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5 Black-Scholes Differential Equation
Fig. 5.1 The price (left) and delta (right) of a European call option at maturity, 6, 12, and 24 month
before maturity. The annual growth rate and volatility is assumed to be 5% and 30%, respectively
Fig. 5.2 The price (left) and delta (right) of a European put option at maturity, 6, 12, and 24 month
before maturity. The annual growth rate and volatility is assumed to be 5% and 30%, respectively
The plot on the right-hand side in Fig. 5.1 shows = ∂c/∂ S, the hedging ratio
according to (5.3). It is near zero, if the share price S is much below the strike price K
reflecting that there is a finite, but small chance that the option ends up in the money.
Therefore a small number of shares is required to hedge the small probability that
the option writer has to hand over the shares to the option owner at maturity. As the
share price gets closer and eventually exceeds the strike price the hedging ratio
approaches unity, indicating a high probability that the owner will exercise the option
at maturity. Thus, the option writer better acquires shares to cover her commitment
towards the option buyer. The fraction of shares in the portfolio thus reflects the
probability to deliver the shares at maturity. In Fig. 5.2 we show the corresponding
plots for a European put option but will not further discuss the details here.
Note that the of call and put options can be calculated explicitely by differentiating c with respect to S in (5.17). The equivalent expression for the put option is
given by the corresponding derivative of p in (4.42), where we have to replace ρ by
the risk-free rate r f . Explicitely performing the calculations, we find
5 Black-Scholes Differential Equation
Fig. 5.1 The price (left) and delta (right) of a European call option at maturity, 6, 12, and 24 month
before maturity. The annual growth rate and volatility is assumed to be 5% and 30%, respectively
Fig. 5.2 The price (left) and delta (right) of a European put option at maturity, 6, 12, and 24 month
before maturity. The annual growth rate and volatility is assumed to be 5% and 30%, respectively
The plot on the right-hand side in Fig. 5.1 shows = ∂c/∂ S, the hedging ratio
according to (5.3). It is near zero, if the share price S is much below the strike price K
reflecting that there is a finite, but small chance that the option ends up in the money.
Therefore a small number of shares is required to hedge the small probability that
the option writer has to hand over the shares to the option owner at maturity. As the
share price gets closer and eventually exceeds the strike price the hedging ratio
approaches unity, indicating a high probability that the owner will exercise the option
at maturity. Thus, the option writer better acquires shares to cover her commitment
towards the option buyer. The fraction of shares in the portfolio thus reflects the
probability to deliver the shares at maturity. In Fig. 5.2 we show the corresponding
plots for a European put option but will not further discuss the details here.
Note that the of call and put options can be calculated explicitely by differentiating c with respect to S in (5.17). The equivalent expression for the put option is
given by the corresponding derivative of p in (4.42), where we have to replace ρ by
the risk-free rate r f . Explicitely performing the calculations, we find