Financial Engineering & Physical Sciences

How to find a good fund manager

Monkeys vs. Fund Managers

10-Week Solicitation

Pitfall of Survivorship

Need 25 years to prove a 6% premium with 95% confidence (assuming 20% volatility).

파생상품 (Derivatives)

Derivative’s value depends on the values of other assets.

The price of a derivative can be estimated with relatively reliable theories.

Financial engineering can be used to design or analyze complex derivative instruments.

Examples Forwards

Osaka Rice Exchange in early 1700’s Agree on price now, pay later.

Swaps Exchange assets now; return them later; pay differential

rent in the meantime.

Options Calls – Agree on price now; if option buyer wants, he

buys asset later. Puts – Agree on price now; if option buyer wants, he

sells asset later

Buy Call

Sell Call

Buy Put

Sell Put

Strangle (OTM Call + OTM Put)

Condor (DITM Call – ITM Call – OTM Call + DOTM Call)

Back Spread (2 OTM Calls – 1 ATM Call)

Strap (2 ATM Calls + 1 ATM Put)

Financial Engineering

Roles of Derivatives and Financial Engineering To provide a wider set of future states (in a convenient way) To satisfy investors with different expectation on the future To manage risks To cope with legal and tax constraints

Two main streams in Financial Engineering industry Econometrics (buy side)

Analysis of past information Future prediction based on past information (time series

analysis)

Applied physics (sell side) Grid-based calculation of PDEs Monte Carlo simulations (variance reduction) Stochastic calculus Theoretical approaches (martingales and measures)

Model of the Behavior of Stock Prices

Fluctuation of stock prices is modeled with stochastic process.

Markov process Only the present value of a variable is relevant for

predicting the future (the market is efficient).

Wiener process (Brownian motion) A particular type of Markov process with =0, 2=1 Scales with t1/2

z = t1/2

( = random drawing from a Gaussian with [0,1])

Proof of z = t1/2

z = z(T)-z(0) = 1N Gi(0,f(t))

(G = random drawing from a Gaussian with [0,f(t)])

N steps of t

Var[z(T)-z(0)] = N Var[G(0,f(t))] = N f(t)2

1 step of Nt

Var[z(T)-z(0)] = N Var[G(0,f(t))] = N f(t)2

N f(t)2 = f(Nt)2

f(x) = x1/2

z = t1/2

Generalized Wiener process

Ito process

Geometric Ito process

dztSdttSS

dS),(),(

dztSdttSdS ),(),(

dzdtdS

Chain Rule Let S = S(t,z).

In deterministic calculus,

In stochastic calculus,

dzz

Sdt

z

S

t

S

dzz

Sdzz

Sdtt

SdS

2

2

22

2

2

1

)(2

1

dzz

Sdtt

SdS

Ito’s Lemma

Let S follow an Ito process:

Then, f(S,t) follows the following process:

dztSdttSdS ),(),(

dzS

fdt

S

f

t

f

S

f

dtS

fdtt

fdzdt

S

f

dzdtdzdtS

fdtt

fdzdt

S

f

dSS

fdtt

fdS

S

fdf

22

2

22

2

222

2

22

2

2

1

2

1)(

]2)()[(2

1)(

)(2

1

Justification ofGeneralized Wiener Process

Generalized Wiener process can be obtained from Conditional Probability Density Functions, which have the following properties:

When the process is Markovian,

This is known as the Chapman-Kolmogorov equation.

121212

111

)|()||()|(

)()|()(

kkkkkkkk

kkkkk

dSSSpSSSpSSp

dSSpSSpSp

12112 )|()|()|( kkkkkkk dSSSpSSpSSp

If we require that the process is continuous, the C-K equation becomes the Fokker-Planck equation, or the Forward Kolmogorov equation:

where

and the initial condition is

)]|()([2

1)]|()([

)|(02

2

00 SSpSb

SSSpSa

St

SSptttt

t

ttttttttt

t

ttttttttt

t

dSSSpSSt

Sb

dSSSpSSt

Sa

)|()(1

lim)(

)|()(1

lim)(

2

0

0

)|()|( 00 SSSSp tt

The solution to the F-K equation is

where the process w0 has the following properties:

