Download What is Financial Mathematics? 1

What is Financial
Mathematics?
1
Introduction
• Financial Mathematics is a collection of
mathematical techniques that find applications 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
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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 established as a separate academic discipline
only since the late eighties, with a number 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
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Preliminary Notions
Discounting and Financial Instruments
• Finance may be defined as the study of
how people allocate scarce resources over
time.
• The outcomes of financial decisions (costs
and benefits) are
– spread over time
– not generally known with certainty ahead
of time, i.e. subject to an element of
risk
• Decision makers must therefore
– be able to compare the values of cashflows at different dates
– take a probabilistic view
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Discounting
• The time value of money: R1.00 in the
hand today is worth more than the expectation of receiving R1.00 at some future
date.
• Thus borrowing isn’t free: the borrower
pays a premium to induce the lender to
part with his/her money. This premium
is the interest.
• We shall make the simplifying assumptions that
– There is only one interest rate: All
investors can borrow and lend at this
(riskless) rate.
– The interest rate is constant over time.
– The same rate applies for all maturities.
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• Let r denote the continuously compounded
interest rate, so that one unit of currency deposited in a (riskless) bank account 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.
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Returns
• The return on an investment S is defined
by
S
R = ln T
S0
i.e.
ST = S0eRT
The random variable R is essentially the
“interest” obtained on the investment,
and may be negative.
• Investors attempt to maximize their expected return.
Fundamental relationship in finance:
•
E[Return] = f (Risk)
where f is an increasing function.
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Securities
• Securities are contracts for future delivery of goods or money, e.g. shares, bonds
and derivatives.
• One distinguishes between underlying (primary) and derivative (secondary) instruments.
• Examples of underlying instruments are
shares, bonds, currencies, interest rates,
and indices.
• A derivative (or contingent claim) is a
financial instruments whose value is derived from an underlying asset.
• Examples of derivatives are forward contracts, futures, options, swaps and bonds.
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• There are two main reasons for using derivatives: Hedging and Speculation.
• Thus derivatives are essentially tools for
transferring risk, and will allow one to
diminish or increase one’s exposure to uncertain 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 underlying) for an agreed amount K (the strike
price) on a specified future date T (maturity).
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• Thus the payoff at expiry is
max{S(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.
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Risk-Neutral Valuation
• Consider a toy model with just two trading 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.
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• We model this using the tuple
(Ω, P, F , T, F, (At, St)t∈T)
• Here Ω = {Up, Down}, and P is a probability measure on Ω.
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• 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?
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• Two possibilities come to mind:
– METHOD I. Calculate the expected
value of the future payoff, and discount this to the present.
Thus
1
[P(UP) · 11 + P(Down) · 0]
1.1
= 10 · P(UP)
C0 =
∗ 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.
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• The correct price can be determined by
an arbitrage argument, as follows:
• Consider a portfolio θ = (θ0, θ1) containing an amount θ0 in the bank and a quantity 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
DOWN
VT (θ) = 1.1θ0 + 22θ1 = 11
VT (θ) = 1.1θ0 + 5.5θ1 = 0
i.e.
10 2
θ= − ,
3 3
2 shares,
and
buy
• Thus if you borrow 10
3
3
the resulting portfolio has the same cashflows at maturity as the call option.
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• To exclude arbitrage, the initial value
of the option must be the same as the
initial value of the portfolio, i.e.
10
2
10
C(0) = −
+ 10 · =
3
3
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• 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.
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• 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 measure P, then the share itself is priced “incorrectly”.
– Its value ought to have been
1 1
1
S0 =
(22) + (5.5) = 12.5
1.1 2
2
– but the real price is S0 = 10.
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• This reflects the fact that investors are
risk averse. In order to take on the risk
of the share, investors require a risk premium Rp:
1
10 = S0 =
EP[ST ]
1 + r + Rp
1
1
1
=
[ · 22 + · 5.5]
1.1 + Rp 2
2
• Suppose that we now change the probability measure to a new measure Q under 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.
1
10 =
[Q(UP) · 22 + (1 − Q(UP)) · 5.5]
1.1
1,
which implies that Q(UP) = 3
and Q(DOWN) = 2
3.
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• If we price the option using the discounted
expected value under the risk–neutral measure Q, we get
1 1
2
10
C(0) =
( · 11 + · 0) =
1.1 3
3
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 under a risk–neutral probability measure, which we can calculate.
– And not under the “real–world” probability measure, which we can never
know.
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Modelling Stock Prices
• Any model of stock price behaviour must
be stochastic, i.e. incorporate the random nature of price behaviour. The simplest such models are random walks.
