framework for pricing derivatives

Ch. 19J. Hull, Options, Futures and

Other Derivatives

Zvi Wiener

02-588-3049http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

Framework for pricing derivatives

Zvi Wiener Hull - 19 slide 2

The Market Price of Risk

Observable underlying process, for example stock, interest rate, price of a commodity, etc.

sdzmdtx

dx

Here dz is a Brownian motion.

We assume that m(x, t) and s(x, t).

x is not necessarily an investment asset!


dzdtf

df

dzdtf

df

222

2

111

1

Suppose that f1 and f2 are prices of two

derivatives dependent only on x and t. For example options. We assume that prior to maturity f1 and f2 do not provide any cashflow.

The same!


zftff

zftff

22222

11111

Form an instantaneously riskless portfolio consisting of 2f2 units of the first derivative

and 1f1 units of the second derivative.

211122 ffff


An instantaneously riskless portfolio must earn a riskless interest rate.

211122 ffff

tffff 12222121

tr

211122 ffff


This implies

121221 rr

or

2

2

1

1 rr

is called the market price of risk of x.


If x is a traded asset we must also have

But if x is not a financial asset this is not true.

For all financial assets depending on x and time a similar relation must held.

s

rm


Volatility

Note that can be positive or negative, depending on the correlation with x.

| | is called the volatility of f.

If s>0 and f and x are positively correlated, then >0, otherwise it is negative.


Example 19.1

Consider a derivative whose price is positively related to the price of oil. Suppose that it provides an expected annual return of 12%, and has volatility of 20%. Assume that r=8%, then the market price of risk of oil is:

2.02.0

08.012.0


Example 19.2

Consider two securities positively dependent

on the 90-day IR. Suppose that the first one

has an expected return of 3% and volatility of

20% (annual), and the second has volatility of

30%, assume r=6%. What is the market price

of interest rate risk? What is the expected

return from the second security?


Example 19.2

15.02.0

06.003.0

015.03.015.006.0

The market price of IR risk is:

The expected return from the second security:


Differential Equation

Finally leading to:

2

222

2

1

x

fxs

t

f

x

fmxf

x

fsxf

rfx

fxs

x

fxsm

t

f

2

222

2

1)(

f is a function of x and t, we can get using Ito’s lemma:


Differential Equation

rfx

fxs

x

fxsm

t

f

2

222

2

1)(

Comparing to BMS equation we see that it is similar to an asset providing a continuous dividend yield q=r-m+s.

Using Feynman-Kac we can say that the expected growth rate is r-q and then discount the expected payoff at the risk-free rate r.


Risk-neutral approach

True dynamics sdzmdtx

dx

Risk-neutral dynamics

*)( sdzdtsmx

dx


Example 19.3

Price of copper is 80 cents/pound. Risk free r=5%. The expected growth rate in the price of copper is 2% and its volatility is 20%. The market price of risk associated with copper is 0.5. Assume that a contract is traded that allows the holder to receive 1,000 pounds of copper at no cost in 6 months. What is the price of the contract?


Example 19.3

m=0.02, =0.5, s=0.2, r=0.05;

the risk-neutral expected growth rate is

08.02.05.002.0 sm

The expected (r-n) payoff from the contract is

63.76880.0000,1 5.008.0 e

Discounting for six months at 5% we get

65.74963.768 5.005.0 e


Derivatives dependent on several state variables

State variables (risk factors):

iiii

i dzsdtmx

dx

Traded security

n

iiidzdt

f

df

1


Multidimensional Risk

Here i is the market price of risk for xi.

This relation is also derived in APT (arbitrage pricing theory), Ross 1976 JET.

n

iiir

1


Pricing of derivatives

To price a derivative in the case of several risk factors we should

• change the dynamics of xi to risk neutral

• derive the expected (r-n) discounted payoff

TrT feEprice ˆ

TrT fEeprice ˆIf r is deterministic


Example 19.5

Consider a security that pays off $100 at time T if stock A is above XA and stock B is above

XB. Assume that stocks A and B are

uncorrelated.

The payoff is $100 QA QB, here QA are QB are

r-n probabilities of stocks to be above strikes.


Example 19.5

T

TrXSNQ

T

TrXSNQ

B

BBBB

A

AAAA

2ln

2ln

2

2


Derivatives on Commodities

The big problem is to estimate the market price of risk for non investment assets. One can use futures contracts for this.

