Lévy processes in finance and risk management

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1 Lévy processes in finance and risk management Peter Tankov Laboratoire de Probabilités et Modèles Aléatoires Université Paris-Diderot World Congress on Computational Finance Abstract We start with an accessible practitioner s introduction to Lévy processes and jump-diffusion models. Next, we discuss the calibration of exponential Lévy models from traded option prices. Without going into details of every specific algorithm we focus on different approaches for determining the qualitative properties of the model. Finally, we review two recent applications which emphasize the importance of jumps in stock price modeling, namely construction of optimal hedging portfolios and computation of risk measures for dynamically insured portfolios in presence of jumps in asset prices. Both examples show that Lévy-based models provide a better understanding of risk while preserving a high level of mathematical tractability. Keywords: Lévy process, jump-diffusion, calibration, hedging, risk management, CPPI 1 Introduction Starting with Merton s seminal paper [25] and up to the present date, various aspects of models with jumps have been studied in the academic finance community (see [12] for a list of almost 400 references on the subject). In the last decade, also the research departments of major banks started to accept jumpdiffusions and Lévy processes as a valuable tool in their day-to-day modeling. This increasing interest to jump models in finance is mainly due to the following reasons. First, in a model with continuous paths like a diffusion model, the price process behaves locally like a Brownian motion and the probability that the stock moves by a large amount over a short period of time is very small, unless one fixes an unrealistically high value of volatility. Therefore, in such models the prices of short term OTM options should be much lower than what one observes in real markets. On the other hand, if stock prices are allowed to jump, even 1

2 7500 Taux de change DM/USD /09/1992 2/10/1992 Figure 1: Jumps in the trajectory of DM/USD exchange rate, sampled at 5- minute intervals. when the time to maturity is very short, there is a non-negligible probability that after a sudden change in the stock price the option will move in the money. Second, from the point of view of hedging, continuous models of stock price behavior generally lead to a complete market or to a market, which can be made complete by adding one or two additional instruments, like in stochastic volatility models. Since in such a market every terminal payoff can be exactly replicated, options are redundant assets, and the very existence of traded options becomes a puzzle. The mystery is easily solved by allowing for discontinuities: in real markets, due to the presence of jumps in the prices, perfect hedging is impossible and options enable the market participants to hedge risks that cannot be hedged using the underlying only. From a risk management perspective, jumps allow to quantify and take into account the risk of strong stock price movements over short time intervals, which appears non-existent in the diffusion framework. The last and probably the strongest argument for using discontinuous models is simply the presence of jumps in observed prices. Figure 1 depicts the evolution of the DM/USD exchange rate over a two-week period in 1992, and one can see at least three points where the rate moved by over 100 bp within a 5-minute period. Price moves like these ones clearly cannot be accounted for in a diffusion model with continuous paths, but they must be dealt with if the market risk is to be measured and managed correctly. 2 A primer on Lévy processes The two basic building blocks of every Lévy process are the Brownian motion (the diffusion part) and the Poisson process (the jump part). The Brownian motion is a familiar object to every option trader since the appearance of the Black-Scholes model, but a few words about the Poisson process are in order. 2

3 The proofs of the statements below can be found in [12, chapter 2]. The Poisson process Take a sequence {τ i } i 1 of independent exponential random variables with parameter λ, that is, with cumulative distribution function P[τ i y] = e λy and let T n = n i=1 τ i. The process N t = n 1 1 t Tn is called the Poisson process with parameter λ. For example, if the waiting times between buses at a bus stop are exponentially distributed, the total number of buses arrived up to time t is a Poisson process. The trajectories of a Poisson process are piecewise constant (right-continuous with left limits or RCLL), with jumps of size 1 only. The jumps occur at times T i and the intervals between jumps (the waiting times) are exponentially distributed. At every date t > 0, N t has the Poisson distribution with parameter λt, that is, it is integer-valued and λt (λt)n P[N t = n] = e. (1) n! The Poisson process shares with the Brownian motion the very important property of independence and stationarity of increments, that is, for every t > s the increment N t N s is independent from the history of the process up to time s and has the same law as N t s. The processes with independent and stationary increments are called Lévy processes after the French mathematician Paul Lévy. Characteristic function The notion of characteristic function of a random variable plays an essential role in the study of jump-diffusion processes: often we do not know the distribution function of such a process in closed form but the characteristic function is known explicitly. The characteristic function of a random variable X is defined by For the Poisson process, this gives φ X (u) E[e iux ]. E[e iunt ] = exp{λt(e iu 1)}. Here, the computation can be done directly using equation (1). Compound Poisson process For financial applications, it is of little interest to have a process with a single possible jump size. The compound Poisson process is a generalization where the waiting times between jumps are exponential but the jump sizes can have an arbitrary distribution. More precisely, let N be a Poisson process with parameter λ and {Y i } i 1 be a sequence of independent random variables with law f. The process N t X t = 3 i=1 Y i

