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with imperfections’, involves a class of problems that we introduce
in the next chapter.
15
Options in real markets
In Chapter 14, we considered the option-pricing problem in ideal friction-
less markets. Real markets are often efficient, but they are never ideal. In
this chapter, we discuss how the complexity of modeling financial markets
increases when we take into account aspects of real markets that are not
formalized in the ideal model. These aspects are addressed in the literature
as market microstructure [26] or market imperfections [127].
The terminology used in the economics literature suggests a clear parallel
with similar scenarios observed in physical sciences. For example, it is much
easier to construct a generalized description of the motion of a mechanical
system in an idealized world without friction than in the real world. A similar
situation is encountered when we compare equilibrium and non-equilibrium
thermodynamics. In this chapter, we show that knowledge of the statistical
properties of asset price dynamics is crucial for modeling real financial
markets. We also address some of the theoretical and practical problems
that arise when we take market imperfections into account.
15.1 Discontinuous stock returns
The existence of a portfolio containing both riskless and risky assets –
replicating exactly the value of an option – is essential in determining
the rational price of the option under the assumption that no arbitrage
opportunities are present. Whether a portfolio is replicating or not depends
on the statistical properties of the dynamics of the underlying asset. In the
previous chapter, we saw that a replicating portfolio exists when the price of
the underlying asset follows a geometric Brownian motion, but we also saw
that this case cannot be generalized. For example, when the asset dynamic
follows a jump-diffusion model [121], a simple replicating portfolio does not
exist. A jump-diffusion model is a stochastic process composed of a diffusive
123
124 Options in real markets
term (as in geometric Brownian motion) plus a second term describing jumps
of random amplitudes occurring at random times.
Roughly speaking, the presence of two independent sources of randomness
in the asset price dynamics does not allow the building of a simple replicating
portfolio.† It is not possible to obtain the rational price of an option just by
assuming the absence of arbitrage opportunities. Other assumptions must be
made concerning the risk aversion and price expectations of the traders.
Taking a different perspective, we can say that we need to know the
statistical properties of a given asset’s dynamics before we can determine the
rational price of an option issued on that asset. Discontinuity in the path of
the asset’s price is only one of the ‘imperfections’ that can force us to look
for less general option-pricing procedures.
15.2 Volatility in real markets
Another ‘imperfection’ of real markets concerns the random character of
the volatility of an asset price. The Black & Scholes option-pricing formula
for an European option traded in an ideal market depends only on five
parameters: (i) the stock price Y at time t, (ii) the strike price K , (iii) the
interest rate r, (iv) the asset volatility rate σ, and (v) the maturity time T . Of
these parameters, K and T are set by the kind of financial contract issued,
while Y and r are known from the market. Thus the only parameter that
needs to be determined is the volatility rate σ.
Note that the volatility rate needed in the Black & Scholes pricing formula
is the volatility rate of the underlying security that will be observed in the
future time interval spanning t = 0 and t = T . A similar statement can be
made about the interest rate r, which may jump at future times.
We know from the previous analysis that the volatility of security prices is
a random process. Estimating volatility is not a straightforward procedure.
15.2.1 Historical volatility
The first approach is to determine the volatility from historical market
data. Empirical tests show that such an estimate is affected by the time
interval used for the determination. One can argue that longer time intervals
should provide better estimations. However, the local nonstationarity of the
volatility versus time implies that unconditional volatility, estimated by using
very long time periods, may be quite different from the volatility observed
in the lifetime of the option.
† For a more rigorous discussion of this point, see [44, 70].
15.2 Volatility in real markets 125
Fig. 15.1. Schematic illustration of the problems encountered in the determination
of historical volatility. The nonstationary behavior of the volatility makes the deter-
mination of the average volatility depend on the investigated period of time. Long
periods of time are observed when the daily volatility is quite different from the
mean asymptotic value (solid line).
An empirical rule states that the best estimate of volatility rate is obtained
by considering historical data in a time interval t1 − t2 chosen to be as long
as the time to maturity T of the option (Fig. 15.1).
15.2.2 Implied volatility
A second, alternative approach to the determination of the volatility is to
estimate the implied volatility σimp, which is determined starting from the
options quoted in the market and using the Black & Scholes option-pricing
formula (14.20). The implied volatility gives an indication about the level of
volatility expected for the future by options traders.
The value of σimp is obtained by using the market values of C(Y , t) and
by solving numerically the equation
C(Y ,T − t) = Y N(d1)−Ke−r(T−t)N(d2), (15.1)
126 Options in real markets
Fig. 15.2. Schematic illustration of the implied volatility as a function of the differ-
ence between the strike price K and the stock price Y . The specific form shown is
referred to as a volatility smile.
where now the time is expressed in days from maturity, and
d1 ≡ ln(Y /K) + (r + σ
2
imp/2)(T − t)
σimp
√
T − t , (15.2)
and
d2 ≡ d1 − σimp
√
T − t. (15.3)
In a Black & Scholes market, a determination of the implied volatility
rate would give a constant value σ for options with different strike prices
and different maturity. Moreover, the value of the implied volatility should
coincide with the volatility obtained from historical data.
In real markets, the two estimates, in general, do not coincide. Implied
volatility provides a better estimate of σ. Empirical analysis shows that σimp
is a function of the strike price and of the expiration date. Specifically, σimp
is minimal when the strike price K is equal to the initial value of the stock
price Y (‘at the money’), and increases for lower and higher strike prices.
This phenomenon is often termed a ‘volatility smile’ (Fig. 15.2). The implied
volatility increases when the maturity increases. These empirical findings
confirm that the Black & Scholes model relies on assumptions that are only
partially verified in real financial markets.
When random volatility is present, it is generally not possible to determine
the option price by simply assuming there are no arbitrage opportunities. In
some models, for example, the market price of the volatility risk needs to be
15.4 Extension of the Black & Scholes model 127
specified before the partial differential equation of the option price can be
obtained.
15.3 Hedging in real markets
In idealized financial markets, the strategy for perfectly hedging a portfolio
consisting of both riskless and risky assets is known. In real markets, some
facts make this strategy unrealistic: (i) the rebalancing of the hedged portfolio
is not performed continuously; (ii) there are transaction costs in real markets;
(iii) financial assets are often traded in round lots of 100 and assume a degree
of indivisibility.
It has been shown that the presence of these unavoidable market imper-
fections implies that a perfect hedging of