A� Seco Math�Risk Page

Luis A. Seco

To The Reader

These are the notes for the course on The Mathematics of Risk Management, given at the Annual SIAM meeting for 1998 in Toronto.

Comments, suggestions, etc. will be welcome at any time. They can be forwarded to

Acknowledgements

Thanks to the RiskLab team at the University of Toronto, who provided expert comments, errors, examples, etc. RiskLab is a joint initiative of the University of Toronto and Algorithmics Inc.

Trivia

First known case of a mathematician in nance?

Thales bought options on the use of mills, in a year when a huge olive harvest was expected. Made a ton of money.

Risk:

If you know that a certain investment is going to lose 10%, there is no risk.

If you know a certain investment is guaranteed to earn be-tween 10 and 20 percent, there is risk.

Besides banks, Math-Finance theories are also relevant in

Government nance programs

utility companies (debt renancing).

Example 1.

In 1986, the Olympic Committee announces that Barcelona will hold the 1992 Games. The local organizing committee needs to prepare a budget (in USD) to submit for approval. The expenses will take place over a multi-year period, and will be in Spanish Pesetas (ESP).

In 1986, the USD exchanges at a rate of 1 USD = 142 ESP

The OC obtains a contract to purchase USD at a xed rate of 114 ESP over the next six years. In 1992, USD has dropped to 90 ESP.

Q1 (Pricing). How much does it cost to purchase

such a contract?.

Q2 (Pricing). At what rate does it come free? (Is

it 114?).

Q3 (Hedging). Did the nancial institution that

sold the contract loose money?

Q4 (Risk Management). How can the nancial

Financial Instruments { Equity

European Options

Expire at a preset future time. Their pay-o f depends on the price of the underlying S at

expi-ration.

Call options with strike K have pay-o given by

f(S) = (S ; K) +

:

Put options have pay{o given by

F(S) = (K ; S) +

:

American Options

Can be exercised at any time in the future. Their pay-o is a function of the value of the under-lying at that time.

Call options with strike K have pay-o given by

f(St) = (S(t) ; K) +

:

Put options have pay{o given by

f(St) = (K ; S(t)) +

Asian Options

Their price depends on the average value of the underlying. Can be issued with a European or American style.

Financial Instruments { Monetary

Bonds

They pay a xed amount (e.g., $1) at a future time. They are sold at a discount their price determines interest rates. They usually pay coupons every few months or every year.

Bond Options

Bonds can be bought or sold any time be-fore they expire. Their price will uctuate. As a consequence, they can be used as nancial underlying for options. They are quite similar to equity, except for the fact that at the time of expiry of the bond, options make no sense. This in fact have very important implications.

Caps

They are contracts that oer protection against time dependent interest rates rising over a certain ceiling, by pay-ing the correspondpay-ing exceedpay-ing interest on a xed notional.

Floors

They charge the corresponding missing interest on a xed notional. They have negative value.

Swaps

They exploit the dierent interest rates that dier-ent parties will be charged for xed and oating rate loans a swap is a contract that exchanges future payments at xed and oating rates.

Swaptions

When a swaps is viewed as an underlying, op-tions are issued on them.

Cross Currency swaps

Same as swaps, but the exchange is between payments in two currencies.

Common Terminology

Long Positions.

Term used when the number of units of a certain instrument is positive.

Short Positions.

Term used when the number of units of a certain instrument is negative.

Pricing Theories

Contents

Heuristic considerations ; One period ; Multiperiod ; Continuous Pricing Theory

; One Period

; Stochastic Calculus Numerical Methods

; Monte Carlo

; Principal Components ; Low Discrepancy

Hedging

Example 2

Example.

(Ignore interest rates). Call Option. Pays

f 0(

S) = (S ; $1) +.

$ 2

$ 0.50 $ 1

p

1-p

1{Period Stock Tree

p

1-p

$ 1

$ ???

$ 0

Option Value = V

Assume p = 95%. Is V = 0.95?

Answer:

No!.

V = 1=3$:

Problems:

Discounted Values

Time is money.

Assume the existence of a bond with con-stant interest rate r.

We build the following portfolio : = 2

3 Stock units + ( ;

1

3) bond

p

1-p

$ 1

$ 0 $ 2/3 - 1/3 B

Portfolio Values

p

1-p

$ 1

$ ???

$ 0

Option Value

No matter what

p

is

, absence of arbitrage implies

Option Price = 2 3

; 1 3

B

= 2 3

; 1 3

e ;rT

:

where T is the time to expiration and r is the (constant)

Implied Probabilities

We can still achieve

Option Price = E ;

e ;rT

f 0

= pe ;rT

by selecting

p = 2 3

e rT

; 1 3 :

In other words, we can construct a probability measure P for

the stock process, such that

Option Price = E P

; B

;1 T

f 0

:

More generally, if we dene the (arbitrage-free) price to equal the discounted pay-o

V = B ;1 T

f 0

then, there exists a measure P under which V is a martingale:

Implied Market Data

Example:

Assume the previous call option is sold for $0.50.

2 3

; 1 3

e

;r = 0 :5:

Hence, the risk-free rate must equal

r = ;ln2:

Example.

Assume the stock valued at $1 today, can be worth

S = 8 > > < > > :

$2 $1 $0:5

after a year. How can we price the call option with strike 1?. Two possibilities:

Another derivative price is known

Multiperiod Pricing

Assume

80

120

60

180

80

72

36

A call option with strike $75 can be priced as follows (r=0):

105

5

0

0 45

0 15

So its value today is $15.

Passage to the continuum

We think of innitesimal time intervals dt.

Brownian motion moves up or down with probability 1 2, by

an amount of p dt:

dW =

p

dt E(dW) = 0:

It is distributed at time t according to

P(xt) = 1p

2t

exp

;

x 2

2t

:

Innitesimal stock movements will be

dS = S (dt +dW):

Note that

dlogS = ( ; 1 2

2)

dt +dW:

Ito's Lemma:

@ t

f(St) = @ S

f(St)dS + @ t

f(St)dt ; 1 2

2

Expected Discounted Payo

European call option with payo f 0(

S), expiration at time T:

f(St) = e

;r(T;t) Z

1 ;1

f 0

S0e

(; 2

2 )(T;t) +x

P (

xT;t)dx

P (

xt) =

1

p

2t 2

exp

;

x 2

2t 2

:

Problems

What is ?. What is ?

Can we replicate the price?

