Conclusion: What Is an Option?
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. Mechanics of Options
8
APPENDIX 8-1
In this appendix, we derive formulas for delta and gamma. The relatively lengthy derivation is
for delta.
Derivation of Delta
The Black-Scholes formula for a plain vanilla European call expiration T , strike, K, is given
by
log
C(St , t) = St
St
+(r+ 1 σ 2 )(T −t)
K
√ 2
σ T −t
−∞
log
1 2
1
√ e− 2 u du − e−r(T −t) K
2π
St
+(r− 1 σ 2 )(T −t)
K
√ 2
σ T −t
−∞
(102)
1 2
1
√ e− 2 u du
2π
Rearrange and let xt =
St
,
Ke−r(T −t)
C(xt , t) = Ke−r(T −t) xt
to get
log xt + 1 σ 2 (T −t)
√2
σ T −t
−∞
1 2
1
√ e− 2 u du −
2π
log xt − 1 σ 2 (T −t)
√2
σ T −t
(103)
−∞
1 2
1
√ e− 2 u du
2π
(104)
Now differentiate with respect to xt :
⎡ log xt + 1 σ2 (T −t)
⎤
√2
σ
T
−t
1 2
1
dC(xt , t)
1
√ e− 2 u du⎦ + √
= Ke−r(T −t) ⎣
dxt
σ
T
−t
2π
−∞
⎡
−1
⎣ √1 e 2
2π
log xt + 1 σ 2 (T −t)
√2
σ T −t
⎡
1
1 − 12
√ e
√
−⎣
xt σ T − t 2π
2
(105)
⎤
⎦
log xt − 1 σ 2 (T −t)
√2
σ T −t
(106)
2
⎤
⎦
Now we show that the last two terms in this expression sum to zero and that
⎤
⎡
2
log xt + 1 σ 2 (T −t)
log xt − 1 σ 2 (T −t)
2
1
√
√2
−
− 12
1
1
1
1
2
σ T −t
σ T −t
⎦=
⎣√ e
√ e
√
√
σ T −t
xt σ T − t 2π
2π
(107)
2
(108)
To see this, on the right-hand side, use the substitution:
1
= e− log xt
xt
and then rearrange the exponent in the exponential function.
(109)
Appendix 8-1
243
Thus, we are left with
⎡
∂C(xt , t)
= Ke−r(T −t) ⎣
∂xt
log xt + 1 σ 2 (T − t)
2
√
σ T −t
−∞
⎤
1 − 1 u2 ⎦
√ e 2 du
2π
(110)
Now use the chain rule and obtain
⎡
∂C(St , t) ⎣
=
∂St
log xt + 1 σ 2 (T −t)
√2
σ T −t
−∞
⎤
1 2
1
√ e− 2 u du⎦
2π
(111)
= N (d1 )
(112)
Derivation of Gamma
Once delta of a European call is obtained, the gamma will be the derivative of the delta. This
gives
∂ 2 C(St , t)
1 − 12
1
√ e
√
=
2
∂St
St σ T − t 2π
with xt =
St
Ke−r(T −t)
log xt + 1 σ 2 (T −t)
√2
σ T −t
2
(113)
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8
APPENDIX 8-2
In this appendix we review some basic concepts from stochastic calculus. This brief review can
be used as a reference point for some of the concepts utilized in later chapters. Øksendal (2003)
is a good source that provides an introductory discussion on stochastic calculus. Heuristics can
be found in Neftci (2000).
Stochastic Differential Equations
A Stochastic Differential Equation (SDE), driven by a Wiener process Wt is written as,
dSt = a(St , t)dt + b(St , t)dWt
t ∈ [0, ∞)
(114)
This equation describes the dynamics of St over time. The Wiener process Wt has increments
ΔWt that are normally distributed with mean zero and variance Δ, where the Δ is a small time
interval. These increments are uncorrelated over time. As a result, the future increments of a
Wiener process are unpredictable given the information at time t, the It .
The a(St , t) and the b(St , t) are known as the drift and the diffusion parameters. The drift
parameter models expected changes in St . The diffusion component models the corresponding volatility. When unpredictable movements occur as jumps, this will be referred as a jump
component.
