MSC FINANCIAL ENGINEERING PRICING I, AUTUMN LECTURE 9: LOCAL AND STOCHASTIC VOLATILITY RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK

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1 MSC FINANCIAL ENGINEERING PRICING I, AUTUMN LECTURE 9: LOCAL AND STOCHASTIC VOLATILITY RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK The only ingredient of the Black and Scholes formula which is not directly observable is the parameter σ, the volatility of the stock return. In a sense, it is more a property of the log-normal statistical model we try to fit to the time-series of prices rather than an intrinsic property of the latter. It is a measure of the instantaneous variability of the stock s return and, as such, closely related to what is called the quadratic variation of a stochastic process. Its constancy in the basic Black and Scholes (Bachelier - Samuelson) model has been challenged on empirical grounds, both for the time series of returns themselves (through studies of historical volatility) and as far as its implications to option prices are concerned (through the observed non-constancy of implied volatility). As alternatives, practitioners use asset price models with a varying volatility parameter, which can either be a function of the asset price and time (local volatility) or given by a stochastic process driven by a second Brownian motion (stochastic volatility). There are also classes of models in which volatility is not directly a parameter or prime driver anymore, notably pure-jump Lévy processes. Several types of volatility which can be distinguished: Historical volatility (HV); Implied Volatility (IV); Local Volatility or (LV); Stochastic Volatility (SV). We will discuss each of these in turn. 1. Historical Volatility In the simplest case this is computed using daily returns R j = log(s j+1 /S j ) and a moving time-window of T days. At any given day t it then is simply the square root of sample variance of observed past 1

2 2 R. BRUMMELHUIS returns (R t, R t 1,..., R t T +1 : explicitly 1 (1) σhv 2 (t; T ) = 1 T 1 (R t j µ) 2, T 1 where µ = µ(t; T ) is the sample mean: (2) µ(t; T ) = 1 T j=0 T 1 R t j. The unit of volatility is (time) 1/2. Note that here we are computing daily volatility: to annualize it (i.e. get yearly volatility) we have to multiply by 365 or 250 (depending on how week-ends are treated). Since, empirically, daily returns are small, one often uses the approximation j=0 (3) σhv 2 (t; T ) 1 T 1 T 1 j=0 R 2 t j. More generally, if we observe returns with a higher-than-daily frequency (or lower) of ( t) 1 : R t t = log(s t /S t t ), then the (square of the) instantaneous volatility at this frequency, measured over a moving time window of T t and expressed in units of year 1/2 if time is measured in fractions of a (banking) year, becomes (4) σ 2 HV (t; t; T ) 2 = 1 T 1 (T 1) t j=0 (R t t j t µ(t; t, T )) 2, with µ(t; t, T ) again the sample mean. If the Black and Scholes model were correct, then Rt t N(µ, σ 2 t), and the different Rt j t would be iid. The sample means and sample variances above would then converge to their population values µ and σ 2 as T, independently of t, with a sampling error of the size of T 1/2. Empirical studies show a different behavior. In practice, one uses other time-series models to model daily (or higher frequency) returns, most popularly GARCH models 2. Implied volatility We have seen that the vega of the Black and Scholes price of a call is always strictly positive: (5) C BS σ > 0, 1 the (T 1) 1 is to get a consistent estimator in the iid case - cf. econometrics

3 PRICING I, LECTURE 9 3 where C BS = C BS (S; K, τ, σ, r). As a consequence, for European call with strike K, maturity T traded in the market at time t for a price of C Market (t; K, T ) there exists a unique σ = σ IV such that (6) C Market (t; K, T ) = C BS (S; K, T t, σ IV ), where S is the (observable) price of the underlying at t, and where we ave suppressed the variable r from the notation. This unique σ IV, which will be a function of time t, underlying S, time-to-maturity τ = T t and strike K: (7) σ IV = σ IV (S, t, K, τ), is called the implied volatility at time t. It is usually presented, for fixed t and S, as a function of τ and of moneyness K/S or log-moneyness log(k/s). Were the Black and Scholes model a good description of reality, σ IV would be constant across all strikes and maturities, and also independent of the time t at which it is computed. As is wellknown. it isn t: it shows variations with strike ( volatility smile ), with time-to-maturity ( term stricture of volatility ) and with time ( dynamic volatility surface ). Correctly modelling it is somewhat the Holy Grail of option pricing theory. Remark. The market price of the option will depend on S. By simple homogeneity considerations, C Market (S, t; K, T ) = KC Market (S/K, t; 1, T ), and similarly of course for the Black and Scholes price. It follows that K 1 C Market (S, t; K, T ) is a function of moneyness K/S alone. The same is of course true for the Black and Scholes price. If we put y := log(k/se rτ ) and x := σ IV τ, the latter is found by solving C Market (S/K, t; 1, T ) = C BS (S/K; 1, τ, σ IV ) = e y Φ ( yx + 12 ) x Φ( ( yx 12 ) x, σ IV will be a function of y, and therefore of moneyness. Observe incidentally that the Se rτ in the definition of y above is the forward price of S with delivery at T. To draw the implied volatility curve, one can take the pragmatic approach of best-fitting a quadratic curve y = ax 2 + bx + c. For certain asset classes (FX) one plots σ IV (or y) against = ΦΦ ( y x x) instead of x. This is for two reasons: hedgers are more interested in anyhow, and it provides better graphs if one is primarily interested in what happens close to at-the-money (where changes rapidly for small changes of y).

