Smile Pricing Explained

By P. Austing

Smile Pricing Explained provides a clear and thorough explanation of the concepts of smile modelling that are at the forefront of modern derivatives pricing. The key models used in practice are covered, together with numerical techniques and calibration.

• Language: English
• Publisher: Palgrave Macmillan
• Release date: Aug 29, 2014
• ISBN: 9781137335722

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Smile Pricing Explained

Peter Austing

Imperial College, London

© Peter Austing 2014

All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission.

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The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988.

First published 2014 by

PALGRAVE MACMILLAN

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ISBN: 978–1–137–33571–5

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To Bertie

Contents

List of Symbols

Acknowledgements

Preface

1 Introduction to Derivatives

1.1 Hedging with Forward Contracts

1.2 Speculation with Forward Contracts

1.3 Arbitrage

1.4 Vanilla Options

1.5 Interest Rates

1.6 Valuing a Forward Contract

1.7 Key Points

1.8 Further Reading

2 Stochastic Calculus

2.1 Brownian Motion

2.2 Stochastic Model for Stock Price Evolution

2.3 Ito’s Lemma

2.4 The Product Rule

2.5 Log-Normal Stock Price Evolution

2.6 The Markov Property

2.7 Term Structure

2.8 Ito’s Lemma in More than One Dimension

2.9 Key Points

2.10 Further Reading

3 Martingale Pricing

3.1 Setting the Scene

3.2 Tradeable Assets

3.3 Zero Coupon Bond

3.4 Rolling Money Market Account

3.5 Choosing a Numeraire

3.6 Changing the Measure

3.7 Girsanov’s Theorem

3.8 Martingales

3.9 Continuous Martingales

3.10 Black–Scholes Formula for a Call Option

3.11 At-the-Money Options

3.12 The Black–Scholes Equation

3.13 An Elegant Derivation of the Black–Scholes Formula

3.14 Key Points

3.15 Further Reading

4 Dynamic Hedging and Replication

4.1 Dynamic Hedging in the Absence of Interest Rates

4.2 Dynamic Hedging with Interest Rates

4.3 Delta Hedging

4.4 The Greeks

4.5 Gamma, Vega and Time Decay

4.6 Vega and Volatility Trading

4.7 Key Points

4.8 Further Reading

5 Exotic Options in Black–Scholes

5.1 European Options

5.2 Asian Options

5.3 Continuous Barrier Options

5.3.1 The Reflection Principle

5.3.2 The Reflection Principle with Log-Normal Dynamic

5.3.3 Valuing Barrier Options in Black–Scholes

5.3.4 Discretely Monitored Barrier Options

5.4 Key Points

5.5 Further Reading

6 Smile Models

6.1 The Volatility Smile

6.2 Smile Implied Probability Distribution

6.3 The Forward Kolmogorov Equation

6.4 Local Volatility

6.5 Key Points

6.6 Further Reading

7 Stochastic Volatility

7.1 Properties of Stochastic Volatility Models

7.2 The Heston Model

7.2.1 What Makes the Heston Model Special

7.2.2 Solving for Vanilla Prices

7.2.3 The Feller Boundary Condition

7.3 The SABR Model

7.4 The Ornstein–Uhlenbeck Process

7.5 Mixture Models

7.6 Regime Switching Model

7.7 Calibrating Stochastic Volatility Models

7.8 Key Points

7.9 Further Reading

8 Numerical Techniques

8.1 Monte Carlo

8.1.1 Monte Carlo in One Dimension

8.1.2 Monte Carlo in More than One Dimension

8.1.3 Variance Reduction in Monte Carlo

8.1.4 Limitations of Monte Carlo

8.2 The PDE Approach

8.2.1 Stable and Unstable Schemes

8.2.2 Choice of Scheme

8.2.3 Other Ways of Improving Accuracy

8.2.4 More Complex Contracts in PDE

8.2.5 Solving Higher Dimension PDEs

8.3 Key Points

8.4 Further Reading

9 Local Stochastic Volatility

9.1 The Fundamental Theorem of On-smile Pricing

9.2 Arbitrage in Implied Volatility Surfaces

9.3 Two Extremes of Smile Dynamic

9.3.1 Sticky Strike Dynamic

9.3.2 Sticky Delta Dynamic

9.4 Local Stochastic Volatility

9.5 Simplifying Models

9.5.1 Spot–Volatility Correlation

9.5.2 Term Structure Vega for a Barrier Option

9.5.3 Simplifying Stochastic Volatility Parameters

9.5.4 Risk Managing with Local Stochastic Volatility Models

9.6 Practical Calibration

9.7 Impact of Mixing on Contract Values

9.8 Key Points

9.9 Further Reading

10 Volatility Products

10.1 Overview

10.2 Variance Swaps