4 Figure 2: Left: sample path of a compound Poisson process with Gaussian distribution of jump sizes. Right: sample path of a jump-diffusion process (Brownian motion + compound Poisson). is called compound Poisson process. Its trajectories are RCLL and piecewise constant but the jump sizes are now random with law f (cf. Fig.2). The compound Poisson process has independent and stationary increments. Its law at a given time t is not known explicitly but the characteristic function is known and has the form E[e iuxt ] = exp{tλ (e iux 1)f(dx)}. R Jump-diffusions and Lévy processes Combining a Brownian motion with drift and a compound Poisson process, we obtain the simplest case of a jumpdiffusion a process which sometimes jumps and has a continuous but random evolution between the jump times (cf. Fig.2): N t X t = µt + σb t + Y i. (2) The best known model of this type in finance is the Merton model [25], where the stock price is S t = S 0 e Xt with X t as above and the jumps {Y i } have Gaussian distribution. The process (2) is again a Lévy process and its characteristic function can be computed by multiplying the CF of the Brownian motion and that of the compound Poisson process (since the two parts are independent): { E[e iuxt ] = exp t (iµu σ2 u 2 )} + λ (e iux 1)f(dx). 2 The class of Lévy processes is not limited to jump-diffusions of the form (2): there exist Lévy processes with infinitely many jumps in every interval (infinite i=1 R 4

5 intensity of infinite activity Lévy processes). Most of such jumps are very small and there is only a finite number of jumps with absolute value greater than any given positive number. One of the simplest examples of this kind is the gamma process, a process with independent and stationary increments such that for all t, the law p t of X t is the gamma law with parameters λ and ct: p t (x) = λct Γ(ct) xct 1 e λx. The gamma process is an increasing Lévy process (also called subordinator). Its characteristic function has a very simple form: E[e iuxt ] = (1 iu/λ) ct. The gamma process is the building block for a very popular jump model, the variance gamma process [22, 20], which is constructed by taking a Brownian motion with drift and changing its time scale with a gamma process: Y t = µx t + σb Xt. Using Y t to model the logarithm of stock prices can be justified by saying that the price is a geometric Brownian motion if viewed on a stochastic time scale given by the gamma process [17]. The variance gamma process is another example of a Lévy process with infinitely many jumps and has characteristic function E[e iuyt ] = (1 + σ2 u 2 κt iµκu). 2 The parameters have the following (approximate) interpretation: σ is the variance parameter, µ is the skewness parameter and κ is responsible for the kurtosis of the process. In general, every Lévy process can be represented in the form X t = γt + σb t + Z t, where Z t is a jump process with (possibly) infinitely many jumps. A detailed description of this component is given by the Lévy-Itô decomposition which is beyond the scope of this introductory paper. The characteristic function of a Lévy process is given by the Lévy-Khintchine formula: { E[e iuxt ] = exp t (iγu σ2 u 2 )} + (e iux 1 iux1 x 1 )ν(dx), (3) 2 R where ν is a positive measure on R describing the jumps of the process: the Lévy measure. If X is compound Poisson, then ν(r) < and ν(dx) = λf(dx) but in the general case ν need not be a finite measure. It must satisfy the constraint (1 x 2 )ν(dx) < R and describes the jumps of X in the following sense: for every closed set A R with 0 / A, ν(a) is the average number of jumps of X in the time interval [0, 1], whose sizes fall in A. 5

