Bloomberg L.P.. Director, Masters Program in Math Finance. Courant Institute, NYU. Carr and Dupire on Volatility. Tuesday July 4, 2006. London, England.
Option Pricing Under Stochastic Volatility
by Peter Carr Quantitative Financial Research Bloomberg L.P. Director, Masters Program in Math Finance Courant Institute, NYU
Carr and Dupire on Volatility
Tuesday July 4, 2006
London, England
File reference: C:/Texfiles/Conferences/LeadingLights/OptionPricingUnderSV/OptionPricingUnderSV.tex
Overview Fischer Black once wrote: Suppose we use the standard deviation . . . of possible future returns on a stock . . . as a measure of its volatility. Is it reasonable to take that volatility as constant over time? I think not. • In this session, I provide a selective survey of some models for valuing European options on a single underlying which has stochastic volatility. • There are now several books on this subject, so I apologize in advance to authors whose work I miss. • There are two kinds of models covered in my talk: 1. Models that are Markovian in Stock Price and Time 2. Models with a Second Stochastic Factor. 2
Models that are Markovian in Stock Price and Time
• We review ≤ 4 approaches for pricing options in a Markovian framework: 1. Path Independent Diffusions: Assumes that stock price is a function of a contemporaneous tractable diffusion and time, possibly only after a path-independent measure change and deterministic time change. 2. Local vol: a generalization of the above, which maintains price continuity, but only requires that instantaneous volatility be a function of stock price and time. 3. L´evy processes: stock returns have stationary independent increments. 4. Local L´evy: a generalization of the above, where the jump compensator is the product of the local aggregate arrival rate and a parametrically specified arrival rate specific to the jump size. • This list is not exhaustive! For example, combinations of the above are allowed. 3
Standard Assumptions • Unless indicated otherwise, we will always assume: – frictionless markets, eg. no transactions costs, no b/a spreads – no arbitrage (and hence the existence of an EMM Q) – constant dividend yield q (often 0) – constant riskfree rate r (often 0) – a continuous time stochastic process for the stock price S.
4
Approach #1: Path-Independent Diffusions
• In this class of models, we specify a tractable diffusion as a driver and then we assume that the underlying stock price is a function of the driver and time, perhaps only after picking a convenient numeraire and clock. • The only diffusion drivers that we will cover are standard Brownian motion (SBM), and a Bessel process. • For example, the Black Scholes model assumes that the stock price S is the following exponential function of a standard Brownian motion B and time t: 2 µt σBt − σ2 t St = S0e e , t ≥ 0, where S0 > 0 is the initial stock price, µ ∈ R is the required relative drift and σ ∈ R is the assumed constant (lognormal instantaneous) volatility. 5
Other Models with a Brownian Driver
• Beating Black Scholes by ≥ 70 years, Bachelier described the stock price S by a different function of a standard Brownian motion B and time t: St = eµt [S0 + aBτ(t)],
t ≥ 0,
where S0 > 0 is the initial stock price, µ is the required relative drift, a is −2µt the assumed constant (normal instantaneous) volatility, and τ(t) ≡ 1−e2µ is a deterministic time change. • Taking the total derivative and using the Brownian scaling property leads to the following dynamics: dSt = µdt + adWt , where W is a a standard Brownian motion.
6
t ≥ 0,
Bessel Process • A Bessel process is a univariate diffusion R with positive initial level R0 and which uniquely solves the following stochastic differential equation: 2ν + 1 dt + dBt , t > 0, dRt = 2Rt where B is an SBM and ν ∈ R is called the index of the Bessel process. • The Bessel process is always positive and clearly not a martingale. • Suppose ν = δ2 − 1 where δ is a positive integer. Then the Bessel process R is the radial distance to the origin of the sum of R0 > 0 and a δ−dimensional standard Brownian motion. • A possible reason for the name “Bessel Process” is that the modified Bessel function Iν(·) appears in the closed form solution for the transition probability density function. This also explains the non-intuitive parametrization of the drift term in the SDE in terms of ν. 7
Constant Elasticity of Variance (CEV) process • From an SDE perspective, the Black Scholes model assumes that the stock price S follows geometric Brownian motion: dSt = µSt dt + σSt dBt ,
t > 0,
while the Bachelier model instead assumes an OU process: dSt = µSt dt + adBt ,
t > 0.
• In an effort to generalize both processes while introducing a skewness parameter consistent with a lower bound of zero on the stock price, Cox (1975) introduced the following CEV process: dSt = µSt dt + σStpdBt ,
8
t > 0, where p ≤ 1].
CEV as a Function of a Bessel Driver • Recall that a Bessel process R with index ν ∈ R solves the following SDE: 2ν + 1 dt + dBt , t > 0, dRt = 2Rt where R0 > and B is an SBM, while the CEV process instead solves: dSt = µSt dt + σSt dBt , p
t > 0, where S0 > 0 and p ≤ 1.
• As shown in Delbaen and Shirakawa (2002) and Cox (1975), if we let: 1−ν
2 2ν µt σ −1 e µt −2ν , Sˆt = e Rτ(t) , where τ(t) ≡ 2ν (1 − ν)µ then from Itˆo’s formula and the Brownian scaling property:
d Sˆt = µSˆt dt − σSˆt
ν+1 2ν
dWt ,
where W is an SBM. • Hence, if we set p ≡ ν+1 , then Sˆ has the same law as the solution to the 2ν SDE for the CEV process. 9
Models with Affine Drift • So far all of the models have required that the stock price have constant proportional drift. • A simple way to let the stock price drift be affine instead of proportional is to let the cushion Ct ≡ St − F be a drifting geometric Brownian motion: 2 /2)t+σB t
Ct = C0 e(µ−σ
,
t > 0,
where B is a standard Brownian motion. • Using Itˆo’s formula, the stock price’s drift and volatility are both affine: dSt = µ(St − F)dt + σ(St − F)dWt ,
t > 0.
