BSE Details June 2011

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Black-Scholes WorldThis is a set of notes to be read together with the Black-Scholes lecture.

As mentioned during the lecture, there is a large amount of working in

deriving the closed form solutions of the Black-Scholes equation. This

is covered in detail in the following pages together with additional com-

mentary and some worked examples. The other reason for writing these

is that the full working is usually not found in texts.

Any questions or feedback can be e-mailed to [email protected]

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IntroductionThe eld of mathematical nance has become particularly prominent

due to the much celebrated Black-Scholes equation written in 1973

by Fisher Black, Myron Scholes and Robert Merton, for which theywere awarded the Nobel prize for economics, in 1997. The origins of

quantitative nance can however be traced back to the start of the

twentieth century. Louis Jean-Baptiste Alphonse Bachelier (March 11,

1870 - April 28, 1946) is credited with being the rst person to derivethe price of an option where the share price movement was modelled by

Brownian motion, as part of his PhD, entitled The Theory of Speculation

(published 1900). Thus, Bachelier is considered a pioneer in the study

of nancial mathematics and one of the earliest exponents of BrownianMotion.

In this section we will derive the Black-Scholes equation(s) and nd

formulae for vanilla call and put options.

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This work is fundamental to pricing in the Black-Scholes environment.

Here we present the classical BSE model and derivation. By classical

we mean in the sense that it is the original 1973 derivation and arguably

the best.

Best in terms of exibility (easy to adapt to dierent situations, models

and contracts).

The ideas discussed here keep returning time and time again in equity

derivatives, exotics, xed income and credit.

The assumptions used in the derivation are essentially incorrect, but

despite this the Black-Scholes Model is robust. The fashion these days

is to criticize this model.

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When we talk about the Black-Scholes derivation framework, the fol-

lowing points spring to mind:

1. Model - set of assumptions

2. Equation - classic Nobel prize winning PDE

3. Formulae - famous closed form pricing formulas for calls and puts

expressed in terms of the standardized Normal cumulative distrib-

ution function N(x) :

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Notation

Consider an options contract

V(S; t; ; ; E; T; r):

Semi-colons separate dierent types of variables and parameters.

Sand t are variables;

and are parameters associated with the asset price;

E and Tare parameters associated with the particular contract;

r is a parameter associated with the currency.

For the moment just use V(S; t) to denote the option value.

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There are no dividends on the underlying (this assumption)

Delta hedging is done continuously (we can do discrete hedging)

The market is frictionless/perfect liquidity, i.e. there are no trans-action costs on the underlying, no taxes or limits to trading (when

you delta hedge stock must be bought and sold - which costs) - we

can also study transaction costs.

There are no arbitrage opportunities. (A portfolio consisting ofan option and stock is constructed. Delta hedging eliminates risk

hence it can only grow at the risk free rate)

The resulting PDE is essentially the Binomial Model in a continuous

time setting.

Black Scholes Model

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Constructing the portfolio

A call option will

( risefall

in value if the underlying asset

( risesfalls

positive correlation

A put option will

( risefall

in value if the underlying asset

( fallsrises

negative correlation

Set up the following portfolio special portfolio consisting of one long

option position and a short position in some quantity , Delta, of the

underlying asset:

=V(S; t) S:

The value ofV is what we wish to nd; we have a model for S;and

we can choose. So the asset evolves according to the SDE

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dS=Sdt + SdX:

The obvious question we ask is how does the value of the portfolio

change over one time-step dt? That is as t ! t + dt:

d =dV dS:

We hold xed during the time step and change when rehedging. It

forV(S; t) gives

dV =@V

@t +

@V

@SdS+

1

2

@2V

@S2dS2:

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and using the form for dSyields

dV =

@V

@t + S

@V

@S +

1

22S2

@2V

@S2

!dt + S

@V

@SdX:

Substituting in d gives the following portfolio change

d =

@V

@t + S

@V

@S +

1

22S2

@2V

@S2

!dt +

S

@V

@SdX (Sdt + SdX)So we note that the change contains risk which is present due to

S@V

@SdX

(SdX) ;

i.e. coecients ofdX: Ideally we want this expression to vanish,

S@V

@SdX

(SdX) = 0;

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which gives

=@V

@S:

This choice of renders the randomness zero. The beauty of this iswe do no have to worry about things like the evaluation of risk or how

much the market wants to be compensated for taking risk, etc.

