9 Why Black–Scholes is Still Important
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OPTION PRICING IN CONTINUOUS TIME
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other than a standard GBM. Black–Scholes usually appears as a component of
the option prices for these option types; for example, for American options.
Should we abandon Black–Scholes (1973) as a hopelessly antiquated option
pricing model or, should we keep it around as a useful tool in the ﬁnancial
engineer’s toolbox? In answering this question, there is at ﬁrst a logical
problem.
It is logically possible that Black–Scholes is an inadequate stand-alone model
and, at the same time, may be important as a potential component of more
complex models. Indeed, this is what the research literature suggests.
For example, Black–Scholes constitutes a lower bound on option prices in
some complex ‘mixed diffusion’ models. Further, even in certain SVOL
models—Hull and White (1987)—Black–Scholes reappears in some form.
Heston’s model also resembles Black–Scholes.
Black–Scholes (1973), which is a continuous time, continuous state space
model even shows up in some jump option pricing models, such as Merton’s
(1976) jump model. Thus, Black–Scholes may indeed survive as a component
of some option pricing models. We should probably keep it around for these
reasons.
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KEY CONCEPTS
1. Arithmetic Brownian Motion (ABM).
2. Shifted Arithmetic Brownian Motion.
3. Pricing European Options under Shifted Arithmetic Brownian Motion
(Bachelier).
4. Theory (FTAP1 and FTAP2).
5. Transition Density Functions.
6. Deriving the Bachelier Option Pricing Formula.
7. Deﬁning and Pricing a Standard Numeraire.
8. Geometric Brownian Motion (GBM).
9. GBM (Discrete Version).
10. Geometric Brownian Motion (GBM), Continuous Version.
11. Itô’s Lemma.
12. Black–Scholes Option Pricing.
13. Reducing GBM to an ABM with Drift.
14. Preliminaries on Risk-Neutral Transition Density Functions.
15. Black–Scholes Pricing from Bachelier.
16. Volatility Estimation in the Black–Scholes Model.
17. Non-Constant Volatility Models.
18. Why Volatility is not Constant.
19. Economic Reasons for why Volatility is not Constant, the Leverage
Effect.
20. Modeling Changing Volatility, the Deterministic Volatility Model.
21. Modeling Changing Volatility, Stochastic Volatility Models.
22. Why Black–Scholes is Still Important.
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END OF CHAPTER EXERCISES FOR CHAPTER 16
1. In this exercise, you will calculate the historical sigma based on 30 days
of Google’s stock price data. We went to nasdaq.com to download the
price data for the period September 2–September 30, 2014.
This was done through ‘basic charting’. Copy the following data into
Microsoft Excel. Note that you will have to ﬁrst reverse it in time order
to calculate ln(Si /Si–1).
a. Calculate the price relatives for the period, (Si /Si–1).
b. Calculate the log price relatives for the period, ln(Si /Si–1).
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591
c. Calculate the mean of the daily log price relatives, E{ln(Si /Si–1)}.
d. Calculate the standard deviation of the daily log price relatives, ͡ daily.
e. Calculate the historical estimator, ͡ annual.
TABLE 16.1
Price Data for End of Chapter 16, Exercise 1
Date
Close/Last
Volume
9/30/2014
577.36
1,617,320
9/29/2014
576.36
1,278,274
9/26/2014
577.1
1,439,687
9/25/2014
575.06
1,918,179
9/24/2014
587.99
1,723,438
9/23/2014
581.13
1,464,386
9/22/2014
587.37
1,684,861
9/19/2014
596.08
3,724,109
9/18/2014
589.27
1,438,201
9/17/2014
584.77
1,687,731
9/16/2014
579.95
1,475,668
9/15/2014
573.1
1,593,030
9/12/2014
575.62
1,594,177
9/11/2014
581.35
1,215,910
9/10/2014
583.1
9/9/2014
581.01
1,283,678
9/8/2014
589.72
1,426,597
9/5/2014
586.08
1,626,806
9/4/2014
581.98
1,454,229
9/3/2014
577.94
1,211,507
9/2/2014
577.33
1,574,096
972,057
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OPTIONS
2. Let St=$60, K=$50, =3 months, annualized r=8%, and annualized =20%.
a. Calculate the current (time=t) Black–Scholes European call option
price, Ct.