The solution to the F-K equation with a=0 & b=1 is

These suggest that we will be interested in processes of the form

twtSaSSt 000 )(

)(][

0][

020

0

SbwE

wE

t

z

tzzp t

t 2exp

2

1)|(

2

0

dztSdttSdS ),(),(

Risk

Risk-Free Assets vs. Risky Assets

Price of Risk

Risk Preference Risk-averse (caused mainly by capital limit) Risk-neutral Risk-loving

Risk-Neutral Valuation

1. Assume that the expected return of the underlying asset is the same as the riskless return ( = r).

2. Calculate the expected payoff from the derivative at its maturity.

3. Discount the expected payoff at the riskless return r.

Won 1997 Nobel Prize in Economics!

- 20

- 10

0

10

20

30

40

50

60

70

80

50 60 70 80 90 100 110 120 130 140 150Stock Price

Payoff from Call Option

Probability Distribution of Stock Price

Black-Scholes Pricing Formula for European Call & Put OptionsCall

Put

Where N(x) is the cumulative standard normal distribution, and

)()(

)]0,[max(E

210 dNKedNS

KSecrT

TrT

)()(

)]0,[max(E

102 dNSdNKe

SKeprT

TrT

Tdd

T

TrKSd

12

221

01

)()/ln(

The Black-Scholes-Merton Differential Equation

Assume the stock has a geometric Brownian motion

Let f(S,t) be the price of a derivative contingent on S.

dzdtS

dS

dzSS

fdtS

S

f

t

fS

S

fdf

222

2

2

1

Consider a portfolio composed of stocks of amount a and derivatives of amount b. Then the value of the portfolio is:

By having , the Wiener process can be eliminated. The value of this portfolio is

dzSS

fbadtS

S

fb

t

fbS

S

fbSad

fbSa

222

2

2

1and ba Sf

tSS

f

t

f

fSS

f

222

2

2

1

Since this portfolio is now riskless, there would be riskless arbitrage opportunity unless

By equating the last two equations,

This is the Black-Scholes-Merton differential equation.

tSS

ffr

tr

frSS

fSr

S

f

t

f

tSS

ffrtS

S

f

t

f

222

2

222

2

2

1

2

1

Expectation & the B-S-M Equation

Consider the following boundary value problem:

Let S satisfy the stochastic differential equation:

Apply Ito’s lemma:

)(

02

1

,

22,2

,2

,,,

SF

SS

FSr

S

F

t

F

ST

StSt

StStSt

dzdtS

dStt StSt

t

t,,

ttStSt

tStSt

tStStSt

St dzSS

FdtS

S

FSr

S

F

t

FdF

t

t

t

t

t

tt

t ,,22

,2

,2

,,,

, 2

1

Integrate from t = 0 to T :

As far as (i.e., Ito integral is definable), one has

If F solves the PDE, taking expectations gives

This is the Feynman-Kac formula, a fundamental connection between PDEs and SDEs. It shows that stochastic calculus can solve PDEs for us.

T

tStSt

tStStSt

SST dtSS

FSr

S

F

t

FFF

t

t

t

tt

T 0

22,2

,2

,,,

,0, 2

10

T

SF dt

0

2)(

00 ,

,0

T

ttStSt dzSS

FE

t

t

]|[]|[ 00,,0 0SESFEF

TT SSTS

T

ttStSt dzSS

Ft

t

0 ,,

Now, by defining , the boundary value problem becomes the B-S-M equation:

Thus, solving the B-S-M equation is equivalent to finding the expectation:

tt Strt

St feF ,,

rfS

fS

S

fSr

t

f

S

fSe

S

fSrefre

t

fe rtrtrtrt

2

222

2

222

2

1

02

1

]|[

]|[

0,,0

,,

0SfeEf

SfeEfe

T

Tt

STrT

S

tSTrT

Strt

Summary Ways Derivatives (and Financial Engineering)

Are Used To hedge risks To speculate (take a view on the future direction of

the market) To lock in an arbitrage profit To change the nature of a liability To change the nature of an investment without

incurring the costs of selling one portfolio and buying another