Lecture on Advanced topics on Stochastic Differential Equations - PowerPoint PPT Presentation

Lecture on Advanced topics on Stochastic Differential Equations Erik Lindström

Recap t X t d W t F X d W t d t T F XX 2 1 F X F t d Y t Then (4) C 1 2 F t X t Y t (3) t X t d t (5) Shorthand notation 0 0 d X t (1) d X t t X t d t t X t d W t (2) We also introduced the Ito formula. Assume ◮ We defined Stochastic Integral Equations ∫ t ∫ t X ( t ) = X ( 0 ) + µ ( s , X ( s )) d s + σ ( s , X ( s )) d W ( s )

Recap (3) F X d W t d t T F XX 2 1 F X F t d Y t Then (4) C 1 2 F t X t Y t t X t d W t t X t d t d X t We also introduced the Ito formula. Assume (2) (1) 0 0 (5) ◮ We defined Stochastic Integral Equations ∫ t ∫ t X ( t ) = X ( 0 ) + µ ( s , X ( s )) d s + σ ( s , X ( s )) d W ( s ) ◮ Shorthand notation d X ( t ) = µ ( t , X ( t )) d t + σ ( t , X ( t )) d W ( t )

Recap (4) Then o formula. Assume (2) (1) (3) 0 0 ◮ We defined Stochastic Integral Equations ∫ t ∫ t X ( t ) = X ( 0 ) + µ ( s , X ( s )) d s + σ ( s , X ( s )) d W ( s ) ◮ Shorthand notation d X ( t ) = µ ( t , X ( t )) d t + σ ( t , X ( t )) d W ( t ) ◮ We also introduced the It ¯ d X ( t ) = µ ( t , X ( t )) d t + σ ( t , X ( t )) d W ( t ) Y ( t ) = F ( t , X ( t )) ∈ C 1 , 2 ( ) d Y ( t ) = F t + µ F X + 1 2 σσ T F XX d t + σ F X d W ( t ) (5)

Applications in Finance - Main result This also holds for derivatives, where the Contract (7) t B T T S T E B t t S t function is a function of the traded assets Option valuation is based on this theorem! Risk-Neutral Valuation Formula : Assume that the (6) free from arbitrage. Then market [ B ( t ) , S ( t ) ] consists of traded assets and it is [ S ( T ) ] S ( t ) B ( t ) = E Q B ( T ) |F ( t ) for some equivalent probability measure Q .

Applications in Finance - Main result (6) (7) function is a function of the traded assets Risk-Neutral Valuation Formula : Assume that the This also holds for derivatives, where the Contract free from arbitrage. Then Option valuation is based on this theorem! market [ B ( t ) , S ( t ) ] consists of traded assets and it is [ S ( T ) ] S ( t ) B ( t ) = E Q B ( T ) |F ( t ) for some equivalent probability measure Q . [ π ( T , S ( T )) ] π ( t , S ( t )) = E Q |F ( t ) B ( t ) B ( T )

Market A B t is a S t discounted price process Z t and the measure if it is equivalent to is an equivalent martingale A measure (10) 0 A 0 ) if and only if equivalent (written A are called A and Two measures Then we introduce the discounted portfolio (9) (8) martingale. ◮ Define the market as d S ( t ) = α S ( t ) d t + σ S ( t ) d W ( t ) d B ( t ) = rB ( t ) d t Z ( t ) = S ( t ) / B ( t ) , leading to dZ ( t ) = ( α − r ) Z ( t ) d t + σ Z ( t ) d W ( t )

Market (9) (10) Then we introduce the discounted portfolio martingale. (8) ◮ Define the market as d S ( t ) = α S ( t ) d t + σ S ( t ) d W ( t ) d B ( t ) = rB ( t ) d t Z ( t ) = S ( t ) / B ( t ) , leading to dZ ( t ) = ( α − r ) Z ( t ) d t + σ Z ( t ) d W ( t ) ◮ Two measures P ( A ) and Q ( A ) are called equivalent (written Q ∼ P ) if and only if P ( A ) = 0 ⇐ ⇒ Q ( A ) = 0 . ◮ A measure Q is an equivalent martingale measure if it is equivalent to P and the discounted price process Z ( t ) = S ( t ) / B ( t ) is a

Some definitions (11) 0 0 (12) This reflects that no money is added or subtracted from the portfolio. ◮ Any portfolio is given by V ( t , h ) = h 1 ( t ) B ( t ) + h 2 ( t ) S ( t ) ◮ A portfolio strategy h is self-financing if ∫ t ∫ t V ( t , h ) = V ( 0 , h ) + h 1 ( u ) d B ( u ) + h 2 ( u ) d S ( u ) .

