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Transcript of Zvi WienerContTimeFin - 4 slide 1 Financial Engineering The Valuation of Derivative Securities Zvi...
Zvi Wiener ContTimeFin - 4 slide 1
Financial Engineering
The Valuation of Derivative Securities
tel: 02-588-3049
Zvi Wiener ContTimeFin - 4 slide 2
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.
Zvi Wiener ContTimeFin - 4 slide 3
Derivative Security
Black-Scholes, Merton 1973
Options, Forwards, Futures, Swaps
Real Options
Zvi Wiener ContTimeFin - 4 slide 4
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.
Zvi Wiener ContTimeFin - 4 slide 5
Arbitrage Valuation
Primary security X:
dX = (X,t)dt + (X,t) dZ
Derivative security V = V(X,t):
dV = VxdX + 0.5Vxx(dX)2 - Vdt
Zvi Wiener ContTimeFin - 4 slide 6
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.
Zvi Wiener ContTimeFin - 4 slide 7
Arbitrage Valuation
Arbitrage portfolio: P = V + hX.
dP = dV + h dX
dP = (Vx+h) dX + 0.5Vxx(dX)2- Vdt
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.
Zvi Wiener ContTimeFin - 4 slide 8
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
Zvi Wiener ContTimeFin - 4 slide 9
Arbitrage Valuation
dP = rPdt = r(V-XVx) dt
0.5Vxx(dX)2- Vdt = r(V-XVx) dt
0.5 2Vxx+ rXVx - rV - V = 0
the general valuation for derivatives
Zvi Wiener ContTimeFin - 4 slide 10
Arbitrage Valuation
0.5 2Vxx+ rXVx - rV - 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.
Zvi Wiener ContTimeFin - 4 slide 11
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.)
Zvi Wiener ContTimeFin - 4 slide 12
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
Zvi Wiener ContTimeFin - 4 slide 13
The Forward Contract
0.5 2X2Vxx+ rXVx - rV - V = 0
V(X,0) = X
This equation was described in Chapter 2.
a = 0.52 b = r c = - r
d = 0 e = 0 m = 1
n = 0
Zvi Wiener ContTimeFin - 4 slide 14
The Forward Contract
0.5 2X2Vxx + rXVx - rV - 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.
Zvi Wiener ContTimeFin - 4 slide 15
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.
Zvi Wiener ContTimeFin - 4 slide 16
The European Call Option
Strike E.
Time to maturity .
Value of the option at maturity is: Max(X-E,0).
X
V
E
Zvi Wiener ContTimeFin - 4 slide 17
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)
Zvi Wiener ContTimeFin - 4 slide 18
The European Call Option
0.52X2Vxx+ rXVx - rV - V = 0
V(X,0) = Max(X-E,0)
Can be solved with the Laplace transform.
Zvi Wiener ContTimeFin - 4 slide 19
The European Call Option
12
2
1
21
21
ln
)()(),(
dd
rEX
d
dNEedXNXV r
Zvi Wiener ContTimeFin - 4 slide 21
Put Call Parity
0.52X2Vxx+ rXVx - rV - V = 0
V(X,0) = Max(E-X,0)
E X
VPut Option
Zvi Wiener ContTimeFin - 4 slide 25
Put Call Parity
X = Call - Put + Ee-r
Synthetic market portfolio
Zvi Wiener ContTimeFin - 4 slide 27
Hedging
CC
XXd
dtX
C
X
C 22
2
2
1
2
2
2
2
1dX
X
C
X
CdX
X
C
C
XdX
Riskless if volatilitydoes not change.
Zvi Wiener ContTimeFin - 4 slide 28
Greeks
Delta of an option isX
C
Gamma of an option is2
2
X
C
Theta of an option is
Rho of an option is
Vega of an option is
t
C
r
C
C
Zvi Wiener ContTimeFin - 4 slide 29
BMS Formula and BMS Equation
Delta of an option is )( 1dNX
C
Gamma of an option is equal to vega.
T
t
dtT )(2
When = (t) the BMS can be modified by
Zvi Wiener ContTimeFin - 4 slide 30
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.
Zvi Wiener ContTimeFin - 4 slide 31
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
Zvi Wiener ContTimeFin - 4 slide 32
Equilibrium Valuation
If the rate of return on X is below the equilibrium rate, i.e. dX = (-)Xdt + XdZ
0.52X2Vxx+ (r-)XVx - rV - V = 0
Can be solved by a substitution and change
of a numeraire.
Y = Xe- V(X, ) = W(Y, )
Zvi Wiener ContTimeFin - 4 slide 33
The American Option
dX = (-)Xdt + XdZ
While the option is alive it satisfies the PDE:
0.52X2Vxx+ (r-)XVx - rV - V = 0
Optimal exercise boundary: Q()
high contact condition = smooth pasting condition
Zvi Wiener ContTimeFin - 4 slide 34
The American OptionWhen X < Q, the equilibrium equation:
0.52X2Vxx+ (r-)XVx - rV - V = 0
When X > Q, the following equation:
0.52X2Vxx+ (r-)XVx - rV - V = rE- X
is derived by substituting V = X-E in the lhs.
V and Vx are continuous at X=Q.
0.52X2Vxx- V is discontinuous at X=Q.
Zvi Wiener ContTimeFin - 4 slide 35
Exercise 3.1V 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?
Zvi Wiener ContTimeFin - 4 slide 36
Solution 3.1XdZXdtdX
22
)(1
2
11ln dX
XdX
XXd
dZdtdtXd 2
)(ln2
tt ZtaXX
2lnln
2
0
tZtt eXX )5.0(
0
2
Zvi Wiener ContTimeFin - 4 slide 37
Solution 3.1
a. E(Xt)=X0et.
b. F0= E(Xt)e (r-)t.
c. r = , or = 0.
Zvi Wiener ContTimeFin - 4 slide 38
Exercise 3.2What are the effects of carrying costs, convenience yields, and dividends?
Zvi Wiener ContTimeFin - 4 slide 39
Solution 3.2
r - risk free rate,
c - carrying cost,
d - dividend yield,
y - convenience yield.
All variables represent proportions of costs or benefits incurred continuously.
tydcrt eXXE )(
0)(
Zvi Wiener ContTimeFin - 4 slide 40
Exercise 3.3Suppose 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?
Zvi Wiener ContTimeFin - 4 slide 41
Exercise 3.4Suppose 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.
Zvi Wiener ContTimeFin - 4 slide 42
Exercise 3.8Value 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.