RISK-NEUTRAL VALUATION AND STATE SPACE FRAMEWORK. JEL Codes: C51, C61, C63, and G13

Transcription

1 RISK-NEUTRAL VALUATION AND STATE SPACE FRAMEWORK JEL Codes: C51, C61, C63, and G13 Dr. Ramaprasad Bhar School of Banking and Finance The University of New South Wales Sydney 2052, AUSTRALIA Fax R.Bhar@unsw.edu.au Prof. Carl Chiarella School of Finance and Economics University of Technology, Sydney P.O. Box 123, Broadway NSW 2007, AUSTRALIA Fax Carl.Chiarella@uts.edu.au Abstract: The main parameter required in pricing of derivative securities is the volatility of the underlying security. This may typically be estimated from historical prices or as implied volatility from derivative prices. It is however not clear how to estimate the volatility of the underlying securities over a given sample period, say six months. In this paper we propose an estimation technique in state space framework which can model unobserved component effectively. This approach allows combining in an optimal fashion the information contained in the prices of the underlying security and its derivatives over the sample period.

2 2 BACKGROUND Probably the single most important innovation in the area of pricing derivative securities is the concept of risk-neutral valuation. This implies that if a derivative security depends only on the price of a traded security then the differential equation for the price of the derivative security does not involve any parameter that depends upon the risk preferences of the traders. In that sense it is safe to discount any cash flow due to the derivative security at the risk-free rate. This assumption greatly simplifies the procedure for pricing such derivative securities. It should, however, be emphasised that this approach is only applicable for derivative securities and it so happens that the price obtained this way is the same as that would be applicable under the assumption of risk-averse investors. The main parameter required as input into the derivative pricing model is the volatility of the stochastic process for the underlying security. Typically practitioners will estimate the so-called implied volatility by finding the volatility that makes the derivative pricing model (typically the Black-Scholes model) fit the most recently observed derivative price. The implied volatility varies considerably day-to-day and a range of time series techniques may be applied to forecast its evolution. Our interest in this paper however is to estimate the volatility of the underlying security over a given sample period, say of three to six months. Such estimates are often required for empirical studies focussing on a longer-run view of market behaviour eg. studies of portfolio performance etc. Of course one may use historical volatility based upon time series of the securities price itself. However, we wish to also use the information contained in the prices of derivatives on the security, which probably have impounded in them the market s expectation about the evolution of volatility. This would suggest use of implied volatility which as we have just pointed out varies considerably over a given sample period. The financial modeller is thus faced with the task of finding some way to average the information contained in the historical volatilities (based on the security prices) and the implied volatilities (based on the derivative security prices). In the present paper we propose an estimation technique that resolves this conflict in some optimal fashion by using in an efficient manner the information contained in both the prices of the underlying security and its derivatives. In particular we apply filtering methods (Jazwinski (1970), Bar-Shalom and Li (1993)) to obtain an estimate of the volatility of the security price over a given sample period. In recent years several authors have adapted filtering techniques based on state space modelling to problems in economics and finance particularly where the system involves unobserved components. Some of these studies include Bhar and Chiarella (1997), Lund (1997), Cheung (1993) and Wolff (1987). Naturally, the question arises as to how the method reconciles the fact that the observations of the asset prices are made under the historical probability measure and not under the risk-neutral measure. We show how to adapt the state-space approach to model such price movements in the unobserved risk-neutral world. Here we focus on equity options within the Black-Scholes risk-neutral framework for ease of exposition. The concepts discussed here are directly applicable to the foreign exchange and interest rate models as well.

3 3 MODEL BUILDING Consider the standard problem of pricing a European call option on a common stock. The aim is to derive the price of the option at time t 0 and the option matures at time T ( > t 0 ). In the Black-Scholes model it is assumed that the stock prices move according to the stochastic differential equation (SDE), ds = S dt + σ S dw, S ( t0 )= S 0, µ (1) where µ, the instantaneous expected return on the stock, reflects the risk-premium of the stock, σ is the volatility of the stock price process and dw are the increments of a standard Wiener process. The SDE that the value of a call option on this stock follows is obtained by application of Ito's lemma and is given by, dc =( 2 2 µ S C +C +0.5 σ S C ) dt + σ S C dw. (2) s t The subscripts in the above equation denote the appropriate partial derivative. ss s By invoking the risk-neutral argument (Hull (1997)) the option value is given by the stochastic differential system, + dc = rcdt + σs dw ( t), C( S, T) = ( S ( T ) x) (3) d S = = rsdt + σsdw ( t), S ( t0 ) S0 (4) where r denotes the risk-free rate of interest, x is the exercise price, T is the maturity of the option, C s, and d W is the Wiener increment under the risk-neutral probability measure. Note that the quantity S is the stock price process under the risk-neutral measure generated by W ( t is related to W (t) via the relationship the stochastic process in equation (4). Note that ) W ( t) = W ( t) λ( s) ds t 0, where λ(t) is the market price of risk of the stock at time t. The tilde sign is used to emphasise the stock price process under the risk-neutral measure. It is