• Partition the interval [0, T ] into subintervals of length ∆t
0 = t0 ≤ t1 ≤ · · · ≤ tN = T
N =
T
∆t
• Let Xtn , n = 1, 2, . . . N be a family of random variables, and let S0 be the stock
price at t = 0. We might (naively) attempt to model the stock price process
by
Stn+1 = Stn + Xtn+1
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• Thus
St = S0 +
t
X
Xu
u=1
• 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.
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• Fact: If Xn are independent random variables, then
X
var(
n
Xn) =
X
n
var(Xn)
• Thus if the Xn are independent, identically 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.
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• 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) = ∆S 2
Var(ST ) =
N
X
Var(Xtn )
n=1
= N ∆S 2
∆S 2
=
T
∆t
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• How large should the jumps in stock price
be? To ensure that Var(ST ) goes to neither 0 nor ∞ as ∆t → 0, we must have
√
∆S = o( ∆t)
• Note that for differentiable functions f (t),
we have ∆f ≈ f 0(t)∆t, i.e.
∆f = o(∆t)
• This shows that St cannot be differentiable!
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• To build a continuous version of our model,
we use the Central Limit Theorem: If
Xn is a largish family of iid random variP
ables, then n Xn is approximately normally distributed.
• Thus: After a largish number of shocks,
the stock price in our naive random walk
model will be approximately normally distributed.
• 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.
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Brownian motion
• Brownian motion is a continuous–time
stochastic process Bt, t ≥ 0 with the following 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.
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GBM
• For stock prices, the Brownian motion
model is inadequate. We expect the change
in price to be proportional to the current
price.
• A better model for share prices is given
by the stochastic differential equation
dSt = µSt dt + σSt dBt
• Here µ is the drift, i.e. the rate at which
the share price increases in the absence
of risk. The differential dBt models the
randomness (risk), and the parameter σ,
known as the volatility, models how sensitive the share price is to these random
events.
• This share price process is called a geometric Brownian motion.
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Value process
• Consider a market with a share St whose
price 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 satisfies
the DE
dAt = rAt dt
At is the riskless asset. It has drift r
and zero volatility.
• Given a dynamic portfolio θt = (θt0, θt1),
the value process Vt(θ) is defined by
Vt(θ) = θt0 At + θt1 St
• It satisfies the SDE
dVt = θt0 dAt + θt1 dSt
= (rθt0 At + µθt1 St) dt + θ1 σSt dBt
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• The value of the portfolio at time T is
therefore
Z
T
VT (θ) = V0 (θ) +
Z0T
+
[rθt0 At + µθt1 St] dt
θ1 σSt dBt
0
• We now see that we need to be able to
evaluate integrals of the form
Z
T
f (t, ω) dBt(ω)
0
• The obvious method would be to regard
the above as a Riemann–Stieltjes (or
Lebesgue–Stieltjes) integral.
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Stochastic Calculus
Naive 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 0 (xi)∆x +
+
1 00
f (xi)(∆x)2
2!
1 000
f (xi)(∆x)3 + terms involving ∆x4 , ∆x5 , . . .
3!
• Thus
f (b) − f (a) =
n−1
X
i=0
n−1
X
[f (xi+1 ) − f (xi)]
n−1
1 X 00
0
=
f (xi)∆x +
f (xi)(∆x)2 + . . .
2 i=0
i=0
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• As ∆x → 0, we get
f (b) − f (a) = lim
X
∆x→0
f 0 (xi)∆x
i
X
1
+
lim
f 00 (xi)(∆x)2 + . . .
2 ∆x→0 i
Z b
Z b
1
=
f 00 (x) (dx)2 + . . .
f 0 (x) dx +
2 a
a
• 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 variation
of any “ordinary” function is zero, i.e.
lim
X
(∆g)2 = 0
∆x→0
for any “ordinary” function g.
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• But Brownian motion is different: Consider ∆B = Bt+∆t − Bt. This is a normally 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
X
2
E(∆B) = lim
∆t→0
X
∆t = T
where T is the total elapsed time. Thus
the quadratic variation of Brownian motion is non–zero.
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• Also
lim
∆t→0
X
4
E(∆B) = 2 lim
X
(∆t)2 = 0
∆t→0
because g(t) = t is an “ordinary” function, with quadratic variation zero.
• Hence we cannot ignore the second–order
term
1
2
Z
b
f 00 (x) (dx)2
a
in the case that x = B.
• But we can ignore all higher–order terms.
• We thus have the following rules for stochastic calculus:
(dBt)2 = dt
dBt · dt = (dt)2 = 0
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• Suppose that f (t, x) is a C 1,2–function,
and let Xt = f (t, Bt). Applying these
rules to a second order Taylor series, we
obtain:
Theorem: (Ito’s Formula)
dXt =
1 ∂ 2f
∂f
+
∂t
2 ∂B 2
!
dt +
∂f
dBt
∂B
• Ordinary calculus shows that for a function f (t, x) we have
df =
∂f
∂f
dt +
dx
∂t
∂x
• In stochastic calculus, we get another term,
due to the non–zero quadratic variation
of Brownian motion.
