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Figure 11.1: Call Option Price and Delta Approximation

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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Figure 11.1: Call Option Price and Delta Approximation

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