The Black-Scholes Formulas

The price of a European call option on a stock S, valued

today at S

0, maturing at time

T with strike K, (constant)

volatility and interest rate r is given by

V (tKr) = S 0

N(d 1)

; K e

;r(T;t)

N(d 2)

where N(d) is the cumulative normal

N(d) = Z

d ;1

e

;x2=2 dx p

2

and

d

1 = ln( S

0

=K) + (r + 1 2

2)(

T ;t)

p

T ;t

d

2 = ln( S

0

=K) + (r ; 1 2

2)(

T ;t)

The Black-Scholes Theory

Assume an option has price f(tS), at any given point in

time, and any possible value of the underlying. Let's set up the following arbitrage free argument:

At any point in time t, build a portfolio consisting of

a = ;@ S

f(St)

units of stock, and the option. Using Ito's formula,

d

t = d

t

f + adS

= 1 2 2 S 2 @ 2 S

f + @ t

f + @ S

f dS + adS

= 1 2 2 S 2 @ 2 S

f + @ t

f:

This is a risk-free investment. Hence, it must earn risk-free interest and we obtain:

8 > < > : @f @t = ; 1 2 2 S 2 @ 2 f @S 2

; rS @f @S

+ rf

f(ST) = f 0(

S):

Pricing Theory (One Period)

Implied probabilities can be obtained, not only from prices dictated by arbitrage arguments, but also from market prices. The implications of this is that a probabilistic approach to pricing is more useful than might have seemed from the con-siderations above.

In this section we assume there is a probability space for the payos of N securities available for trading,

A security is characterized by its cost now, and its payo

after one unit of time.

The

cost of the

i

-th security

, i = 1:::N, is q i.

The

payo

is given by the random variable D i(

!).

The

expected payo

of a security is E(D i(

!)).

A

portfolio

is a vector = ( 1 :::

N)

2 R

N, which

rep-resents the holdings of each security. i can be positive or

negative. If i is positive, our position is said to be long. If

i is negative, our position is said to be short.

A market is said

complete

if Span

D(!) 2 R N

= L 2(

):

and markets are usually assumed to be complete. In a com-plete market, for any payo there is a portfolio with that payo.

The

cost

of a portfolio is q .

If a portfolio has nonzero cost, i.e. q 6= 0, one denes its

return

to be

R (

!) =

D(!) q

:

In a real market, there are hedgers (people trying to min-imize risk), speculators (people trying to maxmin-imize return) and arbitrageurs (people detecting market ineciencies). We say that there is an

arbitrage opportunity

if there is a portfolio such that

q 0 and D 0 a:e:

and D > 0 with non{zero probability.

Theorem. (Riesz representation)

If are linear func-tionals of the payos L

2(

), then there exists a random

vari-able (!) such that

p = E( D) all 2 R

N

: (1)

If markets are complete, is unique. If there are no arbitrage

opportunities, > 0.

In the case that we consider the cost as that linear functional, we obtain that the cost of a portfolio is the expectation of its payo with probabilistic weight (!), which is called the

state{price deator

. The name comes from the fact that

E(R

) = 1 (2)

for all portfolios .

We always assume that D 0(

!) is constant for all ! 2 . This

is a

savings account

.

A

riskless bond

is a portfolio 0 of constant payo i.e. such

that D(!) = D(!

0) for all

!! 0

2 : It always exists:

put = (10:::0). Then from (2) we nd

R 0

E(R

0) = 1 E()

The

riskless interest rate

is given by

r = ;

1

T

lnE(R 0)

:

Theorem.

A price deator exists if and only if there is no arbitrage.

Proof.

If a price deator exists, then (0) = E( (T)).

Since is positive as a functional on L, if (T) > 0 then

(0) > 0 and if (T) = 0 then (0) = 0.

On the other hand, let us suppose that there is no arbitrage. Let us consider the price-payo vector space V = R L.

The (cost , pay-o) hyperplane is

M = f(; q P) : 2 R N

g:

The cone K = R +

L

+ contains all securities of

non-positive price and non-negative payo. If there is no arbi-trage, then K \ M = f0g.

By the separating hyperplane theorem, there exists a func-tional

F : V ! R

The Riesz representation of ( ) is

F(vc) = v + E( c):

In terms of and , we have that

; q + E( ( P)) = 0

for all 2 R

N. Hence

1-Period Summary

There is a measure P, given by the up-ward probability

p = e rT S now ; S down S up ; S down :

An option with pay-o f 0(

S) at time T has a price f given

by

f = E P(

B ;1 T

f

0) Martingale condition :

The option can be replicated with a portfolio consisting of

a = f 0( S up) ; f 0( S down) S up) ; S down

units of stock

B ;1 (

f ; a S)units of bonds:

Moreover, the random variable B ;1 T

S is also a martingale:

B ;1

S

now =

p S

up + (1

; p) S down

Binomial Trees

General facts:

fF

i

g, i = 012 = n. Filtration (-algebras).

f

i g

n

i=0 is a process if

i is F

i{measurable for all i. The conditional expectation is

E P( j jF i)

is the projection of

j onto L

2( F

i).

Given a measure P, is a martingale when E

P(

j jF

i) =

i

i j:

Trivial Fact:

Given P, and any function X on L 2(

F

n), the

process given by E P(

XjF

i) is a martingale.

Theorem Binomial Representation Theorem:

Given two processes fS

i

g and fP i

g a binomial tree which are

mar-tingales with respect to the same measure P, there exists a

process f i

g such that

Arbitrage Pricing (Multi-Period)

An option is an F

T{measurable function.

The stock process generates a measure P under

which it is a martingale.

The arbitrage-free option price is then given by the

new martingale

P i =

E P

; B

;1 T

XjF i

:

The replication strategy is given by the Binomial

Representation Theorem.

P i =

P 0 +

i X

k

(S i

; S i;1)

Continuous Pricing

Stock prices follow the Ito Process

dS S

= dt+ dW t

:

Pricing theories will be an extension of the multi-period case. Agenda

Elementary review of stochastic process. Brownian motion

Stochastic integrals { Stochastic dierential

equa-tions.

Martingale representation

Stochastic Processes

A Filtration is fF t

g,

F

t is a

{algebra for all t.

F

t

F

s when

t < s.

Take a function 2 L 2(

F

t). Its conditional expectation at

time s < t dened as

E s

= P L2(F

s )

where the right hand side denotes its projection onto L 2(

F s).

A Stochastic Process

t is such that

t is F

t{measurable

A Stochastic Process t

2 L 1(

F

t) is a martingale when E

t(

s) =

t

for s t:

We dene the quadratic variation to be adapted process

jj 2

t = lim n!1 2 n ;1 X j=0 ; 2 ;n (j+1)t ; 2 ;n jt 2 : M

2 denotes the class of martingales with nite quadratic

Predictable Processes

An adapted process

t is left continuous if

lim

s"t

s =

t

almost surely.

Consider the ltration F

0 generated by all left{continuous

adapted process.

A process is predictable if it is adapted to F 0.

Equivalently, it is approximated by processes

t which are

constant on intervals (t i

Brownian Motion

A process W

t is a

P{Brownian motion when

W

t is continuous in

t and W

0 = 0.

W

t

; W

s is distributed, under

P, as a normal

dis-tribution with mean 0 and variance t ; s.

For 0 t 0

< < t n

< 1, we have that B

t0 and

all B t

k

; B t

k ;1 are all independent.

