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Dynamic Hedging and Equilibrium PDEs. Note: There are several differences between the notation and equation numbers used here and those found in the text). 1. We want to show that the Black-Scholes option pricing model can be derived from a hedge portfolio and no arbitrage.

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Dynamic hedging and equilibrium pdes

Dynamic Hedging and Equilibrium PDEs

Note: There are several differences between the notation and equation numbers used here and those found in the text)


1. We want to show that the Black-Scholes option pricing model can be derived from a hedge portfolio and no arbitrage.

Assume a stock that pays no dividends with price S(t). Its price follows the diffusion process

If an amount B(t) is invested in the risk free asset, its value follows the process

Now consider a call option that matures at date T


What process does c s t t follow prior to maturity
What process does c(S(T),T) follow prior to maturity? model can be derived from a hedge portfolio and no arbitrage.

By applying Itô’s lemma, we find that the call option c(S(T),T) exhibits the same risk as the stock, that risk is dz.


The hedge portfolio
The Hedge Portfolio model can be derived from a hedge portfolio and no arbitrage.

Now form a portfolio by selling one call option and hedging that call liability by purchasing theunderlying stock and borrowing at the risk-free rate.

  • This portfolio has a zero net investment

  • It is self-financing (any surplus or deficit of funds is made up by either borrowing or lending at the risk-free rate)

    If the number of shares purchased is w(t), the zero net investment restriction implies that the amount invested in the risk-free asset must satisfy, B(t) = c(t) – w(t)*S(t).


The value of the hedge portfolio h t
The value of the hedge portfolio, H(t) model can be derived from a hedge portfolio and no arbitrage.

Substituting for dS from (9.1) and for dc from (9.4) into (9.5), we get

Assume the investor chooses to hold w(t) = ∂c/∂S units of stock. Call ∂c/∂S the hedge ratio.


Dynamic trading leads to complete markets and the pricing of contingent claims
Dynamic trading leads to complete markets and the pricing of contingent claims

Since c(S,t) is a nonlinear function of S and t, w(t) will vary as S and t change, the portfolio will have to be continuously rebalanced to maintain w(t) = ∂c/∂S.

Now substitute this position into (9.6) to get a portfolio that has a riskless rate of return - - the portfolio is riskless since both μ (drift) and dz (volatility) drop out of the equation.


No arbitrage implies the black scholes pde
No Arbitrage Implies the Black-Scholes PDE contingent claims

To avoid arbitrage, the instantaneous rate of return on the hedge portfolio must equal r (the risk-free rate of return). In addition, a zero net investment requires that H(t=0) = 0.


The call option value must satisfy the pde subject to the boundary condition in 9 10
The call option value must satisfy the PDE subject to the boundary condition in (9.10)

The solution to (9.9) and (9.10) is the Black-Scholes call option pricing equation

where N(·) is the standard normal distribution function ~ N(0,1) and


By put call parity the european put option price is
By put-call parity the European put option price is boundary condition in (9.10)

Taking partial derivatives of (9.11) and (9.13) with respect to S(t), we can derive the hedge ratios.

Thus, (0 < ∂c/∂S < 1) and (-1 < ∂p/∂S < 0).


2 we want to show that an equilibrium term structure model can be used to price bonds
2. We want to show that an equilibrium term structure model can be used to price bonds.

A “one-factor” bond pricing model - -

  • assume that one underlying (state) variable affects the prices of all bonds.

  • assume a continuous time stochastic process, continuous trading, and no arbitrage

  • derive the equilibrium relationship between bonds with different maturities


Vasicek model
Vasicek model can be used to price bonds.

Assume that the single factor is the instantaneous yield on a short maturity bond, r(t).

Let P(t, τ) be the price of a pure discount (zero-coupon) bond that pays $1 at τ periods in the future. So, 0< P(t, τ) ≤ 1.


The instantaneous yield follows a process
the instantaneous yield follows a process can be used to price bonds.


Ornstein uhlenbeck process
Ornstein-Uhlenbeck process can be used to price bonds.


Bond price process
Bond price process can be used to price bonds.

If r(t) is the current yield, we can write the bond’s price as P(r(t),τ), and apply Itô’s lemma.

Recall the bond yield process is:

The resulting bond price process exhibits GBM:


The hedge portfolio dz r drops out
The Hedge Portfolio ( can be used to price bonds.dzr drops out)


Portfolio instantaneous rate of return is riskless
Portfolio instantaneous rate of return is riskless can be used to price bonds.

Using (9.18) and (9.19) and the approach we used for stock, we can write the hedge portfolio’s instantaneous rate of return as

The portfolio return is riskless, so the absence of arbitrage implies that the expected rate of return must equal the instantaneous riskless rate, r(t).


Rewriting 9 20 with e p r t
Rewriting (9.20) with E[ can be used to price bonds.μp( )] = r(t)

Equating terms on the first and second lines in (9.21) and rearranging terms, the implication is that in equilibrium the market interest rate risk premium can be written as


Bond prices and the interest rate risk premium
Bond prices and the interest rate risk premium can be used to price bonds.

To derive equilibrium prices of bonds we need to specify the form of the risk premium. */

Initially, assume that the market price of bond risk is a constant over time and equal to q, so that for any bond with maturity (τ) the return can be derived from

as

*/ Cox, Ingersoll and Ross (“A Theory of Term Structure,” Econometrica 1985) show that the bond risk premium can be derived from individual preferences and technology variables.


or can be used to price bonds.


Solving 9 27 subject to the boundary condition that at 0 the bond price equals 1 we get
Solving (9.27) subject to the boundary condition that at τ = 0 the bond price equals $1,we get

where


Using (9.28) it is possible to derive values for = 0 the bond price equals $1,q from the implied bond yield curve: define R( ) as the continuously compounded YTM with maturity =


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