1. Numerical option pricing in the Barndorff-Nielsen - Shephard stochastic volatility model. Martin Groth Centre of Mathematics for Applications University of Oslo Joint work with Fred Espen Benth

2. The Model Consider the market with a bond R t with risk free rate of return r and a risky asset S t, the latter evolving according to the stochastic volatility model: dS t = (  +  Y t )S t dt +  (Y t )S t dB t, S 0 = s > 0 dY t = - Y t dt + dL t,Y 0 = y > 0. where B t is Brownian motion, L t is a subordinator and  2 (y)= y. Hence, the volatility is modelled as a mean reverting Ornstein-Uhlenbeck process with positive jumps given by a Lévy process. The model has the advantage of capturing both the heavy tails and the dependency structure observed in financial return data. Tell me more about the background So, where’s the numerics?

3. Indifference Pricing Let us consider an investor who two has choices, trying to maximize her exponential utility. She can either Enter in to the market by her own account, or Issue a derivative and invest her incremental wealth after collecting the premium. The utility indifference price is then the price of the claim, for a given risk aversion , at which the investor is indifferent between the two different alternatives. Hey, do I spot a PDE down there?

4. References Benth, F.E. and Meyer-Brandis, T., The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps. Finance and Stochastics (2005), vol IX, no. 4. Benth, F.E. and Groth, M., The minimal entropy martingale measure and numerical option pricing for the Barndoff-Nielsen - Shephard stochastic volatility model. To be submitted. Nicolato, E. and Venardos, E., Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type, Mathematical Finance (2003), Vol. 13, No. 4,

5. The Measure The price in the zero risk aversion limit is known to coincide with the price under the Minimal entropy martingale measure (MEMM). Under this measure the dynamics of the processes S t and Y t are changed to dS t * =  (Y t * )S t dB t *, S 0 * = s > 0 dY t * = - Y t * dt + dL * t,Y 0 * = y > 0. where the subordinator is transformed to a pure jump Markov process with jump measure Back to the PDEs

6. The PDE The option prices we calculate is given by a parabolic PDE with an integral term, here assuming r = 0: where  is the option price. The non-local integral term comes from the subordinator L t having jump measure (dz). We observe that H(t,y) appears as a measure change in the integral, giving a coupled system of PDEs. What is H here? Coupled system? Prices under what measure? Now, where is the numerics?

7. Case 1: No Claim Issued Through a dynamic programming approach Benth & Meyer-Brandis derived an integro-PDE for the value function of the investor in the case she enter directly into the market: with (t,y)  [0,T)  R +. We can assume that H(t,0) = 0 and also that the solution will approach the explicit solution, given by Benth & Meyer-Brandis, for the special case  = 0, as y  . Do you solve this ghastly thing? But this is not the option prices, right? OK, but what about the other case?

8. The Plot, H(T,Y) We solve this equation with the finite difference method giving results looking like this: Did I miss something? But why do you care about H(t,y)?

9. Indifference Pricing Text här Åt höger Åt vänster Nedåt

10. Discretizing the PDEs To solve the coupled system of PDEs we use the finite difference method. We discretize the equation and restrict the problem to a finite domain. We use dimensional splitting for the two spatial dimensions of the option prices and derive an implicit Lax-Wendroff scheme for the equation in y. We first have to solve for H(t,y) since this appears in the integrand of the option price. We also have to numerically evaluate the non- local integral term on the whole solution space using a simple trapezoidal rule. Maybe I need to know about the problem anyway. What about the integral then?

11. Case 2: Claim Issued The indifference price for the option comes from the value function in the case the investor issues a claim. Going to the zero risk aversion limit Benth & Meyer-Brandis derived the following PDE for the option price: with terminal conditions  (t,y,s) = f(s), where f(s) is the payoff function of the option. Observe that H(t,y) appears in the integrand. Nice, but what is H doing here? Let’s get our hands dirty with numerics!

12. The Role of H(t,y) The fraction H(t,y+z)/H(t,y) appears as a measure change in the integrand of the option price. It will scale the jumps from the subordinator, making the jump measure of the subordinator time- inhomogeneous and state-dependent. We see that as we approach zero the fraction will approach infinity. Hence for small volatilities the volatility process will have a large probability to jump up. The smaller the volatility the higher intensity and size of the jumps, and thus, we will quickly jump to higher volatilities. How does this effect the option? What does this H-function look like?

13. Results We simulate prices for a European call option using parameters from Nicolato & Venardos. The stationary distribution of the volatility process is assumed to be inverse Gaussian, giving option prices which are approximately normal inverse Gaussian distributed. From Eberlein we know that prices in a exponential Lévy model with a normal inverse Gaussian Lévy process give a characteristic “W”-shape compared to Black& Scholes prices. To compare we need to choose volatility for the to Black & Scholes prices. We compare prices under MEMM at y equal to the expected value of the stationary distribution with Black & Scholes prices with the same volatility. Hit me with a plot!

14. The Integral There are a few issues concerning the integral we need to address. Restricting to a finite domain we need to handle the cut-off for large y. This is taken care of by realizing that the option prices will adjust toward the Black & Scholes price for large volatilities. We can use this to integrate beyond the boundary. The modelling of the volatility process might give a jump measure of the subordinator which has infinite activity. To integrate numerically from zero is then futile and to compensate for the influence from the small jumps we add a drift term to the PDE. Did anyone say boundary condition? Come on, results now!

15. Boundary Conditions For the finite domain we derive appropriate boundary conditions. We show that for large values of s and y the prices will approach Black & Scholes prices with integrated volatility. If s = 0 the asset will be worthless throughout the whole lifetime of the asset and the option value will equal the payoff. On the last side, where y = 0 we show heuristically that it is reasonable to assume that we have a Neumann condition on the boundary. Nice, but does it converge? Let’s go straight to the results!

16. The Plot, Convergence From numerical test it appears to converge: B.C. But you can’t show it, right?

17. The Plot, Option Prices I Our results displays the characteristic “w”-shape as expected: Does it look like option prices? Explain a bit more!

18. The Plot, Option Prices II Plotting the option price as a function of first t and s and then s and y we see that the result in much resembles the B & S price: Does it smile? And compared to B & S prices?

19. The Volatility Smile Using a stochastic volatility model we expect a better fit to observed volatility smiles than given by the Black & Scholes model. Indeed, the results produce a smile in the implied volatility Nice plot! But what about convergence? Can I see what the results look like?

20. Convergence We have so far not proved that our numerical schemes are converging. This analysis will be carried out in further research. We are confident that our solver will fit into a larger framework of convergence analysis for integro-PDEs. At his point we can only rely on numerical justification of the convergence, illustrated in the plot above. Look, what a marvellous plot! Show me some numerical evidence