. Two fund Theorem. For the extended Markowitz model which includes

a risk-free asset, the optimal solution is

x∗µ =1

d

((r − µ1lM)>C−1(r − r01l)

(µ− r0)C−1(r − r01lM)

)

where

d = (r − r01lM)>C−1(r − r01lM).

All efficient portfolios are affine combinations of the risk-free portfolio

1

0...

0

and the market portfolio x+, where

x+ =

(0

C−1(r−r01l)

1l>M C−1(r−r01lM )

).

Proof. Introducing the Lagrange multipliers, one finds that the necessary conditions are

given by the following equations

Cx− λr − γ1lM = 0

λr0 + γ = 0

rT x + r0x0 = µ

1lTM x + x0 = 1

36

Setting γ = −λr0 and assuming that C is invertible, one gets

x = λC−1(r − r01l).

Using the other equations, the result is

x∗(µ) =

(x∗0(µ)

x∗(µ)

), x∗(µ) =

µ− r0

(r − r01lM)T C−1(r − r01lM)C−1(r−r01lM), x∗0(µ) = 1−1lTM x∗(µ).

2

The return Yµ of the optimal portfolio x∗µ is of course equal to µ, its variance is

Var(Yµ) = x∗>µ Cx∗µ =(µ− r0)

2

(r − r01lM)>C−1(r − r01lM).

The standard deviation is a linear function in µ:

Std(Yµ) =√

x∗(µ)>Cx∗(µ) =µ− r0√

(r − r01lM)>C−1(r − r01l)

Denote by Y+ = x>+ · ξ the return of the market portfolio. It has expectation

µ+ := E(Y+) = r>x+ =r>C−1(r − r01l)

1l>M C−1(r − r01lM).

Its variance is

σ2+ := Var(Y+) =

(r − r01lM)>C−1(r − r01lM)

[1l>M C−1(r − r01lM)]2.

If x is any portfolio, then

Cov(x> · ξ, x+ · ξ) = x>Cx+ = x>(

0(r−r01lM )

1l>M C−1(r−r01lM )

)=

(0

x>(r−r01lM )

1l>M C−1(r−r01lM )

).

Thus Cov(x> ·ξ, x+ ·ξ) a linear function in x. Setting Yx the return of portfolio x and using

the fact that E(Yx) = x> · r = x0r0 + x> · r, i.e. x>(r− r01lM) = E(Yx)− x0r0− r0(1− x0)

one gets a linear relation between Cov(Yx, Y+) and E(Yx)

Cov(x> · ξ, x>+ · ξ) =x>(r − r01lM)

1l>M C−1(r − r01lM)

=E(Yx)− x0r0 − (1− x0)r0

1l>M C−1(r − r01lM)

=E(Yx)− r0

1l>M C−1(r − r01lM)

37

Now, using the fact that

µ+ − r0

σ2+

=r>C−1(r − r01l)− r01l

>C−1(r − r01l)

1l>C−1(r − r01l)

1l>C−1(r − r01l)2

(r − r01l)>C−1(r − r01l)

= 1l>C−1(r − r01l)

we get the relation

E(Yx) = r0 +µ+ − r0

σ2+

Cov(x> · ξ, x∗Tm · ξ) (17)

or, denoting by β(x) = Cov(x> · ξ, x>+ · ξ)/σ2+ the regression coefficient of the return of Yx

w.r.t. to the return Y+ of the market portfolio, one gets

E(Yx) = r0 + β(x) · (µ+ − r0). (18)

The quantity β(x) is called the beta coefficient of the portfolio x. The expected return

r0 + β(x)(µ+ − r0) is called the alpha coefficient of the portfolio.

Thus we arrive at the following proposition.

Proposition. For any portfolio x

(i) E(Yx) = r0 + β(x) · (µ+ − r0).

(ii) Var(Yx) ≥(E(Yx)−r0

µ+−r0

)2

σ2+. Equality holds only if Corr(Yx, Y+) = 1.

Proof. Only the second assertion has to be proved. For any portfolio x, the portfolio

(1 − λ)

(1

0

)+ λx+ with λ = E(Yx)−r0

µ+−r0has the same expectation as Yx and lies on the

efficient frontier. Therefore Var(Yx) ≥(E(Yx)−r0

µ+−r0

)2

σ2+. From (i) one gets

(E(Yx)−r0

µ+−r0

)2

=

Cov2(Yx,Y+)

σ4+

≤ Var(Yx)

σ2+

. Thus equality can hold only if Cov2(Yx, Y+) = Var(Yx) · Var(Y+),

i.e. if Corr(Yx, Y+) = 1. 2

References

[1] P. Wilmott. Introduces Quantitative Finance. J. Wiley and Sons, 2001

[2] D. Luenberger. Investment Science. Oxford University Press, 1997

38

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1.002

1.004

1.006

1.008

1.01

12

3

4

5

6

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1

7

Figure 3: A variance efficient frontier including a risk free asset.

39