4 4 important to realise that S is an unobserved process in contrast to the historical stock price process S, which is observed. However both processes start from the same initial value S0. It should also be noted that the volatility of both stock price processes is the same. The representation of the option value via the stochastic differential system (3) and (4) can then be re-expressed by use of the martingale concept, in the more familiar form C ( S 0,t 0 )= e + E t [ (S ( T ) - x ) ] (4 ) -r (T - t0 ) where E t represents the expectation operation calculated under the risk-neutral measure ie. under the stock price distribution calculated according to (4). Performance of the integration in (4 ) yields the Black-Scholes equation. (Hull (1997)). We now pose the estimation problem of determining the volatility, σ, using the equations (3) and (4) and a given set observations of option prices over a given sample period. The option price is jointly determined by these two equations of which only C is observed and only the starting value of S is known. To put this in the state-space framework (as in the engineering literature) the state vector X [ S,C ] evolves following the two SDE's in continuous time and only C is observed at discrete points in time. The state equation may be written in following form for ease of later reference, dx = F( X ) dt + V ( X ) dw ( t) (5) where F [ r S,r C ] and V [ σ S, σ S ]. This formulation is known as a continuous-discrete filtering problem (Jazwinski (1970)). To complete the specification of the state-space representation, an observation equation (also known as the measurement equation) can be introduced as, S ' ck = [ 0,1 ] [,C ] k + ε k, k = 1,2,3,...; t k -1> t k t0. (6) The observation error is assumed to be characterised by εk N(0, R), R > 0 and the error sequences are assumed independent. The error in the measurement equation may be thought to be due to bid-ask spread, thin trading etc. The stochastic dynamical system given by (5) and (6) can be solved by application of filtering theory. The filter operates on the data sequentially giving new estimates of the state vector, as new observations become available. The filter also generates estimate of the error covariance indicating the uncertainty in the estimates. To be able to implement the scheme the SDE's have to be discretised using a suitable scheme. For the purposes of illustration the simple scheme due to Euler-Maruyama (Kloeden and Platen (1992)) is adopted here. The time discretisation within the interval δ ( t k+1 - t k ) results in, k

5 5 = X + F ( ) δ k +V ( ) δ k η X k +1 k X k X k k (7) where η N(0,1). The first two conditional moments of the state vector are, E ( X k + 1 X k ) = X k + F( X k ) δk, (8) Cov ( X k + 1 X k ) = V ( X k ) V '( X k ) δk. (9) From equation (8) the best forecast of X at t k+1 made at t k (knowing c k at t k ) is, Xˆ k +1 k = Xˆ k k + F ( X k k ) δ k (10) and the best forecast of variance of X k+1 is, k +1 k = Pk k + Q, (11) P k + 1 where, Q k+1 is given by (9). The estimation error is, therefore, given by using M [0, 1] c k MXˆ + 1 k + 1 k (12) and the variance of the estimation error is, v = M P M + R. k +1 k +1 k (13) The updating equation for the state vector is Xˆ k +1 k +1 = Xˆ k +1 k + K k +1 [ ck +1 M Xˆ k +1 k ], (14) where K k+1, the Kalman gain matrix, is given by

6 6 K k +1 = P k +1 k M (Note that v k+1 in this case is a scalar). 1 v k + 1. (15) The recursion for the error covariance completes the specification of the Kalman filter updating equations, ' P k + 1 k + 1 = [ I Kk + 1M ] Pk + 1 k [ I Kk + 1M ]' + Kk + 1RKk + 1. (16) The updating equations of the Kalman filter described above are based upon the Gaussian assumption of the measurement error in equation (6) as well as that of the state transition equation. It is well known in the filtering literature that the Kalman filter is optimal only in the Gaussian case and it is the best linear predictor in the non-gaussian case. However, there are other methods available to deal with such non-gaussian cases. Under the assumption that the error sequences in the state transition equation, the error sequences in the measurement equation and the initial state vector X 0 are jointly normal and uncorrelated, the estimation of the parameter ie. the volatility in this case can be obtained by maximum likelihood technique. In order to write the likelihood function it is necessary to examine the innovations from the state space model, e = c - M X k+1 k+1 ˆ k+1 k (17) conditional on X 0. The equations (6) and (17) jointly suggest that the innovations can be represented as, e = M ( X - X )+. k+1 k+1 k+1 k ε k (18) Thus, conditional on c 1,..., c k, the innovations have zero means and covariance matrix given by, k +1= M P k+1 k M + R. ν (19) By assumption these innovations and ε are uncorrelated, so the log likelihood for estimating the parameter s is (for a sample of T discrete observations), T T π k e k. (20) k =1 k =1 T 1 1 ' Ln L = - Ln (2 ) - Ln - -1 νk e ν k

