Essays on Option Pricing, Hedging and Calibration André Manuel da Silva Ribeiro
PhD Thesis
Supervisor:
Rolf Poulsen, University of Copenhagen
Submitted:
May 31, 2015.
Department of Mathematical Sciences Faculty of Science University of Copenhagen
Author:
André Manuel da Silva Ribeiro Department of Mathematical Sciences Universitetsparken 5 DK-2100 Copenhagen
[email protected]
Assessment Committee:
Associate Professor Elisa Nicolato University of Aarhus Aarhus, Denmark Associate Professor Friedrich Hubalek Technische Universität Wien Vienna, Austria Associate Professor Jesper Lund Pedersen University of Copenhagen Copenhagen, Denmark (Chairman)
Financing:
This PhD thesis was co- nanced by Fundação e Tecnologia (SFRH/BD/70389/2010) and by Human Potential gramme of the European Social Fund.
para a Ciência Operating Pro-
Contents
Preface
iii
Summary
v
Sammenfatning
vii
List of papers
ix
1 Introduction
1
2 Approximation Behooves Calibration
11
3 Discretely Sampled Variance Options: A Stochastic Approach
23
Appendices
43
4 Static Hedging for Two-Asset Options
53
A.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . 45 A.2 Proof of Proposition A.1.1 . . . . . . . . . . . . . . . . . . . . . . . 47 A.3 Joint Characteristic Function for the Ornstein-Uhlenbeck process . 49 4.1 4.2
4.3 4.4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Spanning Two-Binary Options . . . . . . . . . . . . . . . . 4.2.1 Assumptions and Notation . . . . . . . . . . . . . . 4.2.2 Probability Density . . . . . . . . . . . . . . . . . . 4.2.3 Spanning 2 Binary options with dierent maturities 4.2.4 Numerical Example . . . . . . . . . . . . . . . . . . Spanning Two-Asset Payos . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendices
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54 56 56 57 58 61 65 73
75
B.1 Two-Binary prices in Black-Scholes model . . . . . . . . . . . . . . 77
5 A Jump Markov-functional Interest Rate Model with Fast Fourier Transform 81 5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 i
5.2 5.3 5.4 5.5 5.6 5.7 5.8
Interest Rate Economy . . . . Markov-functional model . . . 5.3.1 Calibration Procedure Markov process . . . . . . . . Numerical Techniques . . . . Numerics . . . . . . . . . . . Barrier Options . . . . . . . . Conclusions . . . . . . . . . .
Appendices
C.1 Correlation . . . . . . C.1.1 No Jumps . . . C.1.2 Adding Jumps . C.2 Characteristic function C.2.1 Without Jumps C.2.2 With Jumps . .
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86 87 88 92 95 104 110 112
113
115 115 117 119 119 119
6 Pricing and Hedging Correlation Swaps with a Two-Factor Model123 6.1 6.2 6.3 6.4 6.5 6.6
Introduction . . . . . . . . . . . Variance and Correlation Swaps Arbitrage-Free Pricing . . . . . A model for variance portfolios Numerics . . . . . . . . . . . . Conclusions . . . . . . . . . . .
Appendices
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124 126 131 134 143 145
147
D.1 (Xt , Yt )-dynamics under risk-neutral measure . . . . . . . . . . . . . 149 D.2 Process YT − Yt in terms of XT − Xt . . . . . . . . . . . . . . . . . 150
ii
Preface This thesis has been prepared as part of the requirements for the Ph.D. degree at the Department of Mathematical Sciences, Faculty of Science, University of Copenhagen, Denmark. The thesis consists of an introduction and ve chapters on dierent problems on pricing and hedging nancial derivatives. The ve chapters are written as academic papers, thus they are self-contained and they can be read independently.
Acknowledgements I would like to thank my advisor, Rolf Poulsen, for all the support, motivation, suggestions, ideas and discussions during the preparation of this thesis. From February to August 2013 I visited the Department of Mathematics at the University of Chicago. I would like to thank to Professor Roger Lee for this great experience and Hyomin Choi for co-authoring one of the papers in this thesis. I would like to thank to another co-author, Artem Tsvetkov, from ING bank, Amsterdam, for the opportunity he gave me to collaborate within the nance industry. Also, I would like to thank him all his availability during our research project. I would like to thank Elisa Nicolato and David Sloth Pedersen for all their interest in my research and for their invitation to visit the University of Aarhus. I am also grateful to all my o ce mates and colleagues at the Department of Mathematical Sciences for many good academic and non-academic moments. I am grateful to more people than I could possibly list here for their help, support and encouragement over the last years. Furthermore, I acknowledge the Portuguese Foundation for Science and Technology for supporting my Ph.D studies. At last, I thank my family and closest friends for their love and support during this work. André Manuel da Silva Ribeiro Copenhagen, May 2015
iii
Summary
Quantitative nance is concerned about applying mathematics to nancial markets. This thesis is a collection of essays that study dierent problems in this eld:
• How e cient are option price approximations to calibrate a stochastic volatility model? (Chapter 2) • How dierent is the discretely sampled realized variance from the continuously sampled realized variance? (Chapter 3) • How can we do static hedging for a payo with two assets? (Chapter 4) • Can we apply fast Fourier Transform methods to e ciently use interest rate Markov-functional models? Can we extend them to accommodate other types of dynamics? (Chapter 5) • How can we formulate a simple free-arbitrage model to price correlation swaps? (Chapter 6) A summary of the work presented in this thesis:
Approximation Behooves Calibration
In this paper we show that calibration based on an expansion approximation for option prices in the Heston stochastic volatility model gives stable, accurate, and fast results for S&P500-index option data over the period 2005 to 2009.
Discretely Sampled Variance Options: A Stochastic Approximation Approach In this paper, we expand Drimus and Farkas (2012) framework to price variance options on discretely sampled variance. We investigated the impact of their assumptions and we present an adjustment for their formula. Our adjustment provides a better approximation to price discretely sampled realized variance options under dierent market scenarios. v
Static Hedging for Two-Asset Options In this paper we derive a static spanning relation between an option written on two dierent assets and a continuum of two-asset binary options written on the same assets. Our Monte Carlo simulations show that under a continuous price dynamics discretized static hedges are possible with small hedging errors.
A Jump Markov-functional Interest Rate Model with Fast Fourier Transform This paper shows how the fast Fourier Transform (FFT) may be used to calibrate an interest rate Markov-functional (MF) model when the characteristic function of the Markov process is analytically known. We added a compound Poisson process to the standard Markov process to create a new MF model - the
Jump MF model. We show how can we apply the FFT methodology to calibrate the Jump MF model. Finally, we apply the model to price barrier caplets and oorlets.
Pricing and Hedging Correlation Swaps with a Two-Factor Model In this paper, we introduce a new model to price and hedge correlation swaps. The model is based on an approximation for the realized correlation payo, written as the ratio between two quantities, which can be replicated from traded assets in the variance swap markets. The model allow us to derive simple analytical formulas for the price of a correlation swap. Finally, numerical evaluations of correlation swap prices are presented.
