Duality and Derivative Pricing with L evy Processes Jos e Fajardo - PDF document

Duality and Derivative Pricing with L´ evy Processes Jos´ e Fajardo IBMEC Ernesto Mordecki Universidad de la Rep´ ublica del Uruguay 2004 North American Winter Meeting of the Econometric Society 1

Related Works • Margrabe, W., (1978), “The Value of an Option to Exchange One Asset for Another”, J. Finance . • Gerber, H. and W. Shiu, (1996), “Martingale Approach to Pricing American Perpetual Options on Two Stocks”, Math. Finance . • Schroder, Mark. (1999), “Change of Numeraire for Pricing Futures, Forwards, and Options”, Review of Financial Studies . • Peskir, G., and A. N. Shiryaev, (2001), “A Note on The Call-Put Parity and a Call-Put Duality”, Theory Probab. Appl . • Detemple, J. (2001), “American Options: Symmetry Property”, Cambridge University Press. • Carr, P. and Chesney, M. (1998), “American Put Call Symmetry”, Morgan Stanley working paper. • Carr, P., Ellis, K., and Gupta, V., (1998), “Static Hedging of Exotic Options”, J. Finance . 2

Some Relevant Derivatives • Margrabe’s Options: a) f ( x, y ) = max { x, y } Maximum Option b) f ( x, y ) = | x − y | Symmetric Option c) f ( x, y ) = min { ( x − y ) + , ky } Option with Pro- portional Cap • Swap Options f ( x, y ) = ( x − y ) + Option to exchange one risk asset for another. • Quanto Options f ( x, y ) = ( x − ky ) + x is the foreign stock in foreign currency. Then we have the price of an option to exchange one foreign currency for another. 3

• Equity-Linked Foreign Exchange Option (ELF-X Option). Take S : foreign stock in foreign currency and Q is the spot exchange rate. We use foreign market risk measure, then an ELF-X is an invest- ment that combines a currency option with an equity forward. The owner has the option to buy S t with domestic currency which can be converted from for- eign currency using a previously stipulated strike ex- change rate R (domestic currency/foreign currency). The payoff is: Φ t = S T (1 − RQ T ) + Then f ( x, y ) = ( y − Rx ) + . • Vanilla Options. f ( x, y ) = ( x − ky ) + and f ( x, y ) = ( ky − x ) + . 4

Multidimensional L´ evy Processes Let X = ( X 1 , . . . , X d ) be a d -dimensional L´ evy process defined on a stochastic basis B = (Ω , F , {F} t ≥ 0 , P ). This means that X stochastic process with indepen- dent increments, such that the distribution of X t + s − X s does not depend on s , with P ( X 0 = 0) = 1 and trajec- tories continuous from the left with limits from the right. For z = ( z 1 , . . . , z d ) in C d , when the integral is con- vergent (and this is always the case if z = iλ with λ in R d , L´ evy-Khinchine formula states, that E e zX t = exp( t Ψ( z )) where the function Ψ is the characteristic exponent of the process, and is given by � � � Ψ( z ) = ( a, z )+1 e ( z,y ) − 1 − ( z, y ) 1 {| y |≤ 1 } 2( z, Σ z )+ Π( dy ) , R d (1) R d , Π is a positive where a = ( a 1 , . . . , a d ) is a vector in I � measure defined on R d \{ 0 } such that R d ( | y | 2 ∧ 1)Π( dy ) is finite, and Σ = (( s ij )) is a symmetric nonnegative definite matrix, that can always be written as Σ = A ′ A (where ′ denotes transposition) for some matrix A . 5

Market Model and Problem Consider a market model with three assets ( S 1 , S 2 , S 3 ) given by t = e X 1 0 e X 2 0 e X 3 S 1 t , S 2 t = S 2 t , S 3 t = S 3 (2) t where ( X 1 , X 2 , X 3 ) is a three dimensional L´ evy process, and for simplicity, and without loss of generality we take S 1 0 = 1. The first asset is the bond and is usually de- terministic. Randomness in the bond { S 1 t } t ≥ 0 allows to consider more general situations, as for example the pric- ing problem of a derivative written in a foreign currency, referred as quanto option . Consider a function: f : (0 , ∞ ) × (0 , ∞ ) → I R homogenous of an arbitrary degree α ; i.e. for any λ > 0 and for all positive x, y f ( λx, λy ) = λ α f ( x, y ) . In the above market a derivative contract with payoff given by Φ t = f ( S 2 t , S 3 t ) is introduced. Assuming that we are under a risk neutral martingale measure, thats to say, S k S 1 ( k = 2 , 3) are P -martingales, 6

we want to price the derivative contract just introduced. In the European case, the problem reduces to the com- putation of � � e − X 1 0 e X 2 0 e X 3 E T = E ( S 2 0 , S 3 T f ( S 2 T , S 3 0 , T ) = E T ) (3) In the American case, if M T denotes the class of stopping times up to time T , i.e: M T = { τ : 0 ≤ τ ≤ T, τ stopping time } for the finite horizon case, putting T = ∞ for the per- petual case, the problem of pricing the American type derivative introduced consists in solving an optimal stop- ping problem, more precisely, in finding the value function A T and an optimal stopping time τ ∗ in M T such that � � 0 e X 3 e − X 1 0 e X 2 A T = A ( S 2 0 , S 3 τ f ( S 2 τ , S 3 0 , T ) = sup τ 3 ) E τ ∈M T � � e − X 1 0 e X 2 0 e X 3 τ ∗ f ( S 2 τ ∗ , S 3 = E τ ∗ ) . 7

