Pricing and Hedging of European Plain Vanilla Options under Jump Uncertainty
|
|
- Mercy Hodges
- 5 years ago
- Views:
Transcription
1 Pricing and Hedging of European Plain Vanilla Options under Jump Uncertainty by Olaf Menkens School of Mathematical Sciences Dublin City University (DCU) Financial Engineering Workshop Cass Business School, City University of London London, 19 th October 216
2 Pricing and Hedging of Plain Vanilla Options 1 A Quotation Two men are preparing to go hiking. While one is lacing up hiking boots, he sees that the other man is forgoing his usual boots in favor of sporty running shoes. Why the running shoes? he asks. The second man responds, I heard there are bears in this area and I want to be prepared. Puzzled, the first man points out, But even with those shoes, you can t outrun a bear. The second man says, I don t have to outrun the bear, I just have to outrun you. (See Hubbard (29), p. 157/158)
3 Pricing and Hedging of Plain Vanilla Options 2 Outline 1. Literature Review 2. Worst Case Option Pricing 3. Superhedging Strategy 4. Model Calibration 5. Conclusion
4 Pricing and Hedging of Plain Vanilla Options 3 1. Literature Review Option Pricing Some References: Bachelier (19) [ Derivative Pricing using Brownian Motion], Black and Scholes (1973) [ Reference Model for Option Pricing, No Jumps], Cox (1975) [ CEV Model], Heston (1993) [ Stochastic Volatility Model], Madan, Carr, and Chang (1998) [ Variance Gamma Model], Kou (et al.) (22 24, 28) [ Jump Diffusion Model with exponential tails], See Cont and Tankov (24) or Rebonato (24) for more details and for other models as well.
5 Pricing and Hedging of Plain Vanilla Options 4 Worst Case Scenario Optimization Some References: Hua and Wilmott (1997) [ Binomial Model Derivative Pricing], Korn and Wilmott (22), [ Portfolio Optimisation], Mönnig (212), [ Stochastic Target Approach], Belak, M. (216) [ BSDE Approach]. Remark: The worst case scenario optimisation problem is also known as Wald s Maximin approach (Wald 1945, 195), which is a well known concept in decision theory. There, this approach is known as robust optimisation (e.g. Bertsimas et al. (211)) [ usually involves optimisation procedure done by a computer]. Mataramvura and Oksendal (28), Oksendal and Sulem (26, 29, 211) [ Compute optimal strategies directly]. [ parameter uncertainty, perturbation analysis].
6 Pricing and Hedging of Plain Vanilla Options 5 Interpretation of Worst Case Scenarios E [ ln ( X t,x,π,τ,k (T) )] π 2 π 1 Merton approach π 3 WC(π 2 ) WC(π 1 ) ( ( τ (π 1 ),k (π 1) ) τ (π 2 ),k (π 2) ) probability free approach (à la de Finetti) WC(π 3 ) (τ,k)
7 Pricing and Hedging of Plain Vanilla Options 6 2. Worst Case Scenario Option Pricing Consider the initial model with one bond and one risky asset. The aim is to price a contingent claim ξ. Definition 2.1 (Worst-case price; superhedging strategy) (see Belak and M. (216)) The worst-case price V 1 (t;ξ) of ξ at time t [,T] is defined as { V 1 (t;ξ) essinf x L + t : (ζ 1,ζ ) A 1 (t,x) A (ζ 1 ) s.t. X ζ 1,ζ,ϑ t,x (T) ξ ( P (T),P ϑ (T) ) for all ϑ B(t) Furthermore, a strategy (ζ 1,ζ ) A 1 (t,x) A (ζ 1 ) is referred to as a superhedging strategy against ξ if X ζ 1,ζ,ϑ t,x (T) ξ(p (T),P ϑ (T)) for all ϑ B(t). }. We let ξ be a European call option with strike price K >, i.e. ξ(p) = [p K] +.
8 Pricing and Hedging of Plain Vanilla Options 7 It is well-known (see Black and Scholes (1973)) that the fair price V is given by V (t,p) = pφ(d 1 (K,t,p)) Ke r[t t] (d 2 (K,t,p)) with log ( p K ) ] + [r + σ2 2 [T t] d 1 (K,t,p) = σ T t d 2 (K,t,p) = (d 1 (K,t,p)) σ T t,, and where Φ denotes the standard normal cumulative distribution function. Equivalently, the fair price is given as the unique classical solution of the Black-Scholes PDE t V (t,p) rp p V (t,p) σ2 2 p2 2 p 2V (t,p)+rv (t,p) =, V (T,p) = [p K] +.
9 Pricing and Hedging of Plain Vanilla Options 8 In the jump-threatened market, we assume the minimum and maximum jump sizes to be given by constants β D ( 1,] and β U [, ). With this, the pricing PDE for the worst-case price V 1 (t,p) can be written as min { t V 1(t,p) rp p V 1(t,p) σ2 V 1 (t,p) max β {β D,β U } [ 2 p2 2 p 2V 1(t,p)+rV 1 (t,p), V (t,[1+β]p) βp p V 1(t,p) ]} =, which is the pricing PDE obtained in both Mönnig (212) and Belak and M. (216). Notice that we have used the strict convexity of V to replace the supremum over all β [β D,β U ] with the maximum over β D and β U. In a similar fashion, the terminal condition can be written as min { V 1 (T,p) max β {β D,β U } [ [(1+β)p K] + βp p V 1(T,p) V 1 (T,p) [p K] +} =. ],
10 Pricing and Hedging of Plain Vanilla Options 9 We define the constants L := α D/U := K [1+β D ][1+β U ], β2 D/U η D/U (t) := exp K L β U β D 1+β D/U r σ2 2β D/U 1 β D and 1+ 1 β D/U [T t]. The terminal condition can be computed explicitly. Lemma 2.2 (Explicit Formula for the Terminal Condition) Let β D β U. Then the unique solution of (1) is given by V 1 (T,p) = α D p 1 β D 1l {p<l} + [ α U p 1 β U +p K ] 1l {p L}.