Bibliography Drimus, Gabriel and Farkas, Walter. Valuation of options on discretely sampled variance: A general analytic approximation. Working paper available at ssrn.com, 2012.
vi
Sammenfatning Denne afhandling analyserer a edte nansielle aktiver. Mere speci kt prisfastsættelse, afdækning og modelkalibrering involverende disse. Endnu mere speci kt behandles følgende fem spørgsmål i separate kapitler: 1. Hvordan kan ekpansionsapproksimationer til optionspriser bruges til kalibrering af Hestons stokastiske volatilitetsmodel til markedsdata? 2. Hvor stor forskel er der på, om optioner skrives pådiskret eller kontinuert målt realiseret varians? 3. Virker statisk afdækning også, når der er to underliggende aktiver. 4. Hvilken øget eksibilitet giver Markov-funktional-modeller inden for rente(kurve)modellering? 5. Hvordan prisfastsættes korrelationsswapkontrakter påfornuftig vis?
vii
List of papers
This thesis is based on ve papers:
• Ribeiro, A., & Poulsen, R. (2013). Approximation Behooves Calibration. Quantitative Finance Letters, 1 (1), 36-40. • Ribeiro, A., Choi, Hyomin & Lee, Roger (2014), Discretely Sampled Variance Options: A Stochastic Approximation Approach • Ribeiro, A. (2014), Static Hedging for Two-Asset Options • Ribeiro, A. & Tsvetkov, Artem (2015), A Jump Markov-functional Interest Rate Model with Fast Fourier Transform • Ribeiro, A. (2015), Pricing and Hedging Correlation Swaps with a TwoFactor Model
ix
1 Introduction
In this introductory chapter we will review some of the theoretical results on arbitrage free option pricing used throughout this thesis. We will describe the nancial derivatives payos studied in this thesis and we will review some of the standard pricing models.
In the last three decades nancial derivatives have become increasingly important in the world economies. Nowadays, future contracts, options, swaps, forward contracts, and other derivatives are actively traded by nancial institutions, fund managers and companies both in exchanges-regulated markets and in over-thecounter markets. A derivative is a nancial instrument whose value "derives" from the value of other underlying variables. The underlying variable is often the price of a traded asset, such as, the stock price, the interest rate, foreign exchange, etc. But it can be other non traded variable such as temperature. This thesis will only be concerned with derivatives with a tradable variable underlying. Derivatives allow the economic agents ( nancial or non- nancial companies) to trade and manage their risks. These products bene t the entire society by providing better tools to share risks. Risk exposure could be reduced by transferring risks to the economic agents with more conditions to support them. Therefore, defaults and bankrupt became less probable and it promotes nancial and economic stability. Instead of reducing risks, derivatives can also increase them. This can happen whenever the agents decide to trade derivatives for speculation. The main problems needed to solve when derivatives are traded are the following:
• What is a fair price for the contract? 1
2
Introduction
• Suppose we have sold a derivative and thus, we have exposed ourselves to a certain amount of nancial risk. How do we hedge against this risk ? This thesis concerns about the problem of pricing and hedging some types of nancial derivatives.
Types of Derivatives Financial derivatives can be grouped in three main categories, depending on
payo ) in the future: futures and forward contracts, options and swaps. We say that the owner of an option contract exercises it, when she/he their payment (or
requires that the nancial transaction speci ed by the contract is to be carried out immediately between the two parties. If the exercise can only be made in a speci c date in the future, options are called European. In the other hand, owners of
American -style options can exercise at any time before the derivative expires.
In this thesis we will only study European derivatives. We let St denote the price of the relevant asset that the derivative "derives". We called it the underlying security. Table 1.1 gives a list of the names of some option payos used throughout this thesis. The underlying assets used in nancial derivatives can belong to dierent types: 1. Stocks 2. Currencies (e.g., EUR-USD) 3. Interest rates (e.g., euribor, LIBOR, etc) - in this category we can include derivatives on bonds, notes and other government debt instruments. 4. Indexes (e.g., S&P 500, FT-SE100,etc) 5. Commodities (e.g., livestock, energy, metals,etc) - these underlying security are not, as the last ones, nancial assets. They are goods that can be physically purshased and stored. In this thesis we will deal not only with stocks, interest rates and indexes derivatives, but also
statistical derivatives. These are derivatives with a payo
that depends on some statistical quantity that can be computed over a period of time. Examples of such quantities are presented in Table 1.2. Variance swaps and variance options will be studied in Chapter 2 while correlation swaps will be studied in Chapter 6.
Introduction
3
Option Name Digital Option Call Option Put Option Barrier Options Asian Options
Payo at T 1S >K T
max(ST − K, 0) max(K − ST , 0) max(S T − K, 0)1max0≤t≤T St 20 the absolute relative error is lower than 2.5%; and lower than 10% when the payo is near at-the-money (K = 3.5). We also tested all the previously discussed spanning formulas (examples 1 to 5), for dierent values of x and y (not shown). As we expected, the discretized spanning formulas were very accurate as N increases.
Figure 4.6: Relative error for a discrete hedging of a payo V (x, y) = (x − y − K) as a function of the number of replicating instruments N . We consider x = 5, y = 1. The dierent curves correspond dierent values of strike.
Static Hedging for Two-Asset Options
73
4.4 Conclusion Dynamic hedging has been used for hedging multi-asset derivatives. However, the performance of this strategy can deteriorate dramatically as we increase the maturity of the nancial products since the transaction costs can be high. In this paper we propose a new approach for hedging nancial derivatives for a wider class of claims with two assets. Our approach assumes a continuum of two-binary options with dierent strikes. We started by extending Carr and Wu (2013) framework by showing that we can replicate two-binary options with a certain maturity with a portfolio of two-binary options with dierent strikes and shorter maturity. To achieve this result we derive a relation which connects two-binary options with the joint probability density of prices. We provided some Monte Carlo simulations that showed that for dierent scenarios such replications can be achieved with a
10 × 10 grid of two-binary options. This result constitute a practical method to build two-binary options with dierent maturities. In the second part of the paper we provided a new spanning formula for a large class of two-asset nancial instruments which relates a two-asset payo with a portfolio of binary options and two-binary options. Although for many complex payos this spanning formula may require a large number of components, it constitutes a theoretical relation that provide a new way to understand the risks associated with this products. The framework can be applied for risk management problems. Computing two-binary options can be easier than computing more complex payos and thus allow us to compute more complex payos. Future research can be done in dierent directions. One would be to use our new spanning formula to nd new pricing approximation formulas for two-asset derivatives. Another extension would be to decrease the number of components needed in our hedging portfolio by improving the integral discretization method using, for example, a two dimensional Gauss-Hermite quadrature.
