finance ii january 2004docentes.fe.unl.pt/~jna/_private/materiais/gr_derivatives.pdf · jna/feunl...

Derivatives

Finance IIJanuary 2004

JNA/FEUNL Risk management 2

Outline

• Introduction;• Linear and non-linear derivatives;

derivative valuation;• Special cases: futures;• Risk of forward – example;• Special cases: options;• Derivatives on interest rates.


Introduction ...

• A derivative is an asset the value of which depends on other, more fundamental variables. A derivative is also known as a contingent asset.

• A major classification distinguishes between derivatives:– That behave linearly when they expire;– That behave non-linearly.


... Introduction.

• We can graph gains or losses at any moment during the life of a derivative as a function of the underlying asset:

Underlying

G/L

in d

eriv

ativ

Forward

Riskless debt

Value of company

e

Bond

Underlying Underlying

Cal op

tion

Risky d

ebt

G/L

in d

eriv

ativ

e

Interest rate Value of company

Linear behaviour Non-linear behaviour

G/L

in d

eriv

ativ

e


Example of bond ...

• Consider the following bond:– Nominal value: € 100– Coupon: 5%, semi-annual– Maturity: 35 years– Current interest rate (discount rate): 8%– Price: € 66.18 = PVannuity(€2.5, 35, 8%)

+PVfinal payment(€100, 35, 8%)


... Example of bond.Value of bond as a function of the discount rate

0

20

40

60

80

100

120

140

160

180

200

0% 2% 4% 6% 8% 10% 12% 14%

Discount rate

Val

ue o

f bon

d


Derivatives: general concepts ...

• Fundamental derivatives:– Linear:

• Forward – contract to buy or sell a given asset, on a certain date, for a pre-defined price (the forward price) (OTC trading);

• Future – standardized forward, exchange traded.

– Non-linear:• Option – right to buy (call) or to sell (put) a given

asset, on a certain date, for a pre-defined price .


... Derivatives: general concepts ...• Fundamental derivatives (comments):

– Forward price: delivery price, defined on contract date to be applied on maturity date, which makes the current value of the contract equal to zero.

– Other derivatives:• FRA – Forward contract which defines the interest

rate to apply to a pre-defined amount over a certain period of time;

• Swap – Contract to exchange cash flows, over a period of time, according to a pre-defined formula. Swaps are portfolios of forwards.


… Derivados: noções gerais ...

• Fundamental derivatives (comments):– Other cases:

• Caps, floors, collars, swaptions –OTC options on interest rates based on LIBOR market values;

• European and American options; Bermuda options;• Forward start e cliquet; barrier: knock in and knock

out options; chooser and range forward;• Binary or digital options;• Look back, ladder e shout options;• Composed options, exchange, rainbow and basket,

etc.


... Derivatives: general concepts ...

• Valuation of non-contingent assets:– Discount cash flows at risk adjusted coc:

– Certainty equivalent:

PE CF

kk r R R

ii f m i0

1

1=

+= +

~cov ~ , ~d i

b g d i where λ

λ λ λ

λ

cov ~ , ~ cov ~ ,~

cov ~ , ~

~ cov ~ , ~

R R R CF PP P

R CF

PE CF R CF

r

m i m m i

m

f

d i d i

d i d i

=−F

HGIKJ =

=−

+

1 0

0

01 1

1

1


... Derivatives: general concepts.

• Valuation of derivatives or contingent assets:– Arbitrage valuation, by building synthetic

portfolios:– Expected value at synthetic probabilities

πk, discounted at rf:

V VD SP=

VCF

rD

k kk

K

fk

n

K

=×

+==

=

∑∑

ππ1

111 where


Special cases: futures ...

• Contract definition:– Underlying asset, size, delivery terms, quote,

constraints on price changes and individual holdings;

• Margins:– Marking to market; margin account, initial margin and

maintenance margin; clearinghouse.

• Price convergence to spot;• Payment:

– Asset or financial liquidation.


... Special cases: futures.

• Hedging with futures:– Basis risk;– Contract choice;– Choice of hedge ratio;– Rolling over;– Accounting and fiscal issues.


Value of forward ...• Definitions:

– F – forward price;– S – spot price;– K – contractual delivery price;– t – moment 0;– T – contractual expiry date;– r – daily riskless interest rate;– f – value of forward contract;– I – PV on underlying cash flows;– q – underlying daily yield.


... Value of forward ...

• If I = q = 0 I may value the forward using arbitrage pricing:– Forward: commitment to buy asset S, at price K,

on moment T, with zero initial payment.– Synthetic forward: Obtain loan at interest rate r

with maturity T-t; buy asset at spot price S and keep it; at moment T pay .