A jump component would require adding terms such as λ(St , t)dJt to the right-hand side
of the SDE shown above. Otherwise the St will be known as a diffusion process. With a jump
component it becomes a jump-diffusion process.
Examples
The simplest Stochastic Differential Equation is the one where the drift and diffusion coefﬁcients
are independent of the information received over time:
dSt = μdt + σdWt
t ∈ [0, ∞)
(115)
Here, the Wt is a standard Wiener process with variance t. In this SDE, the coefﬁcients μ and
σ do not have time subscripts t, as time passes, they do not change.
The standard SDE used to model underlying asset prices is the geometric process. It is the
model assumed in the Black and Scholes world:
dSt = μSt dt + σSt dWt
t ∈ [0, ∞)
(116)
This model implies that drift and the diffusion parameters change proportionally with St .
An SDE that has been found useful in modelling interest rates is the mean reverting model:
dSt = λ(μ − St )dt + σSt dWt
t ∈ [0, ∞)
(117)
According to this, as St falls below a “long-run mean” μ, the term (μ − St ) will become
positive, which makes dSt more likely to be positive, hence, St will revert back to the mean μ.
Ito’s Lemma
Suppose f (St ) is a function of a random process St having the dynamics:
dSt = a(St , t)dt + b(St , t)dWt
t ∈ [0, ∞)
(118)
Appendix 8-2
245
We want to expand f (St ) around a known value of St , say S0 using Taylor series expansions.
The expansion will yield:
1
f (St ) = f (S0 ) + fs (S0 )[St − S0 ] + fss (S0 )[St − S0 ]2 + R(St , S0 )
2
(119)
where, R(St , S0 ) represents all the remaining terms of the Taylor series expansion.
First note that f (St ) can be rewritten as, f (S0 + ΔSt ), if we deﬁne ΔSt as:
ΔSt = St − S0
(120)
Then, the Taylor series approximation will have the form:
1
f (S0 + ΔSt ) − f (S0 ) ∼
= fs ΔSt + fss ΔSt2
2
(121)
The ΔSt is a “small” change in the random variable St . In approximating the right-hand side,
we keep the term fs ΔSt .
Consider the second term 12 fss (ΔSt )2 . If the St is deterministic, one can say that the term
(ΔSt )2 is small. This could be justiﬁed by keeping the size of ΔSt nonnegligible, yet small
enough that its square (ΔSt )2 is negligible. However, here, changes in St will be random.
Suppose these changes have zero mean. Then the variance is,
2
0 < E [ΔSt ] ∼
= b(St , t)2 Δ
(122)
This equality means that as long as St is random, the right-hand side of (121) must keep the
second order term in any type of Taylor series approximation.
Moving to inﬁnitesimal time dt, this gives Ito’s Lemma, which is the stochastic version of
the Chain rule,
1
df (St ) = fs dSt + fss b(St , t)2 dt
2
(123)
This equation can be regarded as the dynamics of the process f (St ), which is driven by St . The
dSt term in the above equation can be substituted out using the St dynamics.
Girsanov Theorem
Girsanov Theorem provides the general framework for transforming one probability measure
into another “equivalent” measure. It is an abstract result that plays a very important role in
pricing.
In heuristic terms, this theorem says the following. If we are given a Wiener process Wt ,
then, we can multiply the probability distribution of this process by a special function ξt that
depends on time t, and on the information available at time t, the It . This way we can obtain a
˜ t with probability distribution P˜ . The two processes will relate to each
new Wiener process W
other through the relation:
˜ t = dWt − Xt dt
dW
(124)
˜ t is obtained by subtracting an It -dependent term Xt , from Wt .
That is to say, W
Girsanov Theorem is often used in the following way: (1) we have an expectation to calculate,
(2) we transform the original probability measure, such that expectation becomes easier to
calculate, and (3) we calculate the expectation under the new probability.
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Exercises
1. Consider the following comment dealing with options written on the euro-dollar exchange
rate:
Some traders, thinking that implied volatility was too high entered new trades.
