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Unformatted text preview: E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 1 of 34 Copyright Emanuel Derman 2008 Lecture 5:
Static Hedging and Implied Distributions
Recapitulation of Lecture 4:
Plotting the smile against Δ is enlightening and useful.
For a slightly outofthemoney option a fraction J away from atthemoney,
d1
1
1
1 Στ
J
Δ ≈  +  ≈  +  ⎛  –  ⎞
⎝ 2  Σ τ⎠
2
2
2π
2π
Arbitrage constraints on the smile:
implied
volatility
implied
volatility at
index level S allowed range
lower bound on implied volatility
from puts strike
Problems caused by the smile:
The smile manifest in the market values of standard options is inconsistent
with the BlackScholes model. Without the right model, who knows how to
hedge vanilla options or value and hedge exotic options? The errors can be
sizeable. Here are some classes of models:
• dS
Local volatility:  = μ ( S, t ) dt + σ ( S, t ) dZ
S
dS = μ S ( S, V, t ) dt + σ S ( S, V, t ) dZ t • Stochastic volatility: • 2/24/08 upper bound on implied volatility
from calls dV = μ V ( S, V, t ) dt + σ V ( S, V, t ) dW t Jump diffusion 2 V=σ
E [ dWdZ ] = ρ dt Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 2 of 34 Copyright Emanuel Derman 2008 5.1 Static Hedging and Implied Distributions
The BlackScholes formula calculates options prices as the expected discounted value of the payoff over a lognormal stock distribution in a riskneutral world, and – trivially, because a lognormal stock distribution has a single
volatility – produces an implied volatility skew that is flat, independent of
strike level.
We can ask the inverse question: for a fixed expiration, what riskneutral stock
distribution (the socalled implied distribution) matches the observed smile
when options prices are computed as expected riskneutrally discounted payoffs? Let’s look at this when the world has only a discrete and finite number of
possible future states.
At time t, consider a security π i that pays $1 when N the stock is in state i with price S i at a future time
T, and pays zero if the stock price takes any other
value. Suppose you know the market price π i for
each of these securities. i The portfolio that consists of all of these π i is
effectively a riskless bond because it pays off $1 in
every future state, and its value is therefore given by
N ∑ πi 1 1
= exp [ – r ( T – t ) ] ≡ R 1 where r is the continuously compounded riskless rate.
Then the pseudoprobabilities p i ≡ R π i have the characteristics of probabilities because pi
p i = 1 and we can write π i = ∑
R If there is one statecontingent security π i for each state i in the market at time
T, then these securities provide a complete basis that span the space of future
payoffs, and the market is said to be complete. In terms of this basis we can
replicate the payoff of any security V if we know its payoff Vi in all states i.
The replicating portfolio is V = ∑ Vi πi and its current value is V = pi
∑  Vi .
R
i In more elegant continuousstate notation, we can write the current value of a
derivative V in terms of its terminal payoffs V ( S', T ) at time T 2/24/08 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 3 of 34 ∞ V ( S, t ) = e –r ( T – t ) ∫ p ( S, t, S', T ) V ( S', T ) dS' Eq.5.1 Copyright Emanuel Derman 2008 0 2/24/08 Here p ( S, t, S', T ) is the riskneutral (pseudo) probability density.
We define
π ( S, t, S', T ) = e –r ( T – t ) p ( S, t, S', T ) Then π ( S, t, S', T ) dS' is the price at time t of a statecontingent security that
pays $1 if the stock price at time T lies between S' and S' + dS' . Since the integral over all final stock prices of a security that pays $1 at expiration is equivalent to a zerocoupon bond with a face value of $1,
∞ ∫ π ( S, t, S', T ) dS' =e –r ( T – t ) 0 and
∞ ∫ p ( S, t, S', T ) dS' =1 0 an appropriate constraint on a probability density.
If we know the probability density p(S,t,S',T), we can determine the value of all
Europeanstyle payoffs at time T by weighting the probability by the payoff.
In particular, we can write the value of any European option at time T as an
integral over the riskneutral probability density. For a standard call option C
with strike K,
C ( S', T ) = [ S' – K ] + = max ( S' – K, 0 ) = [ S' – K ]θ ( S' – K )
where θ ( x ) is the Heaviside or indicator function, equal to 1 when x is greater
than 0 and 0 otherwise. Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 4 of 34 Therefore Copyright Emanuel Derman 2008 ∞ C K ( S, t ) = e –r ( T – t ) ∫ p ( S, t, S', T ) ( S' – K ) dS' =e –r ( T – t ) ∞ ∫0 d S' ( S' – K )θ ( S' – K ) p ( S, t, S', T ) It turns out that a knowledge of call prices (or put prices) for all strikes K at
expiration time T are enough to determine the density p ( S, t, S', T ) for all S'
Therefore one can statically replicate any known payoff at time T through a
combination of zerocoupon bonds, forwards, calls and puts.
One big caveat. Remember though, that the riskneutral distribution at expiration is insufficient for valuing all options on the underlyer. To value an
option on a stock, one must hedge it; to hedge it, one must hedge against the
changes caused by the stochastic process driving the stock price; the riskneutral distribution at expiration tells you nothing about the evolution of the stock
price on its way to expiration. Hence, implied distributions are not useful in
determining dynamic hedges. Nevertheless, implied distributions are useful for
statically replicating Europeanstyle payoffs at a fixed expiration. 5.1.1 The Heaviside and Dirac Delta functions
The derivative of the Heaviside function is the Dirac delta function:
∂
θ(x) = δ(x)
∂x
δ ( x ) is a distribution, the generic name for a very singular function that only
makes sense when used within an integral. δ ( x ) is zero everywhere except at
x = 0 , where its value is infinite. Its integral over all x is 1.
