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Chapter 16 - Option Valuation
CHAPTER SIXTEEN
OPTION VALUATION
CHAPTER OVERVIEW
This chapter discusses factors affecting the value of an option and presents analytical and
spreadsheet models of option pricing. Put call parity is introduced, manipulating hedge ratios
and portfolio insurance techniques are also presented.
LEARNING OBJECTIVES
After studying this chapter, the student should be able to identify the characteristics that
determine an option’s value and should understand how different values for these variables affect
option prices. The reader should be able to calculate option prices in a two state world (via a
simplified binomial model) and should know how to calculate Black-Scholes put and call option
values when there is no early exercise. Students should be able to calculate put prices from put
call parity and know how to arbitrage a mispriced option. The chapter demonstrates how to
calculate the hedge ratio for an option and students should have a basic understanding of
portfolio insurance.
CHAPTER OUTLINE
1. Option Valuation: Introduction
PPT 16-2 through PPT 16-5
When describing options, intrinsic value refers to the value if the option were immediately
exercised. Exercise value was introduced in Chapter 15 in the Instructor’s Manual because this
helped students understand basic option strategy payoffs. A review is provided below:
Basic boundaries revisited
Ct ≥ 0, Why?
Ct ≥ St – X, Why?
Thus Ct ≥ Max (0, St – X)
where:
A tighter boundary can be developed by considering two different portfolios:
Portfolio 1: Long position in stock at S0
Portfolio 2: Buy 1 at the money call option (C0) and buy a T-bill with a face value = X.
Chapter 16 - Option Valuation
From here we can present the value of a call option at expiration and prior to expiration as
follows:
Chapter 16 - Option Valuation
The difference between the value prior to expiration (curvilinear values) and the exercise or
intrinsic value Max(0,S-X) is the time value of the call. The option is a ‘wasting asset’ that loses
value as expiration approaches. This is the same for put and this is referred to as a – Theta
position. (Writing an option will be a + Theta position.) Going long or buying an option is a play
that the price will move enough before you run out of time value.
2. Binomial Option Pricing
PPT 16-6 through PPT 16-12
A binomial pricing example is developed in the PPT. The example assumes the stock is
currently priced at $100 and will have a value of either $115or $85 at the end of the period. A
call that has an exercise price of $90 will be worth $0 or $25 at the end of the period. The
Chapter 16 - Option Valuation
If one buys H = 0.8333 shares of stock per call written the resulting position will be riskless.
The strategy’s payoff is $70.833 in either state of the economy. Its present value can be found
by discounting $70.833 at the risk free rate of 10% for one period, obtaining $64.39. The current
value of this portfolio HS0 – C0 = $64.39. S0 is known and is equal to $100, so it is trivial to
solve for C0 = $18.94. H is the hedge ratio in the binomial framework and its calculation is
provided above. Conceptually H is roughly analogous to ∆C/∆S.
Chapter 16 - Option Valuation
3. Black-Scholes Option Valuation
PPT 16-13 through PPT 16-24
The Black-Scholes (BS) option pricing model is:
C0 = Current call option value. X (or E) = Exercise Price
S0 = Current stock price, δ = Annual dividend yield on the stock
e = 2.71828, the base of the natural log
r = Risk-free interest rate (annualize with continuous compounding the return on a T-bill with
Including the annual dividend yield is an approximation of a discrete payment, (also technically
the dividend can’t be stochastic). It assumes no early exercise due to the dividend.
The exercise value of the call is S0 – X,
However if the call will not be exercised early the value today is S0 – the present value of X so
this boundary tightens up to S0 – X(e-rT). The cash dividend yield term δ reflects that a dividend
will reduce the stock price thus hurting the value of the call as is in the following: S0e-T – X(e-rT)
The term d1 comes from our assumptions about how stock prices move in continuous trading:
Thus the N(d) terms can be thought of as a measure of the probability of how far in the money
the stock price is likely to be at expiration.
))N(dX(e )N(deS C 2
rT
1
T
00
−− −=
Chapter 16 - Option Valuation
Once the model has been developed it is worthwhile to go over the comparative statics of the
model:
Black-Scholes model with dividends Outputs Formula for Output in Column E
Annual Standard deviation of stock's returns 0.400 d1 0.421466 (LN(B5/B6)+(B4-B7+0.5*B2^2)*B3)/(B2*SQRT(B3))
Maturity in years (365 day year) 0.250 d2 0.221466 E2-B2*SQRT(B3)
Chapter 16 - Option Valuation
The only variables difficult for students to understand are the volatility and the interest rate.
Higher volatility or stock risk increases a call option’s price because the greater volatility
increases the probability that the option will wind up (deeper) in the money by expiration.
Higher volatility also indicates that the stock may not wind up in the money even if it currently
is. However, due to the asymmetric nature of options (one don’t use them if they don’t help)
As noted previously, a call option should not be exercised prior to maturity unless a stock is
about to pay a sufficiently large dividend. In some situations it may be profitable to exercise a
put prior to expiration. This decision is independent of any dividend although in general a
dividend reduces the likelihood of early exercise. At sufficiently low stock prices it makes sense
to go ahead and exercise the put, get the proceeds from the sale and reinvest them until option
maturity and earn the rate r. At low enough stock prices this can have a bigger payoff than any
further possible decline in stock price, and in such cases early exercise is desirable.
A version of the BS model is available for puts:
Determinants of put option values are as follows:
Chapter 16 - Option Valuation
Put call parity can be illustrated with a profit table from a replicating portfolio as follows:
Chapter 16 - Option Valuation
4. Using the Black-Scholes Formula
PPT 16-25 through PPT 16-33
The BS hedge ratio H can be found for a call option on a non-dividend paying stock as:
This means that the call option’s value will move by approximately N(d1) dollars when the
stock’s price moves one dollar. H approaches +1 as a call moves into the money. As a call
moves out of the money H approaches 0. One can use this concept to exploit a call price that
)d(N
S
C
H 1
0
0=
=
Chapter 16 - Option Valuation
If the position is not affected by a change in stock price the position has a delta of zero and is
said to be delta neutral.
Even if the portfolio of stocks remains constant, the deltas change as the stock prices change.
The concept of the delta changing as prices change is shown graphically.
Option elasticity with respect to stock price is high due to option leverage; a 5%-10% change in
option price per 1% change in stock price would not be atypical. Further out of the money
options have greater elasticity, deep out options may have elasticities of 25% or more. Deep in
The B-S model has been heavily tested with the general conclusion that the model generates
option values that are very close to actual market prices. However, there are some problems with
the model
• Stocks with high dividend payouts may lead to early exercise of a call option. The
model does not consider early exercise so B-S prices may be inaccurate in these cases.
One way to handle early exercise is to assume the call will be exercised on the day before
the stock goes ex-dividend. This is an inexact measure because the probability of early
Chapter 16 - Option Valuation
Excel Applications
Chapter 16 has an Excel spreadsheet that is available on the web site. The model allows the