Now

More importantly we term the reduction of risk as hedging. The perfect

elimination of risk, by exploiting correlation between two instruments (in

this case an option and its underlying) is generally calledDelta hedging.

Delta hedging is an example of a dynamic hedging strategy, because@V

@S

is always changing.

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From one time step to the next the quantity @V

@S changes, since it is,

like Va function of the ever-changing variables Sand t.

This means that the perfect hedge must be continually rebalanced.

After choosing the quantity ; i.e. the number of shares we have to

sell, as suggested above, we hold a portfolio whose value changes by the

amount

d =

@V

@t +

1

22S2

@2V

@S2

!dt:

This change is completely riskless.

So having used dynamic delta hedging to eliminate risk we now appeal

to the idea ofno arbitrage. Is there another such portfolio?

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Suppose we put some money in a bank for a time period dt at an

interest rate r:This grows by an amount rdt:

So - if we have a completely risk-free change din the value then it

must be the same as the growth we would get if we put the equivalent

amount of cash in a risk-free interest-bearing account:

d =r dt:

This is an example of the no arbitrage principle. That is, either

1. put money in the bank and getr dt; or

2. buy an option, short some stock and get@V

@t +

1

22S2

@2V

@S2

!dt:

which is riskless.

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Then both portfolios should give exactly the same return, else there

would be arbitrage.

Hence we nd that

@V

@t +

1

22S2

@2V

@S2

!dt = r (V S) dt

= r V S@V@S dt;

i.e. the change in the hedged option portfolio equals the risk-free return

on the same portfolio.

On dividing bydtand rearranging we get the BlackScholes equation

(BSE)for the price of an option,

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@V

@t +

1

22S2

@2V

@S2 + rS

@V

@SrV = 0

The Black-Scholes equation is a linear parabolic partial dierential

equation. This means that

ifV1 and V2 are solutions of the BSE then so is V1+ V2 and

ifVis a solution of the BSE andkis any constant then kV is also

a solution

Two simple solutions of the BSE are

1. Asset V (S; t) =S

2. Cash V (S; t) =S0ert

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Final and Boundary conditions

To solve the Black-Scholes PDE we need to impose suitable boundary

and nal conditions. Until we do so the BSE knows nothing about what

kind of option we are pricing.

If we remind ourselves of the structure of this equation, i.e. rst order

in time and second order in asset price - this tells us that we need one

time condition and two boundary conditions.

1. Final Conditionprovides information on t: This is called thePayo.

2. Boundary Condition tells us something about the underlying for

two values ofS. In this case we choose S= 0 and S! 1 (i.e.when the underlying becomes large).

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Recall that in the absence of such conditions we obtain a general solu-

tion. PDEs (unlike ODEs) are generally solved for particular solutions,

as most equations are obtained from physical situations hence we have

some information about their behaviour. This is dealt with by the nal

condition. We must specify the option value V as a function of the

underlying at the expiry date T. That is, we must prescribe V(S; T),

the payo.

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Options on dividend-paying equities

Now generalise the Black-Scholes model to include dividends. Normally

a dividend D is paid discretely. So a small percentage of the stock is

paid out in dividends continuously, this keeps the model nice and simpleand we get a closed form solution. In one time stepdtthe asset receives

an amount DSdt (assumeD / stock) :

To build this into the derivation of the equation

= V Sd = dV dS DSdt;

because we are short the stock. Using the earlier hedging argument

gives

d =

@V

@t + S

@V

@S +

1

22S2

@2V

@S2

!dt + S

@V

@SdX

(Sdt + SdX) DSdt

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Currency Options

Consider an option on a foreign currencyS:Holding this gives us interest

at the foreign raterf:In one time step the currency receives an amount

rfSdt: So the eect is the same as receiving a continuous dividendyield. The BSE is

@V

@t +

1

22S2

@2V

@S2 +

r rf

S@V

@S rV = 0;

where r and rfare the domestic and foreign rates of interest in turn.from which the BSE is obtained.

We can also write down the SDE for a foreign currency as

dS=

r rf

Sdt + SdX:

Black Scholes Model

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Commodity Options

Commodities have an associated cost of carry. Physical storage of

assets such as grains, oil and metals is not without cost - we have to

pay to hold the commodity.

