The previous chapters have established a framework for constructing the distribution of a portfolio of assets with simple linear payoffs—for example, stocks, bonds, foreign exchange, forwards, futures, and commodities. This chapter is devoted to the pricing of options. An option derives its value from an underlying asset, but its payoff is not a linear function of the underlying asset price, and so the option price is not a linear function of the underlying asset price either. This nonlinearity adds complications to pricing and risk management.

In this chapter we will do the following:

In this chapter, we will mainly focus attention on the pricing of European options, which can only be exercised on the maturity date. American options that can be exercised early will only be discussed briefly. The following chapter will describe in detail the risk management techniques available when the portfolio contains options.

There is enough material in this chapter to fill an entire book, so needless to say the discussion will be brief. We will simply provide an overview of different available option pricing models and suggest further readings at the end of the chapter.

A European call option gives the owner the right but not the obligation (that is, it gives the option) to buy a unit of the underlying asset days from now at the price X. We refer to as the days to maturity and X as the strike price of the option. We denote the price of the European call option today by c, the price of the underlying asset today by and at maturity of the option by .

A European put option gives the owner of the option the right to sell a unit of the underlying asset days from now at the price X. We denote the price of the European put option today by p. The European option restricts the owner from exercising the option before the maturity date. American options can be exercised any time before the maturity date.

We note that the number of days to maturity, , is counted in calendar days and not in trading days. A standard year of course has 365 calendar days but only around 252 trading days. In previous chapters, we have been using trading days for returns and Value-at-Risk (VaR) horizons, for example, referring to a two-week VaR as a 10-day VaR. In this chapter it is therefore important to note that we are using 365 days per year when calculating volatilities and interest rates.

The payoff function is the option's defining characteristic. Figure 10.1 contains four panels. The top-left panel shows the payoff from a call option and the top-right panel shows the payoff of a put option both with a strike price of 1137. The payoffs are drawn as a function of the hypothetical price of the underlying asset at maturity of the option, Mathematically, the payoff function for a call option is

and for a put option it is

The bottom-left panel of Figure 10.1 shows the payoff function of the underlying asset itself, which is simply a straight line with a slope of one. The bottom right-hand panel shows the value at maturity of a risk-free bond, which pays the face value 1, at maturity regardless of the future price of the underlying risky asset and indeed regardless of any other assets. Notice the linear payoffs of stocks and bonds and the nonlinear payoffs of options.

We next consider the relationship between European call and put option prices. Put-call parity does not rely on any particular option pricing model. It states

It can be derived from considering two portfolios: One consists of the underlying asset and the put option and another consists of the call option, and a cash position equal to the discounted value of the strike price. Whether the underlying asset price at maturity, ends up below or above the strike price X, both portfolios will have the same value, namely at maturity and therefore they must have the same value today, otherwise arbitrage opportunities would exist: Investors would buy the cheaper of the two portfolios, sell the expensive portfolio, and make risk-free profits. The portfolio values underlying this argument are shown in the following:

The put-call parity also suggests how options can be used in risk management. Suppose an investor who has an investment horizon of days owns a stock with current value S_t. The value of the stock at the maturity of the option is , which in the worst case could be zero. But an investor who owns the stock along with a put option with a strike price of X is guaranteed the future portfolio value which is at least X. The downside of the stock portfolio including this so-called protective put is thus limited, whereas the upside is still unlimited. The protection is not free however as buying the put option requires paying the current put option price or premium, p.

The key challenge we face when wanting to find a fair value of an option is that it depends on the distribution of the future price of the underlying risky asset (the stock). We begin by making the simplest possible assumption about this distribution, namely that it is binomial. This means that in a short interval of time, the stock price can only take on one of two values, which we can think of as up and down. Clearly this is the simplest possible assumption we can make: If the stock could only take on one possible value going forward then it would not be risky at all. While simple, the binomial tree approach is able to compute the fair market value of American options, which are complicated because early exercise is possible.

The binomial tree option pricing method will be illustrated using the following example: We want to find the fair value of a call and a put option with three months to maturity and a strike price of $900. The current price of the underlying stock is $1,000 and the volatility of the log return on the stock is 0.60 or 60% per year corresponding to per calendar day.