7 7 This is a non-linear function of the unknown parameter and its maximisation will require application of suitable numerical optimisation technique. Estimation of the volatility of the stock price process in this way essentially provides an "average" of the implied volatilities that could be obtained by inversion of the Black-Scholes formula for each of the option prices in the sample together with the corresponding stock prices. The recursive algorithm of the Kalman filter requires suitable starting values. The initial specification of P0 0 ie. the prior covariance for the state vector is a critical one. Since in most application there is only insufficient knowledge about this variable, one of the approaches suggested in the literature is known as the diffuse prior. According to this suggestion P0 0 may be set to κ I where I is the identity matrix and κ is a large but finite integer. Further discussion on choices of the starting values for the filter, see Harvey (1989). EMPIRICAL RESULTS AND CONCLUSIONS Before presenting the empirical result of the procedure discussed here, it may be instructive to point out that the filtering process essentially helps to obtain the conditional transitional probability Pr ( S k+1,ck+1 S k,ck ), which includes the unobserved element S k+ 1. The iterative process is clearly depicted in the Figure 1. We present an intuitive understanding the filtering algorithm. With respect to the two time points k and k+1, we enter the (k+1) th time period with a prior belief about the state vector S k k and the covariance matrix P k k and then the system dynamics produce S k k and P k + 1 k respectively following the equations (14) and (16). At this point equation (12) helps us generate the prediction error, which is used to construct the log likelihood function. Then before leaving the (k+1) th time period the updating equations are applied to compute S k k and P k k. This way we are ready to apply the same procedure in the next time slot. Figure 1 S S C k k+1

8 8 The above procedure has been applied to a set of daily call option prices for an actively traded stock (BHP) from the Australian Stock Exchange covering a period of two months (July and August 1996). The sample contains 39 observations each on closing call option and stock prices for BHP. The option contracts selected are those closest to the money. This option matures on the last Thursday of September As a proxy for the risk-free rate the 13-weeks Treasury note traded on the first day of this sample is selected. The Kalman filter is started with the first pair of prices defining the initial state vector (leaving 38 observations in the sample), κ= 10000, and R = As the discretised state equation allows, the call is calculated at each step using Black-Scholes call delta formula using the prices known at the previous time period and the value of the parameter ie. volatility estimated at that point. At each iteration the call option is priced with respect to the current observed stock price ie. the filter is re-initialised with the observed stock price. The result is depicted in Figure 2. The estimate of the volatility parameter obtained from the Kalman filter is where as the average of the Black-Scholes implied volatility is The closeness of these two figures suggests the viability of the filtering framework although the difference of almost 5% between the two estimates could be significant in empirical work. Another interesting aspect of this approach is that the stock price process for the risk-neutral world is obtained as an output of the filter. This may help in analysing the risk premium attached to the particular stock. Indeed Bhar and Chiarella (1998) have extended the approach developed here to infer market risk premium from derivative prices. Although the example here is based on stock options, the arguments apply equally well to term structure of interest rates problems. In that case, the underlying process is that of the term structure itself and the bond prices are derived from it. In the case of Heath-Jarrow-Morton (1992) model, as identified in Bhar and Chiarella (1997), the system is of higher dimension and the elements of the system matrices are not as simple as in this example. Nevertheless the state space framework discussed here is applicable. Figure 2 Stock price Days observed risk-neutral imp vol Imp vol Implied volatility is obtained from Black-Scholes model. Risk-free rate is p.a.

9 9 REFERENCES Bar-Shalom, Y. and Li, X. (1993), Estimation and Tracking: Principles, Techniques and Software, Artech House, Boston. Bhar, R. and Chiarella, C. (1997), Transformation of Heath-Jarrow-Morton models to Markovian systems, The European Journal of Finance, 3, Bhar, R. and Chiarella, C. (1998), Analysis of Time Varying Financial market Risk Premia, Presented at the HPC Asia 1998, Institute of High Performance Computing, Singapore, ISBN: , pp Jazwinski, A.H. (1970), Stochastic Processes and Filtering Theory, Academic Press, New York and London. Cheung, Y. (1993), Exchange rate risk premiums, Journal of International Money and Finance, 12, Hull, J. (1997), Options, Futures and Other Derivatives, Third Edition, Prentice Hall. Kloeden, P.E. and Platen, E. (1992), The Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin. Lund, J. (1997), Non-Linear Kalman Filtering Techniques for Term-Structure Models, Manuscript, Department of Finance, Aarhus School of Business, Denmark. Wolff, C. C. P. (1987), Forward Foreign Exchange Rates, Expected Spot Rates, and Premia: A Signal-Extraction Approach, The Journal of Finance, Vol. XLII, No. 2,