– To avoid arbitrage • This yields f K r S f S K rT t T t+ × + = ⇒ = − × +− − − −1 1b g b gb g b g

S K r T t= × + − −1b g b gS r T t× + −1b g



• If f = 0:

• Other special cases:

K F S r T t= = × + −1b g

( ) ( )

( ) ( ) ( )tT

tT

tT

rKFf

qrSFq

rISFI

−−

−

−

+×−=

++

×=>

+×−=>

1 :result General

11 ,0 If

.1 ,0 If



• When the TSIR is neither constant nor flat the value of the future contract is not equal to the value of the forward, and the sign of the difference depends on the correlation between the value of the underlying asset and the interest rate:– If ρ > 0 then V(future) > V(forward);– If ρ < 0 then V(future) < V(forward).



• Foreign exchange forward and future contracts: in addition to the cost of financing the transaction in the domestic currency we have the yield of the investment in the foreign currency:– rx – riskless interest rate in foreign currency.

( ) ( ) ( ) ( )

tT

x

tTtTx

rrSF

rKrSf−

−−−−

++

=

+−+=

11

11

:result a As



• In commodity futures we have:– u – daily storage rate;– y – daily yield (convenience yield);– C – cost of carry: combined effect of r, u and y:

tT

yurSF

−

+

++=

11


... Value of forward .

• Future price and expected value of the price of the underlying: it depends on the systematic risk of the underlying asset.

• They would be equal if r = k, where k is the cost of capital of the underlying asset. In theory, it all depends on the covariance between the underlying and the market. – If covi,M>0, then k > r and F < E(ST).


Risk of forward: example ...

• Let us consider the risk of a currency forward, measured as of today, with a monthly time horizon. According to this contract I buy DEM 100 000 m against delivery of USD 70 880 m.

• This transaction has three components:– Obtain dollar loan (1 year, 70 880 m final payment –

interest rate USD 5.8125%);– Buy DEM in spot market (Ex. rate: 0.6962);– Invest 1 year to yield DEM 000 m (interest rate DEM

3.9375%)


...Risk of forward: example ...

• On a monthly basis, the value of this forward contract changes, as the “equivalent transaction” requires changes in the forward price, as a consequence of changes in:– The spot DEM rate;– The USD interest rate;– The DEM interest rate,



• Assume the following assumptions, to be applied to both exchange and interest rates:

– σ

–

σ

ρ DEM DEM y USD yDEM

DEM yUSD y

1 110000 01912 0 0400

1 01912 10000 0 29371 0 0400 0 2937 10000

. . .. . .. . .σ

DEMm

DEM ym

USD ym

- .77% - 0.17%

- 0.28%

3

1

1

α = ≈95% 165z .b g



- 66 9831/(1+5.8125%)- USD70 880

USD loan

66 9831/(1+3.9375%)× 0.6962

DEM100 000

DEM investment

66 983Taxa de câmbio

0.6962DEM

96 212DEM

Present value(USD)

Conversion rate

Notional value

Exposure map: values


...Risk of forward: example.

• Recalling:

•

•

σ σ σ σ ρP n n i j i j iji jn

w w w= +≠∑∑ 2 2 2

σ V

VaR=

= × =

2546165 2546 4202 95%. @ b g

VaR CF CF= ασ ασ' ' 'b g b gΘ


VaR of exposure with hedging...

• Let us reconsider the previous example, assuming the company was committed to delivering DEM 100 000 one year from now. The one-month VaRof the exposure would be USD 66 983 × 3.8% ×1.65 = USD 4202 (α = 95%).

• If the company decides to hedge the exposure with the previous forward contract (according to which I buy DEM 100 000 m against delivery of USD 70880 m) the hedged VaR becomes:


... VaR of exposure with hedging ...

- 66 9831/(1+5.8125%)- USD70 880

USD loan

66 9831/(1+3.9375%)x 0.6962

DEM100 000

DEM investment

00.6962DEM 96 212 - 96 212 = 0

DEM

Present value(USD)

Conversion rate

Notional value

Exposure map with hedging : values


... VaR of exposure with hedging.

•

• This yields a reduction of 92% in the maximum expected loss with a confidence level of 95% (with the corresponding reduction in the equity required to accommodate unexpected price changes).