How to obtain an EMM Define M ( z, t ; h ) = M ( z + h, t ) M ( h, t ) where M ( h, t ) = E ( e h · X ′ t ). Now we will find a vector h ∗ e h ∗· X ′ t such that the probability dQ t = t ) dP t be an EMM, E ( e h ∗· X ′ in other words: S j 0 = E ∗ ( e − rt S j t ) ∀ j, ∀ t take 1 j = (0 , . . . , 1 , . . . , 0),then ���� j − position � M (1 j + h ∗ , 1) � r = log[ M (1 j , 1; h ∗ )] = log M ( h ∗ , 1) Now in our model we need that { S j t } be martingale, as t S 1 S 1 0 = 1, then 0 = E ∗ ( S j S j t ) S 1 t 1 = E ∗ ( e X j t − X 1 t ) Defining ¯ 1 j = ( − 1 , 0 , . . . , 1 , . . . , 0), we have ���� j − position 1 = M (¯ 1 j , 1; h ∗ ) 8

Dual Market Method observe that e − X 1 0 e X 2 0 e X 3 t ) = e − X 1 t + αX 3 0 e X 2 t − X 3 t f ( S 2 t , S 3 t f ( S 2 t , S 3 0 ) . Let ρ = − log E e − X 1 1 + αX 3 1 , that we assume finite. The process Z t = e − X 1 t + αX 3 t + ρt is a density process (i.e. a positive martingale starting at Z 0 = 1) that allow us to introduce a new measure ˜ P by its restrictions to each F t by the formula d ˜ P t = Z t . dP t Denote now by X t = X 2 t − X 3 t , and S t = S 2 0 e X t . Finally, let F ( x ) = f ( x, S 3 0 ) . With the introduced notations, under the change of mea- sure we obtain � � E T = ˜ e − ρT F ( S T ) E � � ˜ e − ρτ F ( S τ ) A T = sup E τ ∈M T 9

The following step is to determine the law of the pro- cess X under the auxiliar probability measure ˜ P . evy process on R d with char- Lemma 1 Let X be a L´ acteristic exponent given in (1). Let u and v be vec- tors in R d . Assume that E e ( u,X 1 ) is finite, and denote ρ = log E e ( u,X 1 ) = Ψ( u ) . In this conditions, introduce the probability measure ˜ P by its restrictions ˜ P t to each F t by d ˜ P t = exp[( u, X t ) − ρt ] . dP t Then the law of the unidimensional L´ evy process { ( v, X t ) } t ≥ 0 under ˜ P is given by the triplet  � a = ( a, v ) + 1 R d e ( u,y ) ( v, y ) 1 {| ( v,y ) |≤ 1 , | x | > 1 } Π( dx ) ˜ 2 [( v, Σ u ) + ( u, Σ v )] +  σ 2 = ( v, Σ v ) ˜ �  R d 1 { ( v,y ) ∈ A } e ( u,y ) Π( dy ) . π ( A ) = ˜ (4) 10

Closed Formulae European derivative Let S 2 T and S 3 T be two risky assets, a contract with pay- T ) + can be priced using The Dual Market off ( S 2 T − S 3 Method : � T ) + � e − rT ( S 2 T − S 3 D = E . � 0 e X 2 0 e X 3 e − rT ( S 2 T − S 3 T ) dP = A Assuming for simplicity S 2 0 = S 3 0 = 1, Then A = { ω ∈ Ω : X 2 T ( ω ) > X 3 T ( ω ) } , we apply the method: � e − rT ( e X 2 T − e X 3 D = T ) dP A � e − rT e X 3 = T ( S T − 1) dP { S T > 1 } where S T = e X T and X = X 2 − X 3 . Now the dual measure: ρ = − log E e − r + X 3 1 = r − log E e X 3 1 , then: P = e X 3 T d � T dP E e X 3 11

With all this: � ( S T − 1) d � D = e − ρT P { S T > 1 } � � S T d � d � D = e − ρT P − e − ρT P { S T > 1 } { S T > 1 } To reduce this expression we need a distribution for X under P , then applying the Lemma 1 we obtain the den- sity of S T under � P . It is worth noting that instead of using a distribution for X , we can make the following change of measure: P = e X 2 T d � T dP E e X 2 we obtain: � T e X 3 T T dP − e − ρT � e X 2 T − X 3 D = e − ρT P ( S T > 1) E e X 3 { S T > 1 } D = e − ρT E e X 2 T P ( S T > 1) − e − ρT � � P ( S T > 1) E e X 3 T 12

American Perpetual Swap Proposition 1 Let M = sup o ≤ t ≤ τ X t with τ an inde- pendent exponential random variable with parameter E e M < ∞ and ρ , then ˜ � � 0 e M − S 3 ˜ 0 ˜ S 2 E ( e M ) E A ( S 2 0 , S 3 0 ) = ˜ E ( e M ) the optimal stopping time is 0 ˜ τ ∗ c = inf { t ≥ 0 , S t ≥ S 3 E ( e M ) } 13

Put-Call Duality Now consider a L´ evy market with only two assets: B t = e rt , r ≥ 0 , S 0 = e x > 0 . S t = S 0 e X t , (5) In this section we assume that the stock pays dividends, with constant rate δ ≥ 0. In terms of the characteristic exponent of the process this means that ψ (1) = r − δ, (6) based on the fact, E e − ( r − δ ) t + X t = e − t ( r − δ + ψ (1)) = 1, con- dition (6) can also be formulated in terms of the charac- teristic triplet of the process X as � � � e y − 1 − y 1 {| y |≤ 1 } a = r − δ − σ 2 / 2 − Π( dy ) . (7) I R 14