Appendices
75
B.1 Two-Binary prices in Black-Scholes model We consider a model with two underlying price processes S 1 and S 2 satisfying under Q the following system of stochastic dierential equations:
dS 1 = rS 1 dt + σ1 S 1
p 1 − ρ2 dW1 + σ1 ρS 1 dW 2
dS 2 = rS 2 dt + σ2 S 2 dW 2 where W 1 and W 2 are independent Brownian motions. Using the Itô formula on the function f (t, x) = log(x) we get
d log S
1
d log S 2
p 1 2 r − σ1 dt + σ1 1 − ρ2 dW1 + σ1 ρdW 2 = 2 1 2 = r − σ2 dt + σ2 dW 2 . 2
Integrating between t and T and rearranging the terms we can write
log
ST1 St1
− r − 21 σ12 (T − t)
=
σ1 log
ST2 St2
− r − 21 σ22 (T − t)
p 1 − ρ2 (W 1 (T ) − W 1 (t)) + ρ(W 2 (T ) − W 2 (t))
= W 2 (T ) − W 2 (t) .
σ2
We can de ne two new variables, x and y as
log
ST1 St1
log
ST2 St2
x: = y: =
− r − 12 σ12 (T − t) √ σ1 T − t − r − 12 σ22 (T − t) √ . σ2 T − t
From the previous relations we can write the following identities
x
dist
=
p 1 − ρ2 1 + ρ2
y
dist
1 .
=
77
78
Static Hedging for Two-Asset Options dist
with = means equality in distribution and where 1 and 2 are two independent standard normal distributed variables with zero mean and unitary variance. Therefore the joint probability density of x, y is given by a bivariate normal distribution,
n(x, y) :=
2 2 1 − x +y −2ρxy p e 2(1−ρ2 ) . 2π 1 − ρ2
In order to nd the joint probability density of ST1 and ST2 , g(ST1 , ST2 ), we do a change of variable,
∂x ∂y dS 1 dS 2 ∂ST1 ∂ST2 T T 1 1 √ √ = × n(x, y)dST1 dST2 1 2 σ1 ST T − t σ2 ST T − t = g(ST1 , ST2 )dST1 dST2 .
n(x, y)dxdy = n(x, y)
From here we can write,
B(St1 , St2 , t; K1 , K2 , T ; Θ) = e−r(T −t) N (d1 , d2 ) with
Z
∞
Z
(B.1)
∞
n(x, y)dxdy
N (a, b) := a
b
and
d1 d2
1 2 1 − r − σ (T − t) log K 1 1 2 St √ = σ1 T − t K2 log S 2 − r − 12 σ22 (T − t) t √ . = σ2 T − t
Observe that we can easily recover the probability density from the price by Theorem 1,
er(T −t)
∂ 1 1 √ √ B(St1 , St2 , t; K1 , K2 , T ; Θ) = ×n (d1 , d2 ) . ∂K1 ∂K2 σ1 K1 T − t σ2 K2 T − t
We can either compete the following partial derivative
∂ ∂S 1 ∂S 2
B(St1 , St2 , t; K1 , K2 , T ; Θ) =
exp(−r(T − t)) 1 √ √ × n (d1 , d2 ) . 1 2 σ1 S T − t σ2 S T − t
Bibliography
Aditya
Bagaria.
FX
Strategist.
Technical report available at
https://research-and-analytics.csfb.com, May 2005. Gurdip Bakshi and Dilip Madan. Spanning and Derivative-Security Valuation.
Journal of Financial Economics, 55(5):205238, 2000. D.T. Breeden and R. H. Litzenberger. Prices of State-Contingent Claims Implicit in Option Prices.
Journal of Business, 51(4):621651, 1978.
P. Carr and A. Chou. Hedging Complex Barrier Options.
Working paper available
at www.math.nyu.edu/research/carrp/papers/pdf/multipl3.pdf, 1997. Peter Carr and Liuren Wu. Static Hedging of Standard Options.
Financial Econometrics, 2013. doi: 10.1093/jj nec/nbs014. Bruno Dupire. Pricing with a Smile.
Journal of
Risk Magazine, pages 1820, 1994.
C. Leuschke and R. Summerbell. The Rise of Multi-Currency Options.
tive Finance, 4(6):327335, 2010.
Quantita-
Morten Nalholm and Rolf Poulsen. Static Hedging of Barrier Options Under General Asset Dynamics: Uni cation and Application.
Journal of Derivatives,
13(4):4660, 2006. Rolf Poulsen. Barrier Options and Their Static hedges: Simple Derivations and Extensions.
Quantitative Finance, 6(4):327335, 2006.
Johannes Siven and Rolf Poulsen. The Long and Short of Static Hedging with Frictions.
Wilmott Magazine, 38:6267, 2008.
Johannes Siven and Rolf Poulsen. Auto-Static for the People: Risk-Minimizing Hedges of Barrier Options. Review of Derivatives Research, 12(3):193211, 2009. 79
5 A Jump Markov-functional Interest Rate Model with Fast Fourier Transform
André Ribeiro and Artem Tsvetkov
Abstract:
1
This paper shows how the fast Fourier Transform (FFT)
may be used to calibrate an interest rate Markov-functional (MF) model when the characteristic function of the Markov process is analytically known. We add a compound Poisson process to the Gaussian Markov process to create a new MF model - the
Jump MF model. We show how
we can apply the FFT methodology to calibrate the Jump MF model. We found that the Jump MF model has small calibration errors and it can include dierent auto-correlation structures for LIBOR's. Finally, we apply the model to price barrier caplets and oorlets and we show how price is aected with dierent jump rates.
Keywords:
Fast Fourier Transform; Interest rate models; LIBOR for-
ward rates; Barrier caplet/ oorlet; Markov-functional Model
Head of MRM Trading Quantitative Analytics, FX, Credit and CVA at ING Bank, Amsterdam, Netherlands -
[email protected] . 1
81
82 A Jump Markov-functional Interest Rate Model with Fast Fourier Transform
5.1 Introduction One of the most demanding problems in option pricing theory is to price and to hedge xed income derivatives. The real challenge in modelling interest rates is the existence of a
yield curve, i.e., a relation between the level of the interest
rates (or the cost of borrowing) and the time to maturity of the debt for a given borrower (also known as the
term ).
The yield curve plays a central role in the economy and re ects expectations of market participants about future changes in interest rates. A yield curve contains information about implied interest rates over future periods of time. These implied future interest rates are referred to as forward interest rates. Forward interest rates over dierent times can be plotted as a
forward curve. Fixed income instruments
typically depend on a segment of the forward curve rather than a single point. Thus, pricing such instruments requires a model that describes the stochastic time evolution of the entire forward curve.
