The Continuous Hedging Argument

Abstract

This chapter develops a continuous hedging argument for derivative security pricing. Following fairly closely the original Black and Scholes (1973) article, we make use of Ito’s lemma to derive the expression for the option value and exploit the idea of creating a hedged position by going long in one security, say the stock, and short in the other security, the option. Alternative hedging portfolios based on Merton’s approach and self financing strategy approach are also introduced.

Appendices

Appendix

Appendix 7.1 Relation Between Stock and Option Betas

Recall the stochastic differential equations for the stock price and the option price which may be written in return form as

$$\displaystyle\begin{array}{rcl} \frac{\mathit{dS}} {S} =\mu \mathit{dt} +\sigma \mathit{dz},& &{}\end{array}$$

(7.45)

$$\displaystyle\begin{array}{rcl} \frac{\mathit{dC}} {C} =\mu _{c}\mathit{dt} +\sigma _{c}\mathit{dz}.& &{}\end{array}$$

(7.46)

We let M denote the value of the market portfolio and assume that it follows the same type of stochastic process as the stock, i.e.

$$\displaystyle{ \frac{\mathit{dM}} {M} =\mu _{m}\mathit{dt} +\sigma _{m}\mathit{dz}. }$$

(7.47)

From portfolio theory the definition of the beta factors is

$$\displaystyle{ \beta = \frac{\mathop{\mathrm{cov}}\nolimits \left (\frac{\mathit{dS}} {S}, \frac{\mathit{dM}} {M} \right )} {\mathop{\mathrm{var}}\nolimits \left (\frac{\mathit{dM}} {M} \right )},\qquad \beta _{c} = \frac{\mathop{\mathrm{cov}}\nolimits \left (\frac{\mathit{dC}} {C}, \frac{\mathit{dM}} {M} \right )} {\mathop{\mathrm{var}}\nolimits \left (\frac{\mathit{dM}} {M} \right )}. }$$

(7.48)

Since

$$\displaystyle{\mathbb{E}\left (\frac{\mathit{dS}} {S} \right ) =\mu \mathit{dt}\;\;\mbox{ and }\;\;\mathbb{E}\left (\frac{\mathit{dM}} {M} \right ) =\mu _{m}\mathit{dt},}$$

it follows that

$$\displaystyle\begin{array}{rcl} \mathop{\mathrm{cov}}\nolimits \left (\frac{\mathit{dS}} {S}, \frac{\mathit{dM}} {M} \right )& =& \mathbb{E}\left [\left (\frac{\mathit{dS}} {S} -\mu \mathit{dt}\right )\left (\frac{\mathit{dM}} {M} -\mu _{m}\;\mathit{dt}\right )\right ] {}\\ & =& \mathbb{E}[(\sigma \mathit{dz})(\sigma _{m}\mathit{dz})] =\sigma \sigma _{m}\mathbb{E}(\mathit{dz}^{2}) =\sigma \sigma _{ m}\;\mathit{dt}. {}\\ \end{array}$$

Also $\mathop{\mathrm{var}}\nolimits \left (\frac{dM} {M} \right ) =\sigma _{ m}^{2}\mathit{dt}$, hence,

$$\displaystyle{ \beta = \frac{\sigma \sigma _{m}\mathit{dt}} {\sigma _{m}^{2}\mathit{dt}} = \frac{\sigma } {\sigma _{m}}. }$$

(7.49)

Similarly

$$\displaystyle{ \beta _{c} = \frac{\sigma _{c}} {\sigma _{m}}. }$$

(7.50)

Eliminating σ _m between (7.49) and (7.50) yields

$$\displaystyle{ \frac{\beta } {\beta _{c}} = \frac{\sigma } {\sigma _{c}}, }$$

(7.51)

which is the result used in the main text.

Problems

Problem 7.1

Rework the hedging argument of Sect. 7.1 but now the hedging portfolio consists of physical positions Q _S in the stock, Q _C in the option and Q _B in risk free bonds. The risk free bond is an instrument whose market value is B and whose return process is given by

$$\displaystyle{\frac{\mathit{dB}} {B} = \mathit{rdt}.}$$

Problem 7.2

• (a) Rework the continuous hedging argument of Sect. 7.1 but now allow the underlying asset to pay a continuously compounded dividend at the rate q.

  Show that in this case the condition of no-riskless arbitrage (7.14) becomes

  $$\displaystyle{\frac{\mu _{c} - r} {\sigma _{c}} = \frac{\mu -(r - q)} {\sigma }.}$$

  Thus show that Eq. (7.16) becomes

  $$\displaystyle{\frac{\partial C} {\partial t} + (r - q)S\frac{\partial C} {\partial S} + \frac{1} {2}\sigma ^{2}S^{2}\frac{\partial ^{2}C} {\partial S^{2}} = \mathit{rC}.}$$

• (b) Adjust the self financing strategy argument of Sect. 7.4 in the case that the underlying asset pays a continuously compounded dividend at the rate q.