26-Aug-2014

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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

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Agenda

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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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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: consultant@quantonline.co.za

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.