smile-lecture12

E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 1 of 22 C opyright E m anuel D erm an 2008 Lecture 12: Jump Diffusion Models of the Smile. 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 2 of 22 C opyright E m anuel D erm an 2008 12.1 Jumps Why are we interested in jump models? Mostly, because of reality: stocks and indexes don’t diffuse smoothly, and do seem to jump. Even currencies some- times jump. As an explanation of the volatility smile, jumps are attractive because they pro- vide an easy way to produce the steep short-term skew that persists in equity index markets, and that indeed appeared soon after the jump/crash of 1987. Towards the end of this section we’ll discuss the qualitative features of the smile that appears in jump models. But jumps are unattractive from a theoretical point of view1 because you can- not continuously hedge a distribution of finite-size jumps, and so risk-neutral arbitrage-free pricing isn’t possible. As a result, most jump-diffusion models simply assume risk-neutral pricing without a thorough justification. It may make sense to think of the implied volatility skew in jump models as simply representing what sellers of options will charge to provide protection on an actuarial basis. Whatever the case, there have been and will be jumps in asset prices, and even if you can’t hedge them, we are still interested in seeing what sort of skew they produce. 1. “So what?” you may say. 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 3 of 22 C opyright E m anuel D erm an 2008 12.1.1 An Expectations View of the Skew Arising from Jumps Assume that there is some probability p that a single jump will occur taking the market from S to K sometime before option expiration T, and that without that jump the future diffusion volatility of the index would have been . Then the expected net future realized volatility contingent on the market jumping to strike K via a jump and a diffusion is approximately If implied volatility is expected future realized volatility, then is also the rational value for the implied volatility of an option with strike K. Below is the implied surface resulting from this picture. It’s not unrealistic for index options, especially for short expirations, and can be made more realistic by allowing the diffusion volatility to incorporate a term structure as well. We choose p to be larger for a downward jump than for an upward jump. σ T( ) σ S K T, ,( ) Tσ2 S K T, ,( ) p S K–( )S----------------- 2× 1 p–( ) Tσ2 T( )×+= σ S K T, ,( ) σ T( ) 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 4 of 22 C opyright E m anuel D erm an 2008 12.2 Modeling Jumps Alone 12.2.1 Stocks that Jump: Calibration and Compensation We’ve spent most of the course modeling pure diffusion processes. Now we’ll look at pure jump processes as a preamble to examining the more realistic mix- ture of jumps and diffusion. Here is a discrete binomial approximation to a diffusion process over time : The probabilities of both up and down moves are finite, but the moves them- selves are small, of order . The net variance is and the drift is . In continuous time this represents the process . Jumps are fundamentally different. There the probability of a jump J is small, of order , but the jump itself is finite. What does this process represent? Let’s look at the mean and variance of the process. Δt ln(S/S0) 0 μΔt σ Δt+ μΔt σ Δt– 0.5 0.5 Δt σ2Δt μ d Sln μdt σdZ+= Δt ln(S/S0) 0 μ'Δt J+ μ'Δt λΔt 1 λΔt– μ' λJ+( )Δt mean E Sln[ ] λΔt μ'Δt J+[ ] 1 λΔt–( )μ'Δt+= μ' λJ+( )Δt= 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 5 of 22 C opyright E m anuel D erm an 2008 The variance of the process is given by Thus, this