11 Pricing and Hedging of Plain Vanilla Options 1 12 The Terminal Boundary Worst Case Boundary Function with Up and Downward Jump No Jump 1 Downward jump of max. size is worst case. 8 Payoff Upward jump of max. size is worst case Risky Asset Price This Figure is plotted assuming β U = β D =.5 and K = 1. The used values are as follows: β U =.5, β D =.5, K = 1.
12 Pricing and Hedging of Plain Vanilla Options The Terminal Boundary Worst Case Boundary Function with Downward Jump No Jump Downward jump of max. size is worst case. Payoff No jump is worst case Risky Asset Price This Figure is plotted assuming β D =.5, β U =, and K = 1. The used values are as follows: β U =.5, β D =.5, K = 1.
13 Pricing and Hedging of Plain Vanilla Options The Terminal Boundary Worst Case Boundary Function with Upward Jump No Jump 1 No jump is worst case. Upward jump of max. size is worst case. 8 Payoff Risky Asset Price This Figure is plotted assuming β D =, β U =.5, and K = 1. The used values are as follows: β U =.5, and K = 1.
14 Pricing and Hedging of Plain Vanilla Options 13 Theorem 2.3 (Explicit Solution for the Worst Case Option Price) For (t,p) [,T) R +, the worst-case price V 1 in a Black-Scholes market with constant minimum and maximum jump sizes is given explicitly as V 1 (t,p) = pφ(d 1 (L,t,p)) Ke r[t t] Φ(d 2 (L,t,p)) (1) +α D η D (t)p β 1 ( D Φ d 2 (L,t,p)+ σ ) T t β D +α U η U (t)p β 1 ( U Φ d 2 (L,t,p) σ ) T t. β U
15 Pricing and Hedging of Plain Vanilla Options 14 This price can be decomposed to: one gap option (with strike K and trigger L) plus α D number of short standard power gap put options (where the standard power option is defined in Haug (27)) with strike and trigger L, and plus α U number of standard power gap call options with strike and trigger L, where these latter three options live in the underlying Black Scholes market (that is without jump risk).
16 Pricing and Hedging of Plain Vanilla Options 15 Option Price Payoff Structure Black Scholes Price Worst Case Price Distorted Black Scholes Price with β D Options Prices p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.4, r =.3, β D =.5, β U =, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, T = 1, σ 1 =.4, σ =.4, and β D =.5. Thus, K D = 2.
17 Pricing and Hedging of Plain Vanilla Options 16 Option Price Payoff Structure Black Scholes Price Worst Case Price Distorted Black Scholes Price with β U Options Prices p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.4, r =.3, β D =, β U =.5, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, T = 1, σ 1 =.4, σ =.4, and β U =.5. Thus, K U =
18 Pricing and Hedging of Plain Vanilla Options 17 Theorem 2.4 (Greeks) p V 1 (t,p) = Φ(d 1 (L)) α Dη D p 1 β 1 D Φ ( d 2 (L)+ σ T t β D β D p 2V 1 (t,p) = 1+β D β 2 D t V 1 (t,p) = α Uη U p 1 β 1 U Φ (d 2 (L) σ T t β U β U + 1+β U [ β 2 U r σ2 2β D [ α D η D p 1 β D 2 Φ ( d 2 (L)+ σ β D T t α U η U p β 1 2 U Φ (d 2 (L) σ T t β U ][ 1+ 1 ][ β D + r σ β U β U rke r[t t] Φ(d 2 (L)), ] ( ), ) ) +, ) + α D η D p β 1 D Φ d 2 (L)+ σ T t β ] D α U η U p β 1 ( U Φ d 2 (L) σ T t β U ) ) + +
19 Pricing and Hedging of Plain Vanilla Options 18 σ V 1 (t,p) = σk[t t] β U β D η D + βd V 1 (t,p) = α D β 2 D + [ L σk[t t] η U β U β D η D p 1 β D Φ ( p ] 1 βd Φ ( d 2 (L)+ σ [ L p { βd β U β U β D +σ T t β D T t ] 1 βu Φ (d 2 (L) σ T t β U d 2 (L)+ σ T t β D β D [1+β U ] β U [β U β D ][1+β D ] α Uη U p [ ) d 2 (L) σ T t β D 1 β U Φ 1+β U β U β D σ T tle r[t t] φ(d 2 (L)). ( ) ) ]} +, + d 2 (L) σ β U T t ) +
20 Pricing and Hedging of Plain Vanilla Options 19 1 Delta Delta Payoff Delta Black Scholes Delta Worst Case Delta p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.4, r =.3, β U = β D =.5, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.4, σ =.4, β D =.5, and β U =.5, T = 1.