σ V

VaR=

= × =

189165 192 311 95%. @ b g


Appendix - VaR of forwardValue of forwardf S e Ke

df fS

dS fr

dr fr

dr e dS Se Tdr Ke Tdr

P e Domestic Discount bondP e Foreign Discount bonddP T e dr dP T e dr

df Se dSS

Se dPP

Ke dPP

Long Forward Long spot currency Long foreign discount bond

t tr T rT

xx

r T r Tx

rT

rT

xr T

rTx

r Tx

r T r T x

x

rT

x

x x

x

x

x x

e =

= + +

- -

:= −

=∂∂

+∂∂

+∂∂

= − +

=

=

= − −

= + −

− −

− − −

−

−

− − −

b g b gc h c h c h

Short domestic discount bond


Options: definition.

• Options give the owner the right to buy or to sell the underlying asset. The call is the right to buy; the put is the right to sell.

• Since we can both buy and sell rights, there are four different possible positions when we trade buy or sell rights.


Value of options on expiry dateCall

De r

i va d

o

Put

Underlying

Buy

Sel l

De r

i va t

i ve

De r

Underlying

ivat

i ve

De r

i va d

o

Underlying Underlying


Valuation of options ...

• Option values on underlying assets that pay no dividends depend on five factors:– Price of underlying asset;– Exercise price;– Maturity;– Riskless interest rate;– Volatility.

• Volatility is the only factor that cannot be observed.


… Valuation of options ...• Option valuation, using either arbitrage or synthe-

tic probability approaches, requires prior defini-tion of the stochastic process that generates prices.

• In continuous time, we usually assume rt, the capitalization power, follows a normal distribution with an expected value that is proportional to time and a standard deviation that is proportional to the square root of time. At date T, price is lognormal (p < 0 has zero probability).


… Valuation of options .

• Once the stochastic process that governs the behaviour of prices is defined, it may be possible (in some cases) to derive option valuation formulas.

• The initial result was obtained by Black and Scholes, whose option valuation model applies to European options on underlying assets that pay no dividend.


Black-Scholes model ...

• Define: SXrT

N xcp

- Stock price; - Exercise value of option;

- Riskless interst rate; - Option maturity, in years; - Standard deviation of relative price changes

of the underlying asset; - Normal cumulative distribution function.

- Value of call; - Value of put.

σ

b g


... Black-Scholes model ...

• With these definitions:

c SN d Xe N d

p Xe N d SN d

dS X r T

T

dS X r T

Td T

rT

rT

= −

= − − −

=+ +

=+ −

= −

−

−

1 2

2 1

1

2

2

2

1

2

2

b g b gb g b g

b g c h

b g c h

ln

ln

σ

σ

σ

σσ


... Black-Scholes model .

Date of contract 18-Fev-03Exercise date 18-Jun-03

Current stock price 3.00Exercise price 3.30

Volatility 35%Riskless interest rate 4%

Value of call 0.14

Compute value of call


Binomial model ...

• It is possible to generate binomial stochastic processes that converge to the continuous normal price distribution that was used in the previous model. Using the binomial processes, we can also price contingent assets, by building riskless synthetic portfolios that are composed of the underlying asset and the riskless security.

• Consider the following example:


... Binomial model ...

St×U with probability π

St+1

St×D with probability 1- π

U, D and p may be chosen in a convenient way such that the binomial process progressively converges to the desiredcontinuous equivalent process.


... Binomial model ...

• Consider stock A with today’s price 100. In one year, it can be worth either 127 or 85. The risklessasset has a 6% annual interest rate. How much are you willing to pay for a call on this stock with an exercise price of 110?

17127

VC=?S=100085


... Binomial model .

• Build a synthetic portfolio:

• Use synthetic probabilities:

17 127 1 6%0 85 1 6%

0 40676232 4573

100 0 406762 32 4573 8 0189

= + += + +

RST⇒

== −RST

= × − =

S BS B

SB

VC

b gb g

..

. . .

π

ππ π

=−−

=−−

=

− =

=× + × −

+=

B DU D

VC

106 0 85127 0 85

50%

1 50%17 0 1

1 0 068 0189

. .

. .

..b g


Appendix: VaR of call option

c c S K r S N d Ke N d

dc cS

dS cS

dS cr

dr c d c d

dS dS dr d dt

r= = × − ×

=∂∂

+∂∂

+∂∂

+∂∂

+∂∂

=

= + + + +

−, , , ,τ σ

σσ

ττ

ρ σ

τb g b g b g1 2

12

2

22

12

2 ∆ Γ Λ Θ


Derivatives on interest rates

• Valuation is more difficult than in other cases (stock or currencies) because:– The statistical behaviour of interest rates is very

complex;– The valuation of many products requires

modelling the entire term structure (TSIR);– Volatility changes along the TSIR;– We use interest rates both to compute the

present value of cash flows and to estimate the cash flows.