6 Exponential Lévy models To ensure positivity as well as the independence and stationarity of log-returns, stock prices are usually modeled as exponentials of Lévy processes: S t = S 0 e Xt. (4) In the jump-diffusion case this gives ( ) N t S t = S 0 exp µt + σb t + Y i. Between the jumps, the process evolves like a geometric Brownian motion, and after each jump, the value of S t is multiplied by e Yi. This model can therefore be seen as a generalization of the Black-Scholes model: i=1 ds t S t = µdt + σdb t + dj t. (5) Here, J t is a compound Poisson process such that the i-th jump of J is equal to e Yi 1. For instance, if Y i have Gaussian distribution, S will have lognormally distributed jumps. The notation S t means that whenever there is a jump, the value of the process before the jump is used on the left-hand side of the formula. The forms (4) and (5) are equivalent: for a model of the first kind one can always find a model of the second kind with the same law. In the rest of the paper, unless explicitly stated otherwise, we will use the exponential form (4). For option pricing, we will explicitly include the interest rate into the definition of the exponential Lévy model: S t = S 0 e rt+xt. (6) While the forms (4) and (6) are equivalent, the second one leads to a slightly simpler notation. In this case, under the risk-neutral probability, e Xt must be a martingale and from the Lévy-Khintchine formula (3) combined with the independent increments property we conclude that this is the case if and only if γ + σ2 2 + (e x 1 x1 x 1 )ν(dx) = 0. (7) R The model (6) admits no arbitrage opportunity if there exists an equivalent probability under which e Xt is a martingale. For Lévy processes it can be shown that this is almost always the case, namely an exponential Lévy model is arbitrage-free if and only if the trajectories of X are not almost surely increasing nor almost surely decreasing. If a Brownian component is present, the martingale probability can be obtained by changing the drift as in the Black-Scholes setting. Otherwise, in finite-intensity models, the drift must remain fixed under all equivalent probabilities since it can be observed from a single stock price trajectory. To satisfy the martingale constraint (7), one must therefore change the Lévy measure, i.e. 6

7 the intensity of jumps. To understand how this works, suppose that X is a Poisson process with drift: X t = N t at, a > 0. We can obtain a martingale probability by changing the intensity of N to λ mart = a e 1. If, however, X is a Poisson process without drift (increasing trajectories), one cannot find a value of λ > 0 for which e Xt is a martingale. Beyond Lévy processes Although the class of Lévy processes is quite rich, it is sometimes insufficient for multiperiod financial modeling for the following reasons: Due to the stationarity of increments, the stock price returns for a fixed time horizon always have the same law. It is therefore impossible to incorporate any kind of new market information into the return distribution. For a Lévy process, the law of X t for any given time horizon t is completely determined by the law of X 1. Therefore, moments and cumulants depend on time in a well-defined manner which does not always coincide with the empirically observed time dependence of these quantities [6]. For these reasons, several models combining jumps and stochastic volatility appeared in the literature. In the Bates [5] model, one of the most popular examples of the class, an independent jump component is added to the Heston stochastic volatility model: dx t = µdt + V t dw X t + dz t, S t = S 0 e Xt, (8) dv t = ξ(η V t )dt + θ V t dw V t, d W V, W X t = ρdt, where Z is a compound Poisson process with Gaussian jumps. Although X t is no longer a Lévy process, its characteristic function is known in closed form [12, chapter 15] and the pricing and calibration procedures are similar to those used for Lévy processes. 3 Model calibration The first step in using any model is to calibrate it to the available data, and before choosing a particular parametrization, one must determine the qualitative features of the model, relevant for the particular application one has in mind. In the context of Lévy-based models, the important questions are Is our model a pure-jump process, a pure diffusion process or a combination of both? Is the jump part a compound Poisson process or an infinite intensity Lévy process? 7