21 Pricing and Hedging of Plain Vanilla Options 2 Gamma Gamma Black Scholes Gamma Worst Case Gamma Worst Case Gamma if β U = Worst Case Gamma if β D = p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.3, r =.3, β U = β D =.25, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.3, σ =.3, β D =.25, and β U =.25, T = 1.
22 Pricing and Hedging of Plain Vanilla Options 21 Theta Theta Black Scholes Theta Worst Case Theta Worst Case Theta if β U = Worst Case Theta if β D = p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.3, r =.3, β U = β D =.25, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.3, σ =.3, β D =.25, and β U =.25, T = 1.
23 Pricing and Hedging of Plain Vanilla Options 22 Vega Vega Black Scholes Vega Worst Case Vega Worst Case Vega if β U = Worst Case Vega if β D = p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.3, r =.3, β U = β D =.25, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.3, σ =.3, β D =.25, and β U =.25, T = 1.
24 Pricing and Hedging of Plain Vanilla Options 23 Beta Beta Worst Case Beta wrt β D Worst Case Beta wrt β U Worst Case Beta if β U = Worst Case Beta if β D = p Initial Price of the Risky Asset This Figure is plotted assuming K = 1, σ =.3, r =.3, β U = β D =.25, and T = 1. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.3, σ =.3, β D =.25, and β U =.25, T = 1.
25 Pricing and Hedging of Plain Vanilla Options 24 Implied Volatility Surface Implied Volatility t Time to Expiry K Strike of the Option This Figure is plotted assuming σ =.3, r =.3, β U = β D =.25, T =
26 Pricing and Hedging of Plain Vanilla Options 25 Implied Volatility Surface for β U = Implied Volatility t Time to Expiry K Strike of the Option This Figure is plotted assuming σ =.3, r =.3, β D =.25, β U =, T =
27 Pricing and Hedging of Plain Vanilla Options 26 Implied Volatility Surface for β D = Implied Volatility t Time to Expiry K Strike of the Option This Figure is plotted assuming σ =.3, r =.3, β U =.25, β D =, T =
28 Pricing and Hedging of Plain Vanilla Options Superhedging Strategy Define H(t,p;β) V 1 (t,p)+βp p V 1(t,p) V (t,[1+β]p). (2) Observe that H is the value of a portfolio. This portfolio consists of one call option and delta shares of the underlying risky asset hence this is the classical delta hedge of Black Scholes for a plain vanilla call option. H is the value of this portfolio if at time t a jump with jump size β happens and the price of the risky asset is p (just prior to the jump).
29 Pricing and Hedging of Plain Vanilla Options 28 Theorem 3.5 (Superhedging Strategy) One has that [ H(t,p;β) = 1 β ] α D η D p 1 ( β D Φ d 2 (L)+ σ ) T t + (3) β D β [ D + 1 β ] α U η U p 1 ( β U Φ d 2 (L) σ ) T t + β U β U +[1+β]pΦ(d 1 (L)) Ke r[t t] Φ(d 2 (L))+ where [1+β]pΦ ( d 1 ( K 1+β )) +Ke r[t t] Φ ( d 2 ( K 1+β H(t,p;β) for all t [,T],p (, ) and β [β D,β U ] ; (4) and equality holds at least for one (t,p,β). )),
30 Pricing and Hedging of Plain Vanilla Options β * t Time to Expiry p Initial Price of the Risky Asset The best jump size for a European call option with a possible jump in both directions.
31 Pricing and Hedging of Plain Vanilla Options 3 p Price of the Risky Asset β * = β U β * = β D t Time to Expiry The worst jump size (bottom) for a European call option with a possible jump in both directions. The used values are: K = 1, r 1 =.3, r =.3, σ 1 =.3, σ =.3, β D =.25, and β U
32 Pricing and Hedging of Plain Vanilla Options min H p Initial Price of the Risky Asset t Time to Expiry H(t,p,β) for a European call with β = β with a possible jump in both directions.