Assume that the commodity price follows (no mean reversion and constant volatility)

dzdttS

dS )(


Derivatives on Commodities

The expected (r-n) future price of a commodity is its future price F(t).

)(ln)(

)(

tFt

t

dzdttS

dS


Example 19.6Futures pricesAugust 99 62.20Oct 99 60.60Dec 99 62.70Feb 00 63.37Apr 00 64.42Jun 00 64.40

The expected (r-n) growth rate between Oct

and Dec 99 is ln(62.70/60.60)=3.4%, or

20.4% annually.


Convenience Yield

y - convenience yield

u - storage costs, then then r-n growth rate is

m - s = r - y + u


Martingales and Measures

A martingale is a zero drift stochastic process

for example: dx = s dz

an important property E[xT] = x0,

fair game.



Real world fdzfdtdf

Risk-neutral world *fdzrfdtdf In the risk-neutral world the market price

of risk is zero, while in the real world it is

r



By making other assumptions we can define other “worlds” that are internally consistent. In a world with the market price of risk * the drift (expected growth rate) * must be

zfdfdtrdf ~)*(

**

r


Equivalent Martingale Measures

Suppose that f and g are price processes of

two traded securities dependent on a single

source of uncertainty. Define x=f/g.

This is the relative price of f with respect to g.

g is the numeraire.



The equivalent martingale measure result states that when there are no arbitrage opportunities, x is a martingale for some choice of market price of risk.

For a given numeraire g the same market price of risk works for all securities f and the market price of risk is equal to the volatility of g.



fdzfdtrdf ffg )(

gdzgdtrdg gfg )( 2

Using Ito’s lemma

dzdtrfd fffg )2/(ln 2

dzdtrgd gg )2/(ln 2



dzdtg

fd gfgffg )(25.0ln 22

dzg

f

g

fd gf )(

A martingale


Forward risk neutral wrt g

T

Tg g

fE

g

f

0

0

Since f/g is a martingale

T

Tg g

fEgf 00 (19.19) in Hull


Money market as a numeraire

Money market account dg = rgdt

zero volatility, so the market price of risk will be zero and we arrive at the standard r-n world. g0=1 and

T

Tg g

fEgf 00

Trdt

T eg 0

T

rdt

Tg efEf 00


Zero-Coupon Bond as a Numeraire

Define P(t,T) the price at time t of a zero-coupon bond maturing at T. Denote by ET the

appropriate measure.

gT = P(T,T)=1, g0 = P(0,T) we get

TT fETPf ),0(0


Zero-Coupon Bond as a Numeraire

Define F as the forward price of f for a contract maturing at time T. Then

TTT fETP

fF

),0(

In a world that is forward risk neutral with respect to P(t,T) the forward price is the expected future spot price.


Important Conclusion

We can value any security that provides a payoff at time T by calculating its expected payoff in a world that is forward risk neutral with respect to a bond maturing at time T and discounting at the risk-free rate for maturity T.

In this world it is correct to assume that the expected value of an asset equals its forward value.


Interest Rates With a Numeraire

Define R(t, T1, T2) as the forward interest rate as

seen at time t for the period between T1 and T2

expressed with a compounding period T1- T2.

The forward price of a zero coupon bond lasting between T1 and T2 is

),(

),(

1

2

TtP

TtP



),(

),(

),,()(1

1

1

2

2112 TtP

TtP

TTtRTT

A forward interest rate implied forthe corresponding period is

),(

),(),(1),,(

2

21

1221 TtP

TtPTtP

TTTTtR



Setting

),(,),(),(

212

21 TtPgTT

TtPTtPf

We get that R(t, T1, T2) is a martingale in

a world that is forward risk neutral with

respect to P(t,T2).

),,(),,( 211221 TTTRETTtR


Annuity Factor as a Numeraire

Consider a swap starting at time Tn with payment

dates Tn+1, Tn+2, …, TN+1. Principal $1. Denote the

forward swap rate Sn,N(t). The value of the fixed

side of the swap is

N

niiiiNn

NnNn

TtPTTtA

tAtS

),()(

)()(

11,

,,



The value of the floating side is ),(),( 1 Nn TtPTtP

The first term is $1 received at the next payment date and the second term corresponds to the principal payment at the end.

