Stochastic Models in Finance · 2006-09-11 · Stochastic Models in Finance Begleitmaterial zur...
Transcript of Stochastic Models in Finance · 2006-09-11 · Stochastic Models in Finance Begleitmaterial zur...
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.)
?
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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,
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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).
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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
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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
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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