δ(x)
θ(x) 0 2/24/08 Eq.5.2 K x Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 5 of 34 There are three important properties of the delta function: Copyright Emanuel Derman 2008 ∞ 2/24/08 ∫ δ ( x ) dx =1 –∞
∞ ∫ f ( x )δ ( x ) dx = f(0) –∞ xδ(x) = 0
The latter equality holds formally because δ ( x ) is zero everywhere except at
the origin, and x itself is zero there. 5.1.2 Finding the riskneutral probability density from call
prices: the BreedenLitzenberger formula
From Equation 5.2
exp ( r τ ) × C ( S, t, K, T ) = ∞ ∫K d S' ( S' – K ) p ( S, t, S', T )
∞ ≡ ∫ d S' ( S' – K )θ ( S' – K ) p ( S, t, S', T )
0 where τ = T – t .
Now differentiate the equation above with respect to K, taking the derivative
on the right hand side under the integral sign, so that
∞ ∂C
exp ( r τ ) ×
= – ∫ p ( S, t, S', T ) dS' = – ( 1 – F ( K ) )
∂K
K Here we have made use of the identity ( S' – K )δ ( S' – K ) = 0 , and F ( K )
is the cumulative distribution function
K F(K) = ∫ p ( S, t, S', T ) dS'
0 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 6 of 34 Differentiate w.r.t K again to obtain the BreedenLitzenberger formula:
2 Copyright Emanuel Derman 2008 exp ( r τ ) × 2/24/08 ∂C
∂K = p ( S , t , K, T ) 2 Eq.5.3 The second derivative with respect to K of call prices is the riskneutral probability distribution, and hence must be positive. In fact, we know that the second
derivative must be positive from our earlier discussion of the noarbitrage
bounds on the skew.
2 ∂C
∂K 2 is a butterfly spread, proportional to C K + dK – 2 C K + C K – dK with termi nal payoff ~ whose height is dK and whose payoff area is
2 2 ( dK ) .In the limit that dK → 0 , ∂C has a payoff with area 1 if S = K and
2
∂K
zero otherwise; it behaves like a statecontingent security.
Note that at any time t:
∞ ∫ p ( S, t, K, T ) dK
0 =e rτ ∞2 ∂C ∫ ∂ K2 dK
0 =e rτ ∂C
∂K –
∞ ∂C
∂K =1
0 because
• • ∂C
∂K = 0 as the strike gets very large and calls become worthless; and
∞ for K → 0 the call becomes a forward with value S – Ke
∂C
∂K = –e –r τ –r τ , so that . 0 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 7 of 34 Copyright Emanuel Derman 2008 5.2 Static Replication: valuing arbitrary payoffs
at a fixed expiration using implied distributions.
From Equation 5.1 and Equation 5.3 we can write
∞2 V ( S, t ) = ∂C ∫ ∂ K2 ( S, t, K, T ) V ( K, T ) dK Eq.5.4 0 If we know call prices and their derivatives for all strikes at a fixed expiration,
we can find the value of any other Europeanstyle derivative security at that
expiration in terms of its payoff and the derivatives of the call prices. Alternatively, one can use the derivatives of put prices.
Note: this involves no use of option theory at all, and no use of the BlackScholes equation. It just assumes you can get all the option prices you need to
get the market’s statecontingent prices irrespective of any modeling issues. It
works even if there is a smile or skew or jumps. 5.2.1 Replicating by standard options
Equation 5.4 involves calculating the expected value of the Europeanstyle
payoff over the riskneutral density function corresponding to the implied distribution.
You can use integration by parts to show that the integral of any European payoff V over the riskneutral density function can be converted into a sum of portfolios of zero coupon bonds, forwards, puts and calls that together replicate the
payoff of V.
Consider an exotic European payoff W ( K, T ) . Then using the density for puts
below strike A and for calls above strike A, we can write
W ( S, t ) = e –r τ ∫ ρ ( S, t, K, T ) W ( K, T ) dK
0
∞ A =e –r τ ∫ ρ ( S, t, K, T ) W ( K, T ) dK + ∫ ρ ( S, t, K, T ) W ( K, T ) dK
0 A2 = ∂P A
∞2 ∂C ∫ ∂ K2 W ( K, T ) dK + ∫ ∂ K2 W ( K, T ) dK
0 A Now integrate by parts twice to get 2/24/08 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions
A W ( S, t ) = ∂ ∞2 2 ∂W ∫ ∂ K2 W ( K, T ) P ( S, K ) dK + ∫ ∂ K2 C ( S, K ) dK
0 Copyright Emanuel Derman 2008 Page 8 of 34 A ∂P
∂W⎞
= ⎛W
⎝ ∂K – P∂K⎠ K=A
K=0 + ⎛W
⎝ ∂C
∂W
–C ⎞
∂K
∂K ⎠ Eq.5.5 K=∞
K=A where P ( S, K ) is the current value at time t and stock price S of a put with
strike K and expiration T, and C ( S, K ) is the corresponding call value.
We can evaluate all these boundary terms as a function of strike K, using the
following conditions for the current call and put prices.