process has an observed drift and an observed vola- tility .If we observe a drift and a volatility , and we want to obtain them from a jump process, we must calibrate the jump process so that The one unknown is which is the probability of a jump in return of J in per unit time. This described how evolves. How does S evolve? Thus if the stock grows risk-neutrally, for example, then We have to compensate the drift for the jump contribution to calibrate to a total return r. var λΔt J 1 λΔt–( )[ ]2 1 λΔt–( ) JλΔt[ ]2+= 1 λΔt–( )J2λΔt 1 λΔt– λΔt+[ ]= 1 λΔt–( )J2λΔt= J2λΔt as Δt 0→→ μ μ' λJ+( )= σ J2λ= μ σ J σλ-------= μ' μ λσ–= λ Sln S( )ln S S μ'Δt J+( )exp S μ'Δt( )exp λΔt 1 λΔt– μ' λJ+( )Δt mean E S[ ] λΔtS μ'Δt J+( )exp 1 λΔt–( )S μ'Δt( )exp+= S μ'Δt( ) 1 λ eJ 1–( )Δt+[ ]exp= S μ' λ eJ 1–( )+{ }Δt[ ]exp≈ r μ' λ eJ 1–( )+= μ' r eJ 1–( )–= 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 6 of 22 C opyright E m anuel D erm an 2008 In continuous-time notation the elements of the jump can be written as a Pois- son process Here is a jump or Poisson process that is modeled as follows: The increment dq takes the value 1 with probability if a jump occurs and the value with probability if no jump occurs, so that the expected value . 12.2.2 The Poisson Distribution of Jumps Let be the constant probability of a jump J occurring per unit time. Let be the probability of n jumps occurring during time t. The probability of no jumps occurring during time t is given by the limit as of the bino- mial Poisson process we wrote down above, so that where we wrote . Similarly d Sln μ'dt Jdq+= dq dq 0 1 0 λΔt 1 λΔt– λΔt mean λdt 0 1 λdt– E dq[ ] λdt= λ P n t,( ) dt 0→ P 0 t,[ ] 1 λdt–( ) t dt----- 1 λtdtt-----–⎝ ⎠ ⎛ ⎞ t dt----- 1 λtN-----–⎝ ⎠ ⎛ ⎞ N e λt– as→= = = N ∞→ N t dt( )⁄= P n t,( ) N!n! N n–( )!------------------------- λdt( ) n 1 λdt–( )N n–= N! n! N n–( )!------------------------- λt N-----⎝ ⎠ ⎛ ⎞ n 1 λtN-----–⎝ ⎠ ⎛ ⎞ N n–= N! Nn N n–( )! -------------------------- λt( ) n n!------------ 1 λt N-----–⎝ ⎠ ⎛ ⎞ N n–= λt( )n n!------------e λt–→ 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 7 of 22 C opyright E m anuel D erm an 2008 as for fixed . Note that One can easily show that the mean number of jumps during time t is , con- firming that should be regarded as the probability per unit time of one jump. One can also show that the variance of the number of jumps during time t is also . 12.2.3 Pure jump risk-neutral option pricing We can value a standard call option (assuming risk-neutrality, i.e. taking the value as the expected risk-neutral discounted value of its payoffs) for a pure jump model as follows. It is the sum of the expected payoff for all numbers of jumps from 0 to infinity during time to expiration : where is the final stock price after n Poisson jumps, and the payoff of the call is multiplied by the probability of the jump occurring, and N ∞→ n P n t,( ) n 0= ∞ ∑ 1= λt λ λt τ C e rτ– max Seμ'τ nJ+ K– 0,[ ] λτ( ) n n!-------------e λt– n 0= ∞ ∑= Seμ'τ nJ+ μ' r eJ 1–( )–= 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 8 of 22 C opyright E m anuel D erm an 2008 12.3 Modeling Jumps plus Diffusion 12.3.1 Some comments • You can replicate an option exactly by means of a position of stock and several other options if the underlying stock undergoes only a finite num- ber of jumps of known size. But with an infinite number of possible jumps, you cannot replicate; you can only minimize the variance of the P&L. • Merton’s model of jump-diffusion regards jumps as “abnormal” market events that have to be superimposed upon “normal” diffusion. This is in philosophical contrast to Mandelbrot, and to Eugene Stanley and his econophysics collaborators, who regard a mixture of two models for the world as being contrived; ideally, a single model, rather than a “normal” and “abnormal” model, should explain all events. Variance-gamma models also provide a unified view of market moves in which all stock price move- ments are jumps of various sizes. 