implied and local volatility for south african derivatives
DESCRIPTION
Talk on implied and local volatility surfaces for traded index options and listed USDZAR FX options. There is a bit of history on where the Black-Scholes partial differential equation (PDE) originated as well.TRANSCRIPT
26-Aug-2014
1
Implied and Local Volatility
Surfaces in the South African
Derivatives Market
Dr A. A. Kotzé
Financial Chaos Theory
MiF 2014
25 August 2014
Skukuza Kruger National Park
Saggitarius A*: supermassive black
hole at the Milky Way’s center
26-Aug-2014
2
Agenda
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3
Before I came here I was confused
about the subject. Having listened to
your lecture I am still confused. But on
a higher level.Enrico Fermi (1901-1954)
Niels Bohr
and
Albert Einstein
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4
Global Warming
Who was the first person to propose that the earth's atmosphere might
act as an insulator of some kind – today known as the greenhouse
effect?
Jean Baptiste Joseph Fourier
21 March 1768 – 16 May 1830
The theory of probabilities is basically just common sense
reduced to calculus; it makes one appreciate with exactness that
which accurate minds feel with a sort of instinct, often without
being able to account for it. Pierre-Simon Laplace (23 March 1749 – 5 March 1827)
• Fourier was interested in heat transfer
• Laplace (his mentor) was interested in probabilities and the
central limit theorem
• Diffusion implies spreading, either observable (physical), or
abstract and probabilistic (stochastic). Mathematical theory
established by:
• 1812 Laplace monograph: Analytical theory of
probability
• 1807 and 1822 Fourier monographs on the
propagation of heat
Calculated that an object the size of earth, and at its distance from
the sun, should be considerably colder than the planet actually is if
warmed by only the effects of incoming solar radiation.
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Heat Conduction
• Laplace’s elliptic parabolic equation: 𝑑2𝑦
𝑑𝑥2=
𝑑 𝑦
𝑑𝑥′
• 𝑦 𝑥, 𝑥′ is the probability that the sum of 𝑥′ identically distributed random
variables takes on the value 𝑥
• Fourier’s parabolic equation for heat diffusion: K𝜕2𝑇
𝜕𝑥2= 𝐶
𝜕𝑇
𝜕𝑡
• 𝐾 = thermal conductivity; 𝑇 = temperature, 𝐶 = thermal capacity, 𝑥 =
distance along the abscissa and 𝑡 = time.
• Comparing the two: probability 𝑦 ≡ temperature, the magnitude of the sum
of random variables 𝑥 ≡ distance 𝑥 and the number of random variables
𝑥′ ≡ time.
Heat (energy) always moves from warmer
substance to colder substance – direction
involved
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From Lagrange to Navier, Dirichlet and Von Humbolt
• Dirichlet conditions are
sufficient conditions for
a real-valued, periodic
function f(x) to be
equal to the sum of its
Fourier series at each
point where f is
continuous.
• Lagrange was one of the creators of the calculus of variations, deriving the Euler–
Lagrange equations for extrema of functionals. He also extended the method to take
into account possible constraints, arriving at the method of Lagrange multipliers.
Lagrange invented the method of solving differential equations known as variation of
parameters, applied differential calculus to the theory of probabilities and attained
notable work on the solution of equations. He proved that every natural number is a
sum of four squares.
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Fast Forward 200 years
• Lord Rayleigh (theory of sound) – 1880
• Francis Edgeworth (law of error) – 1883
• Louis Bachelier (theory of speculation) – 1900
• Albert Einstein (theory of Brownian motion) – 1905
• Adriaan Fokker and Max Planck (time evolution of probability density function) –
1914/17
• Andrey Kolmogorov (his book, Foundations of the Theory of Probability, laid the
modern axiomatic foundations of probability theory and gave a rigorous
definition of conditional expectation) – 1933. His paper Analytic methods in
probability theory, laid the foundation for Markov processes – 1938.