P [ S, 0 ] = 0
∂
P [ S, 0 ] = 0
∂K
C [ S, ∞ ] = 0
∂
C [ S, ∞ ] = 0
∂K
–r τ P [ S, K ] – C [ S, K ] = Ke – S
∂
∂
–r τ
P [ S, K ] –
C [ S, K ] = e
∂K
∂K
We then obtain
–r τ + W' ( A ) [ S – Ae –r τ A ∞ 0 W = W(A)e A + ∫ P ( K ) W'' ( K ) dK + ∫ C ( K ) W'' ( K ) dK Eq.5.6 This formula1 demonstrates that you can decompose an arbitrary payoff at time
T into a constant riskless payoff discounted like a zerocoupon bond, a linear
part which has the same value as a forward contract with delivery price A, and
a combination of puts with strikes below A and calls with strikes above A, with
densities given by ∂ 2 ∂K 2 W ( K, T ) . 1. Derived in this form by Carr and Madan. 2/24/08 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 9 of 34 Copyright Emanuel Derman 2008 The following figure illustrates the replication of the payoff, where the constant and linear parts of the payoff are replicated without any options, and the
curved parts make use of options. 2/24/08 linear payoff S – A
with slope W'(A) payoff W constant
payoff W(A) A terminal stock price Thus there are two sides to static replication.
1. If you know the riskneutral density ρ then you can write down the value
of W(S,t) as an integral over the terminal payoff, as in Equation 5.4.
2. Alternatively, if you know the second derivative of the payoff W, then you
can write down the value of W(S,t) as an integral over call and put prices
with different strikes, as in Equation 5.6.
The one equation is the complement of the other.
If you can buy every option in the continuum you need from someone who will
never default on their payoff, then you have a perfect static hedge. You can go
home and come back to work only when W expires, confident that the options
C and P that you bought will exactly match its payoff. This hedge does not
depend on any theory at all – it’s pure mathematics (plus faith in your counterparties) that matches one payoff by the sum of a series of different ones.
If, as in life, you cannot buy every single option in the continuum because only
a finite number of strikes are available for purchase, then you have only an
approximate replicating portfolio whose value will deviate from the value of
the target option’s payoff. Picking a reasonable or tolerable replicating portfolio is up to you. There is always some residual unhedged risk. 5.2.2 This works even if there is volatility skew. If you can write the payoff
of an exotic option at time T as a sum over vanilla options, and if you know the
skew BlackScholes implied volatilities Σ(K,T) at that instant for all K – i.e the
prices at which the market instantaneously values options of all strikes at that Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 10 of 34 expiration – then you can value the exotic.A Static Replication Example Copyright Emanuel Derman 2008 in the Presence of a Skew 2/24/08 Consider an option of strike B and expiration T on a stock with price S whose
payoff gives you one share of stock for every dollar the option is in the money.
Its payoff in terms of the terminal stock price s is
V ( s ) = s × max [ s – B, 0 ] = s × ( s – B )θ ( s – B ) Eq.5.7 When it is in the money, this payoff is quadratic in the stock price, but vanilla
calls are linear. We can replicate the payoff of this option by adding together a
collection of vanilla calls with strikes starting at B, and then adding successively more of them to create a quadratic payoff, as illustrated below.
We attempt to replicate the security V by means of a portfolio of call options
C(K) with all strikes X greater than B, so that
∞ ∫ q ( K )θ ( K – B ) C(K)dK V= Eq.5.8 0 where q ( K ) is the unknown density of calls with strike K required to replicate
the payoff of V, and we’ve chosen A in Equation 5.6 to be 0.
Differentiating Equation 5.7 with respect to s leads to
∂
∂V
( s ) = [ s × ( s – B )θ ( s – B ) ]
∂s
∂s
= ( s – B )θ ( s – B ) + s θ ( s – B ) + s ( s – B )δ ( s – B )
= ( s – B )θ ( s – B ) + s θ ( s – B )
2 ∂V
∂s 2 = ( s – B )δ ( s – B ) + 2 θ ( s – B ) + s δ ( s – B )
= 2θ(s – B) + sδ(s – B) Therefore for A = 0
V(0 ) = 0
∂V
(0) = 0
∂s
2 ∂V
∂s 2 ( K) = 2θ( K – B) + Bδ(K – B) Substituting this into Eq.5.8 we obtain the decomposition of the target security
V in terms of call options: Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 11 of 34 ∞ V = B C(B) + ∫ 2C(K)dK
Therefore, the current fair value of V is
∞ V ( S, t ) = BC ( S, t, B, T ) + 2 ∫ C ( S, t, K, T ) dK
B What is this worth in real life? The quadratic payoff is a linear combination of
call payoffs.The figure below shows how well the quadratic payoff as function
of terminal stock price s is approximated by a portfolio of 50 calls with strikes
equally spaced and $1 apart between 100 and 150. The replication becomes
progressively more inaccurate for stock prices greater than 150.
140
120
100 payoff 80 Exotic
50 Vanillas 60 35 Vanillas
20 Vanillas 40
20 0 0 0
17 16 0 0 0 15 14 13 0 0 12 11 0
9 10 0
8 0
7 0 0 0
6 5 4 0 0 3 Copyright Emanuel Derman 2008 B stock price Now we examine the convergence of the value of the replicating formula to the
correct noarbitrage value for two different smiles.
The first smile we consider is described by
Kβ
Σ ( K ) = 0.2 ⎛ ⎞
⎝ 100⎠
Here β = – 0.5 corresponds to a “negative” skew in which implied volatility
increases with decreasing strike; β = 0 corresponds to no skew at all; and
β = 0.5 corresponds to a positive skew. 2/24/08 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 12 of 34 Copyright Emanuel Derman 2008 For β = 0 the fair value of V when replicated by an infinite number of calls is
1033. The graph below illustrates the convergence to fair value of the replicating portfolio as the number of strikes included in the portfolio increases. Wiith
10 strikes the value has virtually converged. 2/24/08 Convergence as we increase number of strikes for flat 20% volatility Now we examine the effect of the skew on the value and convergence of V.