10.2.1 The Variance Swap Contract

10.2.2 Idealised Variance Swap Trade

10.2.3 Valuing the Idealised Trade

10.2.4 Beauty in Variance Swaps

10.2.5 Delta and Gamma of a Variance Swap

10.2.6 Practical Considerations

10.3 Volatility Swaps

10.3.1 Volatility Swap in Stochastic Volatility Models and LSV

10.3.2 Volatility Swap Versus Variance Swap

10.3.3 Valuing a Volatility Swap

10.3.4 Stochastic versus Local Volatility

10.4 Forward Volatility Agreements

10.4.1 Practicalities

10.5 Key Points

10.6 Further Reading

11 Multi-Asset

11.1 Overview

11.2 Local Volatility with Constant Correlation

11.3 Copulas

11.4 Correlation Smile

11.5 Marking Correlation Smile

11.5.1 Common Correlation Products

11.5.2 The Triangle Rule

11.6 Modelling

11.6.1 Local Correlation

11.6.2 Practicalities

11.6.3 Local Stochastic Correlation

11.7 Valuing European Contracts

11.7.1 Special Properties of Best-of Options

11.7.2 Valuing a Best-of Option in Black–Scholes

11.7.3 Construction of a Joint PDF

11.7.4 Using the Density Function for Pricing

11.8 Numeraire Symmetry

11.9 Baskets as Correlation Instruments

11.10 Summary

11.11 Key Points

11.12 Further Reading

Afterword

Appendix: Measure Theory and Girsanov’s Theorem

References

Further Reading

Index

List of Symbols

Acknowledgements

I am indebted to the many friends and colleagues from whom I have learned this trade, and who have generously offered their insight and support including Quentin Adam, Mariam Aitichou, Jennifer Austing, Richard Austing, Guillaume Bascoul, Marko Bastianic, Marouane Benchekroun, Oleg Butkovsky, Iain Clark, Jeremy Cohen, John Darlington, Houman Falakshahi, Gareth Farnan, Markus Fritz, Ian Hamilton, Johnson Han, Duncan Harrison, Robert Hayes, Peter Jäckel, Amy Kam, Piotr Karasinski, Vladislav Krasin, Mark Lenssen, Minying Lin, Alex Lipton, Vladimir Lucic, Arthur Mountain, Jean-Pierre O’Brien, Neil Oliver, Vladimir Piterbarg, Juliette Pubellier, Tino Senge, David Shelton, Peter Spoida, Richard Summerbell, Lin Sun, Neil Waldie, Zoe Wang, Claudia Yastremiz and Mathieu Zaradzki.

Preface

In modern derivatives pricing, Black–Scholes theory is only a starting point. Asset volatilities are not constant, but change with market conditions. Large price moves tend to be associated with periods of high market turbulence and this leads to a smile shaped curve of the volatility implied from vanilla option prices.

Smile pricing is a core area of practice and research in modern quantitative finance. There are a number of models that seek to explain the volatility smile. Two famous examples are Dupire’s local volatility model, and the Heston stochastic volatility model. While they agree on vanilla option prices, their asset dynamics are very different, leading to large disagreement in exotic option prices.

This book aims to provide a clear but thorough explanation of the concepts of smile modelling that are at the forefront of modern derivatives pricing theory. The key models used in practice are covered, together with numerical techniques and calibration.

I have kept the needs of students and time-pressed practitioners very much in mind while writing. Topics are presented succinctly, with unnecessary complexity carefully avoided. Intuition is provided before mathematics so that readers may enjoy the book without having to follow every mathematical detail. Extended calculations are rarely necessary, but where they are (as in the solution of the Heston model for example) guidance is provided for those who wish to understand the result without ploughing through the maths.

Smile Pricing Explained is a self-contained textbook and desktop reference. In addition it tells a story, of which each chapter is an integral part. We start, naturally enough, right at the beginning, by using the principle of no arbitrage to value simple forward contracts. Then models are built up, starting with Black–Scholes and adding complexity and numerical techniques until we can create a full local stochastic volatility model. It is only having developed all this technology that we are able to step back and understand just what it is that makes a derivative pricing model good.