Dierent approaches to price interest rate derivatives Over the last three decades, a large number of models have been used to price interest rates products based on simulations of the yield curve - they are known as
term structure models. They are based on dierent choices of state
variables parameterizing the curve, number of dynamic factors, volatility smile characteristics, etc. They can be classi ed in two categories, depending on which framework are they following:
• Short term interest rate models, in which the stochastic state variable is taken to be the instantaneous forward rate. Historically, these were the earliest successful term structure models (e.g., Hull and White (1990),Vasicek (1977) and Cheyette (1996)). • LIBOR market models, in which the stochastic state variable is the entire forward curve represented as a collection of benchmark LIBOR forward rates. The most popular among practitioners were rst presented by Brace et al. (1997) and Jamshidian (1997). Miltersen et al. (1997) developed a log normal term structure model for forward rates that gives closed formulas for many interest rate derivatives. Short term rate models are computationally e cient and they can be imple-
A Jump Markov-functional Interest Rate Model with Fast Fourier Transform 83 mented using lattice methods. However, they are not exible enough to t the implied volatility smiles and correlations between various rates. In this class of models, calibration is performed for each product alone. In the other hand, LIBOR market models developed by Brace et al. (1997) and Jamshidian (1997) can simulate the entire forward curve. However, an accurate implementation has to be done with a high dimension Monte Carlo method, which make their calibration computationally ine cient. Moreover, LIBOR market models have no smiles embedded in the model, meaning that the models need to be extended to account for them, like it is done for example in SABR-LMM model. Hagan et al. (2002) introduced a model with stochastic volatility, the SABR model, which attempts to capture the volatility smile in derivatives markets. Unlike the LIBOR market model, the SABR model can only account for the individual forward rates. Combining LIBOR market models and SABR models would take advantage of best part of each of the two benchmark models and provide a consistent model for pricing, hedging and risk management of the entire portfolio (Rebonato and White (2009)). Unfortunately the combined model is overwhelmingly complex as it substantially enlarges covariance structure (forward rates/forward rates, forward rates/volatilities and volatilities/volatilities). Thus, the calibration and general hedging processes are computationally di cult. Another class of models that can be calibrated to the observed market rates are known as the Markov-functional models and were developed by Hunt et al. (2000). Markov-functional models combine the computational e ciency of short/forward rate models with exible calibration tting of LIBOR market models. The de ning idea of a Markov-functional model is that each rate and discount factor can be written as a functional of some low-dimensional Markovian driving process (usually one or two). This low dimensionality allows for an e cient implementation of the model on a lattice, which in turn provides a fast pricing computation, when compared with the traditional LIBOR market models. The classic short interest rate models are examples of one dimensional MF models. An alternative to the MF model is Quasi-Gaussian model Andersen and Piterbarg (2010). This model has also a low number of driving factors and can be tted to the smile structure. However, Quasi-Gaussian model cannot be easily calibrated when the driving factors do not follow Brownian motion. We show further in this paper how this limitation can be overcome in Markov Functional
84 A Jump Markov-functional Interest Rate Model with Fast Fourier Transform model.
Markov-functional models need improvement There are number of factors which make the pricing of interest rate derivatives particularly di cult.
• Interest rates have the term structure of forward and discount rates, which model should de nitely match. • Forward rates have a correlation term structure. This is the point where short rate models for example do not provide enough exibility. • Swaption or cap volatility has smile. This factor cannot be reproduced in the standard LIBOR and swap market models. • As in other asset classes, volatility has own smile dynamics, i.e. dependence of smile on the rate movements. In case of interest rates, the latter aspect has never been investigated before, to the best of own knowledge. This is also one of the main motivation for the current paper. Current Markov-functional models can satisfy three out of four points above, while still keeping the number of stochastic factors low. We extend Markovfunctional model to satisfy also the fourth point by applying more general underlying processes. Standard MF models are based on diusion, and explicitly rule out the possibility of jumps. However, there is strong statistical evidence that jumps play an important role in the dynamics of interest rates (Johannes (2004)). A natural way to introduce jumps in the MF model would be to change the dynamics of the embodied Markov process. However, this would aect the calibration procedure e ciency that depends on Gaussian assumptions for the Markov process. Therefore, a better approach to e ciently calibrate a MF model for a larger class of Markov processes is needed. The main contributions of this paper can be summarized as follows:
• First, we propose a new numerical method based on the Fourier transform approach to calibrate and to price interest rates derivatives in a MF model. This method makes the computations faster and it makes possible to include dierent stochastic processes underlying MF model; • Second, we introduce a new model with jumps and investigate the eect on derivative pricing, which to our best knowledge has not been studied before.
A Jump Markov-functional Interest Rate Model with Fast Fourier Transform 85 The paper is organized as follows. Section 5.2 introduces notation and reviews the LIBOR market model. Section 5.3 develops a general one-dimensional MF model, for any Markovian process with known characteristic function. In this section the calibration methodology using the FFT algorithm is described. Also in this section the Jump MF is described and the auto-correlation structure is discussed. In section 5.4 we compare the errors from the calibration of a MF model using the Fourier methodology with a quadrature methodology. Finally in Section 5.5 we compare the standard MF model with the Jump MF model for pricing barrier caplets and oorlets.
86 A Jump Markov-functional Interest Rate Model with Fast Fourier Transform
5.2 Interest Rate Economy In this section, we will describe the interest rate economy and the general ideas under which our models are built upon. We will assume a simpli ed interest market with n + 1 liquid zero-coupon bonds, each with dierent maturities, today = T0 < T1 < · · · < Tn < Tn+1 . Let DtT denote the price of a zero-coupon bond at time t, such that t ≤ T , which pays a unit amount at maturity T . From the zero-coupon bond prices we can compute the LIBOR forward rates. The
LIBOR forward rate contracted at time
t, for the period [Ti , Ti+1 ] with t ≤ Ti , is the interest rate earned in a money-market account for that period and it is de ned as Lit :=
1 DtTi − DtTi+1 , αi DtTi+1
i = 0, 1, ..., n .
(5.1)
where αi := Ti+1 −Ti is de ned as the accrual factor. The graph of Lit as a function of Ti is the forward curve. Our problem consists of how to price payos that depend on the LIBOR rate. As a simple example, an European caplet with maturity Ti and strike K , has a payo function that pays at date Ti , for any i = 1, ..., n + 1,
αi (LiTi−1 − K)+ . A more sophisticated payo example would be an European barrier caplet. An European barrier caplet with barrier level B , strike K and maturity Ti . This caplet would pay at Ti the value
αi (LiTi−1 − K)+ 1max0≤s≤T Lis 15, where the standard deviation of the absolute error is small when compared with its mean. For ρ = 0 the error is null as we expected from equation (6.3). From Figure 6.1, the mean absolute error when n > 15, is never greater than 0.01 while the standard deviation is always bellow 0.005. This only happens in the extreme case when all log-returns are perfectly correlated or perfectly uncorrelated, |ρ| = 1. For lower values of |ρ| the errors are lower. The error is smaller as we increase the number of components in our index. Moreover, our experiments showed that when n is su cient large (n > 15), and when the dierences between the square of variance of each components are lower than 10%, the mean absolute error can be very low of the order of 1e − 4 with a standard deviation of the same order (these results are not shown). Therefore our experiments show that ρ˜I is a good approximation for ρI when we are dealing with an index with more than 15 components and when the variances of index components are similar. In this paper we are going to use this approximation to study how to price and how to hedge correlation swaps.
6.3 Arbitrage-Free Pricing In this section we will study the arbitrage-free prices of variance swaps. Then, we ˜ A . Finally, we will discuss apply those results to price the payos RV D and RV how can we price correlation swaps.