• St can be interpreted as the future value at t in a zero interest rate economy when an initial investment of $S0 is split between $F in a riskless asset and $C0 ≡ S0 − F in a drifting GBM. • Constant Proportional Portfolio Insurance (CPPI) is the special case when S is interpreted as wealth and F < S0 is interpreted as the floor. Since a GBM is always positive, the price process S is always above F. 10
Affine Drift and Bessel Drivers • Recall that for the dynamics on the last slide, the stock price is bounded below by the floor F. Setting F = 0 forces the stock price drift to be proportional. As a result, constant dollar dividends cannot be considered. • In his Nobel prize winning paper, Merton (1973) considered the process: dSt = (µSt − c)dt + σSt dBt ,
t > 0, where S0 > 0,
and where B is an SBM. Here, c ≥ 0 is a constant dividend paid continuously over time until the S process first hits zero. • Merton derived the solution for a perpetual option on such a process, while Lewis (1997) derived the solution for the finite-lived case. • Similarly, in their path-breaking paper, Cox Ross (1976) suggested the SDE: p t > 0. dSt = (µSt − c)dt + σ St dBt , • In both cases, the key is to be able to write the S process as a function of a Bessel process, running on a deterministically time-changed clock. 11
Some Generalizations
• Carr, Lipton, & Madan (2000) assume that the risk-neutral process for a state variable S is: dSt = b(St ,t)dt + a(St ,t)dWt ,
t ∈ [0, T ],
and that the discount rate is level-dependent rt = c(St ,t). • In this general setup, the values of path-independent equity options or interest rate derivatives solve a PDE with variable coefficients. • Under weak restrictions on the arbitrary coefficients {a(S,t), b(S,t), c(S,t)}, a scale change of S to X = f (S,t) can render unit vol. Expressing the price of the overlying (ONLY) in terms of a new (dividend-paying) numeraire can also render zero drift (and keep vol at one). We thus obtain a normalized PDE with one coefficient (called either the killing rate or potential).
12
A Further Generalization (con’d)
• If this killing rate/potential is quadratic in X, then it can be eliminated through deterministic time change, along with further changes in the underlying and overlying, leaving the (backward) heat equation (see Carr, Lipton, Madan 2000). • Alternatively, if this discount rate is a quadratic in X plus a nonzero function of time divided by X 2, then we can reduce to a Bessel PDE (see Linetsky’s IJTAF spectral representation paper for details). • Recently, Carr, Laurence, and Wang (2006) derive an expression which the normal volatility coefficient must satisfy in order that a driftless diffusion without killing/discounting can be reduced to a Bessel process.
13
References
[1] Adjaoute, K., 1994, “Les Modeles D’Evaluation D’Option a Elasticite de Variance Constante,” HEC Lausanne working paper. [2] Ang J. and D. Peterson, 1984, “Empirical Properties of the Elasticity Coefficient in the Constant Elasticity of Variance Model,” Financial Review, Vol. 19, No. 4, 372-80. [3] Beckers, S., 1980 “The Constant Elasticity of Variance Model and its Implications for Option Pricing,” Journal of Finance, 35, 661-73. [4] Black, F., 1976, Studies in Stock Price Volatility Change, in Proceedings of the 1976 Business and Economics Statistics Selection, American Statistical Association, 177–181. [5] Carr, P., M. Tari, and T. Zariphopolou, “Closed Form Option Pricing with Smiles”, Banc of America Securities working paper. [6] Carr, P., A. Lipton, and D. Madan, 2000, “The Reduction Method for Valuing Derivative Securities”, Bloomberg working paper. [7] Carr, P., P. Laurence, and T. Wang, 2006, “Generating Integrable One Dimensional Driftless Diffusions”, Comptes Rendus, forthcoming. 14
[8] Choi, J. and F. Longstaff, 1985, “Pricing Options on Agricultural Futures: An Application of the Constant Elasticity of Variance Option Pricing Model,” Journal of Futures Markets, 5, Spring, 247-58. [9] Christie, A., 1982, “The Stochastic Behavior of Common Stock Variances,” Journal of Financial Economics, 10, 407-32. [10] Christie, A., 1993, “Constant Elasticity of Variance Models and Portfolio Formation,” University of Rochester working paper. [11] Cox, J., 1975, “Notes on Option Pricing I: Constant Elasticity of Variance Diffusions,” Stanford University working paper. [12] Cox, J. and S. Ross, 1976, “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics, 3, 145-166. [13] Delbaen, F., and H. Shirakawa, 2002, “A Note on Option Pricing for the Constant Elasticity of Variance Model”, Asia-Pacific Financial Markets, 9, 85–99. [14] Emanuel, D. and J. MacBeth, 1982, “Further Results on the Constant Elasticity of Variance Call Option Pricing Model,” Journal of Financial and Quantitative Analysis, 17, Nov., 533-54.
15
[15] Feller, W., 1951, “Two Singular Diffusions,” Annals of Mathematics, 54, 1, July, 173-82. [16] G´eman, H. and M. Yor, 1992, “Bessel Processes, Asian Options and Perpetuities, Universit´e Pierre et Marie Curie working paper. [17] Gibbons, M. and C. Jacklin, 1990, “CEV Diffusion Estimation,” Stanford University working paper [18] Goldenberg, D., 1991, “A Unified Method for Pricing Options on Diffusion Processes,” Journal of Financial Economics, 29 Mar., 3-34. [19] Hauser, S., and C. Bagley, 1986, “Estimation and Empirical Evidence on Foreign Currency Call Options: Constant Elasticity of Variance Model,” Temple University working paper. [20] Lewis, A., 1987, “Applications of Eigenfunction Expansion in Continuous Time Finance”, working paper. [21] Linetsky, V., 2004, “The Spectral Decompositon of the Option Value”, International Journal of Theoretical and Applied Finance. [22] MacBeth J., and L. Merville, “Tests of the Black-Scholes and Cox Call Option Pricing Models,” Journal of Finance, 35, 2, 285-300 16
[23] Milonas, N., 1986, “A Note on Agricultural Options and the Variance of Futures Prices,” The Journal of Futures Markets, Vol. 4, No. 4, 671-76. [24] Schroder, M., 1989, “Computing the Constant Elasticity of Variance Option Pricing Formula,” Journal of Finance, 44, Mar., 211-219. [25] Tucker, A., D. Peterson and E. Scott, 1988, “Tests of the Black-Scholes and Constant Elasticity of Variance Currency Call option Valuation Models,” Journal of Financial Research, Vol. 11, No. 3, 201-14. [26] Z¨uhlsdorff, C., 1999, “The Pricing of Derivatives on Assets with Quadratic Volatility”, Bonn University working paper.