The binomial tree approach is very useful because it is so simple to derive and because it allows us to price American as well as European options. A downside of binomial tree pricing is that we do not obtain a closed-form formula for the option price.

In order to do so we now assume that daily returns on an asset be independently and identically distributed according to the normal distribution,

Then the aggregate return over days will also be normally distributed with the mean and variance appropriately scaled as in

and the future asset price can of course be written as

The risk-neutral valuation principle calculates the option price as the discounted expected payoff, where discounting is done using the risk-free rate and where the expectation is taken using the risk-neutral distribution:

where as before is the payoff function and where is the risk-free interest rate per day. The expectation is taken using the risk-neutral distribution where all assets earn an expected return equal to the risk-free rate. In this case the option price can be written as

where x* is the risk-neutral variable corresponding to the underlying asset return between now and the maturity of the option. denotes the risk-neutral distribution, which we take to be the normal distribution so that The second integral is easily evaluated whereas the first requires several steps. In the end we obtain the Black-Scholes-Merton (BSM) call option price

where is the cumulative density of a standard normal variable, and where

Black, Scholes, and Merton derived this pricing formula in the early 1970s using a model where trading takes place in continuous time when assuming continuous trading only the absence of arbitrage opportunities is needed to derive the formula.

It is worth emphasizing that to stay consistent with the rest of the book, the volatility and risk-free interest rates are both denoted in daily terms, and option maturity is denoted in number of calendar days, as this is market convention.

The elements in the option pricing formula have the following interpretation:

Using the put-call parity result and the formula for , we can get the put price formula as

where the last line comes from the symmetry of the normal distribution, which implies that for any value of z.

In the case where cash flows such as dividends accrue to the underlying asset, we discount the current asset price to account for the cash flows by replacing by everywhere, where q is the expected rate of cash flow per day until maturity of the option. This adjustment can be made to both the call and the put price formula, and in both cases the formula for d will then be

The adjustment is made because the option holder at maturity receives only the underlying asset on that date and not the cash flow that has accrued to the asset during the life of the option. This cash flow is retained by the owner of the underlying asset.

We now want to use the Black-Scholes pricing model to price a European call option written on the S&P 500 index. On January 6, 2010, the value of the index was 1137.14. The European call option has a strike price of 1110 and 43 days to maturity. The risk-free interest rate for a 43-day holding period is found from the T-bill rates to be per day (that is, ) and the dividend accruing to the index over the next 43 days is expected to be per day. For now, we assume the volatility of the index is per day. Thus, we have

and we can calculate

which gives

from which we can calculate the BSM call option price as

We now introduce a relatively simple model that is capable of making up for some of the obvious mispricing in the BSM model. We again have one day returns defined as

and -period returns as

The mean and variance of the daily returns are again defined as and We previously defined skewness by . We now explicitly define skewness of the one-day return as

Skewness is informative about the degree of asymmetry of the distribution. A negative skewness arises from large negative returns being observed more frequently than large positive returns. Negative skewness is a stylized fact of equity index returns, as we saw in Chapter 1. Kurtosis of the one-day return is now defined as

which is sometimes referred to as excess kurtosis due to the subtraction by 3. Kurtosis tells us about the degree of tail fatness in the distribution of returns. If large (positive or negative) returns are more likely to occur in the data than in the normal distribution, then the kurtosis is positive. Asset returns typically have positive kurtosis.

Assuming that returns are independent over time, the skewness at horizon can be written as a simple function of the daily skewness,

and correspondingly for kurtosis

Notice that both skewness and kurtosis will converge to zero as the return horizon, and thus the maturity of the option increases. This corresponds well with the implied volatility in Figure 10.2, which displayed a more pronounced smirk pattern for short-term as opposed to long-term options.

We now define the standardized return at the -day horizon as

so that

and assume that the standardized returns follow the distribution given by the Gram-Charlier expansion, which is written as

where , is the standard normal density, and D^j is its jth derivative. We have

The Gram-Charlier density function is an expansion around the normal density function, , allowing for a nonzero skewness, , and kurtosis The Gram-Charlier expansion can approximate a wide range of densities with nonzero higher moments, and it collapses to the standard normal density when skewness and kurtosis are both zero. We notice the similarities with the Cornish-Fisher expansion for Value-at-Risk in Chapter 6, which is a similar expansion, but for the inverse cumulative density function instead of the density function itself.