• Richard Feynman and Mark Kac (Feynman-Kac theorem establishes link
between parabolic partial differential equations and stochastic processes) –
1945-49
• Paul Samuelson (Rational theory of warrant pricing) – 1965
• Black-Scholes-Merton (option pricing theory) – 1973
• Ian Stewart (Emeritus Professor of Maths at the University of Warwick) - 2012
The theory of probability as mathematical discipline can
and should be developed from axioms in exactly the
same way as Geometry and Algebra. Andrey Kolmogorov (25 April
1903 – 20 October 1987)
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In Pursuit of the Unknown: 17 Equations That Changed the
World – Ian Stewart (2012)
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Equation number 17: the MIDAS formula
With the associated stochastic differential equation
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What’s in the 1997 Nobel prize?
Myron Scholes (1941 - )
Robert Merton (1944 - )
Fischer Black (1938 - 1995)
Along the way, it changed the way investors
and others place a value on risk, giving rise to
the field of risk management, the increased
marketing of derivatives, and widespread
changes in the valuation of corporate
liabilities.
The theory "is absolutely
crucial to the valuation of
anything from a company
to property rights“. In my
view, financial
economics deals with
four main phenomena:
time, uncertainty, options and
information.William F. Sharp
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The Black-Scholes Equation
Let 𝜎(𝐾, 𝑇) be the implied volatility
With a constant volatility and constant interest rate and dividend yield, this
equation can be solved exactly by using the Feynman-Kac theorem where
• This theorem establishes the link between parabolic PDEs and
stochastic processes
• It offers a method of solving certain PDEs by simulating random paths
of a stochastic process
• It also justifies the practice of evaluating today’s value of an option as
the discounted expectation of its terminal payoff
• The PDE is a backward parabolic PDE also known as the backward
Kolmogorov PDE
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Implied Volatility
• Rebonato stated: Implied volatility is the wrong number to put
into the wrong formula to get the right price of plain-vanilla
options
• The Black-Scholes price of an option or actually the implied volatility
is just a translation mechanism. Traders use it to talk to one another
and be able to understand one another.
• Traders use the term “market volatility” because this is the volatility
traded in the market and the volatility used in the BS equation
• This leads to the implied volatility surface which is a 3D
representation of the volatility traded in the market
• The volatility skew is the market’s way of getting around Black and
Scholes’s simplifying assumptions about how the market behaves.
• The equity skew illustrates that implied volatility is higher as put
options go deeper in the money. This leads to the formation of a
curve sloping downward to the right.
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ALSI Implied Volatility showing Skew
28 May 2014
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USDZAR Implied Volatility showing Smile
28 May 2014
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ALSI Deterministic Skew
• Safex uses a linear quadratic functional form for its liquid ALSI
surface.
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ALSI Deterministic Surface
• Safex uses the following function for the ATM term structure
• Combing both equations leads to the 3D surface
• Parameters are obtained by fitting or optimising the functional form
to the traded market volatilities using the Nelder-Mead routine. All
options are traded on Nutron, Safex’s trading system
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Relaxing Assumptions
• Let’s assume that the volatility 𝜎 is dependent on the asset
price and time – it is not constant anymore BUT it is
deterministic. Then we have
• Such a volatility we call the “local” volatility. The associated
PDE is then
• This PDE is not solvable analytically
• It is also a backward Kolmogorov PDE
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Features of Local Volatility
• Local volatility models are widely used in the finance industry
• Whereas stochastic volatility and jump-diffusion models introduce new risks
into the modeling process, local volatility models stay close to the Black-
Scholes theoretical framework and only introduce more flexibility to the
volatility
• Local volatility does not give a complete representation of the true stochastic
process driving the underlying asset price
• Local volatility is merely a simplification that is practically useful for
describing a price process with non-constant volatility
• A local volatility model is a special case of the more general stochastic
volatility models – known as restricted stochastic volatility models
• The local volatility function is assumed to be a deterministic function of a
stochastic quantity 𝑆𝑡 and time. Still just one source of randomness
• Ensures that the completeness of the Black-Scholes model is preserved.
