Transcript No Slide Title

Financial Engineering
The Valuation of Derivative
Securities
Zvi Wiener
[email protected]
tel: 02-588-3049
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Derivative Security
A derivative security is one whose value
depends exclusively on a fixed set of asset
values and time.
Derivatives on traded securities can be priced
in an arbitrage setting.
Derivatives on non traded securities can be
priced in an equilibrium setting.
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Derivative Security
Black-Scholes, Merton 1973
Options, Forwards, Futures, Swaps
Real Options
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Derivative Security
 - the proportion of the value paid in cash.
Pure options:  = 1.
Pure Forwards:  = 0.
No arbitrage assumption.
Free tradability of the underling asset.
Otherwise one have to find the equilibrium.
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Arbitrage Valuation
Primary security X:
dX = (X,t)dt + (X,t) dZ
Derivative security V = V(X,t):
dV = VxdX + 0.5Vxx(dX)2 - Vdt
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Arbitrage Valuation
How we pay for a derivative security?
A proportion  is paid now (deposited in a
margin account).
If securities can be deposited in margin
account, then  = 0.
If paid in full,  = 1.
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Arbitrage Valuation
Arbitrage portfolio: P = V + hX.
dP = dV + h dX
dP = (Vx+h) dX + 0.5Vxx(dX)2- Vdt
In order to completely eliminate the risk, we
should choose (Vx+h) = 0.
Such a portfolio has no risk, thus it must earn
the risk free interest.
Important assumption: X is traded.
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Arbitrage Valuation
Set h = -Vx.
dP must be proportional to the investment in
the portfolio P. This investment is
V-Xh = V-XVx
Thus
dP = rPdt = r(V-XVx) dt
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Arbitrage Valuation
dP = rPdt = r(V-XVx) dt
0.5Vxx(dX)2- Vdt = r(V-XVx) dt
0.5 2Vxx+ rXVx - rV - V = 0
the general valuation for derivatives
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Arbitrage Valuation
0.5 2Vxx+ rXVx - rV - V = 0
Note that (X,t) does NOT enter the equation!
In addition to the equation one has to determine
the boundary conditions, and then to solve it.
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The Forward Contract
Agreement between two parties to buy/sell a
security in the future at a specified price.
No payment is made now (forward), thus =0.
Let X be the price of the underlying asset.
Assume that there are no carrying costs
(dividends, convenience yield, etc.)
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The Forward Contract
Assume that X follows GBM:
(X,t) = X
(X,t) = X
The boundary conditions are:
V(X,0)=X
immediate purchase
V(0, ) = 0
zero is an absorbing boundary
Vx(X, ) < 
the hedge ratio is finite
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The Forward Contract
0.5 2X2Vxx+ rXVx - rV - V = 0
V(X,0) = X
This equation was described in Chapter 2.
a = 0.52
b=r
c = - r
d=0
e=0
m=1
n=0
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The Forward Contract
0.5 2X2Vxx + rXVx - rV - V = 0
V(X,0) = X
The Laplace transform is equal X/(s-(1- )r).
The inverse Laplace transform is V(X,)=Xer(1-).
As soon as <1, the forward price is higher than
the spot price.
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The Forward Contract
The hedge ratio is Vx = V/X  1.
A perfectly hedged position holds one forward
contract and is short V/X units of the spot
commodity.
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The European Call Option
Strike E.
Time to maturity .
Value of the option at maturity is: Max(X-E,0).
V
X
E
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The European Call Option
V(X,0) = Max(X-E,0)
V(0, ) = 0
Vx(X, ) < 
Normally the price is paid in full,  = 1.
The PDE becomes:
0.5 2X2Vxx+ rXVx - rV - V = 0
V(X,0) = Max(X-E,0)
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The European Call Option
0.52X2Vxx+ rXVx - rV - V = 0
V(X,0) = Max(X-E,0)
Can be solved with the Laplace transform.
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The European Call Option
 r
V ( X ,  )  XN (d1 )  Ee
N (d 2 )
X 
1 2
ln   r   
E 
2 
d1 
 