4 4 R. BRUMMELHUIS 2.1. Term structure of volatility. At fixed moneyness, σ IV will be a function of τ. We know how to price call options with a time-dependent volatility function σ(t): we just have to replace σ 2 in the Black and Scholes formula by its mean- value up-to-maturity: (8) 1 T t T t σ 2 (s)ds. To fit a time-dependent volatility to market data, we then have to back out σ(t) from (9) (T t)σ 2 IV (T t) = T t σ(s) 2 ds, which can be solved by differentiating both sides with respect to t. In practice, σ IV (τ) is only given for some finite set of maturities τ i, and one would try to fit a piece-wise constant σ(s). 3. Local volatility In local volatility modelling one supposes that the stock price follows a modified diffusion process (10) ds t = µs t dt + σ(s t, t)s t dw t, with a volatility which is not constant anymore but depends on the stock price S t and t. Early examples were the so-called constant elasticity of volatility models 2 One can use identical edging arguments as before to derive the following Black and Scholes equation for derivative prices V = V (S, t): (11) t V σ(s, t)2 2 SV + rs S V = rv. As before, µ is immaterial (we might have let it depend also on S and t: this would have changed nothing). It now turns out that if we would dispose of a complete range of call options prices 3 C = C(S, t; K, T ) =: C(K, T ) across all strikes K and 2 Elasticity, in economics, is the relative % change of some economic variable with respect to the % change of some determining variable. For example, demand elasticity with respect to price is ( D/D)/( P )/P ), where D is demand and P is price, taken as a determining factor of demand. If volatility σ = σ(s) is a function of S, then constant elasticity of volatility means something like which upon letting S 0 becomes an ODE: σ/σ S/S = c, SΣ (S) σ(s) with solution σ(s) = S c. 3 Since the SDE is not time-homogeneous anymore but depends explicitly on t, we cannot expect option prices to be a function of time-to-maturity only = c,

5 PRICING I, LECTURE 9 5 maturities T, then these completely determine σ(s, t) through Dupire s equation: (12) σ 2 (K, T ) = 2( T C + rk K C) K 2 2 K C. This is because one can show that as a function of the forward variables (K, T ), call prices ave to satisfy the following PDE, for any fixed S and t: C (13) T = 1 2 σ2 (K, T )K 2 2 C C rk K2 K, T > t. This PDE is in fact equivalent to the Black and Scholes PDE for a call, with the same boundary conditions: C(S, t; K, T ) = max(s K, 0) for T = t. Given a range of observed option prices for a discrete set (K i, T i ) of maturities and strikes, one can smoothly interpolate these to a twicedifferentiable function of K and T, back out the local volatility function using Dupire s formula, and use this volatility as a basis to price other derivatives (e.g. by solving the PDE with different boundary conditions). One can in particular compute and predict future option prices, and test these predicted prices against future market movements. The local volatility model does not come out very well: empirical studies have shown it has low predictive power relative to just assuming constant volatility. It has also been claimed that for a particular class of local volatility models (those with σ(s) only depending on S), the dynamics of the theoretical implied volatility surface σ IV is different from the ones observed in the markets. 4. Stochastic Volatility Another popular line of modelling assumes that volatility is itself a stochastic process, driven by a second Brownian motion, (Z t ) t 0, correlated to the one driving the stock-price SDE, Specifically: (14) ds t = µs t dt + σ t S t dw t, with (15) dσ t = a(σ t, t)dt + b(σ t, t)dz t, and (16) dz t dw t = ρdt, ρ ( 1, 1). with Z t a second Brownian motion correlated to the one driving the stock-price. It is possible to once more derive a PDE for derivative prices, which will now be a function V (S, σ, t) of underlying S and time t, and of instantaneous volatility σ at time t. The hedging argument is slightly different, since we cannot completely eliminate all risk by trading in the underlying S t only. Volatility σ t is not directly traded, only trough the intermediary of derivative assets with underlying S t,