17
Approach #2: Local Vol • Define a complete implied volatility surface as a C2,1 function whose output is the Black Scholes implied volatility and whose two inputs are some measure of moneyness, eg. strike K and some measure of time to maturity (eg. calendar time to maturity T ). • In the non-parametric approach, one first uses splines or regressions or an arbitrage-free option pricing model to obtain a complete implied volatility surface as in ad hoc BS. One can then convert these implieds to call prices C(K, T ). • Dupire (1994) showed that under no arbitrage and the assumption that: dSt = µt St dt + a(St ,t)dWt ,
t ≥ 0,
the deterministic function a(S,t) can be inferred from the term and strike structure of call prices and hence implied vol’s.
18
Local Vol (Con’d) • At a given time t ≥ 0 and for a fixed spot price level K and a fixed point in calendar time T ≥ t, define at2(K, T ) as the local variance at (K, T ): 2 (S − S ) T +4t T at2(K, T ) ≡ lim E |ST = K, Ft , 4t↓0 4t where Ft is the filtration (information set) at t. • In general, the local variance at2(K, T ) is a nonnegative scalar stochastic process for each fixed (K, T ). • However, Dupire (1994) assumed that the filtration just consists of stock prices and that the stock price is Markovian in itself and time. As a result, local variance is a constant scalar process for each fixed (K, T ): 2 (S − S ) T +4t T |ST = K ≡ a2(K, T ). at2(K, T ) = lim E 4t↓0 4t • The square root of the function on the right is being used when we write: dSt = µt St dt + a(St ,t)dWt , 19
t ≥ 0.
Limiting Normalized Calendar Spreads • Consider the initial price of a calendar spread of 2 calls with the same strike price K: CS0(K, T ) = C0 (K, T + 4T ) −C0(K, T ). • The bigger is 4T , the bigger is the cost of a calendar spread, so suppose that the positions are normalized by dividing by the difference in maturities: C0 (K, T + 4T ) −C0(K, T ) . NCS0(K, T ) = 4T • Now let the difference in maturities approach zero: C0(K, T + 4T ) −C0(K, T ) ∂C0 (K, T ) LNCS = lim ≡ . 4T ↓0 4T ∂T • The result is the initial price of a limiting normalized calendar spread, or more simply a calendar spread. • If one has a mechanism for generating implied volatility surfaces, one can (K,T ) at time 0. observe ∂C0∂T 20
Risk-Neutral PDF • Assume that the UK riskfree rate is zero for simplicity and take one pound as the numeraire. • The first fundamental theorem of asset pricing says that under no arbitrage, there exists a risk-neutral measure Q such that the (pound denominated) price of each non-dividend paying asset is the expected value under Q of its payoff. • Consider a claim that pays one pound at T if the stock price at T is below some constant K. The forward price of this claim is the risk-neutral probability that ST < K. • Let q0(K, T ) denote the risk-neutral probability density function (PDF) observed at time 0. If it exists, this function gives the rate at which the above forward price changes as we vary K: q0(T, K)dK ≡ Q{ST ∈ (K, K + dK)|F0}.
21
Limiting Normalized Butterfly Spreads • From Breeden and Litzenberger (1978), the risk-neutral PDF q0(T, K) can be obtained from the initial market price of a limiting normalized butterfly spread: ∂C0 (K, T ) ∂K 2 C0 (K + 4K, T ) − 2C0(K, T ) +C0(K − 4K, T ) = lim . 4K↓0 (4K)2
q(0, S0; T, K) =
• As with the calendar spread, the initial price of the limiting normalized butterfly spread can be observed initially and is just referred to as a butterfly spread. • In a Markovian setting, one can work more simply with the risk-neutral transition probability density function: q(t, S; T, K)dK ≡ Q{ST ∈ (K, K + dK)|St = S}.
22
The Market’s Forecast of Local Volatility • For simplicity, we assume no dividends in addition to zero interest rates. • Under the assumptions made, the ratio of the calendar spread to the butterfly spread can be interpreted as the market’s perfect forecast of half the absolute variance rate, conditional on the future spot price being at K at T. ∂C0(K,T ) 1 2 ∂T a (K, T ). = ∂2C0(K,T ) 2 2 ∂K
• The next page starts to explains why. • The assumptions imply that the RHS is independent of the date 0 at which the market makes its forecast. • To obtain the market forecast of local relative vol instead, double the ratio, take the square root and divide by K.
23
Why can Local Variance be Forecasted? • Let a2(K, T ) denote the forecast of local variance. Recall that Dupire assumed that this function does not vary through calendar time, so no time subscript on a2(K, T ) is needed. • ∂2C0(K, T ) ∂C0 (K, T ) 1 2 = a (K, T ) , By re-arranging: ∂T 2 ∂K 2 which is called Dupire’s forward PDE. • Consider the payoff from holding the (limiting normalized) calendar spread on the LHS to the earlier maturity T . • Paths which finish strictly below K provide zero payoff. Similarly, paths which finish strictly above K also provide zero payoff. • Paths which finish at K actually have a payoff which is infinite. If we condition on ST = K, then the bigger is the market’s forecast at time 0 of the volatility to be experienced over (T, T + dT ), the larger is the conditional value of the calendar spread at T and hence the larger is the unconditional value initially. 24
More Math (K,T ) 0(K,T ) • Recall Dupire’s forward PDE: ∂C0∂T = 12 a2(K, T ) ∂ C∂K . To prove it, 2 one can integrate the Kolmogorov forward p.d.e. in the forward spatial variable twice (see Dupire’s 1994 Risk paper). • An alternative approach uses a generalization of Itˆo’s lemma to nondifferentiable convex functions such as f (S) = (S − K)+. • Yet, a third approach is to consider a binomial model and take limits. Over the time interval (T, T + 4T ) and conditioning on ST = K, we have: 2
√ K + a(K, T ) 4t *
K
H H H HH H H j H
√
K − a(K, T ) 4t
• To make S a martingale, the risk-neutral probabilities are both 12 . 25
Binomial Derivation √ K + a(K, T ) 4t *
Recall: K
H H H HH H H j H
√
K − a(K, T ) 4t
• Since qu = 12 and r = 0, the calendar spread at T given ST = K has value: √ CT (K, T + 4t) −CT (K, T ) 1 a(K, T ) 4t 1 a(K, T ) = = √ . 4t 2 4t 2 4t √ • If the stock price at T is at K − 2a(K, T ) 4t or below, then the calendar spread struck at K is worthless since both calls in it must expire OTM. √ • If the stock price at T is at K + 2a(K, T ) 4t or above, then the calendar spread struck at K is also worthless since both calls in it must expire ITM. ) √ 1(ST = K). • Thus, the calendar spread struck at K sells at T for 12 a(K,T 4t 26
Binomial Derivation (con’d) • Consider the payoff of a normalized butterfly spread maturing at T : $
1√ a(K,T ) 4t
@ @ @ @ @ @ @ @ @ √ K K + a(K, T) 4t Stock Price at T Figure 1: Payoff at T of a Butterfly Spread.
