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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.
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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:
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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:
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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:
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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)
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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
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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.
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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:
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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.
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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 :
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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)
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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
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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.
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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.
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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.
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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%
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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
).
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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.
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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.
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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
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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 !
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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)
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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