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Table 11.1: Delta of American Put Option

Table 11.1: Delta of American Put Option

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The Binomial Tree Model



16



• A similar formula can be used for European puts as well as

for call options of each style

• Note that delta was already used in Chapter 10 to identify

the number of units in the underlying asset we needed to

buy to hedge the sale of one option

• Delta changes in each point of the tree, which shows that

option positions require dynamic hedging in order to

remain risk free

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



The Gram-Charlier Model



17



• As the delta is a partial derivative of an option pricing

model with respect to the underlying asset price, it is

fundamentally model dependent.

• The preceding deltas were derived from the BSM model,

but different option pricing models imply different

formulas for the deltas

• We saw in the previous chapter that the BSM model

sometimes misprices traded options quite severely

• We therefore want to consider using more accurate option

pricing models for calculating the options delta

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



The Gram-Charlier Model



18



• In the case of the Gram-Charlier option pricing model, we have



• and the partial derivative with respect to the asset price in

this case is



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



The Gram-Charlier Model



19



• which collapses to the BSM delta of (d) when skewness,

11, and excess kurtosis, 21, are both zero

• Again, we can easily calculate the put option delta from



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



GARCH Option Pricing Models



20



• Calculating deltas from the general GARCH option pricing

model, we face the issue that the option price is not available

in closed form but must be simulated

• We have in general



• which we compute by simulation as

• where

is the hypothetical GARCH asset price on

option maturity date

for Monte Carlo simulation

path i, where the simulation is done under the risk-neutral

distribution

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



GARCH Option Pricing Models



21



• The partial derivative of the GARCH option price with

respect to underlying asset price can be shown to be



• where the function 1(*) takes the value 1 if the argument is

true and zero otherwise.

• GARCH delta must also be found by simulation as



• Where

asset price



is again the simulated future risk-neutral



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



GARCH Option Pricing Models



22



• The delta of the European put option can still be derived

from the put-call parity formula

• In the special case of closed-form GARCH process, we have

the European call option pricing formula



• and the delta of the call option is



• The formula for P1 is given in the appendix to the previous

chapter

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



The Portfolio Risk Using Delta



23



• Consider a portfolio consisting of just one (long) call option

on a stock

• The change in the dollar value of the option portfolio,

DVPF,t+1, is then just the change in the value of option

• Using the delta of the option, we have that for small

changes in the underlying asset price

• Defining geometric returns on underlying stock as

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



The Portfolio Risk Using Delta



24



• and combining the previous three equations, we get the

change in the option portfolio value to be



• The upshot of this formula is that we can write the change

in dollar value of the option as a known value t times the

future return of the underlying asset, Rt+1

• Notice that a portfolio consisting of an option on a stock

corresponds to a stock portfolio with  shares

• Similarly, we can think of holdings in underlying asset as

having a delta of 1 per share of underlying asset

• Trivially, the derivative of a stock price with respect to the

stock

price

is

1

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



The Portfolio Risk Using Delta



25



Thus, holding one share corresponds to having =1, and

holding 100 shares corresponds to having =100

• And, a short position of 10 identical calls corresponds to

setting =-10c, where c is delta of each call option

• The delta of a short position in call options is negative, and the

delta of a short position in put options is positive as the delta

of a put option itself is negative

• The variance of the portfolio in delta-based model is



• where 2t+1 is the conditional variance of the return on the

underlying stock

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



The Portfolio Risk Using Delta



26



• Assuming conditional normality, the dollar Value-at-Risk

(VaR) in this case is



• where the absolute value, abs(*), comes from having

taken the square root of the portfolio change variance,

2DV,t+1

• Notice that since DVPF,t+1 is measured in dollars, we are

calculating dollar VaRs directly and not percentage VaRs

• The percentage VaR can be calculated immediately from

the dollar VaR by dividing it by the current value of the

portfolio

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



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Table 11.1: Delta of American Put Option

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