√ K − a(K, T) 4t
27
Binomial Derivation (con’d) • Since the strike K is attainable by the binomial process, the strikes K ± a(K, T )4t are not attainable by the process. • Note that the area under the triangular payoff is one. 1 × Base × Height 2 p 1 1 √ = × 2a(K, T ) 4t × 2 a(K, T ) 4t = 1.
Area =
• The height of the triangular payoff is a(K,T1)√4t . If an investor is long one butterfly spread and the stock price hits K at T , then the investor gets 1√ pounds. If ST 6= K, the investor gets nothing. a(K,T ) 4t
28
Binomial Derivation (Con’d) a2(K,T ) butterfly spreads. If the stock price hits 2 2 ) ) √ × a(K,T1)√4t = 12 a(K,T pounds. If ST 6= gets a (K,T 2 4t
• Suppose an investor holds
K at T , then the investor K, the investor gets nothing. • Recall that the calendar spread liquidated at T pays pounds.
a2 (K,T ) 2
• Hence this payoff can be duplicated by holding from time 0 to time T . 0(K,T ) • It follows from no arbitrage that C0(K,T +4t)−C 4t
) 1 a(K,T √ 2 4t 1(ST
butterfly spreads
1 C0(K + 4K, T ) − 2C0(K, T ) +C0(K − 4K, T ) = a2(K, T ) , 2 (4K)2 √ where 4K = a(K, T ) 4t. • In the diffusion limit, we have Dupire’s forward PDE: ∂C0 (K, T ) 1 2 ∂2C0(K, T ) = a (K, T ) . ∂T 2 ∂K 2 29
= K)
Dupire’s Forward PDE • If we add a deterministic riskfree rate r(t) and a deterministic dividend yield q(t), Dupire’s PDE generalizes to the following forward PDE for European calls: ∂C0 1 2 ∂2C0 ∂C0 = a (K, T ) 2 − [r(T ) − q(T )]K − q(T )C0. ∂T 2 ∂K ∂K • One interpretation of the extra terms is that they represent the stock carrying cost which must be paid when the stock price is above the strike at the earlier maturity. By re-arrangement: ∂C0 K2 2 ∂2C0 ∂C0 + [r(T ) − q(T )]K + q(T )C0 = a (K, T ) 2 . ∂T ∂K 2 ∂K • The LHS is the cost of a ratioed calendar spread. Its value is nonnegative for any stochastic process for the underlying. • The exact same PDE holds for European puts or any payoff with a single kink. 30
Dupire’s Forward PDE (Con’d) • Recall Dupire’s forward PDE for calls under a deterministic riskfree rate r(t) and a deterministic dividend yield q(t): ∂C0 a2(K, T ) ∂2C0 ∂C0 + [r(T ) − q(T )]K + q(T )C0 = . ∂T ∂K 2 ∂K 2 • To express local volatility in terms of option prices, solve the forward PDE for a(K, T ): v n o u ∂C (K,T ) ∂C (K,T ) u2 0 + [r(T ) − q(T )]K 0∂K + q(T )C0(K, T ) u ∂T . a(K, T ) ≡ t ∂2C (K,T ) 0
∂K 2
• To express local volatility in terms of implied volatility rather than option prices, substitute in the Black Scholes formula and carry out the differentiations.
31
Valuation of Derivatives • Since the stock price process is continuous and driven by a single Brownian motion, one can theoretically replicate the payoff on any claim by dynamic trading in stock and bond. • Since the stock price process is Markov in itself and time, the value function V (S,t) for many claims (eg. standard American options or barrier options) solves the following backward PDE: ∂V(S,t) ∂V (S,t) a2(S,t) ∂2V (S,t) + = r(t)V (S,t). + [r(t) − q(t)]S ∂t 2 ∂S2 ∂S • By appending the appropriate boundary conditions, one can numerically solve the PDE using say finite differences.
32
Hedging with Smiles • Recall that V (S,t) is the value function obtained by numerically solving a backward PDE. • To obtain delta, the standard approach is to use the following centered finite difference approximation: V (S0 + 4S,t0) −V (S0 − 4S,t0) ∂ V (S0,t0) = + O(4S2). ∂S 24S • One can alternatively differentiate the backward PDE w.r.t. S to get a PDE for delta. • To obtain the deltas of European options, one can just differentiate the forward PDE. The form of the PDE does not change and one obtains deltas for all strikes and maturities with a single pass. • The same idea holds for thetas and gammas of European options (which theoretically one does not need to know to carry out a delta hedge).
33
Naive Empirical Tests • Both the parametric and non-parametric forms of price-dependent instantaneous volatility models make strong predictions which can be tested. • For example when interpreted literally, both approaches require only knowledge of the stock price and the time in order to predict the entire implied volatility surface at each time after the initial calibration. • In order to test the validity of the model, this prediction can be compared to the implied volatilities obtained from market prices of options. • The general finding from index options is that local vol models predict future short term skews which have weaker dependence on strike than is found in the data.
34
Practical Empirical Tests • In practice, models are re-calibrated each day, and as a result, it is reasonable to ask how well the re-calibrated versions of these models perform. • Models with continuous price dynamics make very specific predictions about the average change in the implied volatility surface for a given change in the stock price and time. When the stock price process is also Markovian in just itself and time, the prediction is not just on average but is made path by path. • To eliminate variance just by dynamic delta hedging, we need no other state variables, no price jumps, and a correct prediction of how the implied volatility surface moves with (small) changes in stock price and time. • For example, suppose that implied volatility slopes down in K at each T . Suppose S = K = 100. If S moves up, then the local vol model predicts that the implied for strike 100 moves down. If this implied actually rises, then using the local vol delta injects more variability into the P&L than a model that correctly predicts surface changes. 35
Approach 3: L´evy Processes
• In this segment, I will quickly survey the link between European option pricing and characteristic functions. • I will then introduce L´evy processes and provide a selective survey of some option valuation models which use non-Gaussian L´evy processes to model returns. • I apologize in advance to authors whose work I miss; to simply list all the work on this subject would fill up the segment. • Much of what I review can be downloaded from www.math.nyu.edu/research/carrp/papers
36
European Option Pricing & RN PDF
• Let S be a positive stock price process. Let s ≡ ln ST denote the log of the terminal stock price and k ≡ ln k denote the log of the strike. • The standard approach to European option valuation expresses the option value c(k; T ) as an expectation of its discounted payoff: c(k; T ) ≡ e−rT
Z∞
es − ek q(s; T )ds.
k
• This approach requires that the risk-neutral density q(s; T ) be known and fast valuation requires that it be simple.