To price European options, we can again write the generic risk-neutral call pricing formula as

Thus, we must solve

Earlier we relied on x* following the normal distribution with mean and variance σ² per day. But we now instead define the standardized risk-neutral return at horizon as

and assume it follows the Gram-Charlier (GC) distribution.

In this case, the call option price can be derived as being approximately equal to

where we have substituted in for skewness using and for kurtosis using We will refer to this as the GC option pricing model. The approximation comes from setting the terms involving σ³ and σ⁴ to zero, which also enables us to use the definition of d from the BSM model. Using this approximation, the GC model is just the simple BSM model plus additional terms that vanish if there is neither skewness nor kurtosis in the data. The GC formula can be extended to allow for a cash flow q in the same manner as the BSM formula shown earlier.

While the GC model is capable of capturing implied volatility smiles and smirks at a given point in time, it assumes that volatility is constant over time and is thus inconsistent with the empirical observations we made earlier. Put differently, the GC model is able to capture the strike price structure but not the maturity structure in observed options prices. In Chapter 4 and Chapter 5 we saw that variance varies over time in a predictable fashion: High-variance days tend to be followed by high-variance days and vice versa, which we modeled using GARCH and other types of models. When returns are independent, the standard deviation of returns at the -day horizon is simply times the daily volatility, whereas the GARCH model implies that the term structure of variance depends on the variance today and does not follow the simple square root rule.

We now consider option pricing allowing for the underlying asset returns to follow a GARCH process. The GARCH option pricing model assumes that the expected return on the underlying asset is equal to the risk-free rate, r_f, plus a premium for volatility risk, λ, as well as a normalization term. The observed daily return is then equal to the expected return plus a noise term. The noise term is conditionally normally distributed with mean zero and variance following a GARCH(1,1) process with leverage as in Chapter 4. By letting the past return feed into variance in a magnitude depending on the sign of the return, the leverage effect creates an asymmetry in the distribution of returns. This asymmetry is important for capturing the skewness implied in observed option prices.

Specifically, we can write the return process as

Notice that the expected value and variance of tomorrow's return conditional on all the information available at time t are

For a generic normally distributed variable , we have that and therefore we get

where we have used This expected return equation highlights the role of λ as the price of volatility risk.

We can again solve for the option price using the risk-neutral expectation as in

Under risk neutrality, we must have that

so that the expected rate of return on the risky asset equals the risk-free rate and the conditional variance under risk neutrality is the same as the one under the original process. Consider the following process:

In this case, we can check that the conditional mean equals

which satisfies the first condition. Furthermore, the conditional variance under the risk-neutral process equals

where the last equality comes from tomorrow's variance being known at the end of today in the GARCH model. The conclusion is that the conditions for a risk-neutral process are met.

An advantage of the GARCH option pricing approach introduced here is its flexibility: The previous analysis could easily be redone for any of the GARCH variance models introduced in Chapter 4. More important, it is able to fit observed option prices quite well.

The option pricing methods surveyed so far in this chapter can be derived from well-defined assumptions about the underlying dynamics of the economy. The next approach to European option pricing we consider is instead completely static and ad hoc but it turns out to offer reasonably good fit to observed option prices, and we therefore give a brief discussion of it here. The idea behind the approach is that the implied volatility smile changes only slowly over time. If we can therefore estimate a functional form on the smile today, then that functional form may work reasonably in pricing options in the near future as well.

The implied volatility smiles and smirks mentioned earlier suggest that option prices may be well captured by the following four-step approach:

Notice that this option pricing approach requires only a sequence of simple calculations and it is thus easily implemented.

While this four-step linear estimation approach is standard, we can typically obtain much better model option prices if the following modified estimation approach is taken. We can use a numerical optimization technique to solve for by minimizing the mean squared error

The downside of this method is clearly that a numerical solution technique rather than simple OLS is needed to find the parameters. We refer to this approach as the modified implied volatility function (MIVF) technique.