First calculate:
d1=______________________
N(d1)=___________________
d2=______________________
N(d2)=___________________
PV(K)=__________________
Ct=______________________
b. State the put-call parity relationship for European options.
c. Calculate the Black–Scholes European put option price using the putcall parity relationship for European options deﬁned in b.
3. In this exercise, you will create a Black–Scholes calculator in Excel. The
input list consists of the usual ﬁve inputs.
a. List the inputs,
1.=
2.=
3.=
4.=
5.=
b. Code up the following outputs in Excel,
1. d1
2. d2
3. N(d1)
4. N(d2)
5. Ct , the Black–Scholes call option price.
c. Calculate the Black–Scholes call option price for the input data in
exercise 2.
d. Check your result in c. against an online Black–Scholes calculator. The
one at http://www.hoadley.net/ is good. This site also produces the
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Black–Scholes greeks, or partial sensitivities, to the option parameters.
These are helpful for managing an options portfolio, assuming Black–
Scholes holds.
You now have three ways to calculate Black–Scholes: by hand, as in
exercise 2; by your own Excel program, as in this exercise 3; and using
online calculators.
4. Deﬁne what is meant by implied volatility and explain how Black–Scholes
could be used to generate estimates of the market’s volatility.
5. Once you have an Excel Black–Scholes calculator, it is easy to generate
a mini-implied volatility calculator.
Step 1
In the Black–Scholes spreadsheet, enter the following values for the spot
price, strike price, risk-free rate and time to maturity.
Input Parameters
Spot Price=490
Strike Price=470
Risk-Free Rate=.033
Volatility=0.2
Maturity=0.08
Also, enter an initial guess value for the volatility. This will give you an
initial call price that is reﬁned in the next step.
Use your Black–Scholes calculator to generate a European call option
value for these parameters, which should be 24.5942, which say is in cell
E5 in the spreadsheet.
a. Verify the call option price of around 24.5942.
Step 2
Go to Data>What If Analysis>Goal Seek. Set the Call value to 30 (say
in cell E5 in the spreadsheet) by changing the volatility (say cell B8 in the
spreadsheet). Goal seek will iterate until it ﬁnds the value of that is
consistent with this call price of 30.
b. Verify the implied volatility which should be around 0.32094.
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SELECTED CONCEPT CHECK SOLUTIONS
Concept Check 2
a.
⎛y − ⎞
Var ( y′) = Var ⎜
⎟
⎝ ⎠
1
= 2 Var ( y − )
1
= 2 Var ( y )
1
= 2 2
= 1.0
Concept Check 6
⎛ ⎞
a. G (Y ) = ln ⎜ Y0 ⎟
0
⎝ Y0 ⎠
= ln (1.0)
=0
⎛ 2 ⎞
b. E (G(YT )|G(Y0 ) = 0) = ⎜r − ⎟T
2 ⎠
⎝
c.
Var (G(YT )|G(Y0 ) = 0) = 2T
CHAPTER 17
RISK-NEUTRAL VALUATION, EMMS,
THE BOPM, AND BLACK–SCHOLES
17.1 Introduction
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17.1.1 Preliminaries on FTAP1 and FTAP2 and
Navigating the Terminology
596
17.1.2 Pricing by Arbitrage and the FTAP2
597
17.1.3 Risk-Neutral Valuation without Consensus
and with Consensus
598
17.1.4 Risk-Neutral Valuation without Consensus,
Pricing Contingent Claims with Unhedgeable
Risks
599
17.1.5 Black–Scholes’ Contribution
601
17.2 Formal Risk-Neutral Valuation without Replication
601
17.2.1 Constructing EMMs
601
17.2.2 Interpreting Formal Risk-Neutral
Probabilities
602
17.3 MPRs and EMMs, Another Version of FTAP2
605
17.4 Complete Risk-Expected Return Analysis of the
Riskless Hedge in the (BOPM, N=1)
607
17.4.1 Volatility of the Hedge Portfolio
608
17.4.2 Direct Calculation of S
611
17.4.3 Direct Calculation of C
612
17.4.4 Expected Return of the Hedge Portfolio
616
17.5 Analysis of the Relative Risks of the Hedge Portfolio’s
Return
618
17.5.1 An Initial Look at Risk Neutrality in the Hedge
Portfolio
620
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OPTIONS
17.5.2 Role of the Risk Premia for a Risk-Averse
Investor in the Hedge Portfolio
17.6 Option Valuation
620
624
17.6.1 Some Manipulations
624
17.6.2 Option Valuation Done Directly by a
Risk-Averse Investor
626
17.6.3 Option Valuation for the Risk-Neutral Investor 631
17.1 INTRODUCTION
17.1.1 Preliminaries on FTAP1 and FTAP2 and Navigating the
Terminology
This chapter is a supplemental chapter that consolidates the topics listed above.
