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This paper extends the option betas presented by Cox and Rubinstein (1985) and Branger and Schlag (2007). In particular, we show how the beta of the underlying asset affects both an option’s covariance beta and its asset pricing beta. In contrast to Branger and Schlag (2007), the generalized option betas coincide if the options are evaluated according to the CAPM option pricing model of Husmann and Todorova (2011). The option betas are presented in terms of Black-Scholes option prices and are therefore easy to use in practice.

In continuous time settings, the option beta of [

Based on Black-Scholes option prices, [

This paper extends the option betas presented by [1,6]. The starting point is the CAPM option pricing model of [

This paper is organized as follows. Section 2 presents the assumptions and notation used. Section 3 summarizes the theoretical results about CAPM option pricing in discrete time. In Section 4, we present closed form solutions for option betas and compare them to the option betas of [1,6]. Section 5 concludes.

The assumptions and notation used in this study are exactly the same as those in [

We use the following notation throughout this paper:

Investor’s planning horizon

Time-to-maturity of an option

Exercise price of an option

Price of an underlying asset S

Price of a call on an asset S

Cash flow of the underlying asset

Cash flow of the call

Instantaneous risk-free rate of interest

Instantaneous rate of return on asset S

Instantaneous rate of return on the market index

Standardized cash flow of the market portfolio ()

Expected instantaneous rate of return on asset S

Expected instantaneous rate of return on the market index

Instantaneous volatility of the rate of return on asset S

Instantaneous volatility of the rate of return on the market index

Coefficient of correlation between r_{s} and r_{m}

Market price per unit risk for the investor’s planning horizon Furthermore, we consider a call option on a non-dividend asset S expiring at time when the investor’s planning horizon extends to time. The investor’s planning horizon may be equal to or shorter than the time-to-maturity of the option. We denote the remaining time-to-maturity of the option as and the length of the planning horizon as. For the difference between the investor’s planning horizon and the time-to-maturity of the call, we write. Our aim is to determine the discrete-time beta of an option at time 0.

As usual in option pricing theory, we assume that options, underlyings, and risk-free assets are traded in the market during an option’s remaining time-to-maturity. However, we assume that investors in general do not aim to replicate an options payoff by continuously trading the underlying and a risk-free asset. Reasons for this might be transactions costs, bid-ask spreads, or insufficient market liquidity; moreover, if equity is characterized as an option on the company’s assets the underlying does not trade at all. Therefore, we assume that options are not traded at Black-Scholes prices but at discrete-time CAPM option prices according to [

To clearly arrange our analytical results, we use the following notation for the Black-Scholes price of a call with time-to-maturity when the price of the current asset is replaced by and the strike price is replaced by:

where

In the following, we use some auxiliary variables. Equation (1) will be evaluated for and four different values of,

Accordingly, , , and are used to describe the resulting values of (1). Notice that Equation (1) is linear in and, that is,

The security market line of the CAPM in discrete time is

With the given parameters, for a bivariate normal distribution of rates of return on the market, and the underlying asset, both with respect to the holding period, two definitions of are possible. We refer to

as the asset’s pricing beta, and

as the asset’s covariance beta2. Note that we define the parameter as the rate of return of the expected value, whereas is also used in the literature to identify the expected rate of return. If the latter definition is preferred, must be replaced with throughout this paper. Of course, since both the asset pricing beta and the covariance beta depend on the expected return of an asset we cannot use (2) to explain returns in a lognormal market. However, setting (3) and (4) equal leads to the lognormal security market line in discrete time. After several conversions, we obtain

Of course, if (5) holds in the market, an asset’s pricing beta (3) and its covariance beta (4) coincide, and they are equal to one if. In the limit, when the investor’s planning horizon becomes very short, we can apply L’Hôspital’s rule to (3) and (4) to show that

Therefore, in continuous-time, the security market line (5) is

the well-known intertemporal CAPM of [

[

For a call option, the certainty equivalent valuation equation of the CAPM is

where

Here, describes the market price per unit risk for the investor’s planning horizon. In the following, it is assumed that the investor’s planning horizon is shorter than or equal to the time-to-maturity of the option. Then, in a lognormal market, the expected cash flow of a call and the covariance between the cash flows of the call and the standardized return of the market portfolio are3