Variance portfolios pricing We assume that the market is arbitrage free and prices of all traded assets are given by expectations with respect to an equivalent pricing measure Q. Under the measure Q, the variance
spot swap rate at any time s, such that s ∈ [t, T ], is given
by
X , VtTX (s) = Es RVt,T
(6.4)
which corresponds to the price of a variance payo at time s. Es [·] denotes the expectation under measure Q, conditioned on the information at time s. We can D use the equation (6.4) to price the dispersion payo RVt,T as a function of the spot
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Pricing and Hedging Correlation Swaps with a Two-Factor Model
variance rate of the index and its components,
D D Vt,T (s) := Es RVt,T n X I k = Vt,T (s) − wk2 Vt,T (s) .
(6.5)
k=1
˜ A can be written as Similarly, the price of payo RV t,T h i A ˜ A (s) := Es RV Vt,T t,T =
n X
k (s) wk Vt,T
−
k=1
n X
k (s) wk2 Vt,T
(6.6)
k=1
Due to the development of listed option markets the implied volatilities for dierent stocks have become observable. Therefore, the variance spot rates can be obtained and the prices for portfolios that depend linearly on variance (as the ones with payo RV D and RV A ) can be found explicitly. Before we discuss correlation swap prices, we should do some comments on arbitrage relations between variance spot swap rates at dierent times. From the de nition of realized variance given by equation (6.1) and from the de nition of variance spot rate given by (6.4), we can write
VtTX (s) =
s−t T −s X RVtsX + V (s) . T −t T − t sT
(6.7)
Equation (6.7) holds for all s ∈ [t, T ] if no arbitrage opportunities are available. X For example, consider that the realized variance payo RVt,T was bought at time
t for a price VtTX (t). If at time s the buyer wants to sell her position, she should sell it for the price VtTX (s), given by equation (6.7).
Correlation Swaps Although implied volatilities are quoted in nancial markets, implied correlations are not observable. Therefore we need to have a methodology to compute it from the information available in the market. The free-arbitrage price of the correlation swap value at time s ∈ [t, T ] is given by
Ct,T (s) = Es
RVtTD RVtTA
.
Pricing and Hedging Correlation Swaps with a Two-Factor Model
133
Thus, as expected, the prices of a correlation swap are not given by the ratio of prices between the right hand side of equations (6.5) and (6.6),
Ct,T (s) 6=
D Vt,T (s) . A Vt,T (s)
Instead, we will need to be able to compute the following expectation:
Ct,T (s) := Es
D Vt,T (T ) A Vt,T (T )
! .
(6.8)
D A This requires a model that allows us to model the dynamics for Vt,T (s) and Vt,T (s)
such that s ∈ [t, T ]. In the next section we explore the problem of nding an explicit formula to price correlation swaps.
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Pricing and Hedging Correlation Swaps with a Two-Factor Model
6.4 A model for variance portfolios Our purpose in this section is to introduce a simpli ed model which could be used to price derivatives on portfolios A and D, such as Ct,T (s) given by equation (6.8). We will start from simple assumptions by modeling directly the prices VtTA (s) and
VtTD (s), for all s ∈ [t, T ]. T will be chosen such that it has the same maturity as the derivative we want to price. In our model, the only underlying tradable assets are portfolios A and D, which at any point in time, are a linear mixture between past realized quantities, RV A and RV D , respectively, and future implied portfolios prices as shown by equation (6.7). This approach does not demand the complexity of other methods. However, it does not address the issue of possible arbitrage with other derivative instruments. Therefore, without requiring to model the index and stocks prices processes, model consistency with vanilla option prices can be ignored in this framework. In its simplest version, this approach will allows us to study correlation swap prices based on closed formulas that depend in a small number of intuitive parameters, similarly to Bossu (2005) approach. In a less mathematical tractable version, it will improve Bossu (2005) model by producing correlation prices that can only be between -1 and 1.
Model Setup Let us consider a market with only two tradable assets: portfolios A and D, with its realized payos that depend on time interval [0, T ]. We make the following economic assumptions:
• There is no bank account; • No arbitrage; • In nite liquidity; • No transaction costs; • Unlimited short-selling; • Continuous ow of information. Consider a bivariate process on a ltered probability space (Ω, Ft , P) such that
Pricing and Hedging Correlation Swaps with a Two-Factor Model
135
the components of the R2 -valued stochastic processes (Xt , Yt ) satis es: X dXt = µX t (Xt , Yt )dt + σt dWt
dYt = µYt (Xt , Yt )dt + σtY dBt where W and B are dependent Brownian motions with the property that in nitesimal increments of them at time t are dependent Gaussian random variables with correlation coe cient ρ,
i.e., E[dWt dBt ] = ρdt. We will also assume that the
coe cients satisfy the necessary conditions for the existence and uniqueness of solution of a system of stochastic dierential equations. D A (t) denote the prices at time t of portfolios A (t) and VtD := V0T Let VtA := V0T
and D respectively, for a payo that depends on the period [0, T ]. In our model, the dynamics of VtA and VtD under the probability measure P, are given by the following equations:
VtA = VsA eXt −Xs VtD = VtA f (Yt − Ys ) ,
(6.9) (6.10)
where 0 ≤ s ≤ t ≤ T . We will assume that f : R → [−1, 1] is a twice dierentiable function. Note that VsA and VsD are the prices at time s of portfolios A and D which can be observed in the market at that time (or obtained using a replication portfolio of put and call options as described by Breeden and Litzenberger (1978)). From equation 6.10, if we chose t = s, then the following condition needs to be sati ed:
f (0) =
VsD := c . VsA
VTA and VTD are the prices of portfolios A and D, respectively at maturity. These A D prices coincide with the realized quantities RV0,T and RV0,T respectively. By the fundamental theorem of asset pricing, the arbitrage price of a claim is given by the expected payo under the neutral-risk probability measure Q. Therefore we need to right the dynamics given by equations (6.9) and (6.10) under the risk-neutral probability measure Q. Under measure Q portfolio prices A and D are martingales for all 0 ≤ s ≤ t ≤ T , which imply the following relations VsA = VsA Es [eXt −Xs ]
(6.11)
VsD = VsA Es [eXt −Xs f (Yt − Ys )] .
(6.12)
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Pricing and Hedging Correlation Swaps with a Two-Factor Model
We can show how to derive the dynamics for X and Y under the risk-neutral measure from conditions imposed by equations (6.11) and (6.12) (see Appendix D.1). Therefore, under the risk-neutral measure Q, the dynamics for X and Y are written as
1 ¯t dXt = − (σtX )2 dt + σtX dW 2 1 Y 2 f 00 (Yt ) X Y ¯t , dYt = − (σt ) 0 − ρσt σt dt + σtY dB 2 f (Yt )
(6.13) (6.14)
¯ t and B ¯t are two Wiener processes under the probability measure Q such where W ¯ t dW ¯ t ] = ρdt. Here we are using Lagrange's notation to denote the rst that E[dB and second derivative with respect to Y . Note that a solution under the riskneutral measure for (Xt , Yt ) is not guaranteed and it will be highly dependent on the properties of function f . We will now follow to discuss possible examples for
σtX , σtY and f (Yt ).