Suppose q is the fraction of the commodity Swhich goes towards pay-

ment of cost of carry, i.e. q/ S:

Then in one time step dt an amount qSdt will be required to nance

the holding, hence

d =dV

dS+ qSdt:

The resulting BSE is

@V

@t +

1

22S2

@2V

@S2 + (r+ q) S

@V

@S rV = 0:

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Calls and Puts

For a call option we use the following:

Payo:

V(S; T) = max(S E; 0):

Boundary Conditions:

S= 0 =) V(S; t) = 0

If we put S= 0 in dS=Sdt + SdXthen the change will be zero.

S! 1 =) V(S; t) S

AsSbecomes very large if we look atmax(S E; 0)then we nd that

S >> E; hence V is approximately similar to S:

Black Scholes Model

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For a put option we use the following:

Payo:

V(S; T) = max(E S; 0):

Boundary Conditions:

S= 0 =) V(S; t) =Eer(Tt)

This is obtained from the put call parity:

C(S; t) P(S; t) =S Eer(Tt):whereCandPrepresent a call and put in turn. We know whenS= 0;

C= 0:

S! 1 =) V(S; t) 0

This is all the information we need to solve the BSE.

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Solving the Equation

The BlackScholes equation is now solved for plain vanilla calls and

puts. Starting with

@V@t

+12

2S2@2

V@S2

+ rS@V@S

rV = 0The three main steps are:

Turn the BSE into a one dimensional heat equation by a series oftransformations.

Use a known solution of the heat equation called the fundamentalsolution.

Reverse the transformations.

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Step 2

As we are solving a backward equation we can write

=T t:

The time to expiry is more useful in an options value than simply the

time. We can use the chain rule to rewrite the equation in the new time

variable

@

@t @

@t

@

@

= @@

Under the new time variable

= 0 =) t=T (expiry)Page 27

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so that now will be increasing from zero. So as t

"

#:

The BSE becomes

@U

@ =

1

22S2

@2U

@S2+ rS

@U

@S;

which is simply the Kolmogorov equation. So V(S; t)is the discounted

solution of the Kolmogorov equation.

Step 3

We now wish to cancel out the variable coecients SandS2:When we

rst started modelling equity prices we used intuition about the asset

price return and the idea of a lognormal random walk. Lets write

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= log S:

Again use the chain rule to write the stock in terms of: With this as

the new variable, we nd that this is equivalent to S=e

@

@S =

@

@S

@

@=

1

S

@

@

@2

@S2 =

@

@S

1

S

@

@

!=

1

S

@

@S

@

@

! 1

S2@

@

= 1S

@@S

@@@

@! 1

S2@

@

= 1

S2@2

@2 1

S2@

@=

1

S2

@2

@2 @

@

!

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Now the BlackScholes equation can be written under this transforma-

tion as

@U

@ =

1

22S2

1

S2

@2

@2 @

@

!U+ rS

1

S

@

@U

which simplies to@U

@ =

1

22

@2U

@2 +

r 1

2

@U

@

We need to eliminate the rst order derivative term in :

Final Step

Perform a translation of the co-ordinate system

x=+

r 1

22

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So we are transforming from (; ) to (x; ) :So apply chain rule I

@

@ =

@x

@

@

@x+

@

@

@

@ =

@

@ +

r 12

2 @

@x@

@ =

@x

@

@

@x= 1:

@

@x=) @

2

@

2 = @2

@x2

SoU =W(x; ):@U

@ = 12

2@2U

@2 +

r 12 @U

@ becomes

@@

+

r 12

2 @

@x

W =

1

22

@2W

@2 +

r 12

@W

@

After this change of variables the BSE becomes

@W

@ =

1

22

@2W

@x2 : (1)

To summarize the steps taken to get this 1D heat equation:,

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V(S; t) =

er(Tt)U(S; t) = erU(S; T ) =erU(e; T )

=erUexr122; T !=erW(x; ):So we will start by solving for W(x; ): The equation for this function

is solved using the similarity reduction method, for the fundamentalsolution Wf(x; ; x

0). This is all familiar methodology. We dene

Wf(x; ; x0) =f

(x x0)

!;

wherex0 is an arbitrary constant, and the parameters and are con-stant, to be chosen shortly. We choose

(xx0)

because it is a constant

coecient problem.

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Note that the unknown function depends on only one variable

= (x x0)=

Again we use a combination of product and chain rule to write the PDE

in terms of an ODE:

Wf(x; ; x0) =f() ; = (x x0)=

Sod

d = 1(x x0); d

dx=

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@W

@

= @

d

f() + 1f()

= df

d

d

d + 1f()

= 1 dfd

:

x x0

+ 1f()

= 1

dfd

+ f()!