6 6 R. BRUMMELHUIS and to price derivatives, it turns out that investors have to decide amongst themselves on a market price for volatility risk, in the sense that they have to specify how much extra return they require per unit of volatility. We give the details Hedging in SV models. Values of derivative assets will now depend on both asset price S t and volatility σ t. If V (S t, σ t, t) is the price of the derivative at t, then by the multi-variable version of Ito s lemma, dv (S t, σ t, t) = t V dt + S V ds t + σ V dσ t + 1 ( 2 2 S V (ds t ) Sσ V dσ t ds t + σv 2 (dσ t ) 2). We evaluate (ds t ) 2, dσ t ds t and (dσ 2 t ) 2 using Ito s multiplication table, using (14) and (15), e.g. (writing a t, b t for a(σ t, t), b(σ t, t)), dσ t ds t = (µs t dt + σ t S t dw t )(a t dt + bdz t ) = σ t b t dw t dz t = ρσ t b t dt, we see that all these contribute to the dt-term of dv. Collecting terms, we can write the final result in the convenient form: (17) dv (S t, σ t, t) = ( t V + L(V )) dt + S V ds t + σ V dσ t, where L is the second order differential operator whose action on a function V = V (S, σ, t) is given by (18) L(V )(S, σ, t) = 1 ( σ 2 S SV + 2ρσb(σ, t)s SσV 2 + b(σ, t) 2 σv ) 2. As usual, in (17) we evaluate L(V ) in (S, σ, t) = (S t, σ t, t) (and similar for the other terms). To simplify notations, we will sometimes leave out all arguments and simply write (17) as dv = L(V )dt + S V ds t + σ V dσ t ; similarly, we will leave out the arguments of the two functions a and b in (18) and write L(V ) = 1 2 (σ2 S 2 2 S V + 2ρσbS 2 Sσ + b2 2 σv ). Since there are now two sources of risk, W t and Z t, we can no longer eliminate all risk in the derivative asset V by hedging with S only. What we can do is hedge V with S and some other option, V 1 : think for example of two European options with different strokes and/or maturities. One way to think about this is that a suitable combination of the options will eliminate the dz t -risk coming from the volatility; all that then remains is the dw t -risk which can be hedged using the underlying. Suppose we want to hedge the derivative V = V (S t, σ t, t) at time t by shorting t underlying S t, and 1 = 1,t units of the other derivative

7 V 1. The portfolio s value at time t then is PRICING I, LECTURE 9 7 Π t = V t S t 1,t V 1, where the two derivatives are evaluated in (S t, σ t, t). Simply writing = t and 1 for 1,t, the change of the portfolio value over the next instant in time, [t, t + dt] then is dπ t = dv ds t 1 dv 1 = { t V + L(V ) 1 ( t V 1 + L(V 1 ))} dt + ( S V 1 S V 1 ) ds t + ( σ V 1 σ V 1 ) dσ t. The risky, stochastic, terms are the ones containing ds t and dσ t and to eliminate these we choose (19) 1 = σ V/ σ V 1, and (20) = S V 1 S V 1 = S V ( σ V/ σ V 1 ) S V 1. With this choice of hedging, the portfolio becomes locally 4 risk-free and, by absence of arbitrage, cannot earn more (or less) than the risk-free rate, that is: dπ t = rπ t dt, or { t V + L(V ) 1 ( t V 1 + L(V 1 ))} dt = r (V S 1 V 1 ) dt. Substituting (19) and (20), and multiplying both sides by σ V 1, this becomes, after re-arranging, σ V 1 ( t V + L(V )) σ V ( t V 1 + L(V 1 )) = r { σ V 1 V σ V V 1 which can be written as t V + L(V ) + rs S V rv (21) σ V ( S V σ V 1 σ V S V 1 ) S }, = tv 1 + L(V 1 ) + rs S V 1 rv 1 σ V 1. Contrary to what happened in the classical Black and Scholes case, or even for the generalisation to S- and t-dependent volatilities, we did not at first sight find a partial differential equation for V, only that the differential expression on the right and side of (21) has to be the same for all derivative assets V. As we will see next, this information can be converted into a PDE, but at the price of one of the coefficients of this PDE, the one of σ V, not being uniquely determined by the underlying model (14), (15), but necessitating the specification of a further functional parameter. 4 locally in time