• Completeness is important, because it guarantees unique prices, thus
arbitrage pricing and hedging
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Importance of LV
• Many exotic derivatives have Black-Scholes like closed-form pricing
formulas like the ones derived by Rubinstein for barrier options
• The Black-Scholes-Merton model is far too simple to model most
exotic and complex derivatives, but the underlying methodology and
insights govern the pricing of all derivatives.
• Complex derivatives are mostly priced by using more sophisticated
numerical models, like binomial and trinomial trees, finite difference
models and Monte Carlo simulations.
• In short: an option that is path-dependent cannot be valued by
assuming a fixed volatility even under a skew. Volatility varies with
time and this influences the price thereof.
• We need a volatility that is dependent on the asset price and time
• Local volatility is one such solution and is much simpler than
stochastic volatility models like the Heston model
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Some other thoughts
• Rubinstein said:
• One of the central ideas of economic thought is that, in properly functioning
markets, prices contain valuable information that can be used to make a wide
variety of economic decisions. At the simplest level, a farmer learns of increased
demand (or reduced supply) for his crops by observing increases in prices, which
in turn may motivate him to plant more acreage. In financial economics, for
example, it has been argued that future spot interest rates, predictions of inflation,
or even anticipation of turns in the business cycle, can be inferred from current
bond prices. The efficacy of such inferences depends on four conditions:
• A satisfactory model that relates prices to the desired inferred information,
• A model which can be implemented by timely and low-cost methods,
• Correct measurement of the exogenous inputs required by the model, and
• The efficiency of markets.
• Indeed, given the right model, a fast and low-cost method of
implementation, correctly specified inputs, and market efficiency,
usually it will not be possible to obtain a superior estimate of the
variable in question by any other method
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Implied Volatility Trees
• Local volatility is not traded and thus is not a measurable quantity
like implied volatility. Local volatility must be calculated somehow.
• The above mentioned thought process led Rubinstein and
Derman & Kani to develop numerical schemes where they were
able to relate the local volatility to the stock price, implied
volatilities and time.
• Both of these methods use the so-called implied trees.
• The basic idea of these tree schemes is to price options in a
standard Cox, Ross and Rubinstein (CRR) tree with a constant
volatility, and then adjust the volatility at the nodes in the tree by
using the given implied volatility skew, to obtain the correct market
prices for the relevant options.
• The disadvantages of these methods are that they are slow and
notoriously unstable while convergence seems to be a problem.
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Local Volatility and Instantaneous Volatility
• Remember, historical volatility is calculated from the time series of
the stock prices. This volatility is thus `backward looking.'
• On the other hand, implied or market volatility is `forward looking‘
i.e., it is an estimate of the future volatility or the volatility that should
prevail from today until the expiry of the option.
• Rational market makers base option prices on these estimates of
future volatility. To them, the Black-Scholes implied volatility is `the
estimated average future volatility' of the underlyer over the lifetime
of the option. In this sense, IV is a global measure of volatility.
• On the other hand, the local volatility represents `some kind of
average‘ over all possible instantaneous volatilities, at a certain point
in time, in a stochastic volatility world.
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Instantaneous Volatility
• Let 𝑥𝑖 = ln 𝑆𝑖+1𝑆𝑖
then 𝜎𝑖𝑛𝑡 𝑡𝑖 = 𝑥𝑖
• We calculate the forward instantaneous volatility.
• Local volatility can be obtained
through simulation.
• We show 2 price paths arriving at
the same stock price at the same
time
• The average is obtained by
simulating many paths like these
and taking the average
LV is not observable and
needs to be calculated
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Dupire and Derman and Kani’s Insight
• They noted that knowing all European option prices merely
amounts to knowing the probability densities of the underlying
stock price at different times, conditional on its current value.
• Further insight came when they realised that under risk neutrality,
there was a unique diffusion process consistent with the risk neutral
probability densities derived from the prices of European options.