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@W

@x

= @

dx

f()

= df

d

d

dx=

df

d

= df

d

@2W

@x2 =

@

dx

dfd

!=

d

d

d

dx df

d!=

d2f

d2

= 2d2f

d2

So

@W@

=1 dfd

+ f()! (2)

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@2W

@x2 =2

d2f

d2 (3)

Substituting(2) ; (3)into (1)gives the 2nd order equation

1 f dfd!=1

222 d2f

d2 (4)

We still have a term in (4) and for similarity reduction we need to

reduce the dimension of the problem. This implies

1 = 2 =) =1

2;to give

f 12

df

d

!=

1

22

d2f

d2

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With the correct choice of; we wantZ11

Wf(x; ; x0)dx= 18

So

Z11 Wf(x; ; x0)dx= ZR fxx0p dx = xx

0pp

d = dxSo the integral becomes

ZR

f()p

d=+1=2ZR

f() d = 1

This implies that +1=2 should equal one, in order for the solution to

be normalised regardless of time. Therefore = 1=2:

f() becomes our PDF.

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The function f now satises

12

f+

df

d

!=

1

22

d2f

d2:

where the left hand side can be expressed as an exact derivative

dd

(f) =2d2f

d2:

This can be integrated

f+ 2 dfd

=A

where the constant A= 0because as becomes large, bothf()and

f0 () tend to zero.

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This is variable separable

f = 2 dfd

Z df

f = 1

2Z d

ln f = 122

2 + K

Taking exponentials of both sides gives

f() =Cexp

2

22

!C is a normalising constant such that

CZR

exp 2

22

d= 1:

Easy to solve by substituting u = p

2! p2du = d; and con-

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verts the integral to

Cp

2ZR

eu2

du = 1

Cp

2p

= 1

C = 1

p2

f() = 1p

2e 2

22 :

Replacing gives us the fundamental solution :

Wf(x; ; x0) = 1

p2 e(xx0)2

22 : (5)

This is the probability density function for a Normal random variablex having mean of x0 and standard deviation

p. For

6= 0; Wf

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represents a series of Gaussian curves. (5)allows us to nd the solution

of the BSE at dierent points (e.g. x0 = 2; x0= 17; etc.).

Properties of The Solution

We have made sure from our solution method that

ZRWf dx= 1

this has been xed. At x0 =x (exp 0 = 1)

Wf(x; ; x0) = 1p

2 :

Then as ! 0 (close to expiration); Wf! 1; the Gaussian curvebecomes taller but the area is conned to unity therefore it becomes

slimmer to compensate. As x moves away from x0; exp(1)! 0:Wf(x; ; x

0)is plotted below for dierent values of :If is large then

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Wf is at, asgets smallerWfis increasingly peaked, and focused on

x0:

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-6 -4 -2 0 2 4 6 8

=0.2

=1.0

=5.0x=x'= 1.0

This behaviour of decay away from one point x0, unbounded growth atthat point and constant area means that Wfhas turned in to a Dirac

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delta function(x0

x) as

!0.

Dirac delta function

This is written x x0= lim!0x x0 ; such that

x x0

=

(1 x=x0

0 x

6=x0Z1

1

x x0

dx= 1

or Z1

0

x x0

dx= 1Ifg (x) is a continuous function then

Z1

1g (x)

x x0

dx=g

x0

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So if we take a delta function and multiply it by any other function -

and calculate the area under this product - this is simply the functiong (x) evaluated at the point x=x0: What is happening here?

The delta function picks out the value of the function at which it is

singular (in this case x0). All other points are irrelevant because we aremultiplying by zero.

In the limit as ! 0 the function Wf becomes a delta function atx=x0. This means that

lim!0

1p

2Z11

e(x0x)2

22 g(x0)dx0 =g(x):

Here we have swapped x and x0 - it makes no dierence due to the(x0

x)2 term hence either can be the spatial variable.

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So

1

p

2e(x0x)2

22

is a delta function and g(x0) will be replaced by the payo function.

The term above is also an example of a Greens function, which allowsus to write down the general solution of the BSE in integral form.