8 Is the data adequately described by a time-homogeneous Lévy process or is a more general model such as (8) required? The answer to these questions will depend on the type of data that we consider. A recent trend in the literature, made possible by the appearance of large databases and fast computers is to look at high frequency historical data. In particular, testing for the presence of jumps is discussed in [1, 3] whereas [2, 18, 23] and others treat the issue of volatility estimation in presence of jumps. In this paper we consider a different problem and concentrate on the calibration of an exponential Lévy model from option prices. Several authors have investigated the presence of a jump component looking at the S&P 500 option data. Carr and Wu [11] test for presence of jumps by examining the speeds at which the prices of OTM and ATM options on the S&P 500 index converge to zero for short maturities. They find that while the presence of a continuous diffusion component is constantly felt, the jump component may only be present on some days in the sample but disappear on others. Given that an infinite variation continuous component is present all the time, it is impossible, using the method of Carr and Wu, to tell whether the jump component is of finite or infinite intensity: the small jumps are completely screened out by the diffusion part. Medvedev and Scaillet [24] answer the same question using the implied volatility surface asymptotics. They confirm the presence of a jump component (looking once again at S&P option data) but argue that it is not of compound Poisson type and suggest a general Lévy specification. The third question (whether a time-homogeneous Lévy specification is sufficient) was answered in [13, 14] using a non-parametric calibration procedure. These authors have shown that Lévy processes reproduce the implied volatility smile for a single maturity quite well, but when it comes to calibrating several maturities at the same time, the calibration by Lévy processes becomes much less precise. This is clearly seen from the three graphs of Figure 3. The top graph shows the market implied volatilities for four maturities and different strikes. The bottom left graphs depicts implied volatilities, computed in an exponential Lévy model calibrated using a nonparametric algorithm to the first maturity present in the market data. One can see that while the calibration quality is acceptable for the first maturity, it quickly deteriorates as the time to maturity increases: the smile in an exponential Lévy model flattens too fast. The same effect can be observed in the bottom right graph: here, the model was calibrated to the last maturity, present in the data. As a result, the calibration quality is poor for the first maturity: the smile in an exponential Lévy model is more pronounced and its shape does not resemble that of the market. It is difficult to calibrate an exponential Lévy model to options of several maturities because due to independence and stationarity of their increments, Lévy processes have a very rigid term structure of cumulants. In particular, the skewness of a Lévy process is proportional to the inverse square root of time and the excess kurtosis is inversely proportional to time [21]. A number of empirical studies have compared the term structure of skewness and kurtosis 8

9 0.28 Implied volatility Maturity Strike Implied volatility Maturity Strike 5500 Implied volatility Maturity Strike Figure 3: Top: Market implied volatility surface. Bottom left: implied volatility surface in an exponential Lévy model, calibrated to market prices of the first maturity. Bottom right: implied volatility surface in an exponential Lévy model, calibrated to market prices of the last maturity. implied in market option prices to the skewness and kurtosis of Lévy processes. Bates [6], after an empirical study of implicit kurtosis in $/DM exchange rate options concludes that while implicit excess kurtosis does tend to increase as option maturity shrinks,..., the magnitude of maturity effects is not as large as predicted [by a Lévy model]. For stock index options, Madan and Konikov [21] report even more surprising results: both implied skewness and kurtosis actually decrease as the length of the holding period becomes smaller. It should be mentioned, however, that implied skewness/kurtosis cannot be computed from a finite number of option prices with high precision. A second major difficulty arising while trying to calibrate an exponential Lévy model is the time evolution of the smile. Exponential Lévy models belong to the class of the so called sticky delta or sticky moneyness models, meaning that in an exponential Lévy model, the implied volatility of an option with given moneyness (strike price to spot ratio) does not depend on time. This can be seen from the following simple argument. In an exponential Lévy model Q, the implied volatility σ of a call option with moneyness m, expiring in τ years, satisfies: e rτ E Q [(S t e rτ+xτ ms t ) + F t ] = e rτ σ2 rτ+σwτ E[(S t e 2 τ ms t ) + F t ] 9