Specifying σtX , σtY and f (Yt ) As t approaches maturity T , the uncertainty on prices VtA and VtD diminishes. To reproduce this behavior, the volatility parameters σtX and σtY should collapse to zero between the inception and the maturity. We are going to chose the simplest collapsing functions where
T −t T T −t = κY T
σtX = κX σtY
where κX and κY are, respectively, the volatility parameters of processes X and Y at inception. To choose function f there are some desirable conditions to satisfy:
• −1 < f (y) < 1, for all y - this is the condition required such that correlation prices will be between −1 and 1; • f 00 (y)/f 0 (y) should be such that a solution of equations (6.14) and (6.13) is possible and unique (existence and uniqueness conditions for stochastic dierential equations). • f (0) = c. If we are looking for a mathematical tractable model, these three conditions are
Pricing and Hedging Correlation Swaps with a Two-Factor Model
137
very di cult to achieve. We are going to present the three functions, each de ning a dierent model - that we will study throughout this paper:
2 Logistic function - f (y) = 2/(1 + e−y+˜c ) − 1 with c˜ = log((1 − c)/(1 + c) This function clearly satis es the third condition, f (0) = c. Furthermore, it also satis es the rst condition, −1 < f (y) < 1 for all values of y such that y ∈ R. The ratio between the second and rst derivative of function f , is given by ec˜ − ey f 00 (y) = . f 0 (y) ec˜ + ey It can be shown that a solution for our stochastic dierential equation exists and is unique. However no straightforward closed solution exist for it and we will need to use numerical methods, such us Monte Carlo method, to nd the price of derivatives. In spite of this weakness, this model is free of arbitrage2 since it does not generate absolute correlation prices above 1, as the following more tractable models. For this model, we will discuss in the next section its correlation prices using Monte Carlo simulations.
2 Linear function - f (y) = y + c - For this function, f (0) = c is satis ed. However it does violate the rst property that we were looking for. In the other hand, the ratio, f 00 (y) = 0, f 0 (y) is simple and it will make the model mathematically tractable. This will allows us to derive closed-formulas for the prices of correlation swaps and other simple derivatives on portfolios D and A.
2 Exponential function - f (y) = cey - Again, f (0) = c is satis ed and, as the linear function, it does violate the rst property. Another problem is that this function will not allow to have paths such that the sign of the process is changed. This is because the exponential function is always positive for all y . In the other hand, the ratio f 00 (y) =1 f 0 (y) is simple and tractable like in the linear function. This property will allows us to derive closed-formulas for correlation prices. Next, we will apply our models to price a call option on portfolio A, a call option on portfolio D and the correlation. 2
Remind that we are ignoring plain vanilla options markets
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Pricing and Hedging Correlation Swaps with a Two-Factor Model
Application I: Call option on portfolio A As an application of the model described, we derive the price of an European call option on VTA with strike K , with payo given by
g(VTA ) = (VTA − K)+ . The price is independent of our choices for the function f . In the risk-neutral world the dynamics of VTA can be written as RT
VTA = VsA e
s 1
= VsA e− 2
dXu RT s
X )2 du+ (σu
RT s
X dW ¯u σu
Because the following identity holds,
Z s
T
T −u T
2 du =
T (T − s)3 , 3 T3
we can write the dynamics of VTA as T
3 2 (T −s) +κ X T3
VTA = VsA e− 6 κX
RT s
T −u ¯ dWu T
. 3
Therefore log(VTA ) follows a normal distribution with mean log(VsA ) − T6 κ2X (TT−s) 3
T 2 (T −s)3 κ . As a consequence the closed-formula to price an Euro3 X T3 A option on VT is identical to Black-Scholes formula and is given by
and variance pean call
VsA N (d1 ) − KN (d2 ) where N (·) is the standard normal cumulative distribution function and
d1
d2
log(VsA /K) + 21 T3 κ2X (TT−s) 3 q = 3 T 2 (T −s) κ 3 X T3 r T 2 (T − s)3 = d1 − . κ 3 X T3
3
It is interesting to compare the dierences in prices given by our model and Black Scholes. Suppose that the price VsA is constant. If we want to compare the prices of European call options on A in our model with Black-Scholes model we should
Pricing and Hedging Correlation Swaps with a Two-Factor Model
139
compare the variance of the log VTA along time. In Black-Scholes model we assume the variance would be given by 13 κ2X (T − s) in order to have the same variance at inception than our model. In our model the variance is given by
T 2 (T −s)3 κ . 3 X T3
The
ratio between the variance in our model with Black-Scholes model is given by
(T − s)2 . T2 At inception, s = 0, the variance of log VTA is by de nition the same in both models. This would imply the same prices at inception. As s increases and approaches T , the ratio decreases. This implies that prices in our model would be lower than the ones given by Black-Scholes model.
Application II: Call option on portfolio D To price a call option on portfolio D we need to compute the following expectation under the risk-neutral probability measure Q
Es [(VsA eXT −Xs f (YT − Ys ) − K)+ ] . To compute it we will need to have the joint probability distribution for (XT , YT ). Instead of computing the overall expectation we will start to compute the conditioned expectation
Es [(VsA eXT −Xs f (YT − Ys ) − K)+ |XT − Xs ] , and then we compute the expectation over all values of X . It can be shown (see Appendix D.2) that the process YT − Ys can be written as
YT − Ys
T (T − s)3 p 1 + 1 − ρ2 κY = − κ2X R + ρκX κY 2 3 T3 κY ρ + (XT − Xs ) , κX
Z s
T
T −u ¯ dBu T
where R = 1 when f (y) = cey and R = 0 when f (y) = y + c. Hence, the distribution of YT − Ys conditioned on XT − Xs follows a normal distribution with mean
my = −
T (T − s)3 κY ρ 1 2 κX R + ρκX κY + (XT − Xs ) 2 3 T3 κX
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Pricing and Hedging Correlation Swaps with a Two-Factor Model
and variance
T (T − s)3 . 3 T3 Let us consider the pricing under the linear and the exponential model: 2 When f (y) = y + c, the conditioned price is given by a formula similar to Bachelier formula for European option prices. Hence, we can write vy = (1 − ρ2 )κ2Y
E [(V A eXT −Xs (YT − Ys ) − K)+ |XT − Xs ] Z s∞ s √ −x2 1 A A √ e− 2 (m V + c + v V x − K) = Y Y T T √ 2π (K−mY VTA −c)/ vY VTA A my VT + c − K mY VTA + c − K √ A + vy φ , = (my VT + c − K)N √ √ vy VTA vy VTA √ 2 where φ(x) = 1/ 2πe−x /2 . Recall that my also depends on XT − Xs . Therefore, the price of a call option on portfolio D is given by the following integral
Es [(VsA eXT −Xs (YT − Ys ) − K)+ ] Z +∞ my VsA eu + c − K my VsA eu + c − K √ A u (my Vs e + c − K)N = + vy φ p(u)du , √ √ vy VsA eu vy VsA eu −∞ (6.15) where p(u) is the probability density distribution for XT − Xs given by a normal distribution with mean
(T − s)3 T mx = − κ2X 6 T3
and variance
T 2 (T − s)3 κ . 3 X T3 Equation (6.15) shows that this model provides a price for the call option on portfolio D and it can be obtained using a numerical integration method. 2 When f (y) = cey , the conditioned price is given by a formula similar to Black-Scholes formula for European option prices. vx =
Es [(VsA eXT −Xs (YT − Ys ) − K)+ |XT − Xs ] = cVTA emy N (d1 ) − KN (d2 )
Pricing and Hedging Correlation Swaps with a Two-Factor Model where
d1 =
log
cVTA K
141
+ my + vy √ vy
√
vy . Under f (y) = cey , the price of the European call option on portfolio D is given by and d2 = d1 −
Es [(cVsA eXT −Xs eYT −Ys − K)+ ] Z +∞ = {cVsA eu emy N (d1 ) − KN (d2 )}p(u)du −∞
where p(u) is the probability density distribution for XT − Xs . Again, the call option on portfolio D can be found using a numerical integration method.