If s < t,

W t = W t ; W s + W s

Hence, since E s(

W t

; W

s) = 0 and E s W s = W s, E s W t = W s

and Brownian motion is a martingale. Moreover, jW

t j

2 =

Stochastic Integrals

Given a stochastic process

t on (

FT P), and a function f(t), the stochastic integral

Z t 0

f(s)d s

is simply dened as a random variable on ( FT P), whose

value for any ! 2 is given by the Stiljes integral

Z t 0

f(s)d s(

!)

Appropriate conditions on the paths s(

!) are required, such

as s(

!) to be an increasing function of s. Note that

Brow-nian motion does

not

satisfy this.

For general processes, we rst note that, if f is piecewise

constant, on intervals (t i

t

i+1], then Z

t 0

f(s)d s =

k X

f(t i)

;

t i

;

t i;1

Brownian Integrals

Let W 2 M

2. Since

jWj 2

t is increasing, we can dene

jjjj 2 S(

t) = Z t 0 2 s djWj 2 s the norm

jjjj 2 S = E Z T 0 2 t djWj 2 t

and the corresponding Hilbert Space L 2

jSj.

If

t is piecewise constant

Variance

Z T 0

dW = E

k X i=1 t i ; W t i ; W t i;1 2 E k X i=1 t i ; W t i ; W t i;1 2 E k X i=1 j t i j 2 jWj 2 t i

; jWj 2 t

i;1

= jjjj 2 S : Hence, R t dW

t can be dened by extension for all

2

L 2

In particular, 0 is normal with mean 0 and variance

E 2 =

Z t 0

2 s

ds:

Hence, we can derive the following formula, which will be of use later:

E

Pexp Z

t 0

t

dW t

=

Z R

e

x;x2=(22)

dx p

2 2

= e 2=2

= e12 E

R t

0 2 s

SDE

A process of the type

X t =

X 0 +

Z t 0

s

dW s +

Z t 0

s

ds

will be written down as

dW t =

t

dW t +

t

dt:

SDE appear when the terms and above are made X

dependent, as

dX t =

(X t

t)dW t +

(X t

t)dt:

Ito's Lemma

Ito's Lemma, in this context, reads as follows:

f(X t) =

f(X 0) +

Z t 0 ; f 0( X t) t dt+ 1 2 f 00( X t) 2 t dt + Z t 0 f 0( X t) t dW t :

We can rewrite it as

d t

f(X t) =

; f

0( X

t) + 1 2 2 t f 00( X t)

dt+ f 0(

X)t)dW t

:

The product rule: if

dX i = i dt + i

dW i = 12

then

d(X 1

X 2) =

Feynman-Kac

The Feynman-Kac formula provides a solution to parabolic PDE's in terms of a stochastic integral. It can be viewed as the inverse of the Black{Scholes equation.

Consider the following simple formulation. Let's try to solve

@ t

(xt) = 1 2

@ 2 x

(xt) ; V (x)(xt)

= ;H:

Dene

X

t = exp

; Z

t 0

V(x + W s)

ds

f(x +W t)

:

Using the product rule and Ito's formula,

dX

t = exp ; Z t 0 V ds ;

;V f + f 0 dW t + 1 2 @ 2 x f dt = exp ; Z t 0 V ds

(;H f + f 0 dW t) : Hence d t

EX = ;E

Therefore, the linear operator dened by

P t

f = ;E

exp

;

Z t 0

V (x+ W s)

ds

f(x+ W t)

:

satises the equation

@ t

P t

= ;P

t(

H) P

0

= :

This yields

P t =

e ;tH

:

and

(xt) = P t

Girsanov Theorem

Assume a measure P and a ltration F. If W t is

P{Brownian

and t is

F{adapted, with the property

E P

e ;

12 R T

0 2 t

dt

< 1

dene Q such that

dQ dP

= exp

;

Z T 0

t

dW t

; 1 2

Z T 0

2 t

dt !

Then, ~W t =

W t +

R t 0

s

Martingale Representation Theorem

If X

t is a martingale, and

t is predictable and bounded,

then R

dX is a martingale. The converse:

Theorem :

Assume M

t is a

Q{martingale with non{zero

volatility t. If

N

t is any other

Q-martingale, there exists a

predictable

t such that

N t =

N 0 +

Z t 0

s

dM s

ds:

Furthermore,

t =

(N t) (M

t) :

Corollary :

If P admits a Brownian motion, all

martin-gales M

t are of the form

dM t =

t

dW t

Arbitrage Pricing (Multi-Period)

An option is an F

T{measurable function.

The stock process generates a measure P under

which it is a martingale.

The arbitrage-free option price is then given by the

new martingale

P t =

E P

; B

;1 T

XjF t

:

The replication strategy is given by the Binomial

Representation Theorem.

P t =

P 0 +

Z t 0

s

dS:

or

t =

@P t @S

:

The Black-Scholes equation follows from the

Continuous Pricing

Stock price process

dS S

= rdt+ dW:

European call option with payo f 0(

S)

f(St) = e

;r(T;t) Z 1 ;1 f 0

S0e( r;

2

2 )(T;t) +x

P (

xT;t)dx

where

P (

xt) =

1 p 2 2 t exp ; x 2 2 2 t :

Multifactor options involve higher dimensional integrals. They are studied using MonteCarlo methods, or Low{dis-crepancy sequences. Also 8 > < > : @f @t = ; 1 2 2 S 2 @ 2 f @S 2

; rS @f @S

+ rf

f(ST) = f 0(

S):

American options give rise to free boundaries.

For Interest rate options, the underlying S 2 R is replaced

The Real World.

Stock prices are not log-normal.

Transaction costs make continuous trading

in-nitely expensive.

No one knows future volatility.

Markets are not liquid, complete or ecient.

The counterparty you are dealing with may not

Greeks

Provide an intuitive basis for understanding price changes, and risk.

Delta.

Change in price as a function of the underlying.

= @ @S :

Gamma.

Change of delta as a function of the underlying. ; = @

2 @S

2 :

Vega.

Change of price as a function of volatility

V = @

@ :

Theta.

Essentially:

Numerical Methods

Consider stocks with price processes given by

d(logS

j) = j dt + j dW j

where the Brownian motions dW

j have correlation

coe-cients V = ( ij).

If the time to expiration is T, the interest rate is r, and the

pay{o of an option is f 0(

S 1

:::S

n), its price is e

;rT p

(2)

n det V Z R n f 0 ; e S1

:::e S n e ;(S;) y V ;1 (S;) d

S

:

With a change of variables, the problem then reduces to com-puting integrals of the type

Z 01]

n

f(x 1

:::x n) dx 1 dx n

for appropriate integrands f.

Multidimensional integrals are not easy to compute. There are three basic methods:

1. Grid Methods.

Grid points.

We put evenly spaced points in the unit cube, and approximate

Z Q

f dx = 1 N

X f(x

i) :

The error in this approximation is like N

;1=n.

It can be improved to the trapezoidal rule

Z Q

f dx = 1 N

X (x

i) f(x

i)

where is the characteristic function of the cube,

(x) = (

1 if x 2 Int Q,

2;k if

x belongs to k faces of Q,

which gives an error comparable to N

;2=n.