This chapter has surveyed some key models for pricing European options. First, we introduced the simple but powerful binomial tree approach to option pricing. Then we discussed the famous Black-Scholes-Merton (BSM) model. The key assumption underlying the BSM model is that the underlying asset return dynamics are captured by the normal distribution with constant volatility. While the BSM model provides crucial insight into the pricing of derivative securities, the underlying assumptions are clearly violated by observed asset returns. We therefore next considered a generalization of the BSM model that was derived from the Gram-Charlier (GC) expansion around the normal distribution. The GC distribution allows for skewness and kurtosis and it therefore offers a more accurate description of observed returns than does the normal distribution. However, the GC model still assumes that volatility is constant over time, which we argued in earlier chapters was unrealistic. Next, we therefore presented two types of GARCH option pricing models. The first type allowed for a wide range of variance specifications, but the option price had to be calculated using Monte Carlo simulation or another numerical technique since no closed-form formula existed. The second type relied on a particular GARCH specification but in return provided a closed-form solution for the option price. Finally, we introduced the ad hoc implied volatility function (IVF) approach, which in essence consists of a second-order polynomial approximation of the implied volatility smile.

The probabilities P₁ and P₂ in the closed-form GARCH (CFG) formula are derived by first solving for the conditional moment generating function. The conditional, time-t, moment generating function of the log asset prices as time is

In the CFG model, this function takes a log-linear form (omitting the time subscripts on )

where

and

These functions can be solved by solving backward one period at a time from the maturity date using the terminal conditions

A fundamental result in probability theory establishes the following relationship between the characteristic function and the probability density function :

where the function takes the real value of the argument.

Using these results, we can calculate the conditional expected payoff as

To price the call option, we use the risk-neutral distribution to get

where we have used the fact that Note that under the risk-neutral distribution, λ is set to and θ is replaced by Finally, we note that the previous integrals must be solved numerically.

This chapter has focused on option pricing in discrete time in order to remain consistent with the previous chapters. There are many excellent textbooks on options. The binomial tree model in this chapter follows Hull (2011) who also provides a proof that the binomial model converges to the BSM model when the number of steps in the tree goes to infinity.

The classic papers on the BSM model are Black and Scholes (1973) and Merton (1973). The discrete time derivations in this chapter were introduced in Rubenstein (1976) and Brennan (1979). Merton (1976) introduced a continuous time diffusion model with jumps allowing for kurtosis in the distribution of returns. See Andersen and Andreasen (2000) for extensions to Merton's 1976 model.

For recent surveys on empirical option valuation, see Bates (2003), Garcia et al. (2010) and Christoffersen et al. (2010a).

The GC model is derived in Backus et al. (1997). The general GARCH option pricing framework is introduced in Duan (1995). Duan and Simonato (1998) discuss Monte Carlo simulation techniques for the GARCH model and Duan et al. (1999) contains an analytical approximation to the GARCH model price. Ritchken and Trevor (1999) suggest a trinomial tree method for calculating the GARCH option price.

Duan (1999) and Christoffersen et al. (2010b) consider extensions to the GARCH option pricing model allowing for conditionally nonnormal returns. The closed-form GARCH option pricing model is derived in Heston and Nandi (2000) and extended in Christoffersen et al. (2006).

Christoffersen and Jacobs (2004a) compared the empirical performance of various GARCH variance specifications for option pricing and found that the simple variance specification including a leverage effect as applied in this chapter works very well compared with the BSM model.

Hsieh and Ritchken (2005) compared the GARCH (GH) and the closed-form GARCH (CFG) models and found that the GH model performs the best in terms of out-of-sample option valuation. See Christoffersen et al. (2008) and Christoffersen et al. (2010a) for option valuation using the GARCH component models described in Chapter 4.

GARCH option valuation models with jumps in the innovations have been developed by Christoffersen et al. (2011) and Ornthanalai (2011).

Hull and White (1987) and Heston (1993) derived continuous time option pricing models with time-varying volatility. Bakshi et al. (1997) contains an empirical comparison of Heston's model with more general models and finds that allowing for time-varying volatility is key in fitting observed option prices. Lewis (2000) discusses the implementation of option valuation models with time-varying volatility.

The IVF model is described in Dumas et al. (1998) and the modified IVF model (MIVF) is examined in Christoffersen and Jacobs (2004b), who find that the MIVF model performs very well empirically compared with the simple BSM model. Berkowitz (2010) provides a theoretical justification for the MIVF approach. Bams et al. (2009) discuss the choice of objective function in option model calibration.