Application III: Correlation Swaps The price of a correlation payo is given by the following expectation
C0,T (s) = Es [f (YT − Ys )] .
(6.16)
Since the probability density for YT − Ys is known for functions f (y) = y + c and
f (y) = cey , to nd the prices of correlation swaps is straightforward. For example, when f (y) = y + c, the price of a correlation payo becomes VsD C0,T (s) = Es YT − Ys + A Vs VsD T (T − s)3 = − 2ρκ κ , X Y VsA 6 T3
(6.17)
where we use the results for mx and my to obtain the nal result. Similarly, when
f (y) = cey , the price of a correlation payo becomes C0,T (s) = Es
VsD YT −Ys e VsA
VsD my − 1 vy 2 e VsA 3 (T −s)3 VsD −2ρκX κY T6 (T −s) − 12 (1−ρ2 )κ2Y T3 T3 T3 = e . VsA
=
(6.18)
In summary, we derived two closed-formula solutions to price correlation payos. We should be aware that they do not constitute a realistic model since they allow
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Pricing and Hedging Correlation Swaps with a Two-Factor Model
to generate meaningless values for correlation prices.
Delta-Hedging Strategy We now study a trading strategy that permits to dynamically replicate the correlation swap. We are going to look for the delta hedging strategy that will permit to oset changes in C0,T (s), for all s ∈ [0, T ], whenever the prices of portfolio A and D change. This will be done by taking the opposite position with
δsA units of the portfolio A and in δsD units of the portfolio D. A change in the correlation payo due to a change in VsD and VsA is approximately given by ∆C0,T (s) ≈
∂C0,T (s) ∂C0,T (s) ∆VsD + ∆VsA . D A ∂Vs ∂Vs
(6.19)
When f (y) = y + c, from equation (6.17), we have
∆C0,T (s) ≈
VsD 1 D ∆VsA . ∆V − s VsA (VsA )2
(6.20)
Therefore, to oset the changes in the price of a correlation swap the trader should have a position with δsD = −1/VsA units of portfolio A and δsA = VsD /(VsA )2 of portfolio D. Each day the price of such strategy will cost
costs = δsD VsD + δsA VsA 1 VD = − A VsD + sA 2 VsA = 0 . Vs (Vs ) The daily cost of such strategy is null. The same conclusion is valid for the exponential model when we assume f (y) = cey . Under this model if we compute the δ 's for it using the partial derivatives in equation (6.18) we get
δsD = − and
δsA = −
3 (T −s)3 ∂C0,T (s) 1 −2ρκX κY T6 (T −s) − 12 (1−ρ2 )κ2Y T3 T3 T3 = − e ∂VsD VsA
3 (T −s)3 ∂C0,T (s) VsD −2ρκX κY T6 (T −s) − 12 (1−ρ2 )κ2Y T3 T3 T3 = e ∂VsA (VsA )2
it is straightforward to verify that a portfolio with δsD units of portfolio D and
δsA of portfolio A is costless. Similar costless strategies were obtained by Bossu
Pricing and Hedging Correlation Swaps with a Two-Factor Model
143
(2005) for his model. Because we do not have closed-formula solutions for the logistic model, we will show an approximation for it. To do this, we start by expanding the logistic function, f (x), around the point x = 0,
f (x) =
2 ec˜ − 1 + 2 x + O(x2 ) . 1 + ec˜ (ec˜ + 1)2
Replacing in the previous equation c˜ by its de nition, x by XT −Xs and computing the expectations, we can write
VD C0,T (s) = sA − Vs
VsD VsA
2
Es [XT − Xs ] − Es [XT − Xs ] + Es [O((XT − Xs )2 )] .
Therefore we can approximate the deltas as
δsD ≈ − and
δsA ≈ −
∂C0,T (s) 1 VsD = − − 2 (...) ∂VsD VsA (VsA )2
∂C0,T (s) VsD (VsD )2 = + 2 (...) . ∂VsA (VsA )2 (VsA )3
Building a portfolio with δsA units of portfolio A and δsD units of portfolio D will be costless as the previous models.
6.5 Numerics In this section we are going to compare the prices of correlation swaps under the three dierent models. For both the linear and exponential models, we have closed-form solutions as shown in the previous section. For the logistic model we will use a Monte Carlo Euler discretization scheme. Each path we discretize it in 1000 steps and we performed 10000 simulations. For all the models we start our experiments by choosing the following parameters
V0D = 0.3 , V0D = 0.35 , κX = 0.5 , κY = 1 , ρ = 0.9 . From the two rst parameters we have V0D /V0A = 0.8571. In Figure 6.2 we represent three paths for VtD /VtA , with T = 2, generated by a Monte Carlo simulation with the same Brownian processes. The rst observation we make is that the
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Pricing and Hedging Correlation Swaps with a Two-Factor Model
Figure 6.2: Three paths for VtD /VtA generated by a Monte Carlo simulation. linear and the exponential model can generate ratios greater than one while the logistic model does not move too much around the initial value 0.8571. This lower volatility in the logistic model is only due to the choices of our parameters as we will see later in this section. The oscillations of paths are very similar between the linear and the exponential formulas as expected by the dynamics of Yt which have the same variance for both of the models. In the other hand, the variance for the
Yt process in the logistic model is lower than the variance for the other models. A natural question that arises is how dierent are correlation prices between the three models. In Figure 6.3 we study the price of a correlation payo along time for our three models. When the correlation between X and Y is high, ρ = 0.9 (graph (a)) correlation prices in the linear and exponential model converge fast. For the logistic model price convergence is immediate for our choice of parameters. This is because the variance of the process for the logistic model is to low and then prices are not expected to change to much. This result would be changed if, for example, we would change the levels of our parameters. For example, in Figure 6.4 we show the correlation prices when κX = κY = 5. The graph shows that the linear model behaves poorly, generating negative price values. Under these conditions the logistic and exponential models give consistent prices. To implement our models in practice, we need to calibrate them. Without no available prices for options written on portfolios A and D, one suggestion to estimate the model parameters κX , κY and ρ would to t our models to the historical prices of portfolio A and D. To do this we could, for example, use
Pricing and Hedging Correlation Swaps with a Two-Factor Model
(a)
145
(b)
Figure 6.3: Correlation price C0,T (t) as a function of t: a) ρ = 0.9; b)ρ = 0.1
Figure 6.4: Correlation price C0,T (t) as a function of t, for ρ = 0.9, κX = κY = 5 Avellaneda et al. (2001) guidelines that describe a general approach for calibrating Monte Carlo models to the market prices of securities.