Pros{Cons

As a function of n, it require an exponentially large number

of points. (Compare with Numerical Recipes).

It can easily be turned into an adaptative grid process.

Monte{Carlo.

Monte{Carlo methods appeared ocially in 1949, but they had been used by the U. S. Defense Department in secret projects for several years before that. The name was a code name used by Von Neumann and Ulam at Los Alamos in projects related to The Bomb (simulation of random neutron diusion in ssionable material).

Applied to integration,

Z Q

f(x)dx = 1 N

N X

i=1 f(x

i)

O N

;1 =2

:

if the x

i are uniformly distributed.

Pros{Cons

Reasonable speed is the same in all dimensions. Not easily turned into an adaptative scheme. Ignores smoothness of the integrand.

It's hard to generate random numbers, but it is otherwise

easy to implement.

Higher Dimensions

Generating random numbers in higher dimensions is a di-cult task. For the case of a multivariate normal, with given variance/covariance matrix V, one proceeds as follows:

Consider the Cholesky decomposition of V,

V = H

y H:

Let y = (y 1

:::y

n) be independent normally-(0,1)

dis-tributed random variables. Then

x = y H

produces random vectors which are normally distributed with covariance given by V.

Indeed,

Cov x = E ;

x y

x

= E ;

H y

y y

yH

Principal Components

The biggest limitation in the algorithm for generation of ran-dom numbers in higher dimensions is the inability to produce clean univariate normally distributed numbers, when the di-mension is very high.

In practice, covariance matrices have directions that will hap-pen with high probability, and others that are very unlikely. Set

V = PDP

y

with D diagonal and P orthogonal.

D = 0 B B @

1

...

n1

1 C C A

Assume 1

2

> . One may choose to trim the

covari-ance matrix, by selecting only the rst few eigenvalues of D.

These are the principal components. The signicance of a selection

1

:::

k is given by

1 +

+ k

Sub{random methods.

Uses other limit theorems (i.e., ergodic theorems and number theory). Consider an example in dimension 1:

If is irrational, and ftg denotes the fractional part of t, Z

1 0

f(x)dx = lim N!1 1 N N X i=1

f (fng):

Moreover, we can estimate the remainder: 1

N N X

i=1

f (fng) = 1 N N X i=1 1 X k=;1 ^

f(k)e

2 ink

=

Z 1 0

f(x)dx

+ X k6=0

^

f(k) 1 N N X n=1 e 2 ink :

The speed of convergence is therefore closely related to the diophantic properties of , and the smoothness of f.

Theorem :

If f is of bounded variation, and is an

alge-braic number, then

Z 1 0

f(x)dx ;

1

N N X

i=1

f (fng) C N ;1 log N:

Higher Dimensions.

The speed of convergence is based on the concept of discrepancy of a nite set of points in the unit cube in dimension d, P 2 &01]

d:

Consider sets of the form

B = d Y i=1

&0u i)

:

The discrepancy is dened as

D(P) = sup B jBj ;

jP \ Bj jPj :

Theorem :

1 N N X i=1 f(x

n) ; Z f

V(f) D(x 1

:::x N)

:

A sequence fx n

g is usually referred to as low discrepancy if D(x

1

:::x

N) = O

; N

;1+"

for all " > 0:

Let = ( 1

:::

d) such that 1

1

:::

d are linearly

inde-pendent over the rationals, and the

HEDGING

Problem: to generate a buy/sell strategy that replicates pay-os.

Used to remove risk from option writing. For frictionless ideal markets:

@f @S

(S(t)t)

units of the stock replicates the option price at all times. In real situations:

INVERSE PROBLEMS (implied parameters)

If certain options are liquid enough,

f(St) = e

;r(T;t) Z

1 ;1

f 0

Se

(r; 2

2 )(T;t) +x

P (

xT ;t)dx

is known. This implies a certain value for the volatility.

Interest rates: term structure

American options: inverse scattering.

Time dependent volatilities. Volatility smiles. Implied volatility surfaces.

Interest Rate Theory.

Bonds, Yields

Consider a bond that, with a payment of P(tT) at time t,

pays $1 at time T (and has no other intermediate payments).

If the interest rate r is assumed constant, then we would have

P(tT) = e

;r(T;t) :

Hence,

r = ;

logP(tT)

(T ;t) :

This is useful since it is P, not r that is observed (not quite:

see next section). When r is not constant, we simply dene

the yield rate

r(tT) = ;

logP(tT)

(T ; t) :

Since r determines P, r determines the entire term structure.

As a function of T, r is smooth. As a function of t, it is a

Forward Rates

Short rate:

the cost of instantaneous borrowing:

r t =

r(tt) = ; @ @T

logP(tT)

T=t :

Similarity with stock prices loss of information.

Forward Rate:

f(tT): \our prediction at t of r T."

Consider the following futures contract:

Agreement date: now (time t).

Product to deliver: a zero-coupon bond B issued at T

1, paying $1 at T

2. Delivery date: T

1. Price: P(tT

1 T

2) (Unknown). Payment date: T

It has the only two cash ows:

At time T 1:

P(tT 1

T

2) (Unknown). At time T

2: $1.

Consider a portfolio of 1 bond unit worth P(tT

2) each,

and ;x bond units worth P(tT

1) each, with

x =

P(tT 2) P(tT

1) :

Because it costs nothing now, it has only two cash ows:

At time T

1, an cash out-ow equal to

;x (or in-ow

of x).

At time T

2, a cash in-ow of $1.

Hence,

P(tT 1

T 2) =

P(tT 2) P(tT

1) :

The corresponding forward yield is, by denition

r(tT 1

T 2) =

;

logP(tT 1 T 2) T 2 ; T 1 = ;

logP(tT 2)

; logP(tT 1) T ; T

The forward rate is dened to be

f(tT) = r(tTT)

= ; @ @T

logP(tT):

The term structure is reconstructed from f as follows:

P(tT) = exp

; Z

T t

f(tu)du !

:

Bootstrapping

In practice, bonds pay coupons. Hence, the calculation of the yield needs to be modied.

Example (J. Hull)

Consider the following set of bonds, with $100 principal and associated prices, with coupons paid every six months as described below:

Time to Maturity annual coupon price

3 mo. $0 $97.5

6 mo. $0 $94.9

1 yr. $0 $90.0

1.5 yr. $8 $96.0 2.0 yr. $12 $101.6 2.75 yr. $10 $99.8

Solving the obvious set of (non{linear equations), we can ar-rive at the yield curve with values given by

Black 76

Consider a caplet: pays at t

i+1 the dierence between the

excess between the yield rate r(t i

t

i+1), and a strike K.

Absence of arbitrage implies that, for valuation purposes, we should treat r(t

i t

i+1) as f(t

0 t

i t

i+1).

