What is FinancialMathematics?

1

Introduction

• Financial Mathematics is a collection of

mathematical techniques that find appli-

cations in finance, e.g.

– Asset pricing: derivative securities.

– Hedging and risk management

– Portfolio optimization

– Structured products

• There are two main approaches:

– Partial Differential Equations

– Probability and Stochastic Processes

2

Short History of Financial Mathematics

• 1900: Bachelier uses Brownian motion as

underlying process to derive option prices.

• 1973: Black and Scholes publish their

PDE-based option pricing formula.

• 1980: Harrison and Kreps introduce the

martingale approach into mathematical

finance.

• Financial Mathematics has been estab-

lished as a separate academic discipline

only since the late eighties, with a num-

ber of dedicated journals.

3

Structure of this talk

• Preliminary notions: Time value of money,

financial securities, options.

• Arbitrage and risk–neutral valuation

via a one–period, two–state toy model.

• Modelling stock price behaviour

• Naive stochastic calculus

• PDE approach to finance

• Martingale approach to finance

• Numerical methods

• Current Research

4

Preliminary NotionsDiscounting and Financial Instruments

• Finance may be defined as the study ofhow people allocate scarce resources overtime.

• The outcomes of financial decisions (costsand benefits) are

– spread over time

– not generally known with certainty aheadof time, i.e. subject to an element ofrisk

• Decision makers must therefore

– be able to compare the values of cash-flows at different dates

– take a probabilistic view

5

Discounting

• The time value of money: R1.00 in thehand today is worth more than the expec-tation of receiving R1.00 at some futuredate.

• Thus borrowing isn’t free: the borrowerpays a premium to induce the lender topart with his/her money. This premiumis the interest.

• We shall make the simplifying assump-tions that

– There is only one interest rate: Allinvestors can borrow and lend at this(riskless) rate.

– The interest rate is constant over time.

– The same rate applies for all maturi-ties.

6

• Let r denote the continuously compounded

interest rate, so that one unit of cur-

rency deposited in a (riskless) bank ac-

count grows to erT units in time T .

• Thus an amount X at time T is the same

as Xe−rT now.

• Discounting allows us to compare amounts

of money at different times.

7

Returns

• The return on an investment S is defined

by

R = lnST

S0i.e. ST = S0eRT

The random variable R is essentially the

“interest” obtained on the investment,

and may be negative.

• Investors attempt to maximize their ex-

pected return.

•

Fundamental relationship in finance:

E[Return] = f(Risk)

where f is an increasing function.

8

Securities

• Securities are contracts for future deliv-

ery of goods or money, e.g. shares, bonds

and derivatives.

• One distinguishes between underlying (pri-

mary) and derivative (secondary) instru-

ments.

• Examples of underlying instruments are

shares, bonds, currencies, interest rates,

and indices.

• A derivative (or contingent claim) is a

financial instruments whose value is de-

rived from an underlying asset.

• Examples of derivatives are forward con-

tracts, futures, options, swaps and bonds.

9

• There are two main reasons for using deriva-

tives: Hedging and Speculation.

• Thus derivatives are essentially tools for

transferring risk, and will allow one to

diminish or increase one’s exposure to un-

certain events.

• An option gives the holder the right, but

not the obligation to buy or sell an asset.

• A European call option gives the holder

the right to buy an asset S (the underly-

ing) for an agreed amount K (the strike

price) on a specified future date T (ma-

turity).

10

• Thus the payoff at expiry is

maxS(T )−K,0

• Since the payoff can never be negative,

but is sometimes positive, options aren’t

free. The premium paid for the option

is related to the risk (“probability”) that

the share price is greater than the strike

at expiry.

11

Risk-Neutral Valuation

• Consider a toy model with just two trad-

ing dates t = 0 and t = T , and just two

financial assets

– A risk–free bank account A paying

a constant simple rate r = 10% over

the interval [0, T ].

– A risky stock S. Today’s stock price

is S0 = 10.

• At time T , there are only two possible

states of the world, UP and DOWN.

12

• We model this using the tuple

(Ω, P,F , T, F, (At, St)t∈T)

• Here Ω = Up, Down, and P is a prob-

ability measure on Ω.

13

• Consider a European call option on S

with strike price K = 11 and maturity

T . At maturity the call option has the

following possible values:

• How would we find “the” fair price C0

for this contract at t = 0?