12.3.2 Merton’s jump-diffusion model and its PDE Merton combines Poisson jumps with geometric Brownian diffusion, as fol- lows Eq.12.1 where J very much resembles a random dividend yield paid on the stock; when a jump occurs the stock jumps (up or down) by a factor J. Later we will model J as a normal random variable. You can derive a partial differential equation for options valuation under this jump-diffusion process, as follows. Let be the value of the option. We construct the usual hedged portfolio by shorting n shares of the stock S. Now dS S------ μdt σdZ Jdq+ += E dq[ ] λdt= var dq[ ] λdt= C S t,( ) π C nS–= 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 9 of 22 C opyright E m anuel D erm an 2008 and We can choose n to cancel the diffusion part of the stock price, so that . Then the change in the value of the hedged portfolio becomes Eq.12.2 The partially hedged portfolio is still risky because of the possibility of jumps. Suppose that despite the risk of jumps, we expect to earn the riskless return on the hedged position (this would be true, for example, if jump risk were truly diversifiable). Then and . Applying this to Equation 13.2 we obtain or where E[ ] denotes an expectation over jump sizes . This is a mixed differ- ence/partial-differential equation for a standard call with terminal payoff . For it reduces to the Black-Scholes equation. We will solve it below by the Feynman-Kac method as an expected discounted value of the payoffs. ΔC Ct 12---CSS σS( ) 2+ dt CS μSdt σSdZ+( ) C S JS+ t,( ) C S t,( )–[ ]dq+ += ndS nS μdt σdZ Jdq+ +( )= Δπ ΔC n μSdt σSdZ JSdq+ +[ ]–= Ct CSμS 12---CSS σS( ) 2 nμS–+ + dt CS n–( )σSdZ+= C S JS+ t,( ) C S t,( )– nSJ–[ ]dq+ n CS= Δπ Ct 12---CSS σS( ) 2+ dt C S JS+ t,( ) C S t,( )– CSSJ–[ ]dq+= E Δπ[ ] rπΔt= E dq[ ] λΔt= Ct 1 2---CSS σS( ) 2+ E C 1 J+( )S t,( ) C S t,( )– CSSJ–[ ]λ+ C SCS–( )r= Ct 1 2---CSS σS( ) 2 rSCS rC–+ + E C 1 J+( )S t,( ) C S t,( )– CSSJ–[ ]λ+ 0= J CT max ST K– 0,( )= λ 0= 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 10 of 22 C opyright E m anuel D erm an 2008 12.4 Trinomial Jump-Diffusion and Calibration Diffusion can be modeled binomially, as in The volatility of the log returns adds an Ito term to the drift of the stock price S itself, so that for pure risk-neutral diffusion one must choose . To add jumps one J needs a third, trinomial, leg in the tree: Just as diffusion modifies the drift of the stock price, so do jumps. 12.4.1 The Compensated Process How must we choose/calibrate the diffusion and jumps so that the stock grows risk-neutrally, i.e. that ? First let’s compute the stock growth rate under jump diffusion. ln(S/S0) 0 μΔt σ Δt+ μΔt σ Δt– 1/2 1/2 σ σ2 2⁄ μ r σ2 2⁄–= ln(S/S0) 0 μΔt σ Δt+ μΔt σ Δt– 1 2--- 1 λΔt–( ) 1 2--- 1 λΔt–( ) μΔt J+ ⎭⎪ ⎪⎬ ⎪⎪ ⎫ diffusion jump λΔt E dS[ ] Srdt= 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 11 of 22 C opyright E m anuel D erm an 2008 One can show by expanding this to keep terms of order that so that, if we want the stock to grow risk-neutrally, we must set Eq.12.3 So, to achieve risk-neutral growth in Equation 13.1, we must set the drift of the diffusion process to We have to set the continuous diffusion drift lower to compensate for the effect of both the diffusion volatility and the jumps, since both the jumps and the dif- fusion modify the expected return and the volatility. E SS0 ----- 1 λΔt–( )2-----------------------e μΔt σ Δt+ 1 λΔt–( ) 2-----------------------e μΔt σ Δt– λΔteμΔt J++ += eμΔt 1 λΔt–( )2----------------------- e σ Δt e σ Δt–+( ) λΔteJ+= Δt E SS0 ----- e μ σ 2 2 ----- λ eJ 1–( )+ + ⎩ ⎭⎨ ⎬ ⎧ ⎫Δt higher order terms+= r μ σ 2 2------ λ e J 1–( )+ += μJD r σ 2 2------ λ e J 1–( )––= diffusion compensation jump compensation 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 12 of 22 C opyright E m anuel D erm an 2008 12.5 Valuing a Call in the Jump-Diffusion Model The process we are considering is Eq.12.4 where where at first J is assumed to be a fixed jump size, but will later be generalized to a normal variable. In order to achieve risk-neutrality, we set Eq.12.5 The value of a standard call in this model is given by Eq.12.6 The risk-neutral terminal value of the stock price is given by Eq.12.7 where is given by Equation 12.5. Now, in Equation 12.6 we have to sum over all the final stock prices, which we can break down into those with 0, 1, ... n ... jumps plus the diffusion, where the probability of n jumps in time is Thus, Eq.12.8 where is the terminal lognormal distribution of the stock price that started with initial price S and underwent n jumps as well as the diffusion. dS S------ μdt σdZ Jdq+ += E dq[ ] λdt= var dq[ ] λdt= μ r σ 2 2------ λ e J 1–( )––= CJD e rτ– E ST K– 0,( )[ ]= ST Se μτ Jq σ τZ+ += μ τ λτ( ) n n!-------------e λτ– CJD e rτ– λτn n!--------e λτ– E max ST n K– 0,( )[ ] n 0= ∞ ∑= ST n 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 13 of 22 C opyright E m anuel D erm an 2008 The expected value in the above equation is an expectation over a lognormal stock price that, after time , has undergone n jumps, and therefore is simply related to a Black-Scholes expectation with a jump-shifted distribution or dif- ferent forward price. In the risk-neutral world of Equation 12.5, the expected return on a stock that started at an initial price S and suffered n jumps is where the last term in the above equation adds the drift corresponding to n jumps to the standard compensated risk-neutral drift , which appears in the Black-Scholes formula via the terms . Thus, since ST is lognormal with a shifted center moved by n jumps, Eq.12.9 where is the standard Black-Scholes formula for a call with strike K and volatility with the drift rate given by Eq.12.10 Equation 12.10 omits the term because the Black-Scholes formula for a stock with volatility already includes the term in the terms as part of the definition of . Combining Equation 12.8 and Equation 12.9 we obtain Eq.12.11 τ μn r σ 2 2------ λ e J 1–( )– nJτ------+–= r σ 2 2------– d1 2, E max ST n K– 0,( )[ ] ernτCBS S K τ σ rn, , , ,( )≡ CBS S K τ σ rn, , , ,( ) σ rn rn μn σ 2 2------+≡ r λ e J 1–( ) nJτ------+–= σ2 2⁄ σ σ2 2⁄ N d1 2,( ) CBS CJD e rτ– λτ( )n n!-------------e λτ– ernτCBS S K τ σ rn, , , ,( ) n 0= ∞ ∑= e rτ– λτ( ) n n!-------------e λτ– e r λ eJ 1–( ) nJτ------+–⎝ ⎠ ⎛ ⎞ τ CBS S K τ σ rn, , , ,( ) n 0= ∞ ∑= e λe Jτ( )– λeJτ( )n n!------------------CBS S K τ σ r nJ τ------ λ e J 1–( )–+, , , ,⎝ ⎠⎛ ⎞ n 0= ∞ ∑= 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 14 of 22 C opyright E m anuel D erm an 2008 Writing as the “effective” probability of jumps, we obtain Eq.12.12 This is a mixing formula. The jump-diffusion price is a mixture of Black- Scholes options prices with compensated drifts. This is similar to the result we got for stochastic volatility models with zero correlation -- a mixing theorem -- but here we had to appeal to the diversification of jumps or actuarial pricing rather than perfect riskless hedging. Until now we assumed just one jump size J. We can generalize, as Merton did, to a distribution of normal jumps in return. Suppose Eq.12.13 describes the normal jump