One example was to sell one-year in-the-money euro Puts with strikes around
USD1.10 and buy one-year at-the-money euro Puts. If the euro is above
USD1.10 at maturity, the trader makes the difference in the premiums. The
trades were put on across the curve. (Based on an article in Derivatives Week).
(a) Draw the proﬁt/loss diagrams of this position at expiration for each option
separately.
(b) What would be the gross payoff at expiry?
(c) What would be the net payoff at expiry?
(d) Why would the traders buy “volatility” given that they buy and sell options?
Don’t these two cancel each other in terms of volatility exposure?
2. Consider the following quote:
Implied U.S. dollar/New Zeland dollar volatility fell to 10.1%/11.1% on
Tuesday. Traders bought at-the-money options at the beginning of the week,
ahead of the Federal Reserve interest-rate cut. They anticipated a rate cut
which would increase short-term volatility. They wanted to be long gamma.
Trades were typically for one-week maturities, in average notionals of USD1020 million. (Based on an article in Derivatives Week).
(a) Explain why traders wanted to be long gamma when the volatility was expected
to increase.
(b) Show your argument using numerical values for Greeks and the data given in
the reading.
(c) How much money would the trader lose under these circumstances? Calculate
approximately, using the data supplied in the reading. Assume that the position
was originally for USD30 million.
3. Consider the following episode:
EUR/USD one-month implied volatility sank by 2.7% to 10% Wednesday as
traders hedged this euro exposure against the greenback, as the euro plunged
to historic lows on the spot market. After the European Central Bank raised
interest rates by 25 basis points, the euro fell against leading to a strong demand
for euro Puts. The euro touched a low of USD0.931 Wednesday. (Based on an
article in Derivatives Week).
(a) In the euro/dollar market, traders rushed to stock up on gamma by buying
short-dated euro puts struck below USD0.88 to hedge against the possibility
that the interest rates rise. Under normal circumstances, what would happen to
the currency?
(b) When the euro failed to respond and fell against major currencies, why would
the traders then rush to buy euro puts? Explain using payoff diagrams.
(c) Would a trader “stock up” gamma if euro-triggered barrier options?
Exercises
247
4. You are given the following table concerning the price of a put option satisfying all
Black-Scholes assumptions. The strike is 20 and the volatility is 30%. The risk-free rate
is 2.5%.
Option price
Underlying asset price
10
5
1.3
.25
.14
10
15
20
25
30
The option expires in 100 days. Assume (for convenience), that, for every month the
option loses approximately one-third of its value.
(a) How can you approximate the option delta? Calculate three approximations for
the delta in the previous case.
(b) Suppose you bought the option when the underlying was at 20 using borrowed
funds. You have hedged this position in a standard fashion. How much do you
gain or lose in four equal time periods if you observe the following price
sequence in that order:
10, 25, 25, 30
(125)
(c) Suppose now that the underlying price follows the new trajectory given by
10, 30, 10, 30
How much do you gain or lose until expiration?
(d) Explain the difference between gains and losses.
5. Search the Internet for the following questions.
(a) Which sensitivities do the Greeks, volga and Vanna represent?
(b) Why are they relevant for vega hedging?
(126)
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9
Engineering Convexity Positions
1.
Introduction
How can anyone trade volatility? Stocks, yes. Bonds, yes. But volatility is not even an asset.
Several difﬁculties are associated with deﬁning precisely what volatility is. For example, from
a technical point of view, should we deﬁne volatility in terms of the estimate of the conditional
standard deviation of an asset price St ?
Et [St − Et [St ]]2
(1)
Or should we deﬁne it as the average absolute deviation?
Et [|St − Et [St ]|]
(2)
There is no clear answer, and these two deﬁnitions of statistical volatility will yield different numerical values. Leaving statistical deﬁnitions of volatility aside, there are many instances
where traders quote, directly, the volatility instead of the dollar value of an instrument. For example, interest rate derivatives markets quote cap-ﬂoor and swaption volatilities. Equity options
provide implied volatility. Traders and market makers trade the quoted volatility. Hence, there
must be some way of isolating and pricing what these traders call volatility in their respective
markets.
We started seeing how this can be done in Chapter 8. Options became more valuable when
“volatility” increased, everything else being the same. Chapter 8 showed how these strategies
can quantify and measure the “volatility” of an asset in monetary terms. This was done by
forming delta-neutral portfolios, using assets with different degrees of convexity. In this chapter, we develop this idea further, apply it to instruments other than options, and obtain some
generalizations. The plan for this chapter is as follows.
First, we show how convexity of a long bond relates to yield volatility. The higher the
volatility of the associated yield, the higher the beneﬁt from holding the bond. We will discuss
the mechanics of valuing this convexity. Then, we compare these mechanics with option-related
convexity trades. We see some close similarities and some differences. At the end, we generalize
the results to any instrument with different convexity characteristics. The discussion associated
with volatility trading itself has to wait until Chapter 13, since it requires an elementary treatment
of arbitrage pricing theory.
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Yield
5.20%
4.94%
1.80%
1 month
5 year
10 year
30-year
Maturity
FIGURE 9-1
2.
A Puzzle
Here is a puzzle. Consider the yield curve shown in Figure 9-1. The 10-year zero coupon bond
has a yield to maturity that equals 5.2%. The 30-year zero, however, has a yield to maturity of
just 4.94%. In other words, if we buy and hold the latter bond 20 more years, we would receive
a lower yield during its lifetime.
It seems a bit strange that the longer maturity is compensated with a lower yield. There are
several economic or institutional explanations of this phenomenon. For example, expectations
for inﬂation 20 years down the line may be less than the inﬂationary expectations for the next
10 years only. Or, the relative demands for these maturities may be determined by institutional
factors and, because players don’t like to move out of their “preferred” maturity, the yield
curve may exhibit such inconsistencies. Insurance companies, for example, need to hedge their
positions on long-term retirement contracts and this preference may lower the yield and raise
the price of long bonds.
But these explanations can hardly fully account for the observed anomaly. Institutional
reasons such as preferred habitat and treasury debt retirement policies that reduce the supply of
30-year treasuries may account for some of the difference in yield, but it is hard to believe that
an additional 20-year duration is compensated so little. Can there be another explanation?
In fact, the yield to maturity may not show all the gains that can be realized from holding a
long bond. This may be hard to believe, as yield to maturity is by deﬁnition how much the bond
will yield per annum if kept until maturity.
Yet, there can be additional gains to holding a long bond, due to the convexity properties of
the instrument, depending on what else is available to trade “against” it, and depending on the
underlying volatility. These could explain the “puzzle” shown in Figure 9-1. The 4.94% paid by
the 30-year treasury, plus some additional gains, could exceed the total return from the 10-year
bond. This is conceivable since the yield to maturity and the total return of a bond are, in fact,
quite different ways of measuring ﬁnancial returns on ﬁxed-income instruments.
3.
Bond Convexity Trades
We have already seen convexity trades within the context of vanilla options. Straightforward
discount bonds, especially those with long maturities, can be analyzed in a similar fashion and
3. Bond Convexity Trades
251
have exposure to interest rate volatility. In fact, a “long” bond and a vanilla option are both
convex instruments and they both coexist with instruments that are either linear or have less
convexity.1 Hence, a delta-neutral portfolio can be put together for long maturity bonds to
beneﬁt from volatility shifts. The overall logic will be similar to the options discussed in the
previous chapter.
Consider a long maturity default-free discount bond with price B(t, T ), with t < T . This
bond’s price at time t can be expressed using the corresponding time t yield, ytT :
B(t, T ) =
1
(1 + ytT )T
(3)
For t = 0, and T = 30, this function is plotted against various values of the 30-year zero-coupon
yield, in Figure 9-2. It is obvious that the price is a convex function of the yield.
A short bond, on the other hand, can be represented in a similar space with an almost linear
curve. For example, Figure 9-3 plots a 1-year bond price B(0, 1) against a 1-year yield y01 . We
see that the relationship is essentially linear.2
The main point here is that, under some conditions, using these two bonds we can put together
a portfolio that will isolate bond convexity gains similar to the convexity gains that the dynamic
hedging of options has generated. Thus, suppose movements in the two yields yt1 and yt30 are
perfectly correlated over time t.3 Next, consider a trader who tries to duplicate the strategy
of the option market maker discussed in the previous chapter. The trader buys the long bond
with borrowed funds and delta-hedges the ﬁrst-order yield exposure by shorting an appropriate
amount of the shorter maturity bond.