It discusses the meaning and the underpinnings of ‘risk-neutral valuation’, and
it clariﬁes how EMMs capture the idea of ‘risk-neutral valuation’, and how
they do not.
Interestingly, ‘risk-neutral valuation’ has at least two, not just one, different
senses corresponding to the two fundamental theorems of asset pricing.
Unfortunately, the notion of EMM tends to be a purely mathematical notion,
while risk-neutral valuation is an economic concept. It is important to make
these alternative approaches mutually consistent, and this is not always easily
accomplished.
The beauty of the fundamental theorems of asset pricing, FTAP1 and FTAP2,
is that they recast the economic notions of pricing in mathematical terms,
thereby unleashing the full power of mathematical methods to analyze pricing.
Speciﬁcally, the lessons we learn from the fundamental theorems of asset pricing
are the relationships between no-arbitrage, the existence of a linear, positive
pricing mechanism, and the uniqueness of the pricing mechanism. However,
we have to be careful in applying these theorems, as there are numerous pitfalls
in doing so.
We already know, from Chapter 15, that the existence of a linear, positive
pricing mechanism that can be used to price all contingent claims (including
the underlying asset), is equivalent to the existence of an EMM for the discounted, underlying price process. (Think of the contingent claim as having
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only one risky security in its replicating portfolio). No-arbitrage, in turn, is
equivalent to the existence of a linear, positive pricing mechanism; the ﬁrst
fundamental theorem of asset pricing encapsulates the economic content of
no-arbitrage in mathematical terms.
Furthermore, no claim of uniqueness can be made based solely upon the
ﬁrst fundamental theorem of asset pricing, without making further assumptions.
Therefore, all that we can prove by using the no-arbitrage assumption only,
is that there exists a linear, positive pricing mechanism that can be used to
price all assets.
Or the existence of an EMM for the discounted underlying price process,
which is the same thing. We can’t make any general statements about uniqueness
based upon no-arbitrage alone. That presumably is part of the content of FTAP2.
17.1.2 Pricing by Arbitrage and the FTAP2
An example of a statement that appears to contradict the above is the claim
that a speciﬁc contingent claim (or that all claims) is (are) ‘priced by arbitrage’.
A reader might easily interpret this as ‘uniquely priced by arbitrage’, since most
readers probably don’t think in terms of non-uniqueness of the pricing
mechanism in a world of no-arbitrage.
However, no claim is uniquely priced by arbitrage, unless it is replicable
(FTAP2). Of course, we can’t do away with no-arbitrage, as discussed, because
in that case we wouldn’t be able to come up with a linear, positive pricing
mechanism at all. Therefore, no-arbitrage enters into everything that we do.
It is the standing assumption.
It would not make a lot of economic sense to replicate a contingent claim
in the presence of arbitrage. In that case, we could not say that the price of
the contingent claim is equal to the price of its replicating portfolio, which
is a direct consequence of the assumption of no-arbitrage (or at least of the
law of one price (LOP)). Knowing that the pricing mechanism is unique if it
exists is nice but rather empty, except in a mathematical sense.
However, also as discussed, if we want to claim unique pricing of a
contingent claim, then we have to assume that it is replicable. Otherwise, it
is not uniquely priced. It makes sense to include the word ‘replicable’ somewhere in the description of the pricing method. So we might say, ‘priced
by replication’ or ‘priced by replication and no-arbitrage’, or even ‘given
replicability, priced by arbitrage’. The blank statement ‘priced by arbitrage’ is
not fully correct, or at least ambiguous and a potential source of confusion.