• This diffusion is unique for any particular stock price and holds for all
options on that stock, irrespective of their strike level or time to
expiration.
• Remember that under the general Black-Scholes theory, the implied
volatility skew infers that one stock should have many different
diffusion processes: one for every strike and time to expiry. This, of
course, cannot be the case.
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Dupire’s Question
• Let’s recap the SDE
• In its most general form, the volatility 𝜎 𝑆𝑡, 𝑡 is the instantaneous
volatility – it is stochastic in nature
• Similar to the definition of the instantaneous forward interest rate we
define the implied volatility as follows.
• Dupire asked: whether it was possible to construct a state-
dependent instantaneous volatility that, when fed into the one-
dimensional diffusion equation above will recover the entire
implied volatility surface 𝝈 𝑲, 𝑻 ?
• This suggests he wanted to know whether a deterministic volatility
function exists that satisfies the SDE?
• The answer is NO
IV averaged across TIME
LV averaged across PRICE
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Dupire’s Answer
• The answer to the previous question becomes true if we transform
the SDE to another SDE with a non-random volatility function –
Gyöngy’s theorem (1986).
• Dupire assumed that the probability density of the underlying asset
𝑆𝑇 at the time 𝑡 has to satisfy the forward Fokker-Planck equation
(also known as the forward Kolmogorov equation) given by
• Here 𝜑 ≡ 𝜑 𝑆𝑇, 𝑇 is the forward transition probability density of the
random variable 𝑆𝑇 (final 𝑆 at expiry time 𝑇 ) in the SDE shown
above
• Further, by using the Breeden-Litzenberger formula
• we can rewrite the above PDE to obtain the Dupire forward equation
in terms of call prices
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Dupire’s Forward Equation
• where 𝜎 𝐾, 𝑇 is continuous, twice-differentiable in strike and once in
time, and the local volatility is uniquely determined by the surface of
call option prices.
• Note how, when moving from the backward Kolmogorov equation
(the Black-Scholes PDE) to the forward Fokker-Planck equation, the
time to maturity 𝑇 has replaced the calendar time 𝑡 and the strike
𝐾 has replaced the stock level 𝑆.
• The Fokker-Planck equation describes how a price propagates
forward in time.
• This equation is usually used when one knows the distribution
density at an earlier time, and one wants to discover how this density
spreads out as time progresses, given the drift and volatility of the
process
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Dupire’s Local Volatility - Calls
Using Breeden & Litzenberger Dupire showed that
• 𝐶 𝐾, 𝜏 are vanilla call option prices
• 𝜎𝑙𝑜𝑐(𝐾, 𝜏) is the local volatility that will prevail at time 𝜏 = 𝑇 − 𝑡when the future stock price is equal to 𝐾 (𝑆𝜏 = 𝐾).
This equation ensures the existence and uniqueness
of a local volatility surface which reproduces the
market prices exactly
An implied volatility surface is arbitrage free if the local
volatility is a positive real number (not imaginary)
where 𝜎𝑙𝑜𝑐 𝐾, 𝑇 ∈ ℝ0+
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Dupire’s Local Volatility - IV
Dupire’s equation more useful in terms of implied
volatilities
𝜎𝑖𝑚𝑝 = 𝜎𝑖𝑚𝑝 𝐾, 𝜏 ; 𝜏 = 𝑇 − 𝑡 such that 𝑡 and 𝑆0 are
respectively the market date, on which the volatility
smile is observed, and the asset price on that date
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ALSI Local Volatility function
• Using a deterministic implied volatility function is very
useful because we have analytical solutions
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ALSI LV and IV
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USDZAR LV and IV
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ALSI Barrier Option
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Contact
Dr Antonie Kotzé
http://www.quantonline.co.za
Email: [email protected]
Disclaimer
This article is published for general information and is not intended
as advice of any nature. The viewpoints expressed are not
necessarily that of Financial Chaos Theory Pty Ltd. As every
situation depends on its own facts and circumstances, only specific
advice should be relied upon.