10 50 30 day ATM options 450 day ATM options /01/1996 2/01/1997 2/01/ /12/1998 Figure 4: Implied volatility of at the money European options on CAC40 index. Due to the independent increments property, S t cancels out and we obtain an equation for the implied volatility σ which does not contain t or S t. Therefore, in an exp-lévy model this implied volatility does not depend on date t or stock price S t. This means that once the smile has been calibrated for a given date t, its shape is fixed for all future dates. Whether or not this is true in real markets can be tested in a model-free way by looking at the implied volatility of at the money options with the same maturity for different dates. Figure 4 depicts the behavior of implied volatility of two at the money options on the CAC40 index, expiring in 30 and 450 days. Since the maturities of available options are different for different dates, to obtain the implied volatility of an option with fixed maturity T for each date, we have taken two maturities, present in the data, closest to T from above and below: T 1 T and T 2 > T. The implied volatility Σ(T) of the hypothetical option with maturity T was then interpolated using the following formula: Σ 2 (T) = Σ 2 (T 1 ) T 2 T T 1 T + Σ2 (T 2 ) T T 1 T 2 T 1. As we have seen, in an exponential Lévy model the implied volatility of an option which is at the money and has fixed maturity must not depend on time or stock price. Figure 4 shows that in reality this is not so: both graphs are rapidly varying random functions. This simple test shows that real markets do not have the sticky moneyness property: arrival of new information can alter the form of the smile. The exponential Lévy models are therefore not random enough to account for the time evolution of the smile. Moreover, models based on additive processes, that is, time-inhomogeneous processes with independent increments, although they perform well in calibrating the term structure of implied volatilities for a given date [12], are not likely to describe the time evolution of the smile correctly since in these models the future form of the smile is still a deterministic function of its present shape [12]. To describe the time evolution of the smile in a consistent 10

11 Portfolio value 0.02 Option price Hedge ratio Figure 5: Evolution of an option position and the corresponding delta-hedging portfolio in presence of stock jumps. manner, one may need to introduce additional stochastic factors (e.g. stochastic volatility) [10, 5, 4]. 4 Hedging the jump risk In the Black-Scholes model, the delta-hedging strategy is known to completely eliminate the risk of an option position. This strategy consists in holding the amount of stock equal to C S, the sensitivity of the option price with respect to the underlying. However, in presence of jumps, delta-hedging is no longer optimal. Suppose that a portfolio contains φ t stock, with price S t, and a short option position. After a jump S t, the change in the stock position is φ t S t, and the option changes by C(t, S t + S t ) C(t, S t ). The jump will be completely hedged if and only if φ t = C(t, S t + S t ) C(t, S t ) S t. Since the option price is a nonlinear function of S, φ t C S and delta-hedging does not offset the jump risk completely. This is illustrated in figure 5 where a single 7% jump in the stock price leads to an important residual hedging error. Thus, to hedge a jump of a given size, one should use the sensitivity to movements of the underlying of this size rather than the sensitivity to infinitesimal movements. Since typically the jump size is not known in advance, the risk associated to jumps cannot be hedged away completely: we are in an incomplete market. In this setting, the hedging becomes an approximation problem: instead of replicating an option, one tries to minimize the residual hedging error. Empirical studies show that strategies using only stock lead to high levels of residual risk, and to obtain realistic hedges, liquid options should be added to the hedging portfolio (gamma-hedging). 11

12 In this section we show how to compute optimal hedging strategies in presence of jumps. First, we treat the case when the hedging portfolio contains only stock and the risk-free asset. Let S t denote the stock price and φ the quantity of stock in the hedging portfolio, and suppose that S satisfies (5) with the Lévy measure of the jump part denoted by ν. Then the (self-financing) portfolio evolves as dv t = (V t φ t S t )rdt + φ t ds t. The forward values of the stock and the portfolio S t = er(t t) S t and V t = e r(t t) V t satisfy T VT = e rt V 0 + φ t dst. 0 We would like to compute the strategy which minimizes the expected squared residual hedging error under the martingale probability: ( ) 2 T φ = arginf E[(V T H T ) 2 ] = arg inf E e rt V 0 + φ t dst H T with H T the option s payoff. Using the Itô formula for jump processes and the isometry relation for stochastic integrals (both are out of scope of the present paper but see [16] for details), the residual hedging error can be expressed as E[(V T H T ) 2 ] = ( e rt V 0 E[H T ] ) T { 2 + E dt(st ) 2 σ 2 φ t C } 2 0 S T + E ν(dz)e 2r(T t) {C(t, S t (1 + z)) C(t, S t ) S t φ t z} 2. 0 R From this formula, three immediate conclusions can be made: The initial capital minimizing the hedging error is 0 V 0 = e rt E[H T ]. (9) If the initial capital is given by (9), the residual hedging error is zero (and the market is complete) only in the following two cases: No jumps in the stock price (ν 0). This case correspond to the Black-Scholes model and the optimal hedging strategy is φ t = C S. No diffusion component (σ = 0) and only one possible jump size (ν = δ z0 (z)). In this case, the optimal hedging strategy is φ t = C(S t(1 + z 0 )) C(S t ) S t z 0. 12