33 Pricing and Hedging of Plain Vanilla Options H(.,.;β * ) p Initial Price of the Risky Asset t Time to Expiry H(t,p,β) for a European call with β = β with a possible jump in both directions. 1
34 Pricing and Hedging of Plain Vanilla Options 33 5 H min (.,.;2.*β * ) p Initial Price of the Risky Asset t Time to Expiry H(t,p,β) for a European call with β = 2β (bottom) with a possible jump in both directions. 1
35 Pricing and Hedging of Plain Vanilla Options European Plain Vanilla Put Now, let ξ be a European put option with strike price K >, i.e. ξ(p) = [K p] +. The boundary condition (1) for a European plain vanilla put writes to { min P 1 (T,p) (K p) +, P 1 (T,p) sup β [β D,β U ] [ (K ) ] + } (1+β)p βp p P 1(T,p) =.(5) It is straightforward to verify (either by direct computation or by using the Put Call Parity (see e.g. Seydel (26, Exercise 1.1, p. 52) or Cont and Tankov (24, p. 356))) that the solution is given by
36 Pricing and Hedging of Plain Vanilla Options 35 Corollary 4.6 P 1 (T,p) = [ α D p 1 β D +K p ] 1l {p L} α U p 1 β U 1l {p>l}, (6) where p (, ), Moreover, one has the following Corollary 4.7 The worst case price of a European plain vanilla call is given by P 1 (t,p) = Ke r[t t] Φ( d 2 (L)) pφ( d 1 (L))+ (7) +α D η D p β 1 ( D Φ d 2 (L)+ σ ) T t + β D +α U η U p β 1 ( U Φ d 2 (L) σ ) T t. β U
37 Pricing and Hedging of Plain Vanilla Options 36 Furthermore, H P (t,p;β) = [ 1 β [ β D ] α D η D p β 1 ( D Φ d 2 (L)+ σ β D T t ) +(8) + 1 β ] α U η U p 1 ( β U Φ d 2 (L) σ T t + β U β U +Ke r[t t] Φ( d 2 (L)) [1+β]pΦ( d 1 (L))+ ( ( )) Ke r[t t] K Φ d β ( ( )) K +[1+β]pΦ d 1. 1+β )
38 Pricing and Hedging of Plain Vanilla Options Model Calibration Calibrated implied volatilities for maturities T =.712 (next slide) and T = (second next slide). The blue circles are the implied volatilities observed in the market while the blue dash dotted lines are the interpolation of the blue circles. The black solid lines give the implied volatility of the worst case option price formula where the parameters have been calibrated using the market data with penalty a = 1 3. In particular, note that the calibration is done in such a way that the calibrated curve (black solid lines) should be greater or equal the curve plotted from market data (blue dash dotted lines). For comparison reasons, the usual calibration (that is without penalty, meaning a = ) is given as well (green dashed lines). T =.712 T =.1479 T =.3973 T =.6466 T = σ β D β U O( )
39 Pricing and Hedging of Plain Vanilla Options T = Implied Volatility K Strike of the Option
40 Pricing and Hedging of Plain Vanilla Options T = Implied Volatility K Strike of the Option
41 Pricing and Hedging of Plain Vanilla Options 4.9 T = Implied Volatility K Strike of the Option
42 Pricing and Hedging of Plain Vanilla Options 41.8 T = Implied Volatility K Strike of the Option
43 Pricing and Hedging of Plain Vanilla Options T = Implied Volatility K Strike of the Option
44 Pricing and Hedging of Plain Vanilla Options Conclusion To summarize, one has the following properties: jumps are not averaged out but are fully taken into account (compare with liability insurance), first explicit non trivial superhedging price and superhedging strategy, it explains the volatility smile (as well as the smirk), and the closed form solution is numerically of the same level as the solution of Black and Scholes.
45 Pricing and Hedging of Plain Vanilla Options 44 Thank you very much for your attention! The corresponding paper can be downloaded from SSRN:
FIN FINANCIAL INSTRUMENTS SPRING 2008
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Greeks Introduction We have studied how to price an option using the Black-Scholes formula. Now we wish to consider how the option price changes, either
More informationLecture Quantitative Finance Spring Term 2015
and Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 06: March 26, 2015 1 / 47 Remember and Previous chapters: introduction to the theory of options put-call parity fundamentals
More informationThe Black-Scholes Model
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh The Black-Scholes Model In these notes we will use Itô s Lemma and a replicating argument to derive the famous Black-Scholes formula
More informationDefinition Pricing Risk management Second generation barrier options. Barrier Options. Arfima Financial Solutions
Arfima Financial Solutions Contents Definition 1 Definition 2 3 4 Contenido Definition 1 Definition 2 3 4 Definition Definition: A barrier option is an option on the underlying asset that is activated
More informationA Worst-Case Approach to Option Pricing in Crash-Threatened Markets
A Worst-Case Approach to Option Pricing in Crash-Threatened Markets Christoph Belak School of Mathematical Sciences Dublin City University Ireland Department of Mathematics University of Kaiserslautern
More informationLecture 9: Practicalities in Using Black-Scholes. Sunday, September 23, 12
Lecture 9: Practicalities in Using Black-Scholes Major Complaints Most stocks and FX products don t have log-normal distribution Typically fat-tailed distributions are observed Constant volatility assumed,
More informationOption Pricing. 1 Introduction. Mrinal K. Ghosh
Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified
More informationMathematics of Finance Final Preparation December 19. To be thoroughly prepared for the final exam, you should
Mathematics of Finance Final Preparation December 19 To be thoroughly prepared for the final exam, you should 1. know how to do the homework problems. 2. be able to provide (correct and complete!) definitions
More informationFrom Discrete Time to Continuous Time Modeling
From Discrete Time to Continuous Time Modeling Prof. S. Jaimungal, Department of Statistics, University of Toronto 2004 Arrow-Debreu Securities 2004 Prof. S. Jaimungal 2 Consider a simple one-period economy
More informationBarrier options. In options only come into being if S t reaches B for some 0 t T, at which point they become an ordinary option.
Barrier options A typical barrier option contract changes if the asset hits a specified level, the barrier. Barrier options are therefore path-dependent. Out options expire worthless if S t reaches the
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationMATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS
MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material.
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationINTEREST RATES AND FX MODELS
INTEREST RATES AND FX MODELS 7. Risk Management Andrew Lesniewski Courant Institute of Mathematical Sciences New York University New York March 8, 2012 2 Interest Rates & FX Models Contents 1 Introduction
More informationSYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives
SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October
More informationThe Uncertain Volatility Model
The Uncertain Volatility Model Claude Martini, Antoine Jacquier July 14, 008 1 Black-Scholes and realised volatility What happens when a trader uses the Black-Scholes (BS in the sequel) formula to sell
More informationStochastic Volatility (Working Draft I)
Stochastic Volatility (Working Draft I) Paul J. Atzberger General comments or corrections should be sent to: paulatz@cims.nyu.edu 1 Introduction When using the Black-Scholes-Merton model to price derivative
More informationA Brief Review of Derivatives Pricing & Hedging
IEOR E4602: Quantitative Risk Management Spring 2016 c 2016 by Martin Haugh A Brief Review of Derivatives Pricing & Hedging In these notes we briefly describe the martingale approach to the pricing of
More informationlast problem outlines how the Black Scholes PDE (and its derivation) may be modified to account for the payment of stock dividends.
