Table 11.1: Delta of American Put Option
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The Binomial Tree Model
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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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 =-10c, 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
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• 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