13 Delta Optimal ratio Figure 6: Delta-hedging strategy and optimal quadratic hedging strategy ratios as a function of stock price. In all other cases, the residual hedging error is non-zero (and the market is incomplete) and is minimized by φ (t, S t ) = σ2 C S + 1 S t ν(dz)z(c(t, St (1 + z)) C(t, S t )) σ 2 + z 2 ν(dz) The optimal quadratic hedging strategy is a weighted sum of two terms: the sensitivity of option price to infinitesimal stock movements, and the average sensitivity to finitely-sized jumps. Note that in a pure-jump Lévy model the first term disappears and the hedge ratio does not involve the derivative of the stock price. In fact, in the variance gamma model, for short maturities, this derivative may not even exist! Figure 6 shows the difference between the optimal strategy and the delta C S. These data were obtained in Merton s jump diffusion model (2) with parameters µ = 0.1, r = 0, σ = 0.2, λ = 1, mean jump of 0.1, jump standard deviation of 0.05, for a European put option with strike K = 1.2 and maturity T = 1 month. As we see, the two strategies are not so different after all. The residual hedging errors are also similar: for delta-hedging it has a standard deviation of 1.7% (of the initial stock price) and for the optimal strategy 1.6%. For comparison, in absence of jump risk, the residual hedging error (due to discrete rebalancing) has a standard deviation of 0.7% and if we do not hedge at all, the error is of order of 16%. In conclusion, Hedging with stock only in presence of jumps eliminates a large part risk but still leads to an important residual hedging error. Performances of delta hedging and of the optimal quadratic hedging with stock only are very similar. To eliminate the remaining hedging error, a possible solution is to introduce liquid options into the hedging portfolio. In the above example, if, in addition. 13

14 to the stock, the hedging portfolio contains a European option with strike K = 1, the standard deviation is 0.76%, that is, the risk due to jumps becomes negligible compared to the one associated to discrete rebalancing. Optimal quadratic hedge ratios in the case when the hedging portfolio may contain options can be found in [16]. 5 Risk management in jump models In this section, we review an application of Lévy processes to computing risk measures of dynamically managed portfolios, developed in [15]. We are interested in one of the most widely used portfolio insurance strategies: the constant proportion portfolio insurance (CPPI) introduced by Black and Jones [8] for equity instruments and by Perold [26] for fixed income instruments, see also [9]. Under this strategy, the exposure to the risky asset is equal to the constant multiple m > 1 of the cushion, i.e., the difference between the current portfolio value and the guaranteed amount. In theory, that is, in the Black-Scholes model with continuous trading, this strategy has no downside risk, whereas in real markets this risk is non-negligible and grows with the multiplier value. We show that admitting downward jumps in the risky asset allows a realistic description of risks while maintaining the analytical tractability. Our method establishes a direct relation between the value of the multiple m and the loss probability of the insured portfolio, and hence allows to choose the multiple depending on the risk tolerance of the investor. In the following we will use the following formal definition of the CPPI strategy (the value of the fund is denoted by V t ): A fixed amount N of capital is guaranteed at maturity T. The CPPI strategy is a self-financing strategy such that at every moment t, a fraction of the portfolio is invested into the risky asset S t and the remainder is invested into zero-coupon bond with maturity T and nominal N, whose price is denoted by B t. If V t > B t, the risky asset exposure (amount of money invested into the risky asset) is given by mc t m(v t B t ), where C t is the cushion and m > 1 is a constant multiplier. If V t B t, the entire portfolio is invested into the zero-coupon. We suppose that the price processes for the risky asset S and for the zerocoupon B may be written as ds t S t = dz t and db t B t = rdt, where Z is a Lévy process with Z t > 1 almost surely. 14