This trader will have to borrow B(0, 30) dollars to buy and fund the long bond position. The
payoff of the portfolio
{Long bond, loan of B(0, 30) dollars}
(4)
is as shown in Figure 9-2b as curve BB . Now compare this with Figure 9-2c. Here we show
the proﬁt/loss position of a market maker who buys an at-the-money “put option” on the yield
yt30 . At expiration time T , the option will pay
P (T ) = max[y030 − yT30 , 0]
(5)
This option is ﬁnanced by a money market loan so that the overall position is shown as the
downward sloping curve BB .4 We see a great deal of resemblance between the two positions.
Given this similarity between bonds and options, we should be able to isolate convexity or
gamma trading gains in the case of bonds as well. In fact, once this is done, using an arbitrage
argument, we should be able to obtain a partial differential equation (PDE) that default-free
1 The short maturity bonds are almost linear. In the case of vanilla options, positions on underlying assets such as
stocks are also linear.
2
In fact, a ﬁrst-order Taylor series expansion around zero yields
B(0, 1) =
1
(1 + y01 )
∼
= (1 − y01 )
if the y01 is “small.”
3 This simplifying assumption implies that all bonds are affected by the same unpredictable random shock, albeit
to a varying degree. It is referred to as the one factor model.
4 The option price is the curve BB . The curve shifts down by the money market loan amount P , which makes
0
the position one of zero cost.
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(a)
Bond price
Current
price
B(0, 30)
ytT
y 030
(b)
Net position
B
Net bond position after borrowing B(0, 30),
and buying the 30-year bond
ytT
y 030
B9
(c)
Option value
Put option with strike K 5y 030
financed by a money market loan
B
Current
premium
ytT
K
B9
FIGURE 9-2
discount bond prices will satisfy. This PDE will have close similarities to the Black-Scholes
PDE derived in Chapter 8.
The discussion below proceeds under some simplifying and unrealistic assumptions. We use
the so-called one-factor model. Our purpose is to understand the mechanics of volatility trading
in the case of bonds and this assumption simpliﬁes the exposition signiﬁcantly. Our context is
different than in real life, where ﬁxed-income instruments are affected by more than a single
common random factor. Thus, we make two initial assumptions:
1. There is a short and a long default-free discount bond with maturities T s and T , respectively. Both bonds are liquid and can be traded without any transaction costs.
3. Bond Convexity Trades
253
Short bond price
0.98
0.96
0.92
0.02
0.04
0.06
0.08
0.1
Yield
FIGURE 9-3
2. The two bond prices depend on the same risk factor denoted by rt . This can be interpreted
as a spot interest rate that captures all the randomness at time t, and is the single factor
mentioned earlier.
The second assumption means that the two bond prices are a function of the short rate rt .
These functions can be written as
B(t, T s ) = S(rt , t, T s )
(6)
B(t, T ) = B(rt , t, T )
(7)
and
where B(t, T s ) is the time-t price of the short bond and the B(t, T ) is the time-t price of the long
bond. We postulate that the maturity T s is such that the short bond price B(t, T s ) is (almost)
a linear function of rt , meaning that the second derivative of B(t, T s ) with respect to rt is
negligible.
Thus, we will proceed as if there was a single underlying risk that causes price ﬂuctuations
in a convex and a quasi-linear instrument, respectively. We will discuss the cash gains generated
by the dynamically hedged bond portfolio in this environment.
3.1. Delta-Hedged Bond Portfolios
The trader buys the long bond with borrowed funds and then hedges the downside risk implied
by the curve AA in Figure 9-4. The hedge for the downside risk will be a position that makes
money when rt increases, and loses money when rt declines. This can be accomplished by
shorting an appropriate number of the short bond.
In fact, the trick to form a delta-neutral portfolio is the same as in Chapter 8. Take the partial
derivative of the functions S(rt , t, T s ) and B(rt , t, T ) with respect to rt , evaluate them at point