If we assume this is log-normally distributed, we are just looking at a call option with f(tt

i t

i+1) as the underlying

risk factor, whose price is given by the usual Black{Scholes formula

Caplet = (F(t 0

t i

t i+1)

N(d +)

; K N(d ;))

P(t 0 t 1) with d = ln(

F=K) 1 2 2( t i ; t 0) p t i ; t 0 :

In practice, caps prices are xed by the market, and the formula above is used in an attempt to extract the implied volatility from the market.

Since caplets involve two future dates, implied vols are a two parameter function, the

volatility surface

or

volatility

0 2

4 6

8

10

0 2 4 6 8 10 10 12 14 16 18 20 22

Decaf Heath{Jarrow{Morton

d t

f(tT) = (tT)dt+ dW t

This SDE has a trivial explicit solution

f(tT) = f(0T) + Z

t 0

(sT)ds+W t

:

Model properties:

At a xed time t, f(tT) is smooth in T. f(tT) ; f(tS) is deterministic.

The only source of randomness comes from t. f (and r

t) can become negative.

The Money Market Account

We invest $1 permanently in the short rate:

dB t =

r t

B t

dt B

0 = 1 :

Since

r(t) = f(0t) + Z

t 0

(st)ds+W t

:

we have

B

t = exp

Z t 0

r s

ds

= exp

Z t

W s

ds+ Z

t

f(0u)du + Z

t Z

t

Replicating Strategies

We have an option X on a bond. The option expires at S,

while the bond matures at a later time T.

We will use the money market account for the discounting factor.

Hence, The discounted bond is given by

Z t =

B ;1 t

P(tT):

A replicating strategy will be given by:

Finding a measure P so Z

t is a martingale.

In this way,

E t =

E P

; B

;1 S

XjF t

:

is a P{martingale. By the martingale representation

theo-rem, there is a process

t such that

dE t =

t

dZ t

The Measure

We had

B

t = exp R t 0 W s ds+ R t

0 f(0u)du+ R t 0 R t s

(su)duds

P(tT) = exp ; R

T t

f(0u)du; R

T t

R t

0 (su)dsdu;(T;t)W t

Hence, Z

t is given by

exp ; R

T

0 f(0u)du; R t 0 R T s

(su)duds;(T;t)W t ; R t 0 W s ds

Using Ito's Lemma, we get

dZ t Z t = ; Z T t

(tu)du ; (T ; t)dW t +

1 2

2(

T ; t) 2

dt:

In order to use Girsanov's theorem, set

dW t =

dW~ t +

(tT)

(tT) = ; 1 2

(T ; t) +

1

(T ; t) Z

T t

(tu)du:

This yields

dZ t =

Z t(

T ; t)dW~ t

:

and Z

Absence of Arbitrage

If dierent durations T give dierent drifts (tT). we would

have arbitrage. Therefore, We must have

@ @T

= 0:

Equivalently, we have the following condition on the drift:

(tT) = 2(

Heath{Jarrow{Morton

For 0 t T,

f(tT) = f(0T) + Z

t 0

(sT)dW s +

Z t 0

(sT)ds

Equivalently,

d t

f(tT) = (tT)dW t +

(tT)dt:

Features:

Short Rate Models

dr(t) = (r t

t)dt + (r t

t)dW t

:

Bond Prices:

;logP(tT) = Z

T t

f(tu)du

= g(r t

tT):

= logE Q

e

; R

T t

r(s)ds j r

r = x

The Hull{White Model

Given a, (assumed constant), and a time-dependent

rever-sion level (t), we write the SDE for the spot rate:

dr

t = ( (

t) ;ar t)

dt+ dW t

:

Assuming bond prices are given by P(rtT), Ito's lemma

tells us d t P = @ r

P ( (t) ; ar t) +

@ t P + 1 2 2 @ 2 r P

dt + @ r

P dW t

= (tT)dt+ (tT)dW t

:

The portfolio

(t) = P(tT) +P(tT 2) where = ; @ r

P(tT) @

r

P(tT 2)

= ;

(tT) (tT

2)

is deterministic. Hence,

d t

P(tT); (tT)d t

P(tT 2) =

which implies

(tr) =

(tT); r(t)P(rtT) (tT)

:

must be independent of T. Subbing above, we obtain

@ r

P ( (t) ; ar t) +

@ t +

1 2

@ 2 r

2 =

rP + (tT)

= rP + @ r

P:

or

@ t

P + ((t) ; ar) @ r

P + 1 2

2

@ 2 r

P = rP

with

(t) = (t) ; (t):

We now make the following assumption:

P(rtT) = A(tT) exp(;B(tT)r):

This implies

( @

t

B ; a(t)B + 1 = 0 B(TT) = 0

( @

t

A ; (t)AB + 1 2

2

AB

2 = 0 A(TT) = 0:

where

B(tT) = a

;1 1 ; e

;a(T;t)

logA(tT) = log

P(0T) P(0t) +

B(tT) f(0t)

;

2 ;

e ;aT

; e ;at

2

; e

2at ; 1

4a 3

Other short rate models

Ho-Lee

dr = t

dt + dW t

Vasicek

dr = ( ; r)dt+ dW t

Cox-Ingersoll-Ross

dr = ( t

;

t

r)dt + t

p

rdW t

Black-Derman-Toy

d(logr) =

0( t) +

@ log(t) @t

logr

dt+ (t)dz:

Black-Karasinski

d(logr) = ( t

;

t log

r)dt + t

Calibration

All these models assume a certain set of parameters, such as the volatility, mean reversion parameters, etc.

There is no well established way of doing this.

A popular method is to observe market prices for standard instruments, and work out the parameters that best t mar-ket data. This process is called

calibration

.

Portfolio theory.

Prices have been established in the comfort of the indierence provided by arbitrage-free arguments.

Markowitz Mean Variance Model

Assume n instruments available for trade, with stochastic

returns given by S i.

Return = Mean:

R i =

E(S i).

Risk = Variance:

2 i =

E(S 2 i )

; R 2 i.

Consider also the variance/covariance matrix

V = f

ij

g

2 ij =

E(S i

S j)

; R i

R j

:

An investment choice is given by a weight vector w =

(w 1

:::w

n). Such investment will have a return equal to N

X i=1

w i

R i

and a variance given by

(Static) Investment decisions

Usual investment choices:

For a given level of return , minimize risk: For a given level of risk , maximize return:

This can be generalized to quantifying the relative impor-tance of risk into a parameter in order that we now seek

max

w ;

w R; w t

Vw

subject to restrictions, which may include a cap the amount of money invested (P

w

i), and taking only long positions

(w i

0).

They can be summarized in a two dimensional risk/return graph.

0 0.5 1 1.5 2 2.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

"out"

All possible investments will give rise to a convex subset of

R 2

+. Its boundary is the Ecient Frontier.

Theorem :

The Ecient Frontier is convex.

