Figure 11.1: Call Option Price and Delta Approximation
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Black-Scholes-Merton Model
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• The Black-Scholes-Merton (BSM) formula for a European
call option price
• where (*) is the cumulative density of a standard normal
variable, and
• Using basic calculus, we can take the partial derivative of
the option price with respect to the underlying asset price,
St, as follows:
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Black-Scholes-Merton Model
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• We refer to this as the delta of the option, and it has the
interpretation that for small changes in St the call option price
will change by (d)
• Notice that as (*) is the normal cumulative density function,
which is between zero and one, we have
• so that the call option price in the BSM model will change
in the same direction as the underlying asset price, but the
change will be less than one-for-one.
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Black-Scholes-Merton Model
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For a European put option, we have the put-call parity stating
that
• so that we can easily derive
• Notice that we have
• so that BSM put option price moves in the opposite
direction of underlying asset, and again option price will
change by less than the underlying asset price
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Black-Scholes-Merton Model
• In the case where a dividend or interest is paid on the
underlying asset at a rate of q per day, deltas will be
• where
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Figure 11.2: The Delta of a Call Option (top) and a
Put Option (bottom)
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Figure 11.3: The Delta of Three Call Options
In-the-money
At-the-money
Out-of-the-money
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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