37
European Option Pricing & Fourier Methods • Fourier methods describe an array of techniques that can be used to obtain option prices. • For fixed maturity T , consider the call value c as a function of the log of its strike k. If its Fourier transform exists, then this function is given by: ψ(v; T ) ≡
Z∞
eivk c(k; T )dk.
−∞
• By the Fourier inversion theorem, the call value is the inverse of its Fourier transform: Z∞ 1 c(k; T ) = e−ivk ψ(v; T )dv. 2π −∞
• This approach requires that the Fourier transform ψ(v; T ) be known and fast valuation requires that it be simple. 38
Characteristic Functions
• Recall that q(s; T ) denotes the risk-neutral (RN) probability density function (PDF) of log spot sT ≡ ln ST . • When q is considered as a function of s for fixed maturity T , then its Fourier transform (FT) is called the characteristic function (CF) of sT : φ(v; T ) ≡ EeivsT =
Z∞
eivsq(s; T )ds.
−∞
• By the Fourier inversion theorem, the PDF is obtained by inverting the characteristic function of sT : 1 q(s; T ) = 2π
Z∞
−∞
39
e−ivsφ(v; T )dv.
Obtaining Option Values at Many Strikes
• Recall that ψ(v; T ) ≡
R∞
eivk c(k; T )dk denotes the Fourier Transform (FT)
−∞
of the call price c(k; T ) and that φ(v; T ) ≡ EeivsT denotes the the characteristic function (CF) of sT ≡ ln ST . • Carr Madan JCF99 shows how to analytically relate the FT of the call price ψ(v; T ) to the CF φ(v; T ). • We then numerically invert this FT using the Fast Fourier Transform (FFT). Thus, the option valuation problem is reduced to determining the CF of the terminal log price. • Our approach outputs a vector of option prices at many strikes, so our approach is particularly fast when this is the output needed.
40
Valuing Options on Exponentials of L´evy Processes
• When the log price is a L´evy process, then its characteristic function is explicitly given by the L´evy-Khintchine Theorem. • Thus one can numerically value all options written on exponential L´evy processes, even if the transition density is unknown. • If the transition density is known, our numerical results show that it may still be faster to work in Fourier space, as some stochastic processes generate terminal random variables whose characteristic function is simpler than its PDF (eg. Variance Gamma).
41
Fourier Transform of Dampened Call Value • Let c(k; T ) be the value of a T maturity call with log strike k. Let q(s; T ) denote the RN PDF of the terminal log price sT . The CF of this PDF is: φ(ω; T ) ≡
Z
∞
eiωsq(s; T )ds.
−∞
• The initial call value c(k; T ) is related to the risk-neutral density q(s; T )by: Z ∞ c(k; T ) ≡ e−rT es − ek q(s; T )ds. k
• The function c is not square integrable in k, so we Fourier transform the dampened call price, eαk c(k; T ), α > 0: ψ(ω; T ) ≡
Z
∞
e −∞
e−rT φ(ω − (α + 1)i; T ) . e c(k; T )dk = . . . = 2 α + α − ω2 + i(2α + 1)ω
iωk αk
• Calls are valued by inverting both the FT and the dampening factor: Z Z exp(−αk) ∞ −iωk exp(−αk) ∞ −iωk c(k; T ) = e ψ(ω; T )dω = e ψ(ω; T )dω. 2π π −∞ 0 42
Terminology 101: Stable vs L´evy Processes
• Probabilists and physicists use the same words to describe different things. I will use probabilistic language in what follows. • In words, L´evy processes are the class of continuous time stochastic processes with stationary independent increments. • In words, stable processes are a subset of L´evy processes which loosely speaking have power law tails. The formal definition expresses the L´evy density as a particular function of the jump size. • Physicists often refer to stable processes as L´evy flights or L´evy processes.
43
Terminology 102: Variance vs Quadratic Variation
• A random variable has a symmetric stable distribution if all odd moments vanish and tail behavior is governed by the tail parameter α ∈ (0, 2]. • If random variable X is distributed symmetric stable(α), then EX p exists for p ∈ [0, α] but not for p > α. A standard normal arises when α = 2. • As usual, one can shift and scale a stable random variable and a skewness parameter β ∈ [−1, 1] can also be introduced. • For α ∈ (0, 2), the variance of X is always infinite, but the quadratic variation of a stable process is finite path by path.
44
L´evy Processes and L´evy Khintchine Theorem • By the L´evy Khintchine theorem, the characteristic function (FT of PDF) of the random variable sT when s is a L´evy process (i.e. right continuous left limits processes with stationary independent increments) is given by: 2 2 σ ω + I1(ω) + I2(ω) , EeiωsT = exp T iˆµω − 2 where: Z iω j I1(ω) ≡ e − 1 `(d j) I2(ω) ≡
Z| j|≥1
0 i φXt (θ) ≡ E eiθ Xt = e−tΨx(θ) , t ≥ 0, θ ∈ Rd where the characteristic exponent Ψx(θ), θ ∈ Rd , is given: Z > 1 > > iθ x > Ψx (θ) ≡ −iµ θ + θ Σθ − − 1 − iθ x1|x| ii ii h h > iθ> XTt iθ Xu v iθ Xu φy (θ) ≡ Ee =E E e =E E e Ft Tt = u = Ee−Tt Ψx (θ) = LT (Ψx(θ))
which is the Laplace transform of the stochastic time Tt = evaluated at the characteristic exponent of X.
Rt
0 v(s−)ds,
• To obtain the Laplace transform of T in closed form, consider its specification in terms of the activity rate: Zt LT (λ) ≡ E exp −λ v(s−)ds 0
analogous to the bond pricing formula if we regard v(t) as analogous to the instantaneous spot interest rate. Hence, we can “borrow” closed form 92
solutions for zero coupon bonds that arise under affine, quadratic term structure models, etc.