So as we get closer to expiration, i.e. ! 0; the delta function picksout the value ofg(x0) at which x0 =x

Now introduce the payo at t=T ( = 0):

V(S; T) =Payo(S):

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Recall x = + r 12

2

; so at expiry = 0 =) x = = log S.HenceS=ex to give

W(x; 0) =Payo(ex

):

The solution of this for >0 is

W(x; ) =Z11

Wf(x; ; x0)Payo(ex

0) dx0:

We have converted the backward BSE to the Forward Equation. Look

at

Payo(ex0) dx0:

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We know

x0 = log S0 =) dx0 = dS0

S0 and

ex0

= S0

therefore Payo(ex

0):dx0 becomes

Payo(S0):dS0

S0This result is important. As log Sdoes not exist in the negative plane

the integral goes from 0 to innity, with the lower limit acting as an

asymptote.

Lets start unravelling some of the early steps and transformations. Re-

turning to our Greens function

1p

2(T

t)

e (x

0x)222(Tt) =

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1p

2(Tt)e

0BBB@ 122(Tt):0BB@log S+ r 122 (T t)| {z }x

log S0| {z }x01CCA

2

1CCCA=

1p

2(Tt)e 1

22(Tt):log SS0+r122(Tt)2So putting this together with the Payo function as an integrand we

have

1p

2(Tt)

Z10

e

1

22(Tt):

log SS0+

r122

(Tt)2

Payo(S0):dS0

S0

V(S; t) = er(Tt)q

2(T t)

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Z1

0

e

log(S=S0)+

r122

(Tt)

2/22(Tt)

Payo(S0)dS0

S0: (6)

This expression works because the equation is linear - so we just need to

specify the payo condition. It can be applied to any European option

on a single lognormal underlying asset.

Equation(6)gives us the risk-neutral valuation. er(Tt) present val-ues to today time t: The integral is the expected value of the payo

with respect to the lognormal transition pdf. The future state is S0; Tand today is (S; t) : So it represents P

(S; t) ! S0; T :

Also note the presence of the risk-free IR r in the pdf. So the expected

payo is as if the underlying evolves according to therisk-neutralrandom

walkdS

S =rdt + dX:

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The real world drift is now replaced by the risk-free return r: The

delta hedging has eliminated all the associated risk. This means that iftwo investors agree on the volatility they will also agree on the price of

the derivatives even if they disagree on the drift.

This brings us on to the idea ofrisk-neutrality.

So we can think of the option as discounted expectation of the payo

under the assumption that Sfollows the risk neutral random walk

V(S; t) =er(Tt)Z1

0ep S; t; S0; TV S0; T dS0

wherep

S; t; S0; T

represents the transition density and gives the prob-

ability of going from (S; t) to S0; T under dSS =rdt + dX:

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So clearly we have a denition for ep;i.e. the lognormal density given byep S; t; S0; T= 1S0q

2(T t)elog(S=S0)+r122(Tt)2/22(Tt):

Two important points

epS; t; S0; T

is a Greens for the BSE. As the PDE is linear we can

write the solution down as the integrand consisting of this function

and the nal condition.

The BSE is essentially the backward Kolmogorov equation whose

solution is the transition density ep S; t; S0; T with S0; T xedand varying (S; t) ;but with the discounting factor.

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Formula for a call

The call option has the payo function

Payo(S) = max(S E; 0):When S < E; max(S E; 0) = 0 therefore

Z10

Z E0

+ Z1E

= Z1E

* Z E0

0 = 0

Expression (6)can then be written as

er(Tt)p

2(T

t) Z

1E

e

log(S=S0)+

r122

(Tt)2

=22(Tt)(S0 E)dS

0

S0

:

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Return to the variablex0 = log S0 =) x0 = log 1=S0so we can writethe above integral as

er(Tt)p

2(Tt) Z1log Eex0+log S+r1

22(Tt)2=22(Tt)(ex0E) dx0

= er(Tt)

p

2(Tt)Z1

log Eex0+log S+r122(Tt)2=22(Tt)ex0 dx0

E er(Tt)p

2(Tt)

Z1log E

ex0+log S+

r122

(Tt)2

=22(Tt)dx0:

Page 53

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Just a couple more steps are required to simplify these messy looking

integrals. Lets look at the second integral

E er(Tt)p

2(Tt)

Z1log E

e12x0+log S+

r122

(Tt)2

=2(Tt)dx0

use the substitution

u =

x0+log S+

r122

(Tt)

p

(Tt)du =

1

p(Tt)dx

0 ! q(T t)du=dx0

and the limits:

x0 = 1 ! u= 1

u = log E! u= log E+log S+r122(Tt)p

(Tt)