Interest rate futures

• Interest rate futures are future contracts on underlying assets the prices of which depend on the level of interest rates;

• The contracts depend on the whole TSIR and not only on individual prices.– To hedge, the company must take into account both the

maturity of the exposure and the maturity of the interest rate to which it is exposed; in addition, it must chose the “best” contract (from within the variety of available contracts) to provide the desired hedge.


Forward rate agreements ...

• FRA are contracts in which both parties agree on the interest rate that applies to a notional value over a pre-determined period of time.

• Nominal value: Starting date: Maturity date: Contractual rate: Spot rate with maturity T Spot rate with maturity T

100TTRrr

k

∗

∗ ∗

::


... forward rate agreements ...

•

• For V(0)=0, the contractual rate must equal the forward rate.

V R r r

V

R r r

Rr

rf

r f

kT T T T

kT T T T

k

T

TT T

TT

NN

nn

n

N

0 100 1 1 1

0 0

1 1 1

1

11

1 1 1

1

b g b g c h b gb gb g c h b g

c hb gb g c h

= × + × + − +LNM

OQP

=

+ × + = +

=+

+− =

+ = +

− − −

− − −

−

+

=∏

* *

* *

*

* *

*

*

*

Note:


... forward rate agreements .

• FRA may be valued as if current forward rates equal future spot rates.

Let be the spot rate at , for maturity The reuqired payment to terminate the contract must be:

At time when and are observed rates at time

This is the formula of the present payment when which means the actual interest rate equals the forward rate.

R T T T

PR

R

t t T r r f t

V tR

fr

f R

kT T

T T

kT T

tT T

T t

t

*

*

.

, , , :

,

*

*

*

*

−

=+

+−

LNMM

OQPP

≤ ≤

=+

+−

LNMM

OQPP × +

=

−

−

−

−

− −

1001

11

0

1001

11 1

b gb gb g

b g b gb g

b g b g


Bond and TB futures ...

• Some details: listed price π actual payment:– For a bond, Payment = listed price + interest due;– The price received by the seller equals

Payment = future listed price ¥ conversion factor + due interest

– The cheapest bond to deliver minimizes the relationship Bond listed price – (future listed price ¥ conversion factor).

– Time differences in determining the price of the future and the price of the bond(wild card play).


... Bond and TB futures ...

• Recalling that is the present value of couponsTime zero is Maturity date is: Riskless interest rate: Price actually received by seller of future contract: Price received by seller of bond:

ItTrFS

F S I r T t= − × + −b g b g1


... Bond and TB futures.

• To compute the future listed price we must:– Compute the price to be received using the

listed price of the cheapest bond to deliver;– Use the previous equation to estimate F;– Compute the future listed price using the price

to be received F;– Divide the listed price by the conversion factor

to adjust the cheapest price to deliver to the standard price of the contractual bond.


Swaps ...

• Swaps are private agreements between two parties to exchange future cash flows in accordance with a pre-established formula. In a plain vanilla: – B is committed to pay A cash flows corresponding to a

pre-determined interest rate to be applied to a notional value over a certain number of years;

– At the same time A commits itself to pay B an indexed interest rate on the same notional amount, over the same period.

– Often, LIBOR is the indexed interest rate. It is resettled at the beginning of each interest period to be paid at the end.


... swaps ...

• Pricing is done by adding basis points to the fixed rate. When the bank pays fixed rate, it pays government debt + x bps; when it receives fixed rate, it requires gd + y bps. y-x>0 is the spread of the swap, and it results from market competition.

• Wharehousing is the institutional practice of taking positions in swaps, hedging interest rate risk (for instance, through the futures market) until another client can be found, interested in the other trade position.


... swaps ...

• Swaps can be viewed as the difference between two bonds, in which transaction a financial institution buys a fixed interest rate bond (variable interest rate) and sells a variable rate bond (fixed rate bond).

• Assume the financial institution receives fixed rate and pays variable rate:


... swaps ...

• V vBBQ Ncfr n

fix

flt

n

−−

−

−−−−

alue of the swap to the financial institution; value of the underlying fixed rate bond; value of the indexed rate bond;

otional value of the swap coupon (fixed rate payment); next indexed rate coupon; period interest rate.

;

1

B cr

Qr

B f Qr

Q

V B B

fixn

nn

N

NN

n

N

flt

fix flt

=+

++

=++

=

= −=

∑1 1 11

1

1b g b g b g e


... swaps ...