93
The General Case of Correlated Time Changes • More generally, the generalized CF of Yt ≡ XTt under measure P can be represented as the “Laplace transform” of Tt under a new complex-valued measure Q(θ), evaluated at the characteristic exponent Ψx(θ) of Xt , h > i h i iθ Yt θ −Tt Ψx (θ) φYt (θ) ≡ E e (1) =E e ≡ LTθt (Ψx(θ)) . • For each θ ∈ D , Q(θ) is absolutely continuous with respect to P and is defined by dQ(θ) |FTt ≡ Mt (θ) ≡ exp iθ>Yt + Tt Ψx(θ) , θ ∈ D , (2) E dP which is a complex valued P-martingale with respect to {FTt |t ≥ 0}, for each θ ∈ D .
94
Intuition and Theorem Proof • Why is Mt (θ) ≡ gale?
E dQ(θ) | dP FTt
= exp iθ Yt + Tt Ψx(θ) , θ ∈ D a P martin>
– Recall the familiar Wald martingale defined on a L´evy process > Zt (θ) ≡ exp iθ Xt + tΨx(θ) . – Time change (i.e. replacing t by Tt ) preserves the martingality. • Theorem proof: h > i h > i iθ Yt iθ Yt +Tt Ψx (θ)−Tt Ψx (θ) E e = E e h i h i −Tt Ψx (θ) θ −Tt Ψx (θ) = E Mt (θ)e =E e ≡ LTθt (Ψx(θ)) . • The complex-valued measure loses its probabilistic interpretation, but the mathematical operation remains valid. 95
• Under measure Q(θ), we can take “expectations” as if there is no correlation ⇒ leverage-neutral measure.
96
Asset Pricing under Time-Changed L´evy Processes • Let St be the time-t price of a limited liability asset under statistical measure: St ≡ S0eϑ
>Y t
,t ≥ 0, for given S0 > 0, where recall Yt ≡ XTt .
• The generalized CF of the log return st ≡ ln(St /S0) is h > i iθst φs (θ) ≡ E e = E eiθϑ Yt = LTθϑ(Ψx(θϑ)), t ≥ 0, θϑ ∈ D ⊆ Cd . • Let Ft (M) be the M maturity forward price at time t ∈ [0, M]. To value most European-style claims maturing at M, specify F as a positive martingale > > Ft (M) ≡ F0(M)eϑ Yt +Tt Ψx(−iϑ) , t ∈ [0, M] under an M−forward measure. Tt is now a vector of stochastic clocks so > > that eϑ Yt +Tt Ψx (−iϑ) is an exponential martingale. 97
• The generalized CF of the terminal log return fM ≡ ln(FM (M)/F0(M)) is φ f (θ) ≡ E eiθ fM = LTθϑ (Ψx(θϑ) − iθΨx(−iϑ)).
98
Specification Analysis Ia L´evy processes and Characteristic Exponents: L´evy Components L´evy Density Π(dx)/dx Characteristic Exponent Ψ(θ) Pure Continuous L´evy component µt + σWt
Merton (76) Kou (99) Eraker (2001)
−iµθ + 12 σ2θ2
—
Finite Activity Pure Jump L´evy components 1 σ2 θ2 2 (x−α) iθα− 1 2 j λ q 2 exp − 2σ2 λ 1−e 2πσ j j 2 |x−k| 1 iθk 1−η λ 2η exp − η λ 1 − e 1+θ2η2 1 x 1 λ η exp − η λ 1 − 1−iθη
99
Specification Analysis Ib L´evy processes and Characteristic Exponents: Infinite Activity Jumps L´evy ComponentsL´evy Density Π(dx)/dx NIG Hyperbolic
CGMY
δα eβx π|x| K1 (α|x|) √ eβx |x|
R∞
2
− 2y+α |x| e √ 0 π2 y J 2 (δ 2y)+Y 2 (δ√2y) dy |λ| |λ| −α|x| +1λ≥0λe −G|x| −Y −1
Ce |x| , x < 0, Ce−M|x| |x|−Y −1, x > 0
Characteristic Exponent Ψ(θ) hp i p 2 2 2 2 −δ α − β − α − (β + iθ) √ λ √ 2 2 δ K α −(β+iθ) λ α2 −β2 √ − ln √ 2 2 2 2 α −(β+iθ)
VG
(µ± = LS
c|x|−α−1,
α2 4λ2
σ2j +2
α ± 2λ , v± = µ2±/λ)
x )σ(Z ) σ(Z t t ii > σ(Zt )σ(Zt ) i j γ(Zt )
= = = = =
b> v Zt + cv, a − κZt , αi + β> i Zt , 0, i 6= j, aγ + b> γ Zt .
Laplace Transform LTt (λ) ≡ E e−λTt exp −b(t)>z0 − c(t) , b0(t) = λbv − κ>b(t) − 12 βb(t)2 −bγ (Lq(b(t)) − 1), c0(t) = λcv + b(t)>a − 12 b(t)>αb(t) −aγ (Lq(b(t)) − 1), b(0) = 0, c(0) = 0.
101
Specification Analysis IIb Activity Rate Dynamics and the Laplace Transform : Generalized Affine: Filipovic (2001) Laplace Transform LTt (λ) ≡ E e−λTt
Activity Rate Specification v(t)
A f (x) = 12 σ2x f 00(x) + (a0 − κx) f 0(x) R
f (x) + f (x) (1 ∧ y)) (m(dy) + xµ(dy)), R a0 = a + R+ (1 ∧ y)m(dy), 0 R 2 R+ (1 ∧ y)m(dy) + 1 ∧ y µ(dy) < ∞. +
( f (x + y) − R+ 0
0
0
102
exp (−b(t)v0 − c(t)), b0(t) = λ − κb(t) − 12 σ2b(t)2 R + R+ 1 − e−yb(t) − b(t)(1 ∧ y) µ(dy), 0 R 0 −yb(t) c (t) = ab(t) + R+ 1 − e m(dy), 0 b(0) = c(0) = 0.
Specification Analysis IIc Activity Rate Dynamics and the Laplace Transform Quadratic: Leippold and Wu (2002) Activity Rate Specification v(t)
Laplace Transform −λT LTt (λ) ≡ E e t
h i > > exp −z0 A (t) z0 − b (t) z0 − c (t) , µ(Z) = −κZ, σ(Z) = I, v(t) = Zt>AvZt + b> v Zt + cv.
A0(t) = λAv − A (t) κ − κ>A (τ) − 2A (t)2 , b0(t) = λbv − κb(t) − 2A (t)> b (t) , c0(t) = λcv + trA (t) − b(t)>b(t)/2, A(0) = 0, b(0) = 0, c(0) = 0.