The Ecient Frontier

The ecient frontier carries a lot of graphical information. As a sample:

Normal Markets:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0 0.2 0.4 0.6 0.8 1 Normal Frontier

Incomplete Markets:

0 0.5 1 1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1 Frontier in Incomplete Markets

Arbitrage Opportunities:

0 0.5 1 1.5 2 2.5 3

Sharpe's ratio

It provides a one-dimensional approach to performance eval-uation. It is dened as

Sharpe0s ratio = (Return)

; (Riskfree rate)

risk :

For pay-o distributions that are not symmetric,

downside-risk q

E(S i

; R i)

2 ;

Regret/Reward Model

(R. Dembo, D. Rosen, D. Saunders.)

Regret.

Replaces the concept of risk.

Let D be the pay-out matrix, and a benchmark portfolio.

Set

Dx ; = y +

; y ;

denote the positive and negative tracking errors with respect to the benchmark.

Regret is the expected underperformance:

R ;

=

p T

y ;

:

Reward.

It is the expected excess prot: The expected prot is

p T(

Dx) ; q T

x

and the expected prot earned by the benchmark is

Reward is

R +

=

p T(

Dx); q T

x; p T

+ c

= p T(

y +

; y ;)

; q T

x + c:

We restrict our attention to portfolios with cost less than or equal to that of the benchmark (i.e., q

T

x c).

Financial Risk

\Even in this age of high-tech computing, the basic architecture of risk management remains primitive | it is as if all that fancy technology is stored in the intellectual equivalent of a wooden shack."

History

Orange County.

Bob Citron (County Treasurer), built a leveraged portfolio of short term loans with long term notes. This proved to be a protable investment, since short term rates were low and long term rates were high.

When interest rates began to rise in 1994, losses occurred a snow ball eect happened when investors found out about the losses, and attempted to withdraw the money.

Such portfolios are usually perfectly hedged against parallel shifts in the yield rate, but show big exposure to tilts in the yield curve.

Barings Bank.

Nick Leeson had derivative positions on the Nikkei, that bet on a stable market. The decline of 15% of the Nikkei, which followed the earthquake of Kobe in 1995, produced large losses. The situation lead to a total loss of $1.3 Billion, after the positions were further increased after the initial losses. Some of the positions were unauthorized. although his su-periors had approved a cash infusion of $1billion to cover margin calls.

The price of shares dropped to $0. Baring's Bank was worth about $1 billion in terms of market capitalization. Bond hold-ers received 5c. for each $1. ING oered to purchase Baring's for 1 British Pound.

Risk Management

Problem: to determine potential losses.

Calibration risk Market risk Credit risk Model risk

J.P. Morgan introduced RiskMetrics about 3 years ago, and CreditMetrics last summer. They are industrial standards now.

Estimating Volatility

Volatilities are the quantication of the risk of the market. Portfolios will have a volatility adapted to them, but esti-mating volatility is at the heart of analyzing the risk of any portfolio.

Three ways of estimating volatility:

Implied volatility methods.

Used for hedging and trad-ing. Possibly inadequate measure of future market move-ments.

Averaging Methods.

Provides a simple estimation based on historical market movements.

Averaging Methods

Consider a historical series of risk factors (stock prices, yield rates, etc.) given by fs

i

g, for i = 0:::n. We adopt the

con-vention that s

0 is today's observation, and the values move

backward in time with increasing i.

If they are log-normally distributed, we can dene

= P n i=1 i log s i;1 s i P n i=1 i

for a parameter < 1 (0.95, for example), which gives more

weight to recent observations. Similarly, we set

2 = P n i=1 i log s i;1 s i ; 2 P n i=1 i

For multifactor series fs (k) i

g, covariances are found in a

sim-ilar way: 2 kl = P n i=1 i log s

(k) i;1 s

(k) i ; k log s

(k) i;1 s

GARCH models

It is a stochastic volatility model. It is popular because it accounts for volatility jumps. The general GARCH process is given by the coupled system of equations

( r

i =

i

i =

! +

i;1 +

r i;1

:

where the label i now goes forward in time.

The model is used as follows: given a known series of log-returns (normalized to mean 0),

r

i = log( s

i)

; log(s i;1)

i = 1:::n

and a choice of parameters, the likelihood of that given choice is given by the expression

ln( t) +

n X

i=1 r

2 t

2 t

We calibrate the GARCH parameters!, and to this series

of returns by maximizing this likelihood.

GARCH list

ARCH

(q)

.

i =

! + q X k=1

k

r i;k

:

GARCH(p,q).

i =

! + q X k=1

k

r

i;k + p X

`=1

` r

i;` :

AGARCH.

i =

! + (r i;1

; ) + r i;1

Value at Risk

VaR

of a portfolio is what the portfolio can loose overnight with a certain probability:

Prob

f(0); (t) >

VaR

g = 5%:

0 0.2 0.4 0.6 0.8 1 1.2

0 0.2 0.4 0.6 0.8 1

"exact" "app0"

Methodologies.

Three methods.

Historical Methods. Monte Carlo.

Analytic. RiskMetrics.

Caution:

Lottery ticket

Historical Methods

Prot and Loss statistics are computed using the value of the actual portfolio under a set of historical scenarios.

Optimization problems:

Portfolio Compression

Replace a portfolio with a smaller one that preserves certain characteristics (price, VaR, sensi-tivities, etc.)

Portfolio Replication

Replace exotic or "unwanted" in-struments by a (probably larger) portfolio of standardized instruments, in a way that certain properties are preserved. Basically, given a target portfolio , and a base basket of instruments

i, for

i = 1:::n, we try to choose the position

numbers i that minimize the expression

; n X

i=1 i

i

where jjjj can denote a variety of things, such as absolute

Monte Carlo

Historical scenarios are replaced by "random" ones. Let the variance/covariance matrix given by V:

Consider the portfolio (

S

), as a function of n risk factors

S

distributed normally according to V, with present value

given by

S

0. Let V

1=2

be the Cholesky decomposition of V.

Let 1

:::

N i.i.d. normal random vectors in R

n, with mean

0 and var/covar equal to the identity. Future scenarios are then given by

i

V

1=2

+

S

0

i = 1:::N:

The P&L of the portfolio is given by i =

i

V

1=2

+

S

0

; (

S

0)

i = 1:::N:

VaR can then be computed in two ways:

Parametric VaR: t the distribution of i to a normal

distri-bution and compute its standard deviation . Then

VaR

= 1:65 :

Non{parametric VaR: order the values of i in an increasing

Watch out!