Page 54

r(T t) 1 1 2

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E er(Tt)p

2(T

t) Z 1

log S=E+r12

2(Tt)p(Tt)e

12u

2: q

(T t)du

= Eer(Tt)p2

Z1log S=E+

r122

(Tt)

p

(Tt)

e12u

2: du

= Eer(Tt)p2

Zlog S=E+r122(Tt)p

(Tt)1

e12u

2du

=

Eer(Tt) 1p2 Z

d2

1e

12u

2du

= Eer(Tt)N(d2)

The rst integral requires similar treatment however before we do that

we complete the square on the exponent. The integrand is

=ex0+log S+

r122

(Tt)2

=22(Tt)ex

0r(Tt)

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Now just work on the exponent, and put = T t temporarily tosimplify working

x0+log S+

r122

2

22 + x0 r

= x0+log S+r1222+2(x0r)222

=

x0+log S+

r122

2

2(x0r)2

22

= 12x0+log S+r12222(x0r)2

2

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Now expand the bracket in the numerator

x0

2+ log2 S+ r22 +

1

422 2x0 log S 2x0r+ x02+ 2rlog S

2log S r22

2x02+ 2r22

x02 + log2 S+ r22 +1422 2x0 log S 2x0r x02+ 2rlog S2log S+ r22

now complete the square

= x0 + log S+ r+

1

2

22

22log S

=x0 + log S+

r+1

22

2

22log S

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Lets return to the integral

1p

2)

Z1log E

e 1

22

x0+log S+r+122222log Sdx0

= S

p2(Tt) Z1log Ee 1

22(T

t)x0+log S+r+

12

2

(Tt)2

dx0and as before use a similar substitution

v = x0+log S+

r+12

2

(Tt)

p(Tt)dv = 1

p

(Tt)dx0 ! q

(T t)dv =dx0

and the limits as before:

x0 = 1 ! u= 1

x0 = log E! u= log E+log S+

r+122

(Tt)

p

(Tt) :

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Following the earlier working reduces this to

S 1p2

Zlog S=E+r+122(Tt)p

(Tt)1

e12v

2dv

= S

1

p2 Z d1

1 e12v

2

dv= SN(d1)

Thus the option price can be written as two separate terms involving

the cumulative distribution function for a Normal distribution:

V (S; t) =SN(d1) Eer(Tt)N(d2)

where

d1=log(S=E) + (r+ 12

2)(T t)p

T

t

and

Page 59

log(S=E) + (r 122)(T t) p

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d2=log(S=E) + (r 2 )(T t)

pT t=d1

p

T

t:

N(x) = 1p

2 Z x

1e

12

2d:

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The diagrams above show

a) The value of a call option as a function of the underlying at a xed

time prior to expiry

b) The value of a call option as a function of asset and time

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Observations:

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Observations:

Call values decrease as the strike increases

Call prices decrease as we get closer to expiry (T t) ! 0:

Call prices increase with volatility

Call prices increase with interest rates.

Page 62

When there is a continuous dividend yield D or the option is on a

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When there is a continuous dividend yield D or the option is on a

currency which receives interest at the foreign rate (replace D byrf),then the call option simply becomes

C(S; t) =SeD(Tt)N(d1) Eer(Tt)N(d2)

where

d1=log(S=E) + (r D+ 122)(T t)

p

T t and

d2= log(S=E) + (r D 122)(T t)p

T t =d1 pT t:

Page 63

At The Money Forward Options: A nice approximation

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At-The-Money-Forward-Options: A nice approximation

Within the FX world At-The-Money-Forward (ATMF) options are the

most heavily traded. When an option is struck ATMF, it means that the

strikeE=Se(rD);where we use the earlier denition of =T

t:

This is because the call and put are equal. The put-call parity when

there is a dividend yield is

C(S; t) P(S; t) =SeD Eerwhich for ATMF is

SeD =Eer:

There exists a very nice approximation for ATMF options near expiry.

Page 64

Begin by writing the call option formula

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Begin by writing the call option formula

C(S; t) = SeDN(d1) Se(rD)erN(d2)= SeD (N(d1) N(d2))

Now simplify d1 and d2

d1 =log

S=Se(rD)+ r D+ 122 p

=

log e(rD) +

r D+ 122

p

= (r D) +

r D+ 122

p

=

12

2

p

=

1

2pSimilar working shows d2 = 12

p :

Page 65

Returning to the earlier denition of the option price

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Returning to the earlier denition of the option price

C(S; t) =SeDN12

p

N1

2p

Consider the CDF for a variable x, i.e. N(x) : This can be approximated

due to Kendall and Stuart (1943)

N(x) =1

2+

1p2

x x

3

6 +

x5

40+ O

x7!