For both positive and negative skews, we plot below
1. the implied volatility as a function of strike;
2. the implied distribution corresponding to the skew; and
3. the convergence of the value of the replicating portfolio for option V to its
fair value as a function of the number of calls included in the portfolio. Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 13 of 34 2/24/08 0.08928626
0.07669802
0.06308335
0.04995406
0.03827049
0.02848595
0.02067526
0.14
0.01467862
0.01022136
0.00699748
0.12
0.00471927
0.00314113
0.00206663
0.00134589
0.1
0.00086868
0.00055628
0.00035378
0.08
0.00022364
0.00014063
8.8029E05
5.4887E05
0.06
3.4107E05 density Copyright Emanuel Derman 2008 Positive skew β = 0.5 0.0051252
0.0087056
0.009736
0.0089463
0.0071781
0.005112
0.003184
0.0016101
0.0004504
0.00032561
0.00078955
0.00102217
0.0010966
0.00107149
0.00099007
0.00088202
0.00076637
0.00065439
0.00055208
0.00046197
0.00038456
0.00031917 density functions beta = 0.5 positive skew
Density Difference
0.02
0.015
0.01
0.005
0
0.005
0.01
0.015 skewed vol
flat vol 0.04 excess probability at
high stock prices 0.99994488 0.02 0
stock price Convergence for a positive skew to a fair value of 1100 is slower and requires
more strikes. Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 14 of 34 2/24/08 density functions beta = 0.5 neg. skew
Density Difference 0.12 0.015
0.01
0.005
0 0.1 0.005
0.01 s1 0.015
0.02 0.08
density Copyright Emanuel Derman 2008 Negative skew β = 0.5 skewed vol 0.06 flat vol 0.04 0.02 excess probability
at low stock prices 0
stock price Convergence for a negative skew to a fair value 996 is faster and requires less
strikes.. Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 15 of 34 Copyright Emanuel Derman 2008 Appendix 5.2: The BlackScholes riskneutral
probability density 2/24/08 In the BS evolution, returns ln S T ⁄ S t are normal with a riskneutral mean
2 r τ – 1 σ τ and a standard deviation σ τ , where τ = T – t .
2 Therefore,
2 ln S T ⁄ S t – ( r τ – 1 σ τ )
2
x = στ Eq.5.9 is normally distributed with mean 0 and standard deviation 1, with a probabil2 –x ⁄ 2 e
ity density h ( x ) =  . The returns ln S T ⁄ S t can range from – ∞ to ∞ .
2π
The BS density function
From Eq.5.9,
dS T
 = σ τ dx
ST
The riskneutral value of the option is given by
1
e C = 2π
rτ ∞ ∫
–d2 ∞ 2 2 –x
1
– x dS T
( S T – K ) exp ⎛  ⎞ dx =  ∫ ( S T – K ) exp ⎛  ⎞ ⎝2⎠
⎝ 2 ⎠ ST
σ 2 πτ
K where
2 –x
exp ⎛  ⎞
⎝ 2 ⎠
2 πτσ S T
is the riskneutral density function to be used in integrating payoffs over S T ,
plotted below. Lecture5.2008.fm Copyright Emanuel Derman 2008 E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions 2/24/08 Page 16 of 34 Let’s work out the value of a call with this BS density and show that it gives
the BS formula. It’s tedious but worth doing once.
When S T = K then x min 12
12
12
12
ln K ⁄ S t – ⎛ r τ –  σ τ⎞
– ln S t ⁄ K – ⎛ r τ –  σ τ⎞
ln S t ⁄ K + ⎛ r τ –  σ τ⎞
ln S F ⁄ K –  σ τ
⎝
⎠
⎝
⎠
⎝
⎠
2
2
2
2
=  =  = –  = –  = – d 2
στ
στ
στ
στ Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 17 of 34 Copyright Emanuel Derman 2008 Box 1. The BS density and the BS formula (zero dividends) 2/24/08 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 18 of 34 Copyright Emanuel Derman 2008 5.3 Static Replication of NonEuropean Options 2/24/08 To replicate an option dynamically, you can in principle own a portfolio of
stock and riskless bonds, and adjust them to obtain achieve exactly the same
returns. To do so, you must continuously alter the weights in the replicating
portfolio according to the formula as time passes and/or the stock price moves.
This portfolio is called the dynamic replicating portfolio. Options traders ordinarily hedge options by shorting the dynamic replicating portfolio against a
long position in the option to eliminate all the risk related to stock price movement.
There are three difficulties with this hedging method. First, continuous weight
adjustment is impossible, and so traders adjust at discrete intervals. This causes
small errors that compound over the life of the option, and result in replication
whose accuracy increases with the frequency of hedging, as we’ve seen previously. Second, the transaction costs associated with adjusting the portfolio
weights grow with the frequency of adjustment and can overwhelm the potential profit margin of the option. Traders have to compromise between the accuracy and cost. Third, the systems you need to carry out dynamic replication
must be sophisticated and are costly.
What can you do about all of this? In this section we describe a method of
options replication that bypasses (approximately) some of these difficulties.
Given some particular exotic target option, we show how to construct a portfolio of standard liquid options, with varying strikes and maturities and fixed
timeindependent weights that will require no further adjustment and will (as
closely as possible) replicate the value of the target option for a chosen range
of future times and market levels. We call this portfolio the static replicating
portfolio.