It then follows that the discounted portfolio is given only if (13) Proof: Lemma 9.1 by V Z ( t , h ) = V ( t , h ) B ( t ) = h 1 ( t ) + h 2 ( t ) Z ( t ) ◮ A portfolio strategy h is self-financing if and d V Z ( t , h ) = h 2 ( t ) d Z ( t ) ◮ If h is a self-financing portfolio, the wealth process V Z ( t , h ) is a Q -martingale.

V Z T h V Z T h V Z 0 h V Z T h Arbitrage V 0 which contradicts the arbitrage condition (15) 0 E V 0 h consequently 0 and 0 1 and 0 we also have 0. V T h (14) arbitrage Proof: Assume that h is an arbitrage portfolio Then since wth 0 1 and V T h 0 0. ◮ An arbitrage is a portfolio with zero initial investment V ( 0 ) = 0 and P ( V ( T ) ≥ 0 ) = 1 , P ( V ( T ) > 0 ) > 0 ◮ Theorem: Assume that there exist a martingale measure Q . Then the model is free from

V Z 0 h V Z T h Arbitrage and V 0 which contradicts the arbitrage condition (15) 0 E V 0 h consequently 0. arbitrage (14) ◮ An arbitrage is a portfolio with zero initial investment V ( 0 ) = 0 and P ( V ( T ) ≥ 0 ) = 1 , P ( V ( T ) > 0 ) > 0 ◮ Theorem: Assume that there exist a martingale measure Q . Then the model is free from ◮ Proof: Assume that h is an arbitrage portfolio wth P ( V ( T , h ) ≥ 0 ) = 1 and P ( V ( T , h ) > 0 ) > 0. Then since Q ∼ P we also have Q ( V Z ( T , h ) ≥ 0 ) = 1 and Q ( V Z ( T , h ) > 0 ) > 0

Arbitrage consequently (14) arbitrage which contradicts the arbitrage condition (15) ◮ An arbitrage is a portfolio with zero initial investment V ( 0 ) = 0 and P ( V ( T ) ≥ 0 ) = 1 , P ( V ( T ) > 0 ) > 0 ◮ Theorem: Assume that there exist a martingale measure Q . Then the model is free from ◮ Proof: Assume that h is an arbitrage portfolio wth P ( V ( T , h ) ≥ 0 ) = 1 and P ( V ( T , h ) > 0 ) > 0. Then since Q ∼ P we also have Q ( V Z ( T , h ) ≥ 0 ) = 1 and Q ( V Z ( T , h ) > 0 ) > 0 and V ( 0 , h ) = V Z ( 0 , h ) = E Q [ V Z ( T , h )] > 0 V ( 0 ) = 0.

Change of measure - Girsanov (17) can perfectly hedge all options [Meta theorem in implies that the market is complete, i.e. that we measure is unique. This We find that the [Solve on the Blackboard] (16) the book by Björk] Motion ◮ Let W P ( t ) be a P Brownian Motion. ∫ t 0 g ( s ) d s + W P ( t ) is a Q Brownian ◮ Then X ( t ) = ◮ Example. Find Q when the market is made up of d S ( t ) = µ S ( t ) d t + σ S ( t ) d W ( t ) d B ( t ) = rB ( t ) d t