• Swaps can also be viewed as a portfolio of FRAs. In fact, in each payment date there is the commitment of a pre-defined payment (or receipt) and receiving (paying) an indexed amount, both applied to the notional value. The value of the swap becomes the summation of the value of the FRAs in the portfolio.

• Since all fixed payments are defines in advance, all “FRA” cannot be simultaneously zero, unless we have a flat term structure.


... swaps.

• FRAs can be valued as if the corresponding forward rate actually materializes, by computing the present value of the differences between payments and receipts at the prevailing interest rates. The value of the swap is the sum of the values of all those FRAs:– Compute forward rates corresponding to each indexed cash flow in

the swap;– Compute all swap cash flows assuming all indexed interest rates

materialize at the level of the estimated forwards;– Compute the present value of each FRA;– Sum the present value of all FRAs.


Interest rate options (IRO)...

• The majority of listed options are buy or sell rights on treasury or euro dollar futures. Future prices go up when bond prices go up and vice-versa.

• The expectation of increases in the stinterest rate suggests the acquisition of puts on euro dollar futures; the expectation of a drop in lt interest rates recommends the acquisition of treasury bond futures.


... IRO: caps, floors, collars, swaptions ...

• Bonds with call options (cash in right held by the acquirer) or put options (repurchase option held by the issuer) are examples of debt instruments that are issued with embedded options.

• Caps are unlisted options that provide hedging against increases in indexed interest rates beyond certain pre-defined limits. Floors impose a lower bound on interest rates to be received. Collars impose both.


... IRO: caps ...

• Caps can be viewed in two alternative ways:– They are a portfolio of call options on the

indexed rate R, with deferred payoffs of the options relative to the moment when the interest rate is negotiated;

– They are a portfolio of put options on discount bonds with the payoff of the puts occurring on the very moment when it is computed.


... IRO: swaptions ...

• Swaptions are options on interest rate swaps. The owner has the right of becoming part of na interest rate swap at a future date.

• When you buy them, they can be viewed as the option to pay fixed interest in a future date. In alternative, they commit to the payment of a future interest rate (with no initial payment).


... IRO: swaptions ...

• Swaps are contracts whereby both parties agree to exchange a variable rate bond for a fixed rate bond.– The value of the indexed bond is always equal

to the notional amount of the swap, in each interest resettlement date; as a result, swaptionsare options to exchange a fixed bond for the notional value of the swap.


... IRO: swaptions.

– When the buyer has the right to pay fix and collect indexed rate, it is a put option on the fixed rate bond with exercise value equal to the notional amount of the swap; otherwise, it is a call option on the fixed rate bond with the same exercise value.


TBill Futures – arbitrage example.

• 45-day T Bill: 10% r45

• 135-day T Bill: 10.5% r135

• T Bill future: 10.6% f4590 v(starting in 45

days, 90 day maturity)•

• The future is cheaper than the implicit forward. Sell future; invest @ BT 135; get loan @ BT 45.

1

110 75%135

135365

45

45365

36590

+

+

L

NMMM

O

QPPP

=r

r

b gb g

.


Bonds: duration hedging ...

• Value of bond: period coupon period principal payment

Yield to maturity: Duration:

e

Bn cn A

yD

B c A y Dn c A y

BB

BD y

y

n

n

n nn

Nn

n nn

Nn

::

= + × + =× + × +

= −+

=

− =

−

∑∑

b g b gb g b g

1

111

1∆ ∆


... Bonds: duration hedging.

Price of future contract and duration of underlying bond: and Value of debt to hedge and duration: and Vertical interest rate change and frequency of capitalization: and Number of contracts to buy:

- e -

- -

F DS D

y mN

S SD y F FD y

SD y N FD y N SDFD

F

S

S F

S FS

F

∆

∆ ∆ ∆ ∆

∆ ∆

*

* *

≈ ≈

= × ⇒ =b g


Bonds: convexity.

•

• As interest rate changes are not vertical, we use piecewise TSIR hedging.

dB B y dy B y By

dy By

dy

By

D dy xx

By

Cn c M

xB

dy xx

B B D xx

C xx

n nn

n

N

= + − =∂∂

+∂∂

∂∂

= − =+

∂∂

= =

×++

=+

= × − ×+

+ × ×+

LNMM

OQPP

=∑

b g b g

b gb g

b g b g

12

2

22

2

2

2

1 22

2

12

2

2

1

11

1 1

e

e

∆

∆

∆∆ ∆

finance ii january 2004docentes.fe.unl.pt/~jna/_private/materiais/gr_derivatives.pdf · jna/feunl...

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