The method is not modelindependent in the way that the static replication of
Europeanstyle options was. The method relies on the assumptions behind the
BlackScholes theory, or any other theory you used to replace it. Therefore, the
theoretical value and sensitivities of the static replicating portfolio are equal to
the theoretical value and sensitivities of the target option. You can use this
static replicating portfolio to hedge or replicate the target option as time passes
and the stock price changes. Often, the more liquid options you use to replicate
the target portfolio, the better you can do.The costs of replication and transaction are embedded in the market prices of the standard options employed in the
replication.
The static replicating portfolio is not unique and usually not perfect. You can
examine a variety of static portfolios available to find one that achieves other
aims as well – minimizing the difference between the volatility exposures of
the target and the replicating portfolio, for example. In general, a perfect static
hedge requires an infinite number of standard options. In some cases, it is possible to find a portfolio consisting of only a small number of options that proLecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 19 of 34 Copyright Emanuel Derman 2008 vides a perfect static hedge. Even so, a static hedge portfolio with only several
options can provide adequate replication over a wide variety of future market
conditions. 2/24/08 To illustrate the method we are going to consider a particular class of barrier
options, namely exotic options. Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 20 of 34 Copyright Emanuel Derman 2008 5.4 Valuing Barrier Options 2/24/08 We begin by illustrating how to value a zerorebate downandout barrier
option under Geometrical Brownian Motion. The valuation method will suggest a replicating portfolio. 5.4.1 GBM with zero stock drift
Start by assuming the current stock price is S and that the Brownian motion has
zero drift. Now consider a downandout option with strike K and barrier B.
K
S
B
S' expiration T time τ Then, for a suitably chosen “reflected” stock price S', the blue trajectory beginning at S and the red trajectory beginning at S' have equal probability of reaching any point on the barrier B at time τ , and then from that point, have equal
probability of taking the future green trajectory that finishes in the money.
Conversely, for any green trajectory finishing in the money, there are two trajectories starting out, one beginning at S and another beginning at S', that have
the same probability of producing the green trajectory.
Thus, if we subtract the two densities corresponding to S and S', then, above
the barrier B, the contribution from every path emanating from S that touched
the barrier at time τ will be cancelled by a similar path emanating from S'.
For arithmetic Brownian motion we can simply subtract the two densities with
initial points S and S'. But GBM is symmetric in log space, not stock space.
The probability to get from S to S' in a GBM world depends only on ln S ⁄ S' ,
so that, intuitively, the reflection S' of S in the barrier B must be a log reflection, that is
2 B
S
B
ln  = ln  or S' = S
B
S' Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 21 of 34 Thus the downandout density is the difference between a lognormal distribution from S to ST and a lognormal distribution from S' to STm, where the mean Copyright Emanuel Derman 2008 2 2/24/08 στ
of the normal distribution of the log returns for zero rates is at – 2
The density for reaching a stock price Sτ a time τ later is therefore
2 2 2 ⎛ ln S τ ⁄ S + 0.5 σ τ⎞
⎛ ln ( S τ S ) ⁄ B + 0.5 σ τ⎞
n' = n ⎜ ⎟ – α n ⎜  ⎟
στ
στ
⎝
⎠
⎝
⎠ Eq.5.10 for some coefficient α , where n ( x ) is a normal distribution with mean 0 and
standard deviation 1, and we want this density to vanish when S τ = B , so that
⎛ ln B ⁄ S + 0.5 σ 2 τ⎞
⎛ ln S ⁄ B + 0.5 σ 2 τ⎞
n ⎜  ⎟ – α n ⎜  ⎟ = 0
στ
στ
⎝
⎠
⎝
⎠
We can solve this equation for α to obtain
S
α = ⎛  ⎞
⎝ B⎠ Eq.5.11 So, integrating over the payoff,
2 S
B
C DO ( S, K ) = C BS ( S, K ) –  C ⎛  , K⎞
⎝S ⎠
B BS Eq.5.12 You can see that the value of this option vanishes on the boundary S = B independent of the time at which it reaches the boundary, and, for S > K at expiration, the second option finishes out of the money. Thus C DO has the correct
boundary conditions. The homework assigned asks that you proved that C DO
also satisfies the BlackScholes PDE. Given the same PDE and the correct
boundary conditions, this is the correct solution. 5.4.2 Nonzero riskneutral drift μ = r – 0.5 σ 2
This is a little trickier. When the drift is nonzero then we can’t use the equality
of the probabilities for reaching B from both S and S', since the drift distorts
the symmetry. So, we try to guess our way into this. Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 22 of 34 Try to pick a superposition of densities and S and the same reflection point
2 Copyright Emanuel Derman 2008 S' = B ⁄ S . (A more careful proof can derive the value of S' too.)
Then the trial downandout density for reaching a stock price Sτ a time τ later
is
2 ln S τ ⁄ S – μτ
⎛ ln ( S τ S ) ⁄ B – μτ⎞
n' = n ⎛  ⎞ – α n ⎜  ⎟
⎝
⎠
στ
στ
⎝
⎠ Eq.5.13 for some coefficient α , where n ( x ) is a normal distribution with mean 0 and
standard deviation 1, and we want this density to vanish when S τ = B , so that
ln B ⁄ S – μτ
ln S ⁄ B – μτ
n ⎛  ⎞ – α n ⎛  ⎞ = 0
⎝
⎠
⎝
στ
στ ⎠
We can solve this equation for α to obtain B
α = ⎛  ⎞
⎝ S⎠ 2μ
2
σ Eq.5.14 Notice that α is independent of the time τ at which the stock prices diffuse to
hit the barrier, and so this trial density vanishes on the boundary for all times,
for a fixed α . Therefore, the value of a downandout call is given by the integration of this density over the payoff, namely
B
C DO = C BS ( S, t, σ, K ) – ⎛  ⎞
⎝ S⎠ 2μ
2
σ 2 B
C ⎛  , t, σ, K⎞
⎝
⎠
BS S Eq.5.15 5.5 First Steps: Some Exact Static Hedges
Under certain limited circumstances, you can statically replicate a barrier
option with a position in stocks and bonds alone, avoiding the need for options.