14

• Two possibilities come to mind:

– METHOD I. Calculate the expected

value of the future payoff, and dis-

count this to the present.

Thus

C0 =1

1.1[P(UP) · 11 + P(Down) · 0]

= 10 · P(UP)

∗ PROBLEM: How do we determine

the measure P?

If we consider both states equally

likely, the value of the call option

will be C(0) = 5

– METHOD II. The price of the option

will be determined by the market, in

particular by supply and demand.

15

16

• The correct price can be determined by

an arbitrage argument, as follows:

• Consider a portfolio θ = (θ0, θ1) contain-

ing an amount θ0 in the bank and a quan-

tity θ1 shares. The initial value of the

portfolio is V0(θ) = θ0 + 10θ1.

• We want to ensure that the portfolio has

the same value as the call option in all

states of the world at expiry.

UP VT (θ) = 1.1θ0 + 22θ1 = 11

DOWN VT (θ) = 1.1θ0 + 5.5θ1 = 0

i.e.

θ =(−

10

3,2

3

)

• Thus if you borrow 103 and buy 2

3 shares,

the resulting portfolio has the same cash-

flows at maturity as the call option.

17

• To exclude arbitrage, the initial value

of the option must be the same as the

initial value of the portfolio, i.e.

C(0) = −10

3+ 10 ·

2

3=

10

3

• Arbitrage is the possibility of making a

profit without the possibility of making a

loss.

• In the preceding example, if the option

costs less than the portfolio, then

– Short the portfolio;

– Use the proceeds to buy the option;

– And put the remainder in the bank.

18

• Note that the option price using discounted

expected values was 5, which is higher

than 10/3. How can this be?

• If we insist on using the probability mea-

sure P, then the share itself is priced “in-

correctly”.

– Its value ought to have been

S0 =1

1.1

[1

2(22) +

1

2(5.5)

]= 12.5

– but the real price is S0 = 10.

19

• This reflects the fact that investors are

risk averse. In order to take on the risk

of the share, investors require a risk pre-

mium Rp:

10 = S0 =1

1 + r + RpEP[ST ]

=1

1.1 + Rp[1

2· 22 +

1

2· 5.5]

• Suppose that we now change the prob-

ability measure to a new measure Q un-

der which investors are risk–neutral, i.e.

under which they do not require a risk

premium.

• In this world, the current value of the

share is its discounted expected value.

10 =1

1.1[Q(UP) · 22 + (1−Q(UP)) · 5.5]

which implies that Q(UP) = 13,

and Q(DOWN) = 23.

20

• If we price the option using the discounted

expected value under the risk–neutral mea-

sure Q, we get

C(0) =1

1.1(1

3· 11 +

2

3· 0) =

10

3

• and this is CORRECT!!!

•

Principle of Risk–Neutral Valuation:

– The t = 0–value of an option is its

discounted expected value.

– However, the expectation is taken un-

der a risk–neutral probability mea-

sure, which we can calculate.

– And not under the “real–world” prob-

ability measure, which we can never

know.

21

Modelling Stock Prices

• Any model of stock price behaviour must

be stochastic, i.e. incorporate the ran-

dom nature of price behaviour. The sim-

plest such models are random walks.

• Partition the interval [0, T ] into subinter-

vals of length ∆t

0 = t0 ≤ t1 ≤ · · · ≤ tN = T N =T

∆t

• Let Xtn, n = 1,2, . . . N be a family of ran-

dom variables, and let S0 be the stock

price at t = 0. We might (naively) at-

tempt to model the stock price process

by

Stn+1 = Stn + Xtn+1

22

• Thus

St = S0 +t∑

u=1

Xu

• The intuition behind this is that the price

at time t + ∆t equals the price at time t

plus a “random shock”, modelled by Xt.

• We also assume that these shocks are

independent.

• Efficient Markets Hypothesis: Stock

price processes are Markov processes.

23

• Fact: If Xn are independent random vari-

ables, then

var(∑n

Xn) =∑n

var(Xn)

• Thus if the Xn are independent, iden-

tically distributed, then the variance of

the sum is proportional to the number of

terms.

• So the variance of the stock price in our

naive random walk model is proportional

to the elapsed time.