224 10 Arbitrage and SDEs last problem outlines how the Black Scholes PDE (and its derivation) may be modified to account for the payment of stock dividends. 10.1 (Calculation of Delta First and Finest
More informationThe Use of Importance Sampling to Speed Up Stochastic Volatility Simulations
The Use of Importance Sampling to Speed Up Stochastic Volatility Simulations Stan Stilger June 6, 1 Fouque and Tullie use importance sampling for variance reduction in stochastic volatility simulations.
More informationHedging Under Jump Diffusions with Transaction Costs. Peter Forsyth, Shannon Kennedy, Ken Vetzal University of Waterloo
Hedging Under Jump Diffusions with Transaction Costs Peter Forsyth, Shannon Kennedy, Ken Vetzal University of Waterloo Computational Finance Workshop, Shanghai, July 4, 2008 Overview Overview Single factor
More informationVolatility Smiles and Yield Frowns
Volatility Smiles and Yield Frowns Peter Carr NYU IFS, Chengdu, China, July 30, 2018 Peter Carr (NYU) Volatility Smiles and Yield Frowns 7/30/2018 1 / 35 Interest Rates and Volatility Practitioners and
More informationInvestment Guarantees Chapter 7. Investment Guarantees Chapter 7: Option Pricing Theory. Key Exam Topics in This Lesson.
Investment Guarantees Chapter 7 Investment Guarantees Chapter 7: Option Pricing Theory Mary Hardy (2003) Video By: J. Eddie Smith, IV, FSA, MAAA Investment Guarantees Chapter 7 1 / 15 Key Exam Topics in
More informationIn this lecture we will solve the final-value problem derived in the previous lecture 4, V (1) + rs = rv (t < T )
MSC FINANCIAL ENGINEERING PRICING I, AUTUMN 2010-2011 LECTURE 5: THE BLACK AND SCHOLES FORMULA AND ITS GREEKS RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK In this lecture we will solve the final-value problem
More informationEuropean call option with inflation-linked strike
Mathematical Statistics Stockholm University European call option with inflation-linked strike Ola Hammarlid Research Report 2010:2 ISSN 1650-0377 Postal address: Mathematical Statistics Dept. of Mathematics
More informationUnified Credit-Equity Modeling
Unified Credit-Equity Modeling Rafael Mendoza-Arriaga Based on joint research with: Vadim Linetsky and Peter Carr The University of Texas at Austin McCombs School of Business (IROM) Recent Advancements
More informationStochastic Differential Equations in Finance and Monte Carlo Simulations
Stochastic Differential Equations in Finance and Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH China 2009 Outline Stochastic Modelling in Asset Prices 1 Stochastic
More informationLIBOR models, multi-curve extensions, and the pricing of callable structured derivatives
Weierstrass Institute for Applied Analysis and Stochastics LIBOR models, multi-curve extensions, and the pricing of callable structured derivatives John Schoenmakers 9th Summer School in Mathematical Finance
More informationEquity correlations implied by index options: estimation and model uncertainty analysis
1/18 : estimation and model analysis, EDHEC Business School (joint work with Rama COT) Modeling and managing financial risks Paris, 10 13 January 2011 2/18 Outline 1 2 of multi-asset models Solution to
More informationOn worst-case investment with applications in finance and insurance mathematics
On worst-case investment with applications in finance and insurance mathematics Ralf Korn and Olaf Menkens Fachbereich Mathematik, Universität Kaiserslautern, 67653 Kaiserslautern Summary. We review recent
More informationCHAPTER 12. Hedging. hedging strategy = replicating strategy. Question : How to find a hedging strategy? In other words, for an attainable contingent
CHAPTER 12 Hedging hedging dddddddddddddd ddd hedging strategy = replicating strategy hedgingdd) ddd Question : How to find a hedging strategy? In other words, for an attainable contingent claim, find
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationVolatility Smiles and Yield Frowns
Volatility Smiles and Yield Frowns Peter Carr NYU CBOE Conference on Derivatives and Volatility, Chicago, Nov. 10, 2017 Peter Carr (NYU) Volatility Smiles and Yield Frowns 11/10/2017 1 / 33 Interest Rates
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationTEACHING NOTE 98-04: EXCHANGE OPTION PRICING
TEACHING NOTE 98-04: EXCHANGE OPTION PRICING Version date: June 3, 017 C:\CLASSES\TEACHING NOTES\TN98-04.WPD The exchange option, first developed by Margrabe (1978), has proven to be an extremely powerful
More informationCHAPTER 9. Solutions. Exercise The payoff diagrams will look as in the figure below.