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While we cannot do without no-arbitrage, in an exactly parallel manner
we also cannot do without replicability (market completeness, in the case of
all claims), if we want to claim uniqueness. Apparently, the importance of
FTAP1 is far better appreciated than the importance of FTAP2. That, of course,
doesn’t make it any less important.
This is also why so many research papers and presentations often assume
market completeness. Even in mathematics, existence and uniqueness are the
sought-after goals of the analysis. From a practical perspective, running a
numerical algorithm, without knowing that there is a single solution to the
problem at hand, is fraught with potential error.
17.1.3 Risk-Neutral Valuation without Consensus and with Consensus
When we turn to attempting to understand risk-neutral valuation, there is
another major potential pitfall. The EMM representation is often called a riskneutral valuation representation. But it is only so in the sense of the ﬁrst
fundamental theorem of asset pricing, and not in the sense of the second
fundamental theorem of asset pricing.
Since the risk-neutral valuation method is often illustrated in the context
of the BOPM, a complete model, one obtains the bonus result that the
BOPM is preference-free. This often leads people to believe that the existence
of an EMM (which is implied by no-arbitrage) leads to a risk-neutral valuation
that is also preference-free. Unfortunately, this is not true.
The fact is that what leads to preference-free, risk-neutral valuation is the
additional assumption (beyond no-arbitrage) of replicability! This is actually
the major contribution of Black–Scholes, which was to show that a European
call (put) option could be dynamically hedged with the appropriate number
of shares of the underlying stock and a risk-free bond.
That is, the risk associated with the long call option could be neutralized
by an opposite, dynamically adjusted position in the appropriate number of
shares of the underlying stock. In other words, the long European call option
position could be replicated. Therefore, no additional risk premium would,
or could, be demanded by risk-averse investors for investing in a European
call (put) option.
This is another way to say that the risk-averse investor would value the
option ‘risk-neutrally’; that is, in the same way that a risk-neutral investor would
value the option. Valuing the option risk neutrally should mean not simply
coming up with an EMM, but also not using subjective preferences for the
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trade-off between risk and expected return in valuing the option. This is what
is, or should be, meant by preference-free, risk-neutral valuation.
It could be called ‘risk-neutral valuation with consensus’. The risk-neutral
investor’s valuation of a European call option would be agreed to by all riskaverse investors, because all risk-averse investors know that the risk(s) associated
with the option could be diversiﬁed away (hedged out) in the hedge portfolio.
We will illustrate this in the case of the BOPM, N=1.
This is similar to the situation in ordinary portfolio analysis, in which no
risk premia are required for the diversifiable risks associated with the stocks in
a portfolio. Since you can diversify away diversiﬁable risks, risk-averse investors
would not require risk premia in order to compensate them for the diversiﬁable
risks in risky securities such as the underlying stock, for example. They would
only have to be compensated for the non-diversiﬁable risks. Note that, in the
context of ordinary stock portfolios, the non-diversiﬁable risks would therefore
have risk premia, and these would be priced into the stock portfolio’s price.
17.1.4 Risk-Neutral Valuation without Consensus, Pricing Contingent
Claims with Unhedgeable Risks
We can extrapolate from this understanding of portfolio analysis to speculate
as to what would happen to the non-hedgeable risks in the option valuation
scenario. The non-hedgeable risks are those that cannot be hedged away by
attempting to replicate the European call option. These are perfectly analogous
to the non-diversiﬁable risks on the underlier.
It is important to note here, that if a contingent claim (say, a European call
option) is not replicable using a portfolio of the underlying(s) and a riskless
bond, then that is because the claim involves risk(s) that cannot be neutralized
by any replicating portfolio. The contingent claim’s risk is not spanned by any
replicating portfolio of underlying(s) and a riskless bond.
This could happen because there are fewer independent sources of risk than
independent securities (see the rule of thumb in Chapter 13, section 13.5.2).
In this case, just as in the portfolio case, these non-hedgeable risks would carry
risk premia that would be priced in the contingent claim.
Risk premia involve risk aversion, which depends on risk preferences and
therefore, even though there are EMMs, the resulting option pricing valuations would be formally risk-neutral (consistent with no-arbitrage), but not
preference-free. Risk-averse investors need not agree with the risk-neutral
investor’s (no risk premia, therefore preference-free) valuation of the