:

Ifx is small then to leading order this becomes N(x)

12+

1p2

x:

So if we are close to expiry, then (=T t) is small hence

N1

2

p

1

2 1

p2 1

2

p

N

1

2p

N1

2p

=

1p2

p

0:4p

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Formula for a put

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Formula for a put

The put option has payo

Payo(S) = max(E S; 0):

A similar working as in the case of a call yields

V (S; t) = SN(d1) + Eer(Tt)N(d2);

with the same d1 and d2. Naturally the more sensible approach is toexploit the put-call parity. If the price of a call and put are denoted in

turn by C(S; t) and P(S; t)

C

P =S

Eer(Tt)

Page 68

hence rearranging, using the formula for a call together with N(x) +

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g g, g g ( ) +

N(x) = 1; givesP = C S+ Eer(Tt)

= SN(d1) Eer(Tt)N(d2) S+ Eer(Tt)

= S(N(d1) 1)| {z }=N(d1)

+ Eer(T

t)

(1 N(d2))| {z }=N(d2)

= SN(d1) + Eer(Tt)N(d2)

Page 69

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The diagrams above show

a) The value of a put option as a function of the underlying at a xed

time prior to expiry

b) The value of a put option as a function of asset and time

Page 70

Observations:

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Put values increase as the strike increases

Put prices increase as we get closer to expiry (T t) ! 0:

Put prices increase with volatility

Put prices increase as interest rates decrease.

Page 71

Binary Options

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Also known as digital options. These have discontinuous payos. Two

general types: cash-or-nothing orasset-or-nothing options.

In the rst type, a xed amount of cash is paid at expiry if option isin-the-money, whilst the second pays out the value of the underlying

asset. The payo is dened in terms of the Heaviside function

H (x) = ( 1 x >00 x E0 otherwiseSoH takes the value one when it has a positive argument and zerootherwise. So if the option

Page 72

The diagram shows the value of a binary call sometime before expiration.

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What is happening here?

Each part of the BSE plays a role here.

Page 73


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The greeks

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We now examine the sensitivity of an option price to the input vari-

ables/parameters.

The Greeks are forms of measurement on options that express thechange of the option price when some parameter changes given every

other parameter stays the same. This is an essential form of risk man-

agement carried out by all option traders. The next table denes some

of the basic greeks.

Page 75

greek symbol Measures change in

@V option price change when underlying price

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delta =

@V

@S

p p g y g p

increases by1

theta = @V@toption price when time to expiry

decreases by 1 day

gamma = @@Sdelta when the stock price

increases by 1vega @V@

option price when volatilityincreases by1% (100 basis points)

rho = @V@roption price when interest rate

increases by 1% (100 basis points)

psi = @V@Doption price when dividend yield

increases by 1%

Page 76

The greeks above are all rst order derivatives with the exception of@2V

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gamma which is@ V@S2 and from the list is the only sensitivity that does not

measure a change in the option price change, but rather, it measures the

change in delta. Theta is the only Greek that is in the negative domain as

it measures decreases in time. There is no greek letter assigned to vega.

vega, and all measure one percent increases (100 basis points),such as the risk-free rate increasing from 3.5% to 6% (an example of

).

Page 77

Delta

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The graph illustrates the behaviour of both call and put option deltas

for Europeans and Binaries as they shift from being OTM to ATM andnally ITM. Note that calls and puts have opposite deltas - call options

are positive and put options are negative. The binary deltas have been

rescaled so they can be observed on the same plot.

Page 78

Here V can be the value of a single contract or of a whole portfolio

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of contracts. The delta of a portfolio of options is just the sum of thedeltas of all the individual positions.

Option delta is represented as the price change given a 1point move in

the underlying asset and is usually displayed as a decimal value.

Delta values range between 0 and 1 for call options and1 to 0 forput options (which means decimal notation). Some traders refer to the

delta as a whole number between 0 to 100 for call options and100to0 for put options. We can write

=V (S+ S;t) V (S; t)

Swhere S is the unit move. Since = (S; t) ; this means that the

number of assets held must be continuously changed to maintain a delta

neutral position, i.e. = 0:This procedure is called dynamic hedging.