distribution. Then Eq.12.14 Incorporating the expectation over this distribution of jumps into Equation 12.12 has two effects: first, J gets replaced by , and second, the variance of the jump process adds to the variance of the entire distribution in the Black-Scholes formula, so that we must replace by because n jumps adds amount of variance. (The division by is neces- sary because variance is defined in terms of geometric Brownian motion and grows with time, but the variance of normally distributed J is independent of time.) λ λeJ= CJD e λτ– λτ( )n n!-------------CBS S K τ σ r nJ τ------ λ e J 1–( )–+, , , ,⎝ ⎠⎛ ⎞ n 0= ∞ ∑= E J[ ] J= var J[ ] σJ2= E eJ[ ] e J 12---σJ 2 + = J 12---σJ 2 + σ2 σ2 nσJ 2 τ---------+ nσJ2 τ--------- τ 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 15 of 22 C opyright E m anuel D erm an 2008 The general formula is therefore: where If so that and the jumps add no drift to the process, then we get the simple intuitive formula in which we simply sum over an infinite number of Black-Scholes distribu- tions, each with identical riskless drift but differeing volatility dependent on the number of jumps and their distribution. CJD e λτ– λτ( )n n!-------------CBS S K τ σ 2 nσJ2 τ---------+ r n J 12---σJ 2 +⎝ ⎠⎛ ⎞ τ--------------------------- λ e J 12---σJ 2 + 1–⎝ ⎠⎜ ⎟ ⎛ ⎞ –+, , , , ⎝ ⎠⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎛ ⎞ n 0= ∞ ∑= λ λe J 12---σJ 2 + = J 12---σJ 2–= E eJ[ ] 1= CJD e λτ– λτ( )n n!-------------CBS S K τ σ 2 nσJ2 τ---------+ r, , , ,⎝ ⎠⎜ ⎟ ⎛ ⎞ n 0= ∞ ∑= 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 16 of 22 C opyright E m anuel D erm an 2008 12.6 The Jump-Diffusion Smile (Qualitatively) Jump diffusion tends to produce a steep realistic very short-term smile in strike or delta, because the jump happens instantaneously and moves the stock price by a large amount. Recall that stochastic volatility models, in contrast, have difficulty producing a very steep short-term smile unless volatility of volatility is very large. The long-term smile in a jump-diffusion model tends to be flat, because at large times the effect on the distribution of the diffusion of the stock price, whose variance grows like , tends to overwhelm the diminishing Poisson probability of large moves via many jumps. Thus jumps produce steep short- term smiles and flat long-term smiles. Recall that mean-reverting stochastic volatility models also produce flat long-term smiles. Jumps of a fixed size tend to produce multi-modal densities centered around the jump size. Jumps of a higher frequency tend to wash out the multi modal density and produce a smoother distribution of multiply overlaid jumps at longer expirations. A higher jump frequency produces a steeper smile at expiration, because jumps are more probable and therefore are more likely to occur in the future as well. Andersen and Andreasen claim that a jump-diffusion model can be fitted to the S&P 500 skew with a diffusion volatility of about 17.7%, a jump probability of = 8.9%, an expected jump size of 45% and a variance of the jump size of 4.7%. A jump this size and with this probability seems excessive when com- pared to real markets, and suggests that the options market is paying a greater risk premium for protection against crashes. σ2τ λ λ 5/31/08 smile-lecture12.fm E4718 Spring 2008: Derman: Lecture 12:Jump Diffusion Models of the Smile. Page 17 of 22 C opyright E m anuel D erm an 2008 12.7 An Intuitive Treatment of Jump Diffusion Let’s work out the consequences of a simple mixing model for jumps. You can think of the process as represented by the figure on the right, with J representing a big instanta- neous jump up with a small probability w, and K repres