A GENERAL FORMULA FOR OPTION PRICES IN A STOCHASTIC VOLATILITY MODEL - PowerPoint PPT Presentation

A GENERAL FORMULA FOR OPTION PRICES IN A STOCHASTIC VOLATILITY MODEL Stephen Chin and Daniel Dufresne Centre for Actuarial Studies University of Melbourne Paper: http://mercury.ecom.unimelb.edu.au/SITE/actwww/wps2009/No181.pdf Followed by “Fourier Inversion Formulas in Option Pricing and Insurance”

1. THE PROBLEM The Black-Scholes model says that the risky asset’s price satis- fies dS t = rS t dt + σS t dW t (1) where W is a standard Brownian motion under the risk-neutral measure. This is often replaced with dS t = rS t dt + V t S t dW t , (2) because observed option prices do not agree with (1). Here, { V t } is a stochastic process, called “stochastic volatility”. Page 2

The probability distribution of S t is usually complicated or un- known in these models. Therefore, the computation of European option prices cannot be done by a simple integration with re- spect to the distribution of S t . PROBLEM: Find an alternative way to compute European put and call prices in such models, i.e. to compute E ( S T − K ) + , E ( K − S T ) + . N.B.: “ E ” correspond to the risk-neutral-measure. Page 3

2. A TOOL: PARSEVAL’S THEOREM Parseval’s theorem gives conditions under which Z ∞ Z ∞ 1 g ( x ) dν ( x ) = g ( − u )ˆ ˆ ν ( u ) du 2 π −∞ −∞ where Z ∞ Z ∞ e iux g ( u ) dx, e iux ν ( dx ) . g ( u ) = ˆ ν ( u ) = ˆ −∞ −∞ In option pricing, this may be applied because a European op- tion price is: Z ∞ E g ( X ) = g ( x ) dµ X ( x ) , −∞ Page 4

where µ X ( · ) is the distribution of X (under the risk-neutral measure). In many cases a damping factor e − αx needs to be used, since g ( u ) may not be defined. Then one writes ˆ g ( x ) dµ X ( x ) = e − αx g ( x ) × e αx dµ X ( x ) = g ( − α ) ( x ) dµ ( α ) X ( x ) . Then Z ∞ 1 g ( − α ) ( − u ) d [ µ ( α ) E g ( X ) = X ( u ) du. 2 π −∞ (Ref.: Dufresne, Garrido and Morales, 2009.) Page 5

3. MAIN RESULT The stock price satisfies dS t = rS t dt + V t S t dW t µ ∂ Z t rt − U t ⇐ ⇒ S t = S 0 exp 2 + V s dW s 0 Z t V 2 where U t = s ds. 0 The next step works if { V t } is independent of { W t } (more com- plicated otherwise): if we condition on V , then Z t p d V s dW s = U t Z, Z ∼ N (0 , 1) ( V, Z indep . ) 0 Page 6

Then e − rT E ( K − S T ) + = e − rT E E [( K − S T ) + | V ] p = e − rT E [ K − S 0 exp( rT − 1 2 U T + U T Z )] + = E g ( U T ) 2 + √ uZ )] + . g ( u ) = e − rT E [ K − S 0 exp( rT + u The function g is the price of a European put in the Black- Scholes model. This leads to the problem of finding the simplified expression for the Laplace transform, in the time variable, of the price of a European call or put in the Black-Scholes model. Page 7

Theorem 1 . Suppose r ∈ R , σ, K, S 0 > 0 , and let β = γ + r σ 2 − 1 r K σ 2 , ρ = 2 , K = S 0 p p ρ 2 + 2 β, ρ 2 + 2 β. µ 1 = ρ + µ 2 = − ρ + (a) If γ > − r , then Z ∞ e − γt e − rt E ( K − S 0 e ( r − σ 2 2 ) t + σW t ) + dt 0  1+ µ 1  S 0 K σ 2 √ if K ≤ S 0   µ 1 (1+ µ 1 ) ρ 2 +2 β ∑ ∏ =  1 − µ 2  2 β 2 β − 2 ρ − 1 + βK 1 S 0 √  K − if K > S 0 . βσ 2 µ 2 ( µ 2 − 1) ρ 2 +2 β (b) (Similar for the LT of a call.) Page 8

Theorem 2 . Let ν be the distribution of U T , so that d ν ( α ) ( u ) = E e ( α + iu ) U T . (a) Suppose that E e α ∗ U T < ∞ for some α ∗ > 0 . Then, for any 0 < α < α ∗ , Z ∞ 1 [ e − rT E ( K − S T ) + = ( − u ) d g ( − α ) ν ( α ) ( u ) du, 1 2 π −∞ where, if k = Ke − rT /S 0 ,  (1+ √ 1+8 α +8 iu ) / 2  if Ke − rT < S 0 S 0 k  ( α + iu ) √ 1+8 α +8 iu [ g ( − α ) ( − u ) = (1 − √ 1   1+8 α +8 iu ) / 2 if Ke − rT ≥ S 0 . S 0 ( k − 1) + S 0 k ( α + iu ) √ 1+8 α +8 iu α + iu (b) (Similar integral for the price a call.) Page 9