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Gamma

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Delta and Gamma are arguably the two most important sensitivities as

they are partial derivatives with respect to the underlying stock. The

change in option price is at the greatest percentage of the option price

when the option is close to a payo of zero. This is when Gamma is atits highest values. It can be thought of as the acceleration of the option

when the stock changes. This information can be used to predict how

much can be made or lost based on the movement of the underlying

position. Since gamma is the sensitivity of the delta to the underlying itis a measure of by how much or how often a position must be rehedged

in order to maintain a delta-neutral position.

Page 81

A list of basic greeks was presented in the lecture. More advanced

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greeks are discussed in Espens lecture and an exhaustive collection canbe found in his book. Here we use basic dierentiation techniques to

demonstrate the simplicity in obtaining (for example) the delta of a

European Put. The idea is to show how straightforward it actually is

to produce complex looking formulae. We have just written the price ofa put

P(S; t) =Eer(Tt)N(d2) SeD(Tt)N(d1)

So we want = @P@S:

Useful results:

IfN(x) = 1p2Z x1

e 2=2d then dNdx = 1p2ex2=2 : Leibniz Rule

Page 82

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d1 =log (S=E) + r D+ 122 (T t)

p

T t ,d2 =d1

pT t

9>>=>>; =)@(d1)

@S =

@(d2)

@S

Another result of importance (messy to prove)

SeD(T

t) 1

p2 exp(d2

1 =2) =Eer(T

t) 1

p2 exp(d 2

2 =2)Write

Eer(Tt)N(d2) (a)

SeD(Tt)N(d1) (b)and

@

@S(a) =Eer(Tt)

@

@SN(d2)

Page 83

now use chain rule

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Eer(Tt) @@d2

N(d2) @(d2)@S

= Eer(Tt) 1p2

exp(d 22 =2)@(d2)

@S

@

@S(b) =eD(Tt)

@

@SSN(d1)

use product rule then chain rule

eD(Tt) N(d1) + S @@S

N(d1)= eD(Tt)

N(d1) + S @

@d1N(d1)

@(d1)@S

!

= eD(Tt)N(d1) S 1p2

exp(d 21 =2)@(d1)@S !

Page 84

So now

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= @@S

(a) @@S

(b)

= Eer(Tt) 1p2

exp(d 22 =2)@(d2)

@S

eD(Tt)N(d1) S 1p2 exp(d 21 =2)@(d1)@S != eD(Tt)N(d1) +

@(d1)

@S SeD(Tt) 1

p2exp(

d 21 =2)

Eer(Tt)

1

p2exp(

d 22 =2)!

= eD(Tt)N(d1) +@(d1)

@S (0)

Using

N(x) + N(x) = 1 =) N(x) = 1 N(x)

= eD(Tt) (1 N(d1))= eD(Tt) (N(d1) 1)

Page 85

Basket Options

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In reality options are often written on several underlyings. This is an

example of higher dimensional problem and leads onto the idea ofmulti-

factor models, as there are now more sources of randomness. Consider

the simplest case of a basket option on two stocks S1 and S2

dSi =iSidt + iSidX; i= 1; 2:

So each asset has its own parameters. dX1 and dX2 make this a

two factor model and we will derive an equation for the option priceV (S1; S2; t) : The pair of random sources mean that we now require

two assets with which to hedge away our risk. The portfolio becomes

=V(S; t) 1S1 2S2:As before keeping delta xed across a time step dt gives

d =dV(S; t) 1S1 2S2:

Page 86

We need to consider It for V (S1; S2; t)

@ @ @ @

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d = @V

@tdt +

1

2

@2V

@S21dS21+

1

2

@2V

@S22dS22+

@2V

@S1@S2dS1dS2

@V

@S1 1!

dS1

@V

@S2 2!

dS2

Note

dS21 =21S

21dt; dS

22 =

22S

22dt; dS1dS2=12S1S2dt

To eliminate risk means we take1 =

@V

@S1; 2 =

@V

@S2

which gives a risk free portfolio, i.e. no arbitrage, hence

d =rdt

Page 87

and the pricing PDE becomes

@V 1 @2V 1 @2V @2V

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@V@t

+12

21S21

@2V@S21

+12

22S22

@2V@S22

+ 12S1S2@2V

@S1@S2

= r

V S1

@V

@S1 S2

@V

@S2

!:

Page 88

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