103
Specification Analysis IIIa Correlations and Measure Changes • Correlation via diffusions: Example Xt = Wt ,
p dv(t) = (a − κv(t))dt + η v(t)dZt ,
dWt dZt = ρdt.
– Measure change:
Z dQ(θ) 1 2 t = exp iθYt + θ v(s)ds . dP Ft 2 0
p – v(t) dynamics under Q(θ): dv(t) = (a−(κ−iθηρ)v(t))dt +η v(t)dZt , which remains affine.
104
Specification Analysis IIIa Correlations and Measure Changes: Correlation via jumps: • Example Xt = Ltα,−1, dv(t) = (a − κv(t)) dt − β1/α dLTα,−1 , t where Ltα,−1 denotes a standard L´evy α-stable motion with tail index α ∈ (1, 2] and maximum negative skewness. • Measure change: dQ(θ) α,−1 = exp iθLTt + Ψx (θ)Tt , dP
Ψx(θ) = − (iθ)α sec
Ft
πα , 2
Im (θ) < 0.
• Under this new (leverage-neutral) measure, Πθ (dx) = eiθxΠ(dx), and v(t) satisfies generalized affine:
LTθt (Ψx (θ)) = exp (−b(t)v0 − c(t)). 105
(3)
Pricing State Contingent Claims • Consider a payoff at a given fixed time M which is any linear combination of the following payoffs: ϑ>YM ΠY (k; a, b, ϑ, c) = a + be 1c>YM ≤k
• Examples of claims covered by the above structure include: – European call with strike K: Π(ln(F0(M)/K); −K, F0(M), ϑ, −ϑ), – European put with strike K: Π(ln(K/F0(M)); K, −F0(M), ϑ, ϑ), – A protected put: max[SM , K] = Π(ln(F0(M)/K); 0, F0(M), ϑ, −ϑ)+Π(ln(K/F0(M)); K, 0 – A binary call: Π(ln(F0(M)/K); 1, 0, 0, −ϑ). > where recall that Ft (M) = F0(M)eϑ Yt is the M maturity forward price of the underlying asset at time t ∈ [0, M]. • State price: Let G(k; a, b, ϑ, c; M) denote the price of such a claim. We can compute G with two transform methods using φY . 106
Transform I • Let G I (z; a, b, ϑ, c) denote a Fourier transform of state price G(k; a, b, ϑ, c), defined as
G (z; a, b, ϑ, c) ≡ I
Z
+∞
eizk dG(k; a, b, ϑ, c),
z ∈ R.
(4)
−∞
• G I (z; a, b, ϑ, c) can be written as an affine function of the generalized Fourier transform of YM :
G I (z; a, b, ϑ, c) = aφY (zc) + bφY (zc − iϑ). • The price G(k; a, b, ϑ, c) can then be obtained by inversion: Z
1 ∞ eizk G I (−z; a, b, ϑ, c) − e−izkG I (z; a, b, ϑ, c) G(k; a, b, ϑ, c) = + dz, 2 2π 0 iz where G0I = G I (0; a, b, ϑ, c) = a + bφY (−iϑ).
G0I
Note that this is a one-dimensional inversion regardless of the dimensionality of the uncertainty YM . 107
Transform II • Define a second transform G II (z; a, b, ϑ, c):
G (z; a, b, ϑ, c) ≡ II
G I (z; a, b, ϑ, c) ≡
Z
+∞
Z−∞ +∞
eizk G(k; a, b, ϑ, c) dk, eizk dG(k; a, b, ϑ, c),
z ∈ C ⊆ C. z ∈ R.
−∞
Note the two differences between G I and G II . • G II , when well-defined, is given by: i G II (z; a, b, ϑ, c) = (aφY (zc) + bφY (zc − iϑ)). z • Inversion: Z 1 izi +∞ −izk II G(k) = e G (z; a, b, ϑ, c)dz. 2π izi −∞ • This inversion can be performed numerically using FFT or Fractional FFT, generating superior computational efficiency. By vectorizing in maturity, the model’s option values at all strikes and maturities can be obtained in one stroke. 108
Transform II: The Dampening Factor The choice of Im z is crucial and depends upon the payoff structure. Contingent Claim
Generalized transform −izϕ(z)
Restrictions on Im z
G(k; a, b, ϑ, c)
aφY (zc) + bφY (zc − iϑ)
(0, ∞)
G(−k; a, b, ϑ, c)
aφY (−zc) + bφY (−zc − iϑ)
(−∞, 0)
eαk G(k; a, b, ϑ, c)
aφY ((z − iα)c) + bφY ((z − iα)c − iϑ)
(α, ∞)
eβk G(−k; a, b, ϑ, c)
aφY (−(z − iβ)c) + bφY (−(z − iβ)c − iϑ)
(−∞, β)
a1φY ((z − iα)c1) + b1φY ((z − iα)c1 − iϑ1) eαk G(k; a1, b1, ϑ1, c1) +eβk G(−k; a2, b2, ϑ2, c2) +a2φY (−(z − iβ)c2) + b2φY (−(z − iβ)c2 − iϑ2) (α, β) (α, β, a, b are real constants with α < β.) 109
Summary • We provide a powerful tool in generating tractable option pricing models. – Apply stochastic time change to L´evy processes – Prove a theorem that facilitates the derivation of the CF. – Price options via transform methods. • Applications: Model design and calibration – Huang and Wu (JF 2004): Specification analysis for equity index options – Carr and Wu (wp, 2004): Stochastic skew in currency options – Bakshi, Carr, and Wu (wp, 2004): Stochastic discount factors in international economies.
110
Approach #3: Time-Changed Markov Processes • Consider a time homogeneous Markov process X with infinitessimal generator G . • Formally, the forward price at time t, maturity T of a contingent claim paying f (XT ) at T is given by: E Q[ f (XT )|Xt = x] = [e(T −t)G f ](x). • Now let φ(x) be an eigenfunction solving:
G φ(x) = λφ(x). • The forward value at time t of a claim with an eigenfunction payoff φ(XT ) paid at T factors into the product of its intrinsic value φ(Xt ) and its time value e(T −t)λ: E Q[ f (XT )|Xt = x] = φ(x)e(T −t)λ .
111
Independent Time Change • Let τt be a stochastic clock, i.e. an increasing random process with τ0 = 0. • Suppose that the random clock τ evolves independently of the time-homogeneous Markov process X. Let Yt be the process arising if X is run on τ: Yt ≡ Xτt ,
t ≥ 0.