4. NUMERICAL EXAMPLE We applied Theorem 2 in the case where the volatility process is a Markov chain with 2 or 3 states. In this case, option prices may be obtained by simulation as well. Results (see paper for details): — Theorem 2 does very well, computation times are much shorter than using simulation; — However, some maturities give small errors, apparently due to the oscillatory integrand. More refined integration would most likely remove those errors (we use “NIntegrate” in Math- ematica without any option). Page 10

FOURIER INVERSION FORMULAS IN OPTION PRICING AND INSURANCE Daniel Dufresne, Jose Garrido, Manuel Morales (Methodology and Computing in Applied Probability, 2009)

1. GOALS Many authors have used Fourier inversion to compute option prices. In particular, Lewis (2001) used Parseval’s theorem to find for- mulas for option prices in terms of the characteristic functions of the log of the underlying. The problem here is to compute (for example) E ( e X − K ) + when E e iuX is known. This talk aims at widening the scope of this idea by deriving: (1) formulas with weaker restrictions, related to classical inver- sion formulas for densities and distribution functions; (2) formulas for expectations such as E ( X − K ) + when E e iuX is known (this situation occurs in option pricing and insurance). Page 2

2 . SOME REFERENCES Among many applications of Fourier inversion in option pricing: Bakshi, G., and Madan, D.B. (2000). Spanning and derivative- security valuation. J. Financial Economics 55 : 205-238. Borovkov, K., and Novikov, A. (2002). A new approach to cal- culating expectations for option pricing. J. Appl. Prob. 39: 889- 895. Carr, P., and Madan, D.B. (1999). Option valuation using the fast Fourier transform. J. Computational Finance 2 : 61-73. Page 3

Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with application to bond and currency op- tions. Review of Financial Studies 6 : 327-343. Lewis, A. (2001). A simple option formula for general jump- diffusion and other exponential L´ evy processes. OptionCity.net publications: http://optioncity.net/pubs/ExpLevy.pdf . Raible, Sebastian (2000). L´ evy Processes in Finance: Theory, Numerics, and Empirical Facts . Doctoral dissertation, Faculty of Mathematics, University of Freiburg. Page 4

3 . THE PROBLEM The no-arbitrage price of a European call option is e − rT E ( S T − K ) + , where the expectation is under the risk-neutral measure. Many models assume X T = log S T is not Brownian motion but a more complicated process ( e.g. a L´ evy processes). In those cases, exact pricing of the option is often done in two steps: (1) find the distribution of X T , and (2) integrate ( e x − K ) + against the distribution. Page 5

It is possible to significantly shorten this, if the Fourier trans- form of X T is known (which is often the case). The expectation E ( e X − K ) + is an instance of Z ∞ E g ( X ) = g ( x ) dµ X ( x ) , ( ∗ ) −∞ where µ X ( · ) is the distribution of the variable X . Parseval’s theorem allows one to compute ( ∗ ) directly from the Fourier transform, without having to find the distribution of X in the first place. Page 6

4 . FOURIER TRANSFORMS Fourier transform of a function : if h ∈ L 1 , Z ∞ h ( x ) e iux dx. ˆ · = h ( u ) · −∞ Fourier transform of a measure : If µ is a measure with | µ | < ∞ , then Z ∞ e iux dµ ( x ) . · = µ ( u ) · ˆ −∞ Page 7

The characteristic function of a probability distribution µ X is then Z ∞ µ X ( u ) = E e iuX = e iux dµ X ( x ) . ˆ −∞ Page 8

5 . FOURIER INVERSION Theorem . Suppose h is a real function which satisfies the fol- lowing conditions: (a) h ∈ L 1 and (b) [omitted technical conditions]. Then Z ∞ 1 e − iux ˆ 2[ h ( x +) + h ( x − )] = h ( u ) du. −∞ N.B . Last integral is a principal value (= Cauchy) integral. Page 9

6 . PARSEVAL’S THEOREM Theorem . If a random variable X has distribution µ X , then Z ∞ Z ∞ 1 E g ( X ) = g ( x ) µ X ( dx ) = g ( − u )ˆ ˆ µ X ( u ) du, 2 π −∞ −∞ provided that (i) g ∈ L 1 , and (ii) [omitted technical conditions, usually satisfied in option pricing]. Page 10

7 . EXPONENTIAL DAMPING (= TILTING) Parseval’s theorem cannot directly be applied to the pricing of calls and puts because the functions g 1 ( x ) = ( e x − K ) + , g 2 ( x ) = ( K − e x ) + are not in L 1 ( ⇒ Parseval’s theorem does not apply). Lewis (2001) shows how this difficulty can be avoided by mod- ifying g 1 (or g 2 ). For now, assume that X has a density f X ( · ). For any function ϕ , let ϕ ( α ) ( x ) = e αx ϕ ( x ) , x ∈ R . “tilted ϕ ” Page 11

The Fourier transform of ϕ ( α ) is denoted d ϕ ( α ) . g ( x ) f X ( x ) = g ( − α ) ( x ) f ( α ) Of course, we have: X ( x ). If it happens that both g ( − α ) and f ( α ) are in L 1 , then Parseval’s X theorem says that Z ∞ g ( − α ) ( x ) f ( α ) E g ( X ) = X ( x ) dx −∞ Z ∞ 1 g ( − α ) ( − u ) d [ f ( α ) = X ( u ) du. 2 π −∞ Lewis (2001) assumes that X has a density, which is not always the case in applications. We thus reformulate Lewis’s result in terms of a general probability distribution µ X : Page 12