Change of measure - Girsanov Motion (16) (17) [Solve on the Blackboard] implies that the market is complete, i.e. that we can perfectly hedge all options [Meta theorem in the book by Björk] ◮ Let W P ( t ) be a P Brownian Motion. ∫ t 0 g ( s ) d s + W P ( t ) is a Q Brownian ◮ Then X ( t ) = ◮ Example. Find Q when the market is made up of d S ( t ) = µ S ( t ) d t + σ S ( t ) d W ( t ) d B ( t ) = rB ( t ) d t ◮ We find that the Q measure is unique. This

Back to pricing discounting Monte Carlo (Chapter 12) Real world dynamics Risk adjustment The contingent claim price is given by Contract function See BenchOp paper on home page for comparisons. (18) π ( 0 , S ( 0 )) = e − rT E Q [ φ ( S ( T )) |F ( 0 )] ∫ = e − rT φ ( S ( T )) q S ( T ) | S ( 0 ) ( S ( T )) d S ( T ) ∫ q S ( T ) | S ( 0 ) e − rT = φ ( S ( T )) ( S ( T )) p S ( T ) | S ( 0 ) d S ( T ) ���� � �� � p S ( T ) | S ( 0 ) � �� � � �� � ◮ Analytical (approximations) ◮ Fourier methods (FFT/COS/GL) (Chapter 9.6.1) ◮ Monte Carlo, Multi Level Monte Carlo, Quasi ◮ PDEs (FD/FE/RBFs)

Black & Scholes formula (19) (20) making it complete and free from arbitrage. A Europan Call option is given by (21) Solved in Section 9.2.1 The Q − market is given by d S ( t ) = rS ( t ) d t + σ S ( t ) d W ( t ) d B ( t ) = rB ( t ) d t , C ( K ) = e − rT E Q [ max ( S T − K , 0 ) |F ( 0 )]

Connections to PDEs F t 0 F X d W s But E 0 F X d W s E (23) 0 F d s E Feynman-Kac: Assume that F solves some PDE F 0 X 0 0 T E F X T Proof: It is clear that diffusion process. (22) 0, and F solves the PDE. F t + A F = 0 and that F is regular enough. Then F ( 0 , X ( 0 )) = E [ F ( X ( T ) , T ) |F ( 0 )] where A is the generator associated with the

Connections to PDEs Feynman-Kac: Assume that F solves some PDE (22) diffusion process. Proof: It is clear that But E (23) F t + A F = 0 and that F is regular enough. Then F ( 0 , X ( 0 )) = E [ F ( X ( T ) , T ) |F ( 0 )] where A is the generator associated with the [∫ ] E [ F ( X ( T ) , T ) |F ( 0 )] = F ( 0 , X ( 0 )) + E ( F t + A F ) d s |F ( 0 ) [∫ ] + E σ F X d W ( s ) |F ( 0 ) [∫ ] σ F X d W ( s ) |F ( 0 ) = 0, and F solves the PDE.

(24) o Similarly, if F t + A F − rF = 0, then F ( t , X ( t )) = e − r ( T − t ) E [ F ( T , X ( T )) |F ( t )] Proof: It ¯

Extensions Many extensions possible (Chapt 9.5), including All but local volatility models can use the Fourier framework. ◮ Stochastic volatility ◮ Jump models ◮ Local volatility ◮ Uncertain volatility

Extra: Carr-Madan framework (25) (26) 0 characteristic function of the log price a PDE! ◮ Let ∫ [ e ius ( T ) ] ψ ( u ) = E = e ius d Q ( s ) . s ( T ) = log S ( T ) , cf Chapt 7.5. ◮ Then it follows that ∫ ∞ C ( k ) = e − α k e − rT ψ ( v − ( α + 1 ) i ) e − ivk α 2 + α − v 2 + i ( 2 α + 1 ) v d v . π ◮ Often easier to compute an integral than solving

References time . Oxford university press. Lindström, E., Milovanovi฀, S., Persson, J., ... & Waldén, J. (2015). BENCHOP–The BENCHmarking project in option pricing. International Journal of Computer Mathematics , 92(12), 2361-2379. http: //dx.doi.org/10.1080/00207160.2015.1072172 ◮ Björk, T. (2009). Arbitrage theory in continuous ◮ von Sydow, L., Josef Höök, L., Larsson, E.,