Cancellation problems Dimensionality problems

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0 0.5 1 1.5 2 2.5 3 3.5 4

"exact"

Analytic Methods

Delta Normal VaR (RiskMetrics)

Approximate

(t) ; (0) @ @t t=0 + n X i=1 i &S i(

t) ; S i(0)] where i = @ @S i t=0 :

VaR is then given by

Prob ( n X i=1 i &S i(

t); S i(0)]

< ;

VaR

)

= 0:05:

If S i(

t) ; S

i(0) is normally distributed, with 0 mean and

variance/covariance matrix given by V, this is equivalent to

Z x<;VaR e ;x y V ;1 x=2 dx p

det(2V) = 0 :05:

We now introduce the Cholesky decomposition of V = HH y,

and change variables

xH

to obtain

Z Hy

y

<;VaR e

;jyj2=2

dy

(2) n

=2 = 0 :05:

Let A be the rotation that sends H into (jHj0:::0), and

change variables y = zA, to obtain

Z

jHjz1<;VaR e

;jzj2=2

dz

(2) n

=2 = 0 :05:

Since

Z

jHjz1<;VaR e

;jzj2=2

dz

(2) n

=2 = Z

;VaR ;1

e

;x2=2 dx p

2

we conclude that

VaR

= ;z p

jHj

= ;z p

T

V ;1

where V is the variance/covariance matrix, and z is the 95%

percentile of the univariate normal distribution.

Simple

Example

Let's attempt to recreate the situation at Orange County. We would have a portfolio (r

1

:::r

n) of interest rate

in-struments, depending therefore on yield rates r

i, for

i ranging

from 1 (overnight lending rate) to 14 (the 30 year rate). The yield curve will have a covariance matrix V that

mea-sures the market volatility with respect to motions of the interest rate curve.

We can nd the principal components of V, which will give

us the directions of movement in the market that will tend to be more pronounced.

In doing this, we may nd that parallel shifts of the yield curve will be most likely, with a 60% chance. Tilts will be next, with a 25% chance.

Our portfolio is perfectly hedged against parallel shifts it is exposed to tilts. In other words, if we map the risk factors

r

i into the principal components, and keep only the rst two

ones (then most likely ones), we may nd

= (050%)

It would be a mistake to think that the portfolio is risk-free because it is insensitive to parallel shifts. Similarly, it would be a mistake to think (as the OC ocials did) that the portfolio is risk-free because if we hold to maturity, it will make money with certainty.

In fact, Delta-Normal-VaR will, in our situation, yield 1:65

p

y

V 40:

Bond { VaR

(joint with P. Fernandez. A. Kreinin) Zero coupon bond with notional N

0. Its price is P(R(t)) = N

0 e

;R (t)

where R(t) is the one-year interest rate.

R = R 0

e

where R

0 is today's interest rate and

is a normal distributed

random variable with mean zero and variance

2 (the daily

volatility of the interest rate).

R

0 and

R today's and tomorrow's interest rates, respectively. -Var is dened through

Prob N 0 e ;R0 ; N 0 e ;R > P = :

We solve this by setting

= P N ;1 0 e R0: = Prob

n

1; e

R0(1;e ) > o = Prob e

> 1 ;

As is a (0 ) random variable, we obtain that

log

1 ;

1

R 0

log(1 ; )

= q

where q

is the

-quantile of the standard normal

distribu-tion, that is

= 1

p

2 Z

1 q

e

;y2=2 dy:

With the above denition of

, we can solve for P

to obtain P

=

N 0

e

;R0 1 ; e

R0(1;e q

)

:

It is important to note that this exact calculation is possible because the increment in the value of the bond P(0) ; P(1)

is an increasing function of the random variable . And this

allows us to invert the relation between both of them.

-3 -2 -1 1 2 3

1 2 3 4 5 0.002

0.004 0.006 0.008

An approximation Approach

Taylor series:

P(R) P(R 0)+ @P @R

R =R0

| {z }

(R;R

0)+ 12 @ 2 P @R 2

R =R0

| {z }

;

(R;R 0)

2

Linear approx.

=

@(N 0 e ;R) @R

R =R0

= ;N 0 e ;R0 : The P&L:

f() = P(R 0)

; P(R) = ;(R ;R 0) =

N 0 R 0 e ;R0 ; e ; 1 :

And, as is an increasing function of ,

P = N 0 e ;R0 R 0( e q

Quadratic attempt.

; = 12@ 2(

N 0

e ;R) @R

2

R =R0

= N 0

e ;R0

:

P&L:

g() = N 0

e ;R0

R 0

e

; 1 ; R

0

2 (e

; 1) 2

:

And this not an increasing function any more. In fact, it attains a maximum at = log(1 + 1=R

0).

-1 -0.5 0.5 1

-0.3 -0.2 -0.1 0.1

RiskMetrics

The principal assumption of RiskMetrics methodology is that the log-returns of the underlying are small, and so, we can estimate the dierence between two consecutive values keep-ing only the rst term of the correspondkeep-ing Taylor series. In our case, that means

R; R 0 = R 0 e ; R 0 R 0 :

With this assumption, we can perform the same calculation as before:

P(R 0)

; P(R) = ;(R ; R 0)

;;(R; R 0) 2 ; ::: = ;R 0( + 2 +

:::) ;;R 2 0(

+ :::) 2 +

:::

For the -approximation, we should only keep the rst order

terms in , that is,

P(R 0)

; P(R) = ; = N 0 e ;R0 R 0 :

Again, this is an increasing function of , and the VaR is

For the second order approximation, we should keep all the terms up to second order in , that is,

P(R 0)

; P(R) = ;R 0

; R

0

2

; ;R 2 0 2 = N 0 e ;R0 R 0 + 2 1 ; R 0 2 : :

And the ; Var ;;,

P

is now calculated as:

= Prob + 1; R 0 2 2 > P N 0 e ;R0 R 0

0.5 1 1.5 2 2.5 3 0.01

0.02 0.03 0.04 0.05 0.06

Quadratic Finance

Consider portfolio of price with S 1

:::S

n as underlyings.

Underlying vector

S

= (S 1

:::S n)

:

Portfolio parameters:

Delta : * = r

S =

@ @S

1

@ @S

n

Gamma : ; = &Hessian]S =

@ 2 @S

i @S

j

Finance Time

* Linear Short Term

; Non{linear Long Term

Quadratic approximation: (t) (0) + * +

1 2

; t

Quad{VaR

Under the quadratic approximation, this becomes

Prob

* + 1 2

; t

< ;

VaR

= 0:05

for

=

S

(t) ;

S

(0)

log{normally distributed. After some elementary analysis, VaR changes into a related quantity K, given by

I 0(

K) = Z

x+12 (x;x) K e

; (xAx)

dx = 0:05

Complexity is independent of number of

instru-ments

Complexity ! number of underlying risk factors. Monte Carlo friendly

Two Analytical Issues

Asymptotic Analysis.

Regard VaR as the solution of

I 0(

K) =

for = 0:05, and solve in the limit ! 0.

Explicit algebraic expressions. Vega friendly.

Portfolio Volatility.

Lemma :

I 0(

K) = Z x+ 1 2x t ;x ;K exp(;xV ;1 x t =2) dx p det2V = Z

12 xDx t

R

exp(;jx; vj 2

=2)

dx

(2) n

=2

where v replaces delta, D replaces V and ;, and R replaces

VaR, given by ;K.