Peter Austing

Imperial College

www.smilepricingexplained.com

1.1   Hedging with Forward Contracts

A derivative is any financial product that is derived from a simpler underlying asset. The simplest derivative is a forward contract. For example, it could be a forward contract on wheat. A grower in England plants a field of wheat, and goes to significant expense in buying seed and fertiliser and maintaining the crop. This makes sense because of the current value of wheat. However, unfortunately, an excellent growing season in France means that there is a glut of wheat on the market just at the time he has harvested and needs to sell, and the price he can achieve is much lower than expected. This could be disastrous.

To avoid this situation, the grower could enter into a forward contract with his client, a cereal producer. At the time of sowing, the grower agrees with the cereal company to sell ten tonnes of wheat at a price K pounds per tonne, to be delivered within a given short interval in, say, six months’ time. The grower is happy because he knows what price he will achieve for the wheat, and can budget appropriately. If the market view on the future price of wheat is too low, the grower can look at the forward price of other crops, and plant something else. If at the delivery time the value of wheat has gone up, he will regret missing out, but on the other hand if the price has dropped he will have avoided ruin.

Meanwhile, the cereal company is happy because it knows in advance it can obtain enough wheat at a reasonable price, even if bad weather conditions lead to low crop yields and high prices at harvest time.

The grower and cereal producer are both acting in the market to hedge their risk. This is the main benefit of using derivatives, and in this way derivatives help to make markets more stable and more efficient. If used wisely, they can prevent bankruptcies and job losses in times of sudden market stress.

This example can also illustrate some of the risks in using derivatives. What if, due to a bad growing season, the grower is unable to deliver ten tonnes of wheat? Then, since he has entered into a firm obligation to deliver the crop, he would have to go into the market and purchase enough to make up his delivery. Assuming other participants in the market had also experienced poor conditions, this could be ruinously expensive. So in reality, the grower ought only to enter into the contract for a quantity of wheat he is absolutely certain he can deliver, even in the worst possible case.

1.2   Speculation with Forward Contracts

A second, entirely different, participant in the market is a wealthy individual who does not own, nor have any capacity to grow, wheat. However, she has the strong view that, due to political problems in an important wheat growing country, the price of wheat will increase. She wishes to profit from the increase in price without any outlay now. So she enters into a forward contract to buy N tonnes of wheat at a price K at a future time T (say one year’s time). If the price has risen, she will immediately sell the wheat and make a profit

where ST is the price of wheat at the expiry time T of the contract. Needless to say, if the price has fallen, she makes a loss.

In reality, the speculator does not want to be bothered with receiving and then delivering N tonnes of wheat, and so she will close out the contract before the delivery date. That is, she will either sell the contract, or enter into an opposite contract. Alternatively, she may have originally entered into a cash settled contract with a third party. This third party will handle the buying and selling of contracts, and at the expiry simply pay the cash profit (or demand payment of the loss). Banks perform this kind of service in huge volumes, not necessarily for wheat contracts, but certainly for contracts in shares (equities), foreign currencies (FX), and precious metals, oil or energy (commodities).

In the jargon of derivatives, St is the spot price or simply spot at time t, namely the value of the asset in the open market at time t. The quantity N is known as the notional amount, since from the point of view of a speculator entering into a cash settled transaction, the cash flow of NK agreed for the commodity never actually takes place. The agreed price in the contract, K, is known as the strike price, since it is the price at which the deal is struck.

1.3   Arbitrage

The third type of participant in the market aims to make money by arbitrage. An arbitrage opportunity arises when it is possible to make money without any risk. In the simplest example, let’s suppose an investor notices that it is possible to buy wheat contracts on one exchange for 59 US dollars and to sell on another for 62 dollars. Then he can make an instant profit by simultaneously buying on one and selling on the other.

If arbitrage exists in a market, then very quickly prices will be forced up in the underpriced asset as buyers charge in, and prices will be forced down on the overpriced asset as more and more people sell on that exchange. This does not mean that in an efficient market arbitrage never exists. Usually though, it only exists for a fleeting moment, until it is eliminated by the most sophisticated investors.

The concept of arbitrage is fundamental in pricing derivatives contracts. A quant¹ working for a bank will often be asked to price a new style of exotic derivative. If the price she comes up with can be demonstrated to be arbitrageable against other liquid assets in the market, then the bank will certainly lose money. Let’s suppose that this particular bank has an electronic dealing system in, say, foreign exchange derivatives. When outsiders spot the arbitrage, they will continually buy the liquid assets and sell the new exotic derivative, or vice versa, mercilessly making money at the expense of the bank (and no doubt the unfortunate quant).