We present and analyze several examples below. 5.5.1 European DownandOut Call
Consider a European downandout call option with time t to expiration on a
stock with price S and dividend yield d. We denote the strike level by K and the
level of the outbarrier by B. We assume in this particular example that B and K 2/24/08 Lecture5.2008.fm Copyright Emanuel Derman 2008 E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 23 of 34 are equal. and that there is no cash rebate when the barrier is hit. There are two
classes of scenarios for the stock price paths: scenario 1 in which the barrier is
avoided and the option finishes inthemoney; and scenario 2 in which the barrier is hit before expiration and the option expires worthless. These are shown
in Figure 5.1 below.
FIGURE 5.1. A downandout European call option with B = K.
scenario 1:
barrier avoided
value = S’  K stock
price B=K knockout barrier
scenario 2 :
barrier hit
value = 0
expiration time In scenario 1 the call pays out S' – K , where S' is the unknown value of the stock
price at expiration. This is the same as the payoff of a forward contract with
delivery price K. This forward has a theoretical value F = Se –dt – Ke – rt ,
where d is the continuously paid dividend yield of the stock. You can replicate
the downandout call under all stock price paths in scenario 1 with a long
position in the forward.
For paths in scenario 2, where the stock price hits the barrier at any time t'
before expiration, the downandout call immediately expires with zero value.
In that case, the above forward F that replicates the barrieravoiding scenarios
of type 1 is worth K e – dt' – Ke – rt' . This matches the option value for all barrierstriking times t' only if r = d. So, if the riskless interest rate equals the dividend
yield (that is, the stock forward is close to spot1), a forward with delivery price
K will exactly replicate a downandout call with barrier and strike at the same
level K, no matter whether the barrier is struck or avoided. 1. In late 1993, for example, the S&P dividend yield was close in value to the shortterm interest
rate, and so this hedge might have been applicable to shortterm downandout S&P options 2/24/08 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 24 of 34 Copyright Emanuel Derman 2008 5.5.2 European UpandOut Put1
Now consider an upandin put with strike K equal to the barrier B, as illustrated below. stock
price
B=K S knockin barrier : expiration time Trajectories like the blue one that hit the barrier generate a standard put
P ( S=K, K, σ, τ ) , whereas red trajectories that avoid the barrier expire worthless. Thus to replicate the upandin put we need to own a security that expires
worthless if the barrier is avoided and has the value of the put P ( K, K, σ, τ ) on
the barrier.
A standard call option C ( S, K, σ, τ ) bought at the beginning will expire
worthless for all values of the stock price below K at expiration. And, on the
boundary S = K, the value C ( S=K, K, σ, τ ) = C ( S=K, K, σ, τ ) if interest
rates and dividend yields are zero. This putcall symmetry follows because of
the symmetry of the density above and below the barrier when rates and dividend yields are zero.
Thus, a standard call C ( S, K, σ, τ ) can replicate a downandin put when
B = K . But notice, when and if the stock price hits the barrier, you must sell
the standard call and immediately buy a standard put, which, theoretically,
from the argument in the previous paragraph, should have the same value. 5.5.3 Hedging Using PutCall Symmetry
In a BlackScholes world, in the special circumstances where r = d = 0 , it’s
possible to create more static hedges for barrier options. 1. Many of these examples come from papers by Peter Carr and collaborators. 2/24/08 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 25 of 34 Copyright Emanuel Derman 2008 Start by working with arithmetic Brownian motion, d S = σ dW . Then, as
illustrated in the figure below, the probability of moving from B up towards K
through a range K – B is the same as the probability of moving from B down
away from K to K' = B – ( K – B ) = 2 B – K , i.e. through a range K – B to
the stock price K' . K
B
2B – K Hence, by symmetry, when the stock is at B, a call struck at K has the same
price as a put struck at K', i.e. C ( B, K ) = P ( B, K' )
So, the portfolio W = C ( S, K ) – P ( S, K' ) for S ≥ B will have the same payoff as an ordinary call struck at K (since the put will expire out of the money
when t he call is in the money), and, will have value zero when S = B. In other
words, W has the same boundary conditions as a downandout call with barrier B.
Now let’s look at geometric Brownian motion. Then the diffusion symmetry is
in the log of S, so that K' is determined by the condition ln K ⁄ B = ln B ⁄ K' or
2 K' = B ⁄ K . However C ( B, K ) ≠ P ( B, K' ) because of the mismatch between
logarithmic symmetry and linear payoff. Instead, because of the homogeneity
of the solution to the BlackScholes equation,
C ( B, K )
P ( B, K' )
B
 = F ⎛ ln K⎞ = F ⎛ ln ⎞ = ⎝ ⎠
⎝ K'⎠
K'
B
B Eq.5.16 Therefore,
K'
B
P ( B, K' ) =  C ( B, K ) ≡  C ( B, K )
B
K
and so, on the barrier B, 2/24/08 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 26 of 34 2 Copyright Emanuel Derman 2008 K
B
C ( B, K ) =  P ⎛ B, ⎞
⎝ K⎠
B 2/24/08 Eq.5.17 So, the portfolio
2 K
B
W = C ( S, K ) –  P ⎛ S, ⎞
B ⎝ K⎠ Eq.5.18 has the payoff of a call at expiration when S > B and vanishes everywhere on
the barrier when S = B, and so is a perfect static hedge.