CHAPTER 9 Solutions Exercise 1 1. The payoff diagrams will look as in the figure below. 2. Gross payoff at expiry will be: P(T) = min[(1.23 S T ), 0] + min[(1.10 S T ), 0] where S T is the EUR/USD exchange
More informationFE610 Stochastic Calculus for Financial Engineers. Stevens Institute of Technology
FE610 Stochastic Calculus for Financial Engineers Lecture 13. The Black-Scholes PDE Steve Yang Stevens Institute of Technology 04/25/2013 Outline 1 The Black-Scholes PDE 2 PDEs in Asset Pricing 3 Exotic
More information2 f. f t S 2. Delta measures the sensitivityof the portfolio value to changes in the price of the underlying
Sensitivity analysis Simulating the Greeks Meet the Greeks he value of a derivative on a single underlying asset depends upon the current asset price S and its volatility Σ, the risk-free interest rate
More informationOptimal robust bounds for variance options and asymptotically extreme models
Optimal robust bounds for variance options and asymptotically extreme models Alexander Cox 1 Jiajie Wang 2 1 University of Bath 2 Università di Roma La Sapienza Advances in Financial Mathematics, 9th January,
More information- 1 - **** d(lns) = (µ (1/2)σ 2 )dt + σdw t
- 1 - **** These answers indicate the solutions to the 2014 exam questions. Obviously you should plot graphs where I have simply described the key features. It is important when plotting graphs to label
More informationCONTINUOUS TIME PRICING AND TRADING: A REVIEW, WITH SOME EXTRA PIECES
CONTINUOUS TIME PRICING AND TRADING: A REVIEW, WITH SOME EXTRA PIECES THE SOURCE OF A PRICE IS ALWAYS A TRADING STRATEGY SPECIAL CASES WHERE TRADING STRATEGY IS INDEPENDENT OF PROBABILITY MEASURE COMPLETENESS,
More informationThe Black-Scholes PDE from Scratch
The Black-Scholes PDE from Scratch chris bemis November 27, 2006 0-0 Goal: Derive the Black-Scholes PDE To do this, we will need to: Come up with some dynamics for the stock returns Discuss Brownian motion
More informationLecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.
Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous
More informationChapter 3: Black-Scholes Equation and Its Numerical Evaluation
Chapter 3: Black-Scholes Equation and Its Numerical Evaluation 3.1 Itô Integral 3.1.1 Convergence in the Mean and Stieltjes Integral Definition 3.1 (Convergence in the Mean) A sequence {X n } n ln of random
More informationMASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS.
MASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS May/June 2006 Time allowed: 2 HOURS. Examiner: Dr N.P. Byott This is a CLOSED
More informationPricing of a European Call Option Under a Local Volatility Interbank Offered Rate Model
American Journal of Theoretical and Applied Statistics 2018; 7(2): 80-84 http://www.sciencepublishinggroup.com/j/ajtas doi: 10.11648/j.ajtas.20180702.14 ISSN: 2326-8999 (Print); ISSN: 2326-9006 (Online)
More informationNear-expiration behavior of implied volatility for exponential Lévy models
Near-expiration behavior of implied volatility for exponential Lévy models José E. Figueroa-López 1 1 Department of Statistics Purdue University Financial Mathematics Seminar The Stevanovich Center for
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2017 20 Lecture 20 Implied volatility November 30, 2017
More informationA Continuity Correction under Jump-Diffusion Models with Applications in Finance
A Continuity Correction under Jump-Diffusion Models with Applications in Finance Cheng-Der Fuh 1, Sheng-Feng Luo 2 and Ju-Fang Yen 3 1 Institute of Statistical Science, Academia Sinica, and Graduate Institute
More informationMath 623 (IOE 623), Winter 2008: Final exam
Math 623 (IOE 623), Winter 2008: Final exam Name: Student ID: This is a closed book exam. You may bring up to ten one sided A4 pages of notes to the exam. You may also use a calculator but not its memory
More informationAN ANALYTICALLY TRACTABLE UNCERTAIN VOLATILITY MODEL
AN ANALYTICALLY TRACTABLE UNCERTAIN VOLATILITY MODEL FABIO MERCURIO BANCA IMI, MILAN http://www.fabiomercurio.it 1 Stylized facts Traders use the Black-Scholes formula to price plain-vanilla options. An
More informationDynamic Hedging in a Volatile Market
Dynamic in a Volatile Market Thomas F. Coleman, Yohan Kim, Yuying Li, and Arun Verma May 27, 1999 1. Introduction In financial markets, errors in option hedging can arise from two sources. First, the option
More informationCalculating Implied Volatility
Statistical Laboratory University of Cambridge University of Cambridge Mathematics and Big Data Showcase 20 April 2016 How much is an option worth? A call option is the right, but not the obligation, to
More informationMFE/3F Questions Answer Key
MFE/3F Questions Download free full solutions from www.actuarialbrew.com, or purchase a hard copy from www.actexmadriver.com, or www.actuarialbookstore.com. Chapter 1 Put-Call Parity and Replication 1.01
More informationCredit Risk and Underlying Asset Risk *
Seoul Journal of Business Volume 4, Number (December 018) Credit Risk and Underlying Asset Risk * JONG-RYONG LEE **1) Kangwon National University Gangwondo, Korea Abstract This paper develops the credit
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationIncorporating Managerial Cash-Flow Estimates and Risk Aversion to Value Real Options Projects. The Fields Institute for Mathematical Sciences
Incorporating Managerial Cash-Flow Estimates and Risk Aversion to Value Real Options Projects The Fields Institute for Mathematical Sciences Sebastian Jaimungal sebastian.jaimungal@utoronto.ca Yuri Lawryshyn
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationLocal vs Non-local Forward Equations for Option Pricing
Local vs Non-local Forward Equations for Option Pricing Rama Cont Yu Gu Abstract When the underlying asset is a continuous martingale, call option prices solve the Dupire equation, a forward parabolic
More informationBasic Concepts in Mathematical Finance
Chapter 1 Basic Concepts in Mathematical Finance In this chapter, we give an overview of basic concepts in mathematical finance theory, and then explain those concepts in very simple cases, namely in the
More informationP VaR0.01 (X) > 2 VaR 0.01 (X). (10 p) Problem 4
KTH Mathematics Examination in SF2980 Risk Management, December 13, 2012, 8:00 13:00. Examiner : Filip indskog, tel. 790 7217, e-mail: lindskog@kth.se Allowed technical aids and literature : a calculator,