When we price a derivative, the price must be arbitrage free. We shall see that in many cases, the arbitrage-free price is unique, and so finding it gives the true value of the derivative.

1.4   Vanilla Options

So far, we have looked at the simplest derivative, a forward contract. This is an agreement made now to buy a quantity of some asset at a future date (the expiry date). The deal is struck at the strike price K, and at the expiry time T the payout is

where N is the notional (the quantity of asset), ST is the spot price (value of the underlying asset) at expiry. At any time, this contract has a certain value for a given strike. This value can be positive or negative depending on whether the strike is above or below the market view of the forward value of the asset. Often, the strike will be chosen so that the value is zero when the contract is entered into, and no exchange of money is required between the buyer and seller at the start.

An investor who believes that the asset price will rise may enter into a forward contract to buy at strike K, but what if the asset price actually falls sharply? Her losses could be enormous. So let’s consider a contract that gives her the option, but not the obligation, to buy N units of the asset at strike price K. If, at the expiry time T, the spot price is larger than K then the investor would exercise the option. She could then immediately sell at the prevailing price ST and make profit N(ST – K). On the other hand, if the spot price ends up smaller than the strike, she would certainly not exercise the option. So the payout at expiry is

which we write as

Figure 1.1   Payouts of vanilla call and put options.

This contract is called a vanilla call option. In finance, call means buy and this is an option to buy the asset. The opposite of call is to put, namely to sell. The term vanilla indicates that these are the most basic option contracts. Supposing the investor entered into a contract giving the option to sell the asset at strike price K, that is the investor bought a vanilla put option, the payout would be

The call and put payouts as a function of the spot price at expiry are plotted in Figure 1.1. In these diagrams, the notional is 1 and the strike price 1. As you can see, if you buy a call option, you have limited loss, but unlimited potential gain if spot ends up very high. On the other hand, if you sell a call option, you have unlimited potential liability.

It is now time to whet your appetite for some of the beautiful facts we will soon be encountering. If you are new to this subject, you might well argue that it is impossible to attach a fair value to the contracts that we have been discussing. Imagine we are a bank selling a forward contract, and we cannot find any other counterparty with whom to do the opposite trade to eliminate the risk. Even if we could know in advance the probability distribution of the spot at expiry, the best we could work out would be the probability distribution of money made or lost. As your argument would go, there is surely no way the bank could prevent itself from being annihilated by a massive unfavourable movement in the market. You would be absolutely wrong.

In fact, there is a simple strategy that the bank can follow when selling a forward contract to ensure they have exactly enough money to pay the client at the end. The value of the forward contract is the cost of following this strategy, and will be completely independent of the random process that the spot follows.

In the case of vanilla options, we do need to know something about the random spot process: its volatility. Armed with this knowledge, we will be able to trade in the underlying asset through the life of the option in just such a way as to have exactly enough to pay the client at the end, and we will know in advance how much it will cost us to do this trading. This statement is stunning. If we know how volatile a stock price is, we can tell a trader just how to trade in that stock over the coming year to replicate a particular payout agreed with a client and, incredibly, we can calculate in advance how much that trading strategy will cost the bank, even though the asset spot price could eventually end up anywhere.²

We will learn that trading vanilla options amounts to trading volatility, and this will give us a handle on pricing exotic options with more general underlying dynamics in which the volatility may not be constant. At the same time, we will learn that there are still challenges in pricing some types of contract that may have simple payout but happen to probe the aspects of asset dynamics that are not easily understood. In short, we will arrive at the forefront of modern derivatives pricing.

1.5   Interest Rates

We cannot proceed without a discussion of interest rates. We assume that at any time, a trader can go into the market and find a counterparty with whom to either borrow or lend money for any fixed period at a fixed interest rate. The interest rate achieved when lending will depend on the counterparty’s credit rating. We would require a much higher interest rate to lend to a company having, in our opinion, a high probability of going bust and defaulting. The rate that we are interested in is the risk free rate. This is the hypothetical rate for a counterparty that has zero risk of defaulting.

The risk free rate is often described as the rate at which we could lend to the central bank for the currency in question. This is considered close to being risk free because the central bank can always print more money if necessary. However, in practice, central banks do not provide such a service. You cannot go to the central bank and demand to lend to it for a given fixed period of your choosing. In any case, there always remains risk that a central bank will default for political reasons.

However, there are many interest rate instruments trading interbank in the market, and these can be used to determine the simple interest rate for a fixed term of borrowing. The instruments that are often considered to give the closest thing to a risk free rate are called overnight indexed swaps (OIS). An OIS is agreed for a fixed period of