Inserting (9) and (10) into (8) yields the CAPM option pricing model of [

In risk-neutral settings,

and Equation (11) equals the call option price of [

The definition of an option’s asset pricing beta with respect to the holding period is

Using and (9) results in

Furthermore, with recourse to (3), (13) yields

[1,6] derive option betas where the options are evaluated at the Black-Scholes option price (1) and not at the more general CAPM valuation Equation (8), which includes [

For the remaining period

must apply if the option is evaluated (risk neutral) at the Black-Scholes option price at the end of the holding period. Therefore, , and (14) results in

Since is linear in and, and

as

applies to the remaining period, (16) yields

Now, let’s consider two special cases of (17). If the planning horizon becomes very short, we can apply L’Hôspital’s rule and (6) to achieve

where, if the lognormal security market line (7) holds,

and is equal to the option’s delta.

Therefore, in continuous time, the beta of an option with respect to the market is given by the beta of its underlying asset times the elasticity of the option price with respect to the stock price. This is a well known result of [

In discrete time, if and the distributions of the rates of return on the market and the underlying asset are bivariate normal, then and. In this case, and (17) is equal to Equation (2.15) of [

who analyze the beta of an option in discrete time with respect to the underlying6. However, if, the bivariate normal distribution of the rates of return on the market and the underlying asset affects both the beta of an option with respect to the underlying and the beta of the underlying, even if options are equal to Black-Scholes prices at the end of the holding period.

The definition of the discrete covariance beta is

where, in a lognormal market,

Using (10) and (22), (21) is equal to

Using (4), we can transform (23) to

Using similar arguments as above (L’Hôspital’s rule and (6)), we obtain the special cases, [

and [

An option’s asset pricing beta can easily be converted to its covariance beta. Inserting (11) into (13) results in7

Using the definition of in (8) and the definition yields

which is equal to the covariance beta (23). Hence, in contrast to [

Adequate risk measures are essential for both hedging and performance evaluation. This paper extends the option betas presented by [1,6]. Whereas [

We utilize a more general, lognormal option pricing equation, which explicitly incorporates the planning horizon and the investors’ expectations about the development of the underlying asset and the market portfolio, and show how the beta of the underlying asset affects both an options’s covariance beta and its asset pricing beta. Furthermore, the two risk measures coincide if the options are evaluated according to the CAPM option pricing model of [

[

where.

To prove the results obtained by [

Proof. Let’s start with the left-hand side. The exponent in (A2) extends to

Using

solves

which also equals the first derivative of the right-hand side of (A2),

Computation of

1^{0}See [

When the planning horizon is shorter that the time-to-maturity, we use the results presented by [

Furthermore, we take into account that the call option price in a lognormal market for the case can be shown to equal1^{0}

where

Thus, we can express the expected value of a call at the end of the holding period as

where

and

are Black-Scholes prices of options with time to maturity of. Splitting (B1) in three integrals and utilizing the solution presented by [

where

and

Computation of.

We compute the covariance between the value of the call and the return on the market portfolio at the end of the holding period using the decomposition theorem. First, to compute, we must integrate

with and defined as in (B2) and (B3), respectively. Using the definition of the conditional density we obtain the following:

Note that the conditional density of the bivariate normal distribution equals the density of the normal distribution with the parameters

such that

(B4) is therefore equal to

For the sake of clarity, we first consider only one term of (B5),

and split it into two separate components,

and

We start with (B8). Its exponent

can be transformed into

Next, we make following substitution

Using, (B8) is equal to

where

Thus, the integral term of (B9) has the same structure as the left-hand side of (A2). Applying

into (B8) yields

Similarly, the exponent of (B7)

can be transformed into

Substituting

and using in (B7) leads to

where

Again, using (A2) yields

such that (B7) takes the following form:

Using (B10) and (B11) in (B6) and multiplying by from (B5) gives

The remaining terms in (B5) can be computed in an analogue way such that takes the simple form

where

The covariance is thus given by