8 8 R. BRUMMELHUIS 4.2. The market-price of volatility risk. The point is that since the expression in (21) is the same for all derivatives V, it can only depend on the variables (S, σ, t), but not on the type of derivative - that is, not on any characteristics of the pay-off, like ((thinking of European calls) strike or time of exercise. It follows that the left hand side of (21) is some function of (S, σ, t), which we will call (provisionally) q(s, σ, t) (the minus-sign is for convenience): (22) t V + L(V ) + rs S V rv σ V or, multiplying out the σ V, and re-arranging: = q(s, σ, t), (23) t V + L(V ) + rs S V + q(s, σ, t) σ V = rv. This is our Black and Scholes equation in a SV-model, and solving it with appropriate boundary conditions (such as V (S, σ, T ) = F (S) for a European option) will give us the price of an option. Two questions remain, though: What, if anything, is the interpretation of this new, unknown function q(s, σ, t)? How do we determine this function in practice (for we clearly need to know it in order to do pricing)? On an abstract level, the second question is answered by (22): if we would know the complete pricing formula of at least one derivative asset, we could compute q(s, σ, t) by (22). A natural choice would be to take the asset itself: V = S, but this does not lead anywhere: the left and side becomes 0/0, which is ill-defined. If we would speculate for a moment that volatility σ would be a directly traded asset, then we might take V = σ, in which case an easy computation shows that the left hand side of (22) would be rσ, so that we would have q = rσ. The coefficient of σ V would then be σs, very similar to the coefficient rs of S V. Our equation would be very much like the Black and Scholes equation, and the coefficient rσ would reflect that volatility risk can be completely hedged away, and that after because of this hedging the rate of return on volatility can, for derivative-pricing purposes, be set equal to the risk-free rate, r. However, its useless to speculate: volatility is, for the moment at least, not a directly traded asset. It is in fact traded precisely through the intermediary of options. To get some hold on the function q, we consider a derivative which is only -hedged using the underlying asset S, and examine by how much the return on such a portfolio differs from the risk-free return, That is, letting 5 Π ph t = V ( S V )S t (V as usual evaluated in (S t, σ t, t)), we compute (24) dπ ph t rπ ph t dt. 5 the superscript ph stands for partially hedged

9 PRICING I, LECTURE 9 9 Since dπ ph t = ( t V + L(V )) dt + σ V dσ t, (the ds t -term cancels out since the portfolio is -hedged against small movements in S t ), (24) becomes ( t V + L(V ) + rs S V rv ) dt + σ V dσ t ( t V + L(V ) + a σ V + rs S V rv ) dt + b σ V dz t, where we used (15). If we now use the pricing equation (24), we see that (25) dπ ph t rπ ph t dt = (a q) σ V dt + b σ V dz t ( ) a q = b σ V dt + dz t. b The residual volatility of Π ph t over [t, t + dt] (volatility conditional on t) due to dσ t is proportional to b( σ V ), since the conditional variance at t of σ V dσ t = a σ V dt + b σ dz t is b 2 ( σ V ) 2. One can therefore interpret (25) as telling us that the market asks for an additional return of λ = λ(σ, t) = a q, b per unit b σ V of volatility risk as reward for taking on the volatility in Π ph coming from the stochasticity of σ t ; λ is called the price of volatility risk, and in terms of λ, q = a λb, The fundamental pricing PDE (23) becomes: t V + L(V ) + rs S V + (a λb) σ V = rv, or, written out in full, t V + 1 ( (26) σ 2 S SV + 2ρσb(σ, t)s SσV 2 + b(σ, t) 2 σv ) 2 +rs S V + (a(σ, t) λ(σ, t)) b(σ, t)) σ V = rv. 5. Popular stochastic volatility models Special choices of the functions a, b and of λ give the special kinds of SV models used in practice. Most assume that instantaneous volatility is mean-reverting around a long-term mean, and volatility, or its square, variance, is typically modelled by an Ornstein-Uehlenbeck process Stein and Stein model: ds t = rs t dt + σ t S t dw t dσ t = α(σ σ t )dt + ηdz t O-U process with constantly correlated Brownian motions: dw t dz t = ρdt (constant correlation is a standard assumption in this field). Observe that:

10 10 R. BRUMMELHUIS σ t mean-reverts to σ : long-term equilibrium value for volatility; α is the mean-reversion rate η: is the volatility of volatility, or vol of vol, parameter It doesn t matter that σ t can become negative with positive probability, since dz t can have either sign (with equal probabilities) To arrive at a solvable PDE one chooses λ(σ) = c 1 + c 2 σ with c 1 and c 2 positive constants. There are analytical formulas available for options like calls and puts, not quite in closed form as for classical Black and Scholes, but at least in integral form (which then has to be numerically evaluated) Heston model. Instead of modelling the stochastic evolution of the instantaneous volatility σ t, we model that of the instantaneous variance, v t = σ 2 t (which is equal to v t = E t (((ds t rs t dt)/s t ) 2, where E t is the conditional expectation given all available price- and volatility information up to time t). We take β(v) := v and suppose v t meanreverts to a long-time equilibrioum variance v : ds t = rs t dt + σ t S t dw t dv t = α(v v t )dt + η v t dz t Some remarks: The process followed by v t is known as a CIR or Cox-Ingersoll- Ross process - first used by these authors for interest rate modelling. By a general property of such processes (proved by Feller in the 1950 s) v t never becomes negative, so we can can take the square-root. Instead of the option price being a function of S and σ, it now becomes a function of S and variance v; specifically: one finds that for the Heston model, V (S, v, t) has to satisfy (27) t V ( vs 2 V V + 2ρηvS S2 v S + η2 V v 2 +rs V S + ( α(v v η vλ(v) ) V v = rv. One chooses the price of volatility-risk to be of the form λ(v) = c v, so that the coefficient of v V will have the form κ(θ v) for suitable new constants κ and θ (e.g. κ = cα, etc.) This model is popular with practitioners since semi-explicit formulas for call and other option prices exist, again in the form of integrals - see for example Heston s paper. Their numerical treatment can be numerically delicate) 5.3. Classical SABR. (Hagan, Kumar, Lesniewski, Woodward). In its original form this equation was actually proposed for derivatives )

11 on futures F t = e r(t t) S t : PRICING I, LECTURE 9 11 df t = α t F β t dw t dα t = να t dz t dz t dw t = ρdt The term SABR comes from Stochastic αβρ - model (but traders fondness for ferocious animals with which they like to compare themselves probably also has something to do with the choice of name). The SABR-model is not explicitly solvable, except in special cases. Hagan, Kumar, Lesniewski, Woodward derived asymptotic expressions for implied vol, etc. using so-called singular perturbation techniques: they replaced α t εα t, ν εν and looked for asymptotic solutions as ε 0 Mathematically, this is equivalent amounts constructing approximate solutions of associated PDE as time-to-maturity τ 0 and there is a close connection with so-called heat-kernel asymptotics for line-bundles on Riemannian manifolds as studied in the field of differential geometry (Henry-Labordère). The asymptotic technique also applicable to other models than SABR, e.g. to Heston, or (even simpler) Local Volatility models. Henry-Labordère s mean-reverting SABR-model adds mean-reverting drift term to evolution of α t We note that long-term asymptotics, τ, of option prices and implied volatility has also been studied, notably by Lewis for Heston model. 6. References Some original papers on SV-models: Patrick S. Hagan, Deep Kumar, Andrew S. Lesniewski, Diana E. Woodward, Managing Smile Risk, Wilmott Magazine Pierre Henry-Labordère, Analysis, Geometry, and Modelling in Finance - Advanced Methods in Option Pricing, Chapman & Hall/CRC Financial Mathematics Series (2009). Pierre Henry-Labordère, A General Asymptotic Implied Volatility for Stochastic Volatility Models. Frontiers in Quantitative Finance, Wiley (2008). S. Heston, A closed Form Solution for Options with Stochastic Volatility wit Applications to Bond and Currency Options. Rev. Fin. Studies 6, J. Hull and A. White, The Pricing of Options with Stochastic Volatilties, J. Finance 42 (1987), E. M. Stein and J. C. Stein, Stock Price Distributions with Stochastic Volatility, Rev. Fin. Studies 4 (1991),