Asset prices with jumps • Geometric Brownian motion has continuous paths. • Stock prices, and prices of other assets, often show jumps caused by unpredictable events or news items. • So geometric Brownian motion can only be an approximation to the real behavior of asset prices. 1
• The times at which jumps occur are often modeled by a Poisson process. • That is, if N t jumps occur in (0 , t ], then for some λ > 0 – for each s ≥ 0 and t > 0, N s + t − N s ∼ Poisson with mean λt ; – for each n ≥ 1 and times 0 ≤ t 0 ≤ · · · ≤ t n , the increments { N t r − N t r − 1 } are independent; – N 0 = 0; – N t is right-continuous in t ≥ 0. • Note the parallel with the definition of Brownian motion. 2
• The parameter λ is the intensity of the process; that is, the expected number of jumps per unit time: E [ N s + t − N s ] = λ. t • If τ i is the time of the i th jump, then the inter-jump times τ 1 , τ 2 − τ 1 , τ 3 − τ 2 , . . . are independently exponentially dis- tributed with mean 1 /λ . 3
• Suppose that an asset price generally follows a GBM, but each jump reduces the asset price by a fraction δ : µ − 1 �� � � 2 σ 2 (1 − δ ) N t . S t = S 0 exp t + σW t • Then S t should satisfy a differential equation like dS t = µdt + σdW t − δdN t . S t • As always, the meaning of this SDE is the corresponding stochastic integral equation � t � t � t S t − S 0 = 0 µS u du + 0 σS u dW u − 0 δS u dN u . 4
• The first integral is conventional, and the second is a stochas- tic integral. • For a continuous function f , the conventional Stieltjes inte- gral is � t N t � 0 f ( u ) dN u = f ( τ i ) . i =1 • Because S t may have discontinuities at the times τ i when N t increases, the third integral is not a conventional Stieltjes integral. 5
• We define it by � t N t � � � 0 f ( u, S u ) dN u = f τ i − , S τ i − . i =1 • With this definition, we have a generalized Itˆ o formula: if dY t = µ t dt + σ t dW t + ν t dN t and f is twice continuously differentiable, then � t � t 0 f ′ ( Y u − ) dY u + 1 0 f ′′ ( Y u − ) du f ( Y t ) − f ( Y 0 ) = 2 N t N t � − f f ′ � � � � � � �� � � + � Y τ i − Y τ i − Y τ i − Y τ i − f Y τ i − . i =1 i =1 6
Girsanov’s Theorem with jumps • Let { W t } t ≥ 0 be a standard P -Brownian motion and { N t } t ≥ 0 a (possibly time-inhomogenous) Poisson process with intensity { λ t } t ≥ 0 under P . That is, � t M t = N t − 0 λ u du is a P -martingale. • We write F t for the σ -field generated by F W ∪ F N t . t • Suppose that { θ t } t ≥ 0 and { φ t } t ≥ 0 are {F t } t ≥ 0 -previsible pro- cesses with φ t > 0, such that � t � t 0 θ 2 s ds < ∞ and 0 φ s λ s ds < ∞ . 7
• Further, let Q be the measure whose Radon-Nikodym deriva- tive with respect to P is d Q � � = L t , � d P � F t where L 0 = 1 and dL t = θ t dW t − (1 − φ t ) dM t . L t − • Then, under Q , the process { X t } t ≥ 0 defined by � t X t = W t − 0 θ s ds is a standard Brownian motion and { N t } t ≥ 0 has intensity { φ t λ t } t ≥ 0 . 8
• Can we use this theorem to find an equivalent martingale measure? • Suppose again that dS t = µdt + σdW t − δdN t , S t • Then the discounted process { ˜ S t } t ≥ 0 satisfies d ˜ S t = ( µ − r ) dt + σdW t − δdN t ˜ S t = ( µ − r + σθ t − δλφ t ) dt + σdX t − δdM t . 9
• Here { X t } t ≥ 0 is Q -Brownian motion, and { M t } t ≥ 0 defined by � t M t = N t − 0 λ s φ s ds is a Q -martingale. • So { ˜ S t } t ≥ 0 is a Q -martingale for any choice of { θ t } t ≥ 0 and { φ t } t ≥ 0 for which µ − r + σθ t − δλφ t = 0 . • Thus there is no unique equivalent martingale measure, and the market is incomplete: there exist F T -measurable claims C T that cannot be replicated, and cannot be hedged. 10
• If we have a second asset whose price is driven by the same { W t } t ≥ 0 and { N t } t ≥ 0 , then both discounted processes become martingales under a unique Q , provided the equations µ ( i ) − r + σ ( i ) θ t − δ ( i ) λφ t = 0 , i = 1 , 2 are nonsingular for all t ≥ 0. • The extended market is therefore complete, and all claims can be replicated. 11
Stochastic Volatility • The time-dependent volatility σ t may have its own dynamics: dS t = µS t dt + σ t S t dW (1) t where � � � ρdW (1) 1 − ρ 2 dW (2) σ t = a ( S t , σ t ) dt + b ( S t , σ t ) + . t t � � � � W (1) W (2) • Here and are independent Brownian t t t ≥ 0 t ≥ 0 motions, and ρ ∈ ( − 1 , 1) is the correlation between the in- crements in the two equations. 12
• As in the case of jumps, equivalent martingale measures may be found but are not unique. • Again, if we have a second asset whose price is governed � � � � W (1) σ (1) by the same and , such as an option t t t ≥ 0 t ≥ 0 on S T , the extended market may have a unique equivalent martingale measure and hence be complete. 13
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