24

• We attempt to build a continuous–time

model of stock price behaviour over an

interval [0, T ]. As a first approximation,

we use Bernoulli shocks every unit time,

i.e. we let

Xt =

+∆S with probability 0.5

−∆S with probability 0.5

• Note that

Var(Xt) = ∆S2

Var(ST ) =N∑

n=1

Var(Xtn)

= N∆S2

=∆S2

∆tT

25

• How large should the jumps in stock price

be? To ensure that Var(ST ) goes to nei-

ther 0 nor ∞ as ∆t → 0, we must have

∆S = o(√

∆t)

• Note that for differentiable functions f(t),

we have ∆f ≈ f ′(t)∆t, i.e.

∆f = o(∆t)

• This shows that St cannot be differen-

tiable!

26

• To build a continuous version of our model,

we use the Central Limit Theorem: If

Xn is a largish family of iid random vari-

ables, then∑

n Xn is approximately nor-

mally distributed.

• Thus: After a largish number of shocks,

the stock price in our naive random walk

model will be approximately normally dis-

tributed.

• We seek a continuous-time version of the

random walk — a stochastic process that

is changing because of random shocks at

every instant in time.

27

Brownian motion

• Brownian motion is a continuous–time

stochastic process Bt, t ≥ 0 with the fol-

lowing properties:

(1) Each change

Bt −Bs = (Bs+h −Bs) + (Bs+2h −Bs+h)

+ · · ·+ (Bt −Bt−h)

is normally distributed with mean 0

and variance t− s.

(2) Each change Bt − Bs is independent

of all the previous values Bu, u ≤ s.

(3) Each sample path Bt, t ≥ 0 is (a.s.)

continuous, and has B0 = 0.

• Brownian motion is a martingale:

EsBt = Bs s ≤ t

where Es denotes the expectation at time

s.

28

GBM

• For stock prices, the Brownian motionmodel is inadequate. We expect the changein price to be proportional to the currentprice.

• A better model for share prices is givenby the stochastic differential equation

dSt = µSt dt + σSt dBt

• Here µ is the drift, i.e. the rate at whichthe share price increases in the absenceof risk. The differential dBt models therandomness (risk), and the parameter σ,known as the volatility, models how sen-sitive the share price is to these randomevents.

• This share price process is called a geo-

metric Brownian motion.

29

Value process

• Consider a market with a share St whoseprice process is a GBM

dSt = µSt dt + σSt dBt

• Let the risk–free interest rate be r, i.e.the risk–free bank account At satisfiesthe DE

dAt = rAt dt

At is the riskless asset. It has drift r

and zero volatility.

• Given a dynamic portfolio θt = (θ0t , θ1

t ),the value process Vt(θ) is defined by

Vt(θ) = θ0t At + θ1

t St

• It satisfies the SDE

dVt = θ0t dAt + θ1

t dSt

= (rθ0t At + µθ1

t St) dt + θ1σSt dBt

30

• The value of the portfolio at time T istherefore

VT(θ) = V0(θ) +

∫ T

0[rθ0

t At + µθ1t St] dt

+

∫ T

0θ1σSt dBt

• We now see that we need to be able toevaluate integrals of the form∫ T

0f(t, ω) dBt(ω)

• The obvious method would be to regard

the above as a Riemann–Stieltjes (or

Lebesgue–Stieltjes) integral.

31

Stochastic CalculusNaive Approach

• Let f(x) be a differentiable function on

an interval [a, b]. Partition this interval:

a = x0 < x1 < x2 < xn = b

where xi+1 − xi = ∆x

• Then by Taylor series expansion, we get

f(xi+1)− f(xi) = f ′(xi)∆x +1

2!f ′′(xi)(∆x)2

+1

3!f ′′′(xi)(∆x)3 + terms involving ∆x4,∆x5, . . .

• Thus

f(b)− f(a) =n−1∑i=0

[f(xi+1)− f(xi)]

=n−1∑i=0

f ′(xi)∆x +1

2

n−1∑i=0

f ′′(xi)(∆x)2 + . . .

32

• As ∆x → 0, we get

f(b)− f(a) = lim∆x→0

∑i

f ′(xi)∆x

+1

2lim

∆x→0

∑i

f ′′(xi)(∆x)2 + . . .

=

∫ b

a

f ′(x) dx +

[1

2

∫ b

a

f ′′(x) (dx)2 + . . .

]

• In ordinary calculus, only the first term

counts (by the Fundamental Theorem of

Calculus), and the other terms are zero.