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2017 9 Lecture 9 9.1 The Greeks November 15, 2017 Let
More informationOptimal Hedging of Variance Derivatives. John Crosby. Centre for Economic and Financial Studies, Department of Economics, Glasgow University
Optimal Hedging of Variance Derivatives John Crosby Centre for Economic and Financial Studies, Department of Economics, Glasgow University Presentation at Baruch College, in New York, 16th November 2010
More informationMFE/3F Questions Answer Key
MFE/3F Questions Download free full solutions from www.actuarialbrew.com, or purchase a hard copy from www.actexmadriver.com, or www.actuarialbookstore.com. Chapter 1 Put-Call Parity and Replication 1.01
More informationarxiv: v1 [q-fin.pm] 13 Mar 2014
MERTON PORTFOLIO PROBLEM WITH ONE INDIVISIBLE ASSET JAKUB TRYBU LA arxiv:143.3223v1 [q-fin.pm] 13 Mar 214 Abstract. In this paper we consider a modification of the classical Merton portfolio optimization
More informationMATH 476/567 ACTUARIAL RISK THEORY FALL 2016 PROFESSOR WANG
MATH 476/567 ACTUARIAL RISK THEORY FALL 206 PROFESSOR WANG Homework 5 (max. points = 00) Due at the beginning of class on Tuesday, November 8, 206 You are encouraged to work on these problems in groups
More informationHomework Assignments
Homework Assignments Week 1 (p 57) #4.1, 4., 4.3 Week (pp 58-6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15-19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9-31) #.,.6,.9 Week 4 (pp 36-37)
More informationLecture 15: Exotic Options: Barriers
Lecture 15: Exotic Options: Barriers Dr. Hanqing Jin Mathematical Institute University of Oxford Lecture 15: Exotic Options: Barriers p. 1/10 Barrier features For any options with payoff ξ at exercise
More informationCompleteness and Hedging. Tomas Björk
IV Completeness and Hedging Tomas Björk 1 Problems around Standard Black-Scholes We assumed that the derivative was traded. How do we price OTC products? Why is the option price independent of the expected
More informationQuadratic hedging in affine stochastic volatility models
Quadratic hedging in affine stochastic volatility models Jan Kallsen TU München Pittsburgh, February 20, 2006 (based on joint work with F. Hubalek, L. Krawczyk, A. Pauwels) 1 Hedging problem S t = S 0
More informationThe Derivation and Discussion of Standard Black-Scholes Formula
The Derivation and Discussion of Standard Black-Scholes Formula Yiqian Lu October 25, 2013 In this article, we will introduce the concept of Arbitrage Pricing Theory and consequently deduce the standard
More informationTime-changed Brownian motion and option pricing
Time-changed Brownian motion and option pricing Peter Hieber Chair of Mathematical Finance, TU Munich 6th AMaMeF Warsaw, June 13th 2013 Partially joint with Marcos Escobar (RU Toronto), Matthias Scherer
More informationNEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 MAS3904. Stochastic Financial Modelling. Time allowed: 2 hours
NEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 Stochastic Financial Modelling Time allowed: 2 hours Candidates should attempt all questions. Marks for each question
More informationLecture 18. More on option pricing. Lecture 18 1 / 21
Lecture 18 More on option pricing Lecture 18 1 / 21 Introduction In this lecture we will see more applications of option pricing theory. Lecture 18 2 / 21 Greeks (1) The price f of a derivative depends
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More informationIEOR E4602: Quantitative Risk Management
IEOR E4602: Quantitative Risk Management Basic Concepts and Techniques of Risk Management Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationIlliquidity, Credit risk and Merton s model
Illiquidity, Credit risk and Merton s model (joint work with J. Dong and L. Korobenko) A. Deniz Sezer University of Calgary April 28, 2016 Merton s model of corporate debt A corporate bond is a contingent
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Spring 2018 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Spring 218 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 218 19 Lecture 19 May 12, 218 Exotic options The term
More informationBluff Your Way Through Black-Scholes
Bluff our Way Through Black-Scholes Saurav Sen December 000 Contents What is Black-Scholes?.............................. 1 The Classical Black-Scholes Model....................... 1 Some Useful Background
More informationCalibration of SABR Stochastic Volatility Model. Copyright Changwei Xiong November last update: October 17, 2017 TABLE OF CONTENTS
Calibration of SABR Stochastic Volatility Model Copyright Changwei Xiong 2011 November 2011 last update: October 17, 2017 TABLE OF CONTENTS 1. Introduction...2 2. Asymptotic Solution by Hagan et al....2
More information( ) since this is the benefit of buying the asset at the strike price rather
Review of some financial models for MAT 483 Parity and Other Option Relationships The basic parity relationship for European options with the same strike price and the same time to expiration is: C( KT
More informationFair Valuation of Insurance Contracts under Lévy Process Specifications Preliminary Version
Fair Valuation of Insurance Contracts under Lévy Process Specifications Preliminary Version Rüdiger Kiesel, Thomas Liebmann, Stefan Kassberger University of Ulm and LSE June 8, 2005 Abstract The valuation
More informationSolving the Black-Scholes Equation
Solving the Black-Scholes Equation An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Initial Value Problem for the European Call The main objective of this lesson is solving
More informationM3F22/M4F22/M5F22 EXAMINATION SOLUTIONS
M3F22/M4F22/M5F22 EXAMINATION SOLUTIONS 2016-17 Q1: Limited liability; bankruptcy; moral hazard. Limited liability. All business transactions involve an exchange of goods or services between a willing
More informationDynamic Relative Valuation
Dynamic Relative Valuation Liuren Wu, Baruch College Joint work with Peter Carr from Morgan Stanley October 15, 2013 Liuren Wu (Baruch) Dynamic Relative Valuation 10/15/2013 1 / 20 The standard approach
More informationValuation of Volatility Derivatives. Jim Gatheral Global Derivatives & Risk Management 2005 Paris May 24, 2005
Valuation of Volatility Derivatives Jim Gatheral Global Derivatives & Risk Management 005 Paris May 4, 005 he opinions expressed in this presentation are those of the author alone, and do not necessarily
More informationAre stylized facts irrelevant in option-pricing?