15 Let τ = inf{t : V t B t }. Then, since the CPPI strategy is self-financing, up to time τ the portfolio value satisfies which can be rewritten as dv t = m(v t B t ) ds t S t + {V t m(v t B t )} db t B t, dc t C t = mdz t + (1 m)dr t, where we recall that C t = V t B t denotes the cushion. Change of numeraire Introducing the discounted cushion C t = Ct B t, we find C t = C 0 E(mL) t, where L t Z t rt and E denotes the stochastic or Doléans-Dade exponential written explicitly as E(X) t = X 0 e Xt 1 2 [X]c t (1 + X s )e Zs. s t, X s 0 After time τ, according to our definition of the CPPI strategy, the process C remains constant. Therefore, the discounted cushion value for this strategy can be written explicitly as C t = C 0 E(mL) t τ, or alternatively ( ) V t V0 = E(mL) t τ. (10) B t B 0 Since the stochastic exponential can become negative, in presence of negative jumps of sufficient size in the stock price, the capital N at maturity is no longer guaranteed by this strategy. Probability of loss A CPPI-insured portfolio incurs a loss (breaks through the floor) if, for some t [0, T], V t B t. The event V t B t is equivalent to C t 0 and since R is continuous and E(X) t = E(X) t (1 + X t ), C t 0 for some t [0, T] if and only if m L t 1 for some t [0, T]. This leads us to the following result (see [15] for details of the proof): Proposition 1 Let L be a Lévy process with Lévy measure ν. Then the probability of going below the floor is given by ( ) P[ t [0, T] : V t B t ] = 1 exp T 1/m ν(dx). 15

16 Series µ σ λ p η + η MSFT CAC Table 1: Corollary 1 Suppose that S follows an exponential Lévy model of the form S t = S 0 e Nt, where N is a Lévy process with Lévy measure ν. Then the probability of going below the floor is given by ( ) P[ t [0, T] : V t B t ] = 1 exp T log(1 1/m) ν(dx). (11) Application to an exponential Lévy model In this example we compute the loss probability of a CPPI-insured portfolio supposing that the risky asset follows the Kou s model [19], that is, an exponential Lévy model where the driving Lévy process has a non-zero Gaussian component and a Lévy density of the form ν(x) = λ(1 p) e x/η+ 1 x>0 + λp e x /η 1 x<0. (12) η + η Here, λ is the total intensity of positive and negative jumps, p is the probability that a given jump is negative and η and η + are characteristic lengths of respectively negative and positive jumps. The parameters of Kou s jump diffusion model were estimated by maximum likelihood for daily time series of the French CAC40 index and of the Microsoft Corporation (MSFT) share price. For both series, 10 years of data, from December 1st 1996 to December 1st 2006 were used, making a total of 2500 data points for each series. The jump intensity parameter λ was bounded from above by 250. The estimated parameter values are shown in table 1. For Kou s exponential Lévy model the equation (11) for loss probability yields ) P[ t [0, T] : V t B t ] = 1 exp ( Tpλη (1 1/m) 1/η. Figure 7 shows the dependence of the loss probability on the multiplier for a CPPI portfolios containing Microsoft stocks and CAC40 as risky asset. References [1] Y. Aït-Sahalia, Telling from discrete data whether the underlying continuous-time model is a diffusion, Journal of Finance, 57 (2002), pp

17 0.15 MSFT CAC Figure 7: Probability of loss as a function of the multiplier. [2], Disentangling diffusion from jumps, Journal of Financial Economics, 74 (2004), pp [3] O. Barndorff-Nielsen and N. Shephard, Econometrics of testing for jumps in financial economics using bipower variation, Journal of Financial Econometrics, 4 (2006), pp [4] O. E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein- Uhlenbeck based models and some of their uses in financial econometrics, J. R. Statistic. Soc. B, 63 (2001), pp [5] D. Bates, Jumps and stochastic volatility: the exchange rate processes implicit in Deutschemark options, Rev. Fin. Studies, 9 (1996), pp [6] D. S. Bates, Testing option pricing models, in Statistical Methods in Finance, vol. 14 of Handbook of Statistics, North-Holland, Amsterdam, 1996, pp [7] D. Belomestny and M. Reiss, Spectral calibration of exponential Lévy models, Finance and Stochastics, 10 (2006), pp [8] F. Black and R. Jones, Simplifying portfolio insurance, Journal of Portfolio Management, 14 (1987), pp [9] F. Black and A. Perold, Theory of constant proportion portfolio insurance, The Journal of Economics, Dynamics and Control, 16 (1992), pp [10] P. Carr, H. Geman, D. Madan, and M. Yor, Stochastic volatility for Lévy processes, Math. Finance, 13 (2003), pp

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