Tải bản đầy đủ - 0 (trang)
Figure 10.3: Implied BSM Volatility from Gram-Charlier Model Prices

Figure 10.3: Implied BSM Volatility from Gram-Charlier Model Prices

Tải bản đầy đủ - 0trang

Allowing for Dynamic Volatility



63



• The GC model is able to capture the strike price structure but not

the maturity structure in observed options prices

• We now consider option pricing allowing for the underlying asset

returns to follow a GARCH process

• The GARCH option pricing model assumes that the expected

return on the underlying asset is equal to the risk-free rate, rf , plus

a premium for volatility risk, λ , as well as a normalization term

• The observed daily return is then equal to the expected return plus

a noise term



Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



Allowing for Dynamic Volatility



64



The noise term is conditionally normally distributed with mean zero

and variance following a GARCH(1,1) process with leverage

• By letting past return feed into variance in a magnitude depending

on the sign of the return, the leverage effect creates an asymmetry in

distribution of returns

• This asymmetry is important for capturing the skewness implied in

observed option prices



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Allowing for Dynamic Volatility



65



• Specifically, we can write the return process as



• Notice that the expected value and variance of tomorrow’s

return conditional on all the information available at time t

are



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Allowing for Dynamic Volatility

• For a generic normally distributed variable

and therefore we get



66



we have



• where we have used

• This expected return equation highlights the role of λ as the

price of volatility risk

Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



Allowing for Dynamic Volatility

We can again solve for the option price using the risk-neutral

expectation as in



• Under risk neutrality, we must have that



• so that expected rate of return on risky asset equals riskfree rate and conditional variance under risk neutrality is

same as the one under original process

Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



67



Allowing for Dynamic Volatility

Consider the following process:



• Here we can check that the conditional mean equals



• which satisfies the first condition

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



68



Allowing for Dynamic Volatility

• Furthermore, the conditional variance under the risk-neutral process equals



• where last equality comes from tomorrow’s variance

being known at the end of today in GARCH model

Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



69



Allowing for Dynamic Volatility



70



The conclusion is that the conditions for a risk-neutral process are

met

• An advantage of the GARCH option pricing approach introduced

here is its flexibility

• The previous analysis could easily be redone for any of GARCH

variance models introduced in Chapter 4

• More important, it is able to fit observed option prices quite well



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Model Implementation

• While we have found a way to price the European option under

risk neutrality, we do not have a closed-form solution available.

• Instead, we have to use simulation to calculate price



• The simulation can be done as follows: First notice that

we can get rid of a parameter by writing



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



71



Model Implementation



72



• For a given conditional variance σ 2t+1 , and parameters

ω, α, β, λ *, we can use Monte Carlo simulation to create future

hypothetical paths of the asset returns

• We can illustrate the simulation of hypothetical daily returns from

day t+1 to the maturity on day

as



• where the

are obtained from a N(0,1) random number

generator and where MC is the number of simulated return

paths

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Model Implementation



73



• We need to calculate the expectation term E*t[*] in the option

pricing formula using the risk-neutral process, thus, we calculate

the simulated risk-neutral return in period t+j for simulation path i

as



• and the variance is updated by

• the simulation paths in the first period all start out

from the same σ2t+1; therefore, for all i, we have



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Tài liệu bạn tìm kiếm đã sẵn sàng tải về

Figure 10.3: Implied BSM Volatility from Gram-Charlier Model Prices

Tải bản đầy đủ ngay(0 tr)

×