• This is because the quadratic variationof any “ordinary” function is zero, i.e.

lim∆x→0

∑(∆g)2 = 0

for any “ordinary” function g.

33

• But Brownian motion is different: Con-

sider ∆B = Bt+∆t − Bt. This is a nor-

mally distributed random variable with E[∆B] =

0 and variance var(∆B) = ∆t.

• Consider next the random variable (∆B)2.This has

E[(∆B)2] = var[∆B] = ∆t

var[(∆B)2] = E[(∆B)4]− (∆t)2 = 2(∆t)2 << ∆t

• Thus the variance of (∆B)2 is ≈ 0, i.e.

though ∆B is a random variable, (∆B)2

is a constant (!! I promise that this can be

made precise.)

• It follows that

lim∆t→0

∑E(∆B)2 = lim

∆t→0

∑∆t = T

where T is the total elapsed time. Thus

the quadratic variation of Brownian mo-

tion is non–zero.

34

• Also

lim∆t→0

∑E(∆B)4 = 2 lim

∆t→0

∑(∆t)2 = 0

because g(t) = t is an “ordinary” func-

tion, with quadratic variation zero.

• Hence we cannot ignore the second–orderterm

1

2

∫ b

a

f ′′(x) (dx)2

in the case that x = B.

• But we can ignore all higher–order terms.

• We thus have the following rules for stochas-

tic calculus:

(dBt)2 = dt

dBt · dt = (dt)2 = 0

35

• Suppose that f(t, x) is a C1,2–function,

and let Xt = f(t, Bt). Applying these

rules to a second order Taylor series, we

obtain:

Theorem: (Ito’s Formula)

dXt =

(∂f

∂t+

1

2

∂2f

∂B2

)dt +

∂f

∂BdBt

• Ordinary calculus shows that for a func-tion f(t, x) we have

df =∂f

∂tdt +

∂f

∂xdx

• In stochastic calculus, we get another term,

due to the non–zero quadratic variation

of Brownian motion.

36

• Since Brownian motion has non-zero quadratic

variation, Brownian sample paths are (a.s.)

of unbounded variation.

• This means that in general the Ito stochas-

tic integral∫ T0 f dBt cannot be interpreted

as a Riemann–Stieltjes integral.

• Nevertheless, the stochastic integral can

be defined with semimartingale integra-

tors (using an approximation in a L2–

space, rather than an (almost) pointwise

limit).

• Fact: The Ito integral

Mt =

∫ t

0f(u, Bu) dBu

is a (local) martingale, i.e.

Es

[∫ t

0f dBu

]=

∫ s

0f dBu

37

Stock price process parameters

• Let’s have another look at volatility. The

GBM model for stock prices is

dSt = µSt dt + σSt dBt

Thus

E[dS

S

]2= σ2 dt

and thus σ2 dt is the variance of the re-

turn of the stock over a small period dt.

• It follows that σ is the standard devia-

tion of the annual return of the stock

S.

• This can be measured from market data.

38

• Can we also measure the drift µ?

No.

• So the correct, real-world dynamics of a

share price are unknowable: We can get

the volatility, but not the drift.

• Amazingly, we don’t care!!

39

Black-Scholes ModelPDE Approach

• Consider again market with a share St

whose price process satisfies the SDE

dSt = µSt dt + σSt dBt

• Let the risk–free interest rate be r, and

let At be the riskless bank account, with

dynamics

dAt = rAt dt

• Let V (t, St) be European–style derivative

whose value depends on both the share

price and time. Consider a portfolio Π

which contains 1 derivative, and n shares,

i.e. its value is

Πt = Vt + nSt

40

• A small amount of time dt later, the share

price has changed. The value of the port-

folio changes by

dΠt = dVt + n dSt

• By Ito’s Formula,

dVt =∂V

∂tdt +

∂V

∂SdS +

1

2

∂2V

∂S2dS2

=

(∂V

∂t+ µS

∂V

∂S+

1

2σ2S2 ∂V

∂S2

)dt + σS

∂V

∂SdBt

• Hence

dΠt =

(∂V

∂t+ µS

∂V

∂S+

1

2σ2S2 ∂V

∂S2+ nµS

)dt

+ σS

(∂V

∂S+ n

)dBt

41

• Now if we take n = −∂V∂S (i.e. the portfo-

lio is short −∂V∂S shares), then the portfo-

lio is unaffected by a random changein the stock price:

dΠt =

(∂V

∂t+

1

2σ2S2 ∂V

∂S2

)dt (1)

• Thus, for a brief instant, the portfoliois risk–free. By a no–arbitrage argu-ment, it must earn the same return asthe risk–free bank account, i.e.

dΠt = rΠt dt = r

(V −

∂V

∂SS

)dt (2)

42

• Equating (1) and (2), we get

∂V

∂t+

1

2σ2S2∂2V

∂S2+ rS

∂V

∂S− rV = 0

• This is the famous Black–Scholes PDE.