This will be true even if the local volatility is not constant, but rather a function
K
K
B
σ = σ ⎛ ⎞ . because then Σ ⎛ ⎞ = Σ ⎛ ⎞ .
⎝ S⎠
⎝ B⎠
⎝ K'⎠ Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 27 of 34 Copyright Emanuel Derman 2008 5.6 Hedging PathDependent Exotics with Standard Options More Generally
References: Derman, Ergener, Kani. Static Options Replication, The Journal of
Derivatives, 24 Summer 1995, pp. 7895. Mark Joshi’s book. Papers by
Poulsen et al.
Consider a discrete downandout call with strike K, a barrier B below the
strike, and an expiration time T; the options knocks out only at n times
{ t 1, t 2, ..., t n } between inception of the trade and expiration. S
K B
t0 t1 t2 t3 tn T We want to create a portfolio of standard options that have the payoff of a call
with strike K at expiration T if the barrier B hasn’t been penetrated, and vanishes in value on the boundary B at time { t 1, t 2, ..., t n } .
We can replicate the payoff of the call at expiration with a standard call
C ( K, T ) , which denotes a security that is a call with strike K and expiration T,
with value C ( S, t, K, T ) at time t and stock price S. Now we want to put this
call into a portfolio V such that the portfolio value is the call payoff at expiration, but vanishes at each intermediate time ti when S = B . The value of these
extra securities added to the portfolio serve to cancel the value of entire portfolio at the points on the barrier, but they must also add have no payoff above B,
else they will not represent the value of the call at expiration, which has no earlier payoffs.
One solution is to use puts P ( S, t, B, t i ) with strike K and expiration time t i ,
because such puts have zero value at expiration when S > B , since they expire
out of the money. There are other possibilities too. For example we could
choose all expirations to be T, and vary the strikes to lie below B. 2/24/08 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 28 of 34 Here we replicate with a payoff of n standard puts P ( S, t, B, t i ) and the call
C ( S, t, K, T ) such that Copyright Emanuel Derman 2008 n 2/24/08 ∑ αj P ( S, t, K, tj ) V ( S, t ) = C ( S, t, K, T ) + Eq.5.19 j=1 where the α j are the number of puts with strike B and expiration t i in the portfolio. Note that since both the call and the put satisfy the BlackScholes equation, so does V, which it should. Only its boundary conditions differ from those
of a standard call or put.
We can now solve for the α j such that the value of this portfolio vanishes at all
the intermediate times t i for i = 0 to n – 1 on the barrier S = B , namely
n V ( B, t i ) = C ( B, t i, K, T ) + ∑ αj P ( B, ti, K, tj ) =0 Eq.5.20 j=1 where P ( B, t i, K, t j ) is the value of a put with strike K and expiration time t j at
time t i . Here we have n equations for the n unknowns α j , which can be solved
in sequential order by imposing Equation 5.20 starting with time t n and working backwards one step at a time.
Note that while the value of any put at expiration is defined by its payoff and is
modelindependent, the value of that put at earlier times depends on the market
(in real life) and on a model (in the theory we are developing here), and so this
method of replication is not truly model independent. The hope is that if we do
the replication in a BlackScholes world, or even better in a model world that
matches the price of all puts to the observed volatility smile, then the perturbations to the value of the portfolio will be insensitive to the details of the model.
By letting the number n of barrier points at times t n increase, we can move
closer and closer to replicating a continuous barrier. The PDE for options valuation dictates that if the boundary conditions are met, the value of the options is
determined. We can extend this method to more complicated boundaries too,
and, importantly, to any valuation model, not just BlackScholes.
When the stock price hits the barrier, the replicating portfolio must be immediately unwound. This assumes that the stock price moves continuously and that
there are no jumps across the barrier. Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 29 of 34 Copyright Emanuel Derman 2008 5.7 A Numerical Example: UpandOut Call
Barrier options have high gamma when the underlying stock price is in the
neighborhood of the barrier. In that region, dynamic hedging is both expensive
and inaccurate, and static hedging is an attractive alternative. Let’s look at an
upandout Europeanstyle call option, described in Table 1. All options values
are completed with the BlackScholes formula.
TABLE 1. An upandout call option.
Stock price: 100 Strike: 100 Barrier: 120 Rebate: 0 Time to expiration: 1 year Dividend yield: 5.0% (annually compounded) Volatility: 25% per year Riskfree rate: 10.0% (annually compounded) UpandOut Call Value: 0.656 Ordinary Call Value: 11.434 There are two different classes of stock price scenarios that determine the
option’s payoff, as displayed in Figure below.
FIGURE 5.2. Stock price scenarios for an upandout European call option with strike
K = 100 and barrier B = 120.
stock
price
scenario 2: barrier struck,
call expires worthless
B scenario 1:
barrier avoided
payoff = max(S’ K, 0) 0
expiration 2/24/08 time Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 30 of 34 Copyright Emanuel Derman 2008 From a trader’s point of view, a long position in this upandout call is equivalent to
owning an ordinary call if the stock never hits the barrier, and owning nothing otherwise. Let’s try to construct a portfolio of ordinary options that behaves like this. 2/24/08 First we replicate the upandout call for scenarios in which the stock price never
reaches the barrier of 120 before expiration. In this case, the upandout call has the
same payoff as an ordinary oneyear Europeanstyle call with strike equal to 100. We
name this call Portfolio 1, as shown in Table 2. It replicates the target upandout call
for all scenarios which never hit the barrier prior to expiration. Table 2: Portfolio 1. Its payoff matches that of an upandout call if the
barrier is never crossed before expiration.