• Recall that φ(x) is an eigenfunction of the generator G of X, i.e. φ(x) solves G φ(x) = λφ(x). • Using a conditioning argument, it can be shown that the forward value at time t of a claim with an eigenfunction payoff φ(YT ) paid at T factors into the product of its intrinsic value φ(Yt ) and its time value E Qeλ(τT −τt ) : E Q[ f (YT )|Yt = y] = φ(y)E Qeλ(τT −τt ) . • Thus, if the Laplace transform of τT − τt is known in closed form, then one can value claims with eigenfunction payoffs. As the eigenfunctions form a basis, one can also value arbitrary European claims by integrating/summing across eigenvalues. 112
Appendix 1: Elimination of the Market Price of Volatility Risk One can derive the following PDE governing the function C (1) (t, S,Y) relating the price of an option to time t, the price of the underlying stock S, and the price of a state variable governing volatility Y : # " ∂C (1) ∂C (1) f 2 (Y ) ∂2C (1) ∂2C (1) β2 ∂2C (1) ∂C (1) (1) + [α(m −Y ) − λβ] + ρ f (Y )β = 0, (5) +r S −C + + 2 ∂t ∂S ∂Y 2 ∂S ∂S∂Y 2 ∂Y 2 where λ is the market price of volatility risk. The above PDE also holds for the price of a second option: # " ∂C (2) ∂C (2) f 2 (Y ) ∂2C (2) ∂2C (2) β2 ∂2C (2) ∂C (2) (2) + [α(m −Y ) − λβ] + ρ f (Y )β = 0. +r S −C + + 2 ∂t ∂S ∂Y 2 ∂S ∂S∂Y 2 ∂Y 2
(6)
Let γ(t, S,C (2)) be the function relating the price of the first option to time t, the underlying stock price S, and the price of the second option: γ(t, S,C (2)) ≡ C (1) (t, S,Y), where C (2) solves (6). Equivalently: C (1) (t, S,Y) = γ(t, S,C (2)(t, S,Y))
(7)
∂C (1) ∂C (2) = γ1 + γ3 ∂t ∂t
(8)
∂C (1) ∂C (2) = γ2 + γ3 ∂S ∂S
(9)
∂C (1) ∂C (2) = γ3 ∂Y ∂Y
(10)
Differentiating once:
Differentiating one more time: ∂2C (1) ∂C (2) = γ + 2γ + γ33 22 23 ∂S2 ∂S 113
∂C (2) ∂S
!2
+ γ3
∂2C (2) ∂S2
(11)
∂2C (1) ∂C (2) ∂C (2) ∂C (2) ∂2C (2) = γ23 + γ33 + γ3 ∂S∂Y ∂Y ∂S ∂Y ∂S∂Y !2 2 (1) (2) 2 (2) ∂C ∂C ∂C = γ33 + γ3 2 ∂Y ∂Y ∂Y 2
(12) (13)
Substituting (8) to (13) in (5): γ1 + γ3 ∂C∂t + r[S[γ2 + γ3 ∂C∂S ] − γ] + [α(m −Y) − λβ]γ3 ∂C∂Y (2) 2 (2) 2C(2) f 2(Y ) ∂C(2) ∂C(2) + 2 [γ22 + 2γ23 ∂C∂S + γ33 ∂C∂S + γ3 ∂ ∂S 2 ] + ρ f (Y)β[γ23 ∂Y + γ33 ∂S (2) 2 2 2 (2) + β2 [γ33 ∂C∂Y + γ3 ∂ ∂YC 2 ] = 0. (2)
(2)
(2)
∂C(2) ∂Y
C + γ3 ∂∂S∂Y ] 2 (2)
Re-arranging terms:
+γ3 +
h
f 2 (Y ) 2 γ22 + γ23
γ1 + r[Sγ2 − γ]
∂C(2) ∂t
h
i 2 (2) 2C(2) β2 ∂2C(2) ∂C(2) ∂C(2) + [α(m −Y ) − λβ] ∂C∂Y + f 2(Y ) ∂ ∂S 2 + ρ f (Y )β ∂S ∂Y + 2 ∂Y 2 i 2 f 2(Y ) ∂C(2) 2 β2 ∂C(2) ∂C(2) ∂C(2) ∂C(2) ∂C(2) = 0. + ρ f (Y )β ∂S ∂Y + γ33 + ρ f (Y )β ∂S ∂Y + 2 2 ∂S ∂Y
(2) + rS ∂C∂S
f 2 (Y ) ∂C∂S
(2)
Substituting (6) into the above simplifies the result to:
+
f 2 (Y ) 2 γ22 + γ23
h
f 2 (Y ) ∂C∂S + ρ f (Y )β ∂C∂S (2)
(2)
γ1 + r[Sγ2 − γ] + rC (2)γ3 i (2) f 2(Y ) ∂C(2) 2 ∂C(2) + γ + ρ f (Y )β ∂C∂S 33 2 ∂Y ∂S
Now the risk-neutral dynamics of S are: dSt = rSt dt + f (Yt )dW1t . Note that the variance rate of S is: Var(dS) = f 2 (Yt )dt. The risk-neutral dynamics of C (2) are given by: dC (2) = rC (2) dt +
∂C (2) ∂C (2) f (Yt )dW1t + βdW2t , ∂S ∂Y 114
∂C(2) ∂Y
+ β2
2
∂C(2) ∂Y
2
= 0.
where dW1t dW2t = ρdt. Note that the covariance rate of C (2) with S is: " # ∂C (2) ∂C (2) ∂C (2) (2) 2 Cov(dC , dS) = f (Y ) + ρ f (Y )β dt, ∂S ∂S ∂Y while the variance rate of C (2) is:
Var(dC (2) ) = f 2 (Y )
∂C (2) ∂S
!2
+ 2ρ f (Y)β
∂C (2)
∂C (2)
∂S
∂Y
+ β2
∂C (2) ∂Y
!2 dt.
Substituting into the above PDE gives: γ1 + r[Sγ2 − γ] + rC (2)γ3 +
1 Var(dS) Cov(dC (2) , dS) 1 Var(dC (2) ) γ22 + γ23 + γ33 = 0. 2 dt dt 2 dt
(14)
If one can exogenously model the volatility structure of C (2) without reference to λ, then one does not need to specify λ. There are many ways to do this.
115