It is a second–order parabolic PDE, i.e.

essentially a heat equation.

• It is now clear why we don’t care about

the drift µ of the underlying asset S: It

does not appear in the BS PDE!!

43

Black–Scholes ModelRisk–Neutral Approach

• Since we don’t care about the drift rate

µ of an underlying asset, we may as well

simplify our asset price dynamics by as-

suming that all assets have the same

drift.

• The riskless asset (bank account) has drift

r, which we can actually see. We thus

assume that all assets have the same re-

turn, namely the risk–free rate r.

• Mathematically, this corresponds to a changeof measure — from a real world, un-knowable probability measure P to a know-able, risk–neutral measure Q. In therisk–neutral world, the dynamics of S are

dSt = rSt dt + σS dBt

Thus we change the drift of the asset

from µ to r.

44

• Mathematically, this is accomplished us-ing the Cameron–Girsanov Theorem:Let

dYt = µ(t, ω) dt + θ(t, ω) dBt

be an Ito process in a filtered probabilityspace (Ω,Ft, P) Suppose that there existsprocesses u(t, ω) and ν(t, ω) such that

θ · u = ν − µ

Define a process M by

Mt = exp(∫ t

0u dBs −

1

2

∫ t

0||u||2 ds)

and define a measure Q by

dQdP

= MT

Then under Q, Y –dynamics are

dYt = ν(t, ω) dt + θ(t, ω) dBt

where Bt is a Q–Brownian motion.

• Amazingly, a change of measure changesonly the drift and not the volatility.

45

• We can calculate option prices in the risk–

neutral world, because the asset price

dynamics/distributions are known.

• But: Prices in the real– and risk–neutral

world are the same! It is just probabilities

that are changed.

•

Fundamental Theorem of Asset Pric-

ing: There are no arbitrage opportunities

if and only if there exists a risk–neutral

measure.

46

PDE = Risk–Neutral

• Consider a European call option C on a

share S with strike K and maturity T .

The volatility of the underlying share S is

σ and the risk–free rate is r.

• We must solve the following BVP:∂V

∂t+

1

2σ2S2∂2V

∂S2+ rS

∂V

∂S− rV = 0

V (T ) = Φ(ST) = maxST −K,0

• Theorem: In the risk–neutral world, thediscounted value process Vt

At= e−rtVt is

a martingale:

e−rTVT = V0 +

∫ T

0e−rtσSt dBt

47

• It follows that the expected value ofe−rtVt at any time is its current value,and thus the value of the call option withstrike K and maturity T is given by

V0 = E0[e−rTVT ] = e−rTE0[maxST −K,0]

• In the same way, the value of any European–style contingent claim V with maturity Tand payoff (boundary condition) V (T, ST ) =Φ(ST ) is simply

V0 = e−rTE[Φ(ST)]

• This is a version of the Feynman–Kac

Theorem, which gives the solution to a

large class of parabolic PDE’s as an ex-

pectation of a diffusion (here with loss of

mass, represented by discounting).

48

Computational Toolbox

• Numerical integration

• Optimization techniques

• Finite difference methods

• Lattice/tree methods

• Monte Carlo and quasi–Monte Carlo meth-

ods

• Statistical techniques: Principal compo-

nent analysis, factor analysis, maximum

entropy

49

• Time series analysis

• Numerical solution of SDE’s

• Dynamic programming

• Stochastic control theory

50

Current Research

• Altenatives to Black–Scholes

– Stochastic volatility models.

– Jump diffusions.

– Levy processes.

• Interest–rate modelling.

• Pricing in incomplete markets.

• Pricing/measuring/hedging credit risk.

51

• Capital adequacy based risk measures

• Real options

• Differential game theory

• Entropy–based option pricing

• Viability theory

• Non-standard finance

52