PROOF:

Change variables for x = yV

1=2

,

I 0(

K) = Z 0 y+ 1 2y t ; 0 y ;K exp(;jyj 2 =2) dy

(2) n

=2

where

*0 = * V

1=2

;

0 = V

1=2

;V

1=2 :

Put

;0 = S

;1 DS

with D diagonal and S orthogonal. Change variables again, z = Sy

I 0(

K) = Z 00 z+ 1 2zDz t ;K

exp(;jzj 2

=2)

dz

(2) n

with

*00 = *0 S

;1 = * V

1=2 S

;1 :

Finally, put

z = (x ;v) v = * 00

D ;1

which yields

I 0(

K) = Z

1

2zDz t

;K+ 00

D ;1

00t

exp(;jz ; vj 2

=2)

dz

(2) n

=2

Q_ED

Remark

Putting

1

2

n1

;

1

;

n2

the risk factor largest responsible for VaR is the one corre-sponding to

Reduction to positive Gamma

"The eleventh commandment was `Thou Shalt Compute' or `Thou Shalt Not Compute'

::: I forget which."

{ Epigrams in Programming, ACM SIGPLAN Sept. 1982

For D as before (real diagonal),

I(R) = Z xDx t R e ;jx;j2 dx: D = D ;1 1 0 0 ;D ;1 2 ! with D 1, D

2 positive diagonal.

I(R) = Z

jx;1j2;jy;2j2 R e

;(xD1x);(yD2y)

dxdy

= Z 1 0

Z

jy;2j2=r2 e

;(y D2y) Z

jx;1j2 r2+K e

;(xD1x) r

n2;1 dr

=

Z 1

I n1(

p r 2 + K) @ @r I n2(

r 2)

r n2;1

Harmonic VaR

(joint work with C. Albanese).

Z jxj R e ; (x;v)D ;1 (x;v) t dx = R n Z R n J n 2

(2Rjj)

jRj n

2 e

; (D) cos(2

v) d p detD ;1 = 1 X kj R n+2k k+j a kj Z R n jj 2k( v) 2j e ; D d p detD ;1 The a

kj are basically the Taylor coecients of the Bessel

functions and the cosine function.

To compute each Gaussian moment, put

f( ) =

det(D + i + i v t

v)

;12 :

This can be easily computed, since D is diagonal, and

f( ) = 0 @ n Y ( j + i) 1 A ;1 =2 0 @1 +

Then Z R n jj 2k( v) 2j e ; D

d = i k+j @ k @ k =0 @ j @ j =0

f( )

Further, @ i+j f @ i @ j (0 0) = Z R2

(2i^) k

2i^

j b

f(^^)d^d:^

A technical point: f is not integrable in , but @

f is. Hence,

F(^^) = Z

R2 e

;2 i(+^ ^ )

@f @

( )dd

G(^) = Z

R e

;2 i^

f( 0)d

which yields

I(R) = R n 2 ( Z 0 ;1

d^ J

n

2 2R p

jj^

G(^)

(2jj^ ) n 4 +i ZZ cos(2 q

2^) ; 1

2 2^

J n

2 2R p

jj^

F(^^)

(2jj^ ) n

4 d^d^ )

Lemma

F(^^) is supported inside

0 ^ tan ;1

jjvjj 2

Visualizing Risk

Large VaR-Large Delta

0

0.5

1 0

0.5

1 0

0.5 1

Low VaR-Large Delta

0

0.5

1 0

0.5

1 0

Large Var -- Low Delta 1.18 0.88 0.584 0.288 -0.00801

0

0.5

1 0

0.5

1

-0.5 0 0.5 1 1.5

Low Var -- Low Delta 0.294 0.22 0.146 0.072 -0.002

0

0.5

1 0

0.5

1

Asymptotic VaR

Joint work with R. Brummelhuis, A. Cordoba, M. Quintanilla We want to solve

I(R) =

for near 0. Note that this is equivalent to the limit R ! 1.

I(R)

p

2 exp(;2R 2 ;1 1 ) n i=2( ;1 i ; ;1 1 ) 1=2 R ;1 1

Hence, VaR is approximated by the solution to the equation

p

2 exp(;2R 2 ;1 1 ) n i=2( ;1 i ; ;1 1 ) 1=2 R ;1 1

= 0:05:

Sketch or proof.

e

;jxj2=2 = (2 )

n=2

0(

x) + 12 Z

1 0

*x( e

;jxj2=2t t

n=2 ) dt

hence

I(R) = 12 Z 1 0 dt t n=2 Z xDxR

*x( e

;jxj2=2) dx

Lemma

Let be a manifold, with closest to then origin. Then,

Z D

*(exp(;jxj 2

=2))dx = e

(;jx0j2=2)

X <N

c

;(n;3)=2; +

O(

;(n;3)=2;N) !

The rst term is

c

0 = 2(2 )

(n;1)=2 jx

0

j det(I + jx 0

jK) ;1=2

where K is the principal curvature matrix at x 0.

Application to Value-at-Risk

0 0.2 0.4 0.6 0.8 1 1.2

0 0.2 0.4 0.6 0.8 1

"exact" "app1" "app0"

Credit risk

Hedging credit risk is dicult, although Credit

Der-ivatives provide a replicating approach to managing credit risk. Hence, arbitrage{free pricing is replaced by risk neutral pricing.

Credit risk assumes an option contract with a party

that is default prone. Credit exposure is based on default probabilities, which we assume to be inde-pendent of the underlying security of the option. This simplies calculations but it is not realistic.

When default occurs, a portion of the value of the

Credit Premium

Losses due to credit risk will follow a certain probability dis-tribution.

Expected Loss

The expected losses under the default probability distribution.

Unexpected Loss

The standard deviation of the losses. As a trader, you want to charge for both.

Problems.

The distribution of losses is not normal. And

non-parametric approaches are hard.

Discounting. A pricing scheme should discount to

Credit Spread

We dene the credit spread s as follows:

let P (

tT) be price of the bond issued by the default-prone

party:

s = 1 T

log P (0

T) P(0T)

:

More generally, if we know the price V

of a contract with

the counterparty with cash ows c

i at times t

i,

V =

X i

c i

P(0t i)

e ;st

i :

Note that the default-free price is

V = X

i c

i

P(0t i)

:

The Hazard Rate

Let T

be the default time, and

F its probability distribution:

F(t) = Prob ft

> tg:

The process h(t) is the hazard rate when

F(t) = E

e ;

R t

0 h(s)ds

:

F can be calibrated through the observed price of coupon

bearing bonds maturing at t

i, as follows:

P j = X i c i j

P(0t j i)

F(t j i) +

P(0T j)

F(T j)

+R X

i

P(0t j i)

F(t j i;1

; F(t j i)

:

R is the recovery rate.

This admits a solution in the form

F(t) = e ;

R t

0 a(s)ds

Hazard Rate Models

We can now use the similarity between the hazard and the short rate, to translate the methodology for interest rate models into the modeling of the hazard rate evolution.

In a general setting, we can consider h(t) = F(X

t), where

dX t =

(X t)

dt +(X t)

dW t

It gives rise to a Markovian model.