Quantity Type Strike Expiration Value 1 year before expiration
Stock at 100 1 call 100 1 year Stock at 120 11.434 25.610 The value of Portfolio 1 at a stock level of 120 is 25.610, much too large when compared with the zero value of the upandout call on the barrier. Consequently, its value
at a stock level of 100 is 11.434, also much greater than the BlackScholes value
(0.657) of the upandout call with a continuous knockout barrier.
Portfolio 1 replicates the target option for scenarios of type 1.
Portfolio 2 in Table 3 illustrates an improved replicating portfolio. It adds to Portfolio
1 a short position in one extra option so as to attain the correct zero value for the replicating portfolio at a stock price of 120 with 6 months to expiration, as well as for all
stock prices below the barrier at expiration. Figure 4 shows the value of Portfolio 2 for
stock prices of 120, at all times prior to expiration. You can see that the replication on
the barrier is good only at six months. At all other times, it again fails to match the upandout call’s zero payoff. Table 3: Portfolio 2. Its payoff matches that of an upandout call if the
barrier is never crossed, or if it is crossed exactly at 6 months to
expiration.
Quantity Type Strike Expiration Value 6 months before expiration
Stock at 100 Stock at 120 1.000 call 100 1 year 7.915 22.767 2.387 call 120 1 year 4.446 22.767 3.469 0.000 Net Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 31 of 34 Copyright Emanuel Derman 2008 Table 4: Value of Portfolio 2 on the barrier at 120. 2/24/08 By adding one more call to Portfolio 2, we can construct a portfolio to match the zero
payoff of the upandout call at a stock price of 120 at both six months and one year.
This portfolio, Portfolio 3, is shown in Table 5. Table 5: Portfolio 3. Its payoff matches that of an upandout call if
barrier is never crossed, or if it is crossed exactly at 6 months or 1 year to
expiration.
Quantity Type Strike Expiration Value for stock price = 120
6 months 1 year 1.000 call 100 1 year 22.767 25.610 2.387 call 120 1 year 22.767 32.753 0.752 call 120 6 months 0.000 7.142 0.000 0.000 Net FIGURE 5.3. .Value of Portfolio 4 on the barrier at 120 Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 32 of 34 ut Copyright Emanuel Derman 2008 You can see that this portfolio does a much better job of matching the zero value of an
upandout call on the barrier. For the first six months in the life of the option, the
boundary value at a stock price of 120 remains fairly close to zero. 2/24/08 By adding more options to the replicating portfolio, we can match the value of the target option at more points on the barrier. Figure 5.4 shows the value of a portfolio of
seven standard options at a stock level of 120 that matches the zero value of the target
upandout call on the barrier every two months. You can see that the match between
the target option and the replicating portfolio on the barrier is much improved. In the
next section we show that improving the match on the boundary improves the match
between the target option and the portfolio for all times and stock prices.
FIGURE 5.4. Value on the barrier at 120 of a portfolio of standard options that is
constrained to have zero value every two months. 5.8 Replication Accuracy
We can see how well the replicating portfolio can match the value of the option
at all stock prices and times before expiration. Let’s look at an option with high
gamma, the upandout Europeanstyle call option defined in Table 6.
Its theoretical value in the BlackScholes model with one year to expiration is
1.913. We can use our method to construct a static replicating portfolio. Table
6 shows one particular example. It consists of a standard Europeanstyle call
option with strike 100 that expires one year from today, plus six additional
options each struck at 120. The 100strike call replicates the payoff at expiration if the barrier is never struck. The remaining six options expire every two
months between today and the expiration in one year. The position in each of
them is chosen so that the total portfolio value is exactly zero at two month
intervals on the barrier at 120. Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 33 of 34 Table 6: An upandout call option. Copyright Emanuel Derman 2008 Stock price: 2/24/08 100 Strike: 100 Barrier: 120 Rebate: 0 Time to expiration: 1 year Dividend yield: 3.0% (annually compounded) Volatility: 15% per year Riskfree rate: 5.0% (annually compounded) UpandOut Call Value: 1.913 The theoretical value of the replicating portfolio in Table 7 at a stock price of
100, one year from expiration, is 2.284, about 0.37 or 19% off from the theoretical value of the target option.
TABLE 7. The replicating portfolio.
Quantity Option
Type Strike 0.16253 Call 120 2 0.000 0.25477 Call 120 4 0.018 0.44057 Call 120 6 0.106 0.93082 Call 120 8 0.455 2.79028 Call 120 10 2.175 6.51351 Call 120 12 7.140 1.00000 Call 100 12 6.670 Total Expiration
(months) Value
(Stock = 100) 2.284 Instead of using six options, struck at 120, to match the zero boundary value on
the barrier every two months for one year, we can use 24 options to match the
boundary value at halfmonth intervals. In that case, the theoretical value of the
replicating portfolio becomes 2.01, only 0.10 away from the theoretical value
of the target option. You can see that the portfolio value varies like that of an
upandout option with barrier at 120. Lecture5.2008.fm E4718 Spring 2008: Derman: Lecture 5:Static Hedging and Implied Distributions Page 34 of 34 30
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264 .00 2/24/08 Call Value vs Stock Price and Time to Expiration 130 Copyright Emanuel Derman 2008 Here’s the behavior over all stock prices and time prior to expiration of a 24option replicating portfolio. You can see it looks a lot like the payoff of an up and out call option. Lecture5.2008.fm ...
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