6.6 Conclusions In this paper, we introduced a new model to price correlation swaps. The motivation to build our model was based on an approximation for correlation payo which allows us to write the correlation payo as a ratio between two tradable assets that can be found in variance swap markets. We then built a model where we characterize the dynamics of these two assets and presented three dierent models that allows us to price and to hedge a corre-
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Pricing and Hedging Correlation Swaps with a Two-Factor Model
lation payo. Two of them, that we called the linear and the exponential models, allows us to compute the prices of correlation swap with closed-formulas. These two models are versatile enough and they provided us with a costless delta-hedging strategy based on the two tradable assets. One main disadvantage of these models is that they both allow for correlation prices above one which will allow to build trading strategies. To overcome this issue, we proposed another model based on the logistic function which only generates correlation prices between -1 and 1. However, it doesn't have a closed-formula solutions for derivatives prices and thus pricing derivatives requires to use numerical-methods. We performed dierent tests to compare our three models. We showed that before maturity prices were very dierent between models. Further research could be done in dierent directions. One would be to improve this model by nding a better function f such that we can have closed-formulas for correlation prices and have correlation prices between -1 and 1. Another research direction would be to study correlation prices considering the whole variance term structure. Finally empirical research should be made to verify the e ciency of our models to dynamically hedge correlation products.
Appendices
147
D.1
(Xt, Yt)-dynamics
under risk-neutral measure
From equation (6.11) the process Es [eXt ] is a martingale with respect to the probability measure Q. Therefore deXt should be driftless. Using Itô Lemma to the function h(x) = ex we have
1 deXt = eXt dXt + eXt (dXt )2 2 1 X 2 X Xt ¯t . µt (Xt , Yt ) + (σt ) dt + σtX dW = e 2
(D.1)
The condition for the process given by equation (D.1) to be driftless is to choose
1 X 2 µX t (Xt , Yt ) = − (σt ) . 2
(D.2)
From equation (6.12) and using the same martingale argument, d(eXt f (Yt )) should be driftless. Using Itô Lemma to the function h(x) = ex f (y) we have
1 d(eXt f (Yt )) = f (Yt )eXt dXt + f (Yt )eXt (dXt )2 2 1 + eXt f 0 (Yt )dYt + f 00 (Yt )eXt (dYt )2 + eXt f 0 (Yt )dXt dYt 2 1 00 Xt Y Y 2 0 X Y 0 = e µt (Xt , Yt )f (Yt ) + f (Yt )(σt ) + ρf (Yt )σt σt dt + 2 Xt X ¯ 0 ¯t + f (Yt )e σt dWt + f (Yt )eXt σtY dB (D.3) Therefore, the condition for the process given by equation (D.3) to be driftless is to choose
µYt (Xt , Yt ) = −
1 f 00 (Yt ) Y 2 (σ ) − ρσtX σtY . 2 f 0 (Yt ) t
(D.4)
Using equations (D.2) and (D.4) in the processes described by equations (6.9) and (6.10) we get the dynamics (6.13) and (6.14) as we wanted to show.
149
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Pricing and Hedging Correlation Swaps with a Two-Factor Model
D.2 Process YT − Yt in terms of XT − Xt The processes (6.14) and (6.13) can be written in terms of two independent Wiener ¯t and Z¯t : processes B
1 (D.5) dXt = − (σtX )2 dt + σtX dZ¯t 2 p 1 f 00 (Yt ) ¯t + ρσtY dZ¯t . (D.6) dYt = − (σtY )2 0 − ρσtX σtY dt + 1 − ρ2 σtY dB 2 f (Yt ) Because functions σtX and σtX given by equations (6.15) and (6.15) are only different by a multiplicative constant, we can rewrite as (D.6) as
dYt = + = +
2 p 1 2 f 00 (Yt ) T − t T −t ¯t − κY 0 dt + 1 − ρ2 κY dB − ρκX κY 2 f (Yt ) T T T −t ρκY dZ¯t T 2 p 1 2 f 00 (Yt ) T −t T −t 2 ¯t − κY 0 − ρκX κY dt + 1 − ρ κY dB 2 f (Yt ) T T 2 ! 1 2 T −t κY dXt + κX dt (D.7) ρ κX 2 T
Let us consider the two cases with f (y) = y and f (y) = ey where we obtain 0 or 1 respectively for the ratio
f 00 (Yt ) . f 0 (Yt )
We call this quantity R. Integrating both
sides of the previous stochastic dierential equation (D.7) between s and T we get
YT − Ys
T (T − s)3 p 1 2 + 1 − ρ2 κY = − κX R + ρκX κY 2 3 T3 κY ρ + (XT − Xs ) . κX
Z s
T
T −u ¯ dBu T
Bibliography
Marco Avellaneda, Robert Bu, Craig Friedman, Nicolas Grandchamp, Nicolas Gr, Lukasz Kruk, and Joshua Newman. Weighted Monte Carlo: A New Technique for Calibrating Asset-Pricing Models.
Applied Finance, 4(1):129, 2001.
International Journal of Theoretical and
Colin Bennett. Trading Volatility, Correlation, Term Structure and Skew.
available at http://www.imperial.ac.uk/ ajacquie/, 2014.
Book
S. Bossu. Arbitrage Pricing of Equity Correlation Swaps. Technical report available
at http://www.sbossu.com/docs/CorrelationSwaps7.pdf, JP Morgan, May 2005.
D.T. Breeden and R. H. Litzenberger. Prices of State-Contingent Claims Implicit in Option Prices.
Journal of Business, 51(4):621651, 1978.
Hans Buehler. Consistent Variance Curve Models.
Finance and Stochastics, 10
(2):178203, 2006. Peter Carr and Jian Sun. A New Approach for Option Pricing under Stochastic
Review of Derivatives Research, 10(2):87150, 2007.
Volatility. Yujin
Chloe
Choi.
Analysis
and
Development
Of
Correlation
Technical report http://erasmus-mundus.univ-paris1.fr/, Lyxor Asset bitrage
Strategies
on
Equities.
Ar-
available at Management,
May 2008. A Jacquier and S Slaoui. Variance Dispersion and Correlation Swaps.
paper available at http://arxiv.org/abs/1004.0125v1, 2010. Walter Rudin.
Working
Real and complex analysis. McGraw-Hill Book Co., New York,
third edition, 1987. 151
152
Pricing and Hedging Correlation Swaps with a Two-Factor Model
Giovanni Salvi and Anatoliy V. Swishchuk. Modeling and Pricing of Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities.
Working paper available at http://arxiv.org/abs/1205.5565, 2012.