The swap rate can be found as),(),()()( 1,, NnNnNn TtPTtPtAtS

)(

),(),()(

,

1, tA

TtPTtPtS

Nn

NnNn



We can apply an equivalent martingale measure by setting P(t,Tn)-P(t,TN+1) as f and

An,N(t) as g. This leads to

)()( ,, TSEtS NnANn For any security f we have

)()(

,,0 TA

fEtAf

Nn

TANn


Multiple Risk Factors

n

iiig

n

iiif

dzrgdtdg

dzrfdtdf

1,

1,



n

iiig

n

iigi

n

iiif

n

iifi

dzgdtrdg

dzfdtrdf

1,

1,

*

1,

1,

*

Equivalent world can be defined as

Where i* are the market prices of risk



Define a world that is forward risk neutral with respect to g as a world where i*=g,i. It

can be shown from Ito’s lemma, using the fact that dzi are uncorrelated, that the process

followed by f/g in this world has zero drift.


An Option to Exchange Assets

Consider an option to exchange an asset worth U to an asset worth V. Assume that the correlation between assets is and they provide no income. Setting g=U, fT=max(VT-UT,0) in

19.19 we get.

T

TTU U

UVEUf

)0,max(00



0,1max00

T

TU U

VEUf

VUVU 2222

The volatility of V/U is

This is a simple option.



Tdd

T

TqqU

V

d

dNUdNVf

VU

12

2

0

0

1

20100

2ln

)()(

VUVU 2222

Assuming that the assets provide anincome at rates qU and qV.


Change of Numeraire

Dynamics of asset f with forward risk neutral measure wrt g and h we have

n

iiif

n

iifih

n

iiif

n

iifig

dzfdtrdf

dzfdtrdf

1,

1,,

1,

1,,

When changing numeraire from g to h we update drifts by

n

iifigih

1,,,


Change of Numeraire

Set v be a function of traded securities. Define v,i as the i-th component of v volatility. The

rate of growth of v responds to a change of numeraire in the same way. Define q=h/g, then h,i- g,i is the i-th component of

volatility of q. Thus the drift update of v is

qvv


Quantos

Quanto provides a payoff in currency X at time T. We assume that the payoff depends on the value of a variable V observed in currency Y at time T.

F(t) - forward value of V in currency Y

PX(t,T) - value (in X) of 1 unit of X paid at T

PY(t,T) - value (in X) of 1 unit of Y paid at T

G(T) forward exchange rate units of Y per X

G(t)= PX(t,T)/PY(t,T)


QuantosIn equation 19.19 set g=PY(t,T), f - be a security

that pays VT units of currency Y at time T.

fT=VT/ST, and gT=1/ST

f0= PY(t,T)EY(VT), no arbitrage means

F(0)=f0/PY(0,T), hence

EY(VT)=F(0)

When we move from X world to Y world the expected growth rate increases by FG


Quantos

This means that approximately

TYX

GFeTFETFE )]([)]([

TTX

GFeFVE )0()(

Since V(T)=F(T) and EY(VT)=F(0)


Example 19.7

Nikkei = 15,000

yen dividend yield = 1%

one-year USD risk free rate = 5%

one-year JPY risk free rate = 2%

The forward price of Nikkei for a contract denominated in yen is

15,000e(0.02-0.01)1=15,150.75


Example 19.7

Suppose that the volatility of the one-year forward price of the index is 20%, the volatility of the one-year forward yen/USD exchange rate is 12%. The correlation of one-year forward Nikkei with the one-year forward exchange rate =0.3.

The forward price of the Nikkei for a contract that provides a payoff in dollars is

15,150.75 e0.3*0.2*0.12=15,260.2


Siegel’s paradoxConsider two currencies X and Y. Define S an exchange rate (the number of units of currency Y for a unit of X).

The risk-neutral process for S is

tSXY SdBSdtrrdS )(

By Ito’s lemma the process for 1/S is

tSSXY dBS

dtS

rrS

d111 2


Siegel’s paradox

The paradox is that the expected growth rate of 1/S is not ry - rX, but has a correction term.

If we change numeraire from currency X to currency Y the correction term is 2=-2.

The process in terms of currency Y becomes

tSXY dBS

dtS

rrS

d111

framework for pricing derivatives

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