d 2  d1   
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Normal Distribution
N(x)
x
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Put Call Parity
0.52X2Vxx+ rXVx - rV - V = 0
V(X,0) = Max(E-X,0)
V
Put Option
X
E
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Put Call Parity
Underlying
V
Call
Put
E
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X
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Put Call Parity
Underlying
V
Call-Put
E
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X
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Put Call Parity
Underlying =
V
Call-Put+Bond
Bond = Ee-r
E
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X
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Put Call Parity
X = Call - Put + Ee-r
Synthetic market portfolio
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Hedging
X - h*Call - riskless
What is h?
1
 C 

 h
 X 
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Hedging
X 

d X 
C 
C 

X C
1 C  C
2


dX 
dX 
dX

2
C X
2 X X
2
1 C  C 2


dt
2
2 X X
2
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Riskless if volatility
does not change.
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Greeks
Delta of an option is
C

X
Gamma of an option is
Theta of an option is
C

t
Vega of an option is
Rho of an option is
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C

r
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C

2
X
2
C


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BMS Formula and BMS Equation
Delta of an option is
C

 N (d1 )
X
Gamma of an option is equal to vega.
When = (t) the BMS can be modified by
 T t 

T

2
( )d
t
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Implied Volatility
The value of volatility  that makes the BMS
formula to be equal to the observed price.
Volatility smile.
Confirms that the BMS formula is more general
than the BMS formula.
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Equilibrium Valuation
This corresponds to the case when the underlying
security does not earn the risk-free rate r.
Example:
dividends are paid (continuously or discrete)
it is not traded
cost-of-carry
(storage, maintenance, spoilage costs)
convenience yield from liquid assets
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Equilibrium Valuation
If the rate of return on X is below the
equilibrium rate, i.e. dX = (-)Xdt + XdZ
0.52X2Vxx+ (r-)XVx - rV - V = 0
Can be solved by a substitution and change
of a numeraire.
Y = Xe-
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V(X, ) = W(Y, )
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The American Option
dX = (-)Xdt + XdZ
While the option is alive it satisfies the PDE:
0.52X2Vxx+ (r-)XVx - rV - V = 0
Optimal exercise boundary: Q()
high contact condition = smooth pasting condition
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The American Option
When X < Q, the equilibrium equation:
0.52X2Vxx+ (r-)XVx - rV - V = 0
When X > Q, the following equation:
0.52X2Vxx+ (r-)XVx - rV - V = rE- X
is derived by substituting V = X-E in the lhs.
V and Vx are continuous at X=Q.
0.52X2Vxx- V is discontinuous at X=Q.
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Exercise 3.1
V is a forward contract on X. X follows a
GBM. Assume that there are no carrying costs,
convenience yield, or dividends. Let the rate of
return on the cash commodity (X) be
 = r+(M-r)
a. Find the expected future cash price.
b. Relationship between the forward price and
the expected cash price.
c. Under what conditions the expectation
hypothesis is correct?
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Solution 3.1
dX  Xdt  XdZ
1
1 1
2
d ln X  
dX 
( dX )
2
X
2 X
d (ln X )  dt 
2
dt  dZ
2
2

 
t  Z t
ln X t  ln X 0   a 
2 

X t  X 0e
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( 0.5 2 ) t Z t
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Solution 3.1
a. E(Xt)=X0et.
b. F0= E(Xt)e (r-)t.
c. r = , or  = 0.
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Exercise 3.2
What are the effects of carrying costs,
convenience yields, and dividends?
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Solution 3.2
E ( X t )  X 0e
( r c d  y ) t
r - risk free rate,
c - carrying cost,
d - dividend yield,
y - convenience yield.
All variables represent proportions of costs
or benefits incurred continuously.
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Exercise 3.3
Suppose that an underlying commodity’s price
follows an ABM with drift  and volatility .
What economic problems will it cause?
What is the value of a forward contract
assuming that a proportion of the price, , is
kept in a zero-interest margin account?
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Exercise 3.4
Suppose that the value of X follows a mean
reverting process:
dX = (-X)dt+XdZ
When this situation can be used?
Value a forward contract on value of X in  periods.
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Exercise 3.8
Value a European option on an underlying index X,
that follows a mean-reverting square root process:
dX = ( -  X)dt+XdZ
When this situation can be used?
Value a forward contract on value of X in  periods.
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Home Assignment
X follows an ABM. Calculate Et(Xs).
X follows a GBM. Calculate Et(Xs).
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