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• Since Brownian motion has non-zero quadratic
variation, Brownian sample paths are (a.s.)
of unbounded variation.
• This means that in general the Ito stochasR
tic integral 0T f dBt cannot be interpreted
as a Riemann–Stieltjes integral.
• Nevertheless, the stochastic integral can
be defined with semimartingale integrators (using an approximation in a L2–
space, rather than an (almost) pointwise
limit).
• Fact: The Ito integral
Z
Mt =
t
f (u, Bu ) dBu
0
is a (local) martingale, i.e.
Z
t
Z
f dBu =
Es
0
s
f dBu
0
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Stock price process parameters
• Let’s have another look at volatility. The
GBM model for stock prices is
dSt = µSt dt + σSt dBt
Thus
dS 2
E
= σ 2 dt
S
and thus σ 2 dt is the variance of the return of the stock over a small period dt.
• It follows that σ is the standard deviation of the annual return of the stock
S.
• This can be measured from market data.
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• 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!!
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Black-Scholes Model
PDE 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
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• A small amount of time dt later, the share
price has changed. The value of the portfolio changes by
dΠt = dVt + n dSt
• By Ito’s Formula,
∂V
∂V
1 ∂ 2V
2
dVt =
dt +
dS +
dS
∂S
2 ∂S 2
∂t
∂V
∂V
∂V
1 2 2 ∂V
=
dt
+
σS
+ µS
+ σ S
dBt
∂t
∂S
2
∂S 2
∂S
• Hence
∂V
∂V
1
∂V
dΠt =
+ µS
+ σ 2 S 2 2 + nµS
∂t
∂S
2
∂S
∂V
+ n dBt
+ σS
∂S
dt
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• Now if we take n = − ∂V
∂S (i.e. the portfolio is short − ∂V
∂S shares), then the portfolio is unaffected by a random change
in the stock price:
dΠt =
∂V
1
∂V
+ σ2S 2 2
∂t
2
∂S
dt
(1)
• Thus, for a brief instant, the portfolio
is risk–free. By a no–arbitrage argument, it must earn the same return as
the risk–free bank account, i.e.
∂V
dΠt = rΠt dt = r V −
S
∂S
dt
(2)
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• Equating (1) and (2), we get
∂V
1 2 2 ∂ 2V
∂V
+ σ S
+ rS
− rV = 0
2
∂t
2
∂S
∂S
• 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!!
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Black–Scholes Model
Risk–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 assuming 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 return, namely the risk–free rate r.
• Mathematically, this corresponds to a change
of measure — from a real world, unknowable probability measure P to a knowable, risk–neutral measure Q. In the
risk–neutral world, the dynamics of S are
dSt = rSt dt + σS dBt
Thus we change the drift of the asset
from µ to r.
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• Mathematically, this is accomplished using the Cameron–Girsanov Theorem:
Let
dYt = µ(t, ω) dt + θ(t, ω) dBt
be an Ito process in a filtered probability
space (Ω, Ft, P) Suppose that there exists
processes u(t, ω) and ν(t, ω) such that
θ·u=ν−µ
Define a process M by
1 t
||u||2 ds)
Mt = exp( u dBs −
2 0
0
and define a measure Q by
Z t
Z
dQ
= MT
dP
Then under Q, Y –dynamics are
ˆt
dYt = ν(t, ω) dt + θ(t, ω) dB
ˆt is a Q–Brownian motion.
where B
• Amazingly, a change of measure changes
only the drift and not the volatility.
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• 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 Pricing: 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
∂V
1 2 2 ∂ 2V
+
rS
+ σ S
− rV = 0
∂t
2
∂S 2
∂S

V (T ) = Φ(ST ) = max{ST − K, 0}
• Theorem: In the risk–neutral world, the
Vt
discounted value process A
= e−rtVt is
t
a martingale:
−rT
e
Z
VT = V0 +
T
e−rtσSt dBt
0
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• It follows that the expected value of
e−rtVt at any time is its current value,
and thus the value of the call option with
strike K and maturity T is given by
V0 = E0 [e−rT VT ] = e−rT E0 [max{ST − K, 0}]
• In the same way, the value of any European–
style contingent claim V with maturity T
and payoff (boundary condition) V (T, ST ) =
Φ(ST ) is simply
V0 = e−rT E[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 expectation of a diffusion (here with loss of
mass, represented by discounting).
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Computational Toolbox
• Numerical integration
• Optimization techniques
• Finite difference methods
• Lattice/tree methods
• Monte Carlo and quasi–Monte Carlo methods
• Statistical techniques: Principal component 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.
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• Capital adequacy based risk measures
• Real options
• Differential game theory
• Entropy–based option pricing
• Viability theory
• Non-standard finance
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