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

Zvi [email protected]

tel: 02-588-3049


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.


Derivative Security

Black-Scholes, Merton 1973

Options, Forwards, Futures, Swaps

Real Options


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.


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


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.


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.


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


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


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.


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.)



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



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



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.



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.


The European Call Option

Strike E.

Time to maturity .

Value of the option at maturity is: Max(X-E,0).

X

V

E



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)



0.52X2Vxx+ rXVx - rV - V = 0

V(X,0) = Max(X-E,0)

Can be solved with the Laplace transform.



12

2

1

21

21

ln

)()(),(

dd

rEX

d

dNEedXNXV r


Normal Distribution

x

N(x)


Put Call Parity

0.52X2Vxx+ rXVx - rV - V = 0

V(X,0) = Max(E-X,0)

E X

VPut Option


Put Call Parity

E X

V

CallPut

Underlying


Put Call Parity

Call-Put

E X

V Underlying


Put Call Parity

E X

V

Bond = Ee-r

Underlying =

Call-Put+Bond


Put Call Parity

X = Call - Put + Ee-r

Synthetic market portfolio


Hedging

X - h*Call - riskless

What is h?

hX

C

1


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.


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


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


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.


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


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, )


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


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.


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?


Solution 3.1XdZXdtdX

22

)(1

2

11ln dX

XdX

XXd

dZdtdtXd 2

)(ln2

tt ZtaXX

2lnln

2

0

tZtt eXX )5.0(

0

2


Solution 3.1

a. E(Xt)=X0et.

b. F0= E(Xt)e (r-)t.

c. r = , or = 0.


Exercise 3.2What are the effects of carrying costs, convenience yields, and dividends?


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)(


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?


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.


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.


Home Assignment

X follows an ABM. Calculate Et(Xs).

X follows a GBM. Calculate Et(Xs).

zvi wienerconttimefin - 4 slide 1 financial engineering the valuation of derivative securities zvi...

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