Are stylized facts irrelevant in option-pricing? Kyiv, June 19-23, 2006 Tommi Sottinen, University of Helsinki Based on a joint work No-arbitrage pricing beyond semimartingales with C. Bender, Weierstrass
More informationGreek parameters of nonlinear Black-Scholes equation
International Journal of Mathematics and Soft Computing Vol.5, No.2 (2015), 69-74. ISSN Print : 2249-3328 ISSN Online: 2319-5215 Greek parameters of nonlinear Black-Scholes equation Purity J. Kiptum 1,
More informationStatistical Methods in Financial Risk Management
Statistical Methods in Financial Risk Management Lecture 1: Mapping Risks to Risk Factors Alexander J. McNeil Maxwell Institute of Mathematical Sciences Heriot-Watt University Edinburgh 2nd Workshop on
More informationAsset-or-nothing digitals
School of Education, Culture and Communication Division of Applied Mathematics MMA707 Analytical Finance I Asset-or-nothing digitals 202-0-9 Mahamadi Ouoba Amina El Gaabiiy David Johansson Examinator:
More informationPricing Dynamic Guaranteed Funds Under a Double Exponential. Jump Diffusion Process. Chuang-Chang Chang, Ya-Hui Lien and Min-Hung Tsay
Pricing Dynamic Guaranteed Funds Under a Double Exponential Jump Diffusion Process Chuang-Chang Chang, Ya-Hui Lien and Min-Hung Tsay ABSTRACT This paper complements the extant literature to evaluate the
More informationHow do Variance Swaps Shape the Smile?
How do Variance Swaps Shape the Smile? A Summary of Arbitrage Restrictions and Smile Asymptotics Vimal Raval Imperial College London & UBS Investment Bank www2.imperial.ac.uk/ vr402 Joint Work with Mark
More informationComputing Bounds on Risk-Neutral Measures from the Observed Prices of Call Options
Computing Bounds on Risk-Neutral Measures from the Observed Prices of Call Options Michi NISHIHARA, Mutsunori YAGIURA, Toshihide IBARAKI Abstract This paper derives, in closed forms, upper and lower bounds
More informationLecture on Interest Rates
Lecture on Interest Rates Josef Teichmann ETH Zürich Zürich, December 2012 Josef Teichmann Lecture on Interest Rates Mathematical Finance Examples and Remarks Interest Rate Models 1 / 53 Goals Basic concepts
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 19 11/20/2013. Applications of Ito calculus to finance
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.7J Fall 213 Lecture 19 11/2/213 Applications of Ito calculus to finance Content. 1. Trading strategies 2. Black-Scholes option pricing formula 1 Security
More informationAll Investors are Risk-averse Expected Utility Maximizers. Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel)
All Investors are Risk-averse Expected Utility Maximizers Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel) First Name: Waterloo, April 2013. Last Name: UW ID #:
More informationApproximation Methods in Derivatives Pricing
Approximation Methods in Derivatives Pricing Minqiang Li Bloomberg LP September 24, 2013 1 / 27 Outline of the talk A brief overview of approximation methods Timer option price approximation Perpetual
More informationMSc Financial Engineering CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL. To be handed in by monday January 28, 2013
MSc Financial Engineering 2012-13 CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL To be handed in by monday January 28, 2013 Department EMS, Birkbeck Introduction The assignment consists of Reading
More information2.3 Mathematical Finance: Option pricing
CHAPTR 2. CONTINUUM MODL 8 2.3 Mathematical Finance: Option pricing Options are some of the commonest examples of derivative securities (also termed financial derivatives or simply derivatives). A uropean
More informationUsing Lévy Processes to Model Return Innovations
Using Lévy Processes to Model Return Innovations Liuren Wu Zicklin School of Business, Baruch College Option Pricing Liuren Wu (Baruch) Lévy Processes Option Pricing 1 / 32 Outline 1 Lévy processes 2 Lévy
More information