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

Lecture on Advanced topics on Stochastic Differential Equations

Erik Lindström

Recap

◮ We defined Stochastic Integral Equations

X(t) = X(0) + ∫ t µ(s, X(s))ds + ∫ t σ(s, X(s))dW(s) (1) Shorthand notation dX t t X t dt t X t dW t (2) We also introduced the Ito formula. Assume dX t t X t dt t X t dW t (3) Y t F t X t C1 2 (4) Then dY t Ft FX 1 2

TFXX

dt FXdW t (5)

Recap

◮ We defined Stochastic Integral Equations

X(t) = X(0) + ∫ t µ(s, X(s))ds + ∫ t σ(s, X(s))dW(s) (1)

◮ Shorthand notation

dX(t) = µ(t, X(t))dt + σ(t, X(t))dW(t) (2) We also introduced the Ito formula. Assume dX t t X t dt t X t dW t (3) Y t F t X t C1 2 (4) Then dY t Ft FX 1 2

TFXX

dt FXdW t (5)

Recap

◮ We defined Stochastic Integral Equations

X(t) = X(0) + ∫ t µ(s, X(s))ds + ∫ t σ(s, X(s))dW(s) (1)

◮ Shorthand notation

dX(t) = µ(t, X(t))dt + σ(t, X(t))dW(t) (2)

◮ We also introduced the It¯

• formula. Assume

dX(t) = µ(t, X(t))dt + σ(t, X(t))dW(t) (3) Y(t) = F(t, X(t)) ∈ C1,2 (4) Then dY(t) = ( Ft + µFX + 1 2σσTFXX ) dt + σFXdW(t) (5)

Applications in Finance - Main result

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

Applications in Finance - Main result

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

Market

◮ Define the market as

dS(t) = αS(t)dt + σS(t)dW(t) (8) dB(t) = rB(t)dt (9) Then we introduce the discounted portfolio Z(t) = S(t)/B(t), leading to dZ(t) = (α − r)Z(t)dt + σZ(t)dW(t) Two measures A and A are called equivalent (written ) if and only if A A (10) A measure is an equivalent martingale measure if it is equivalent to and the discounted price process Z t S t B t is a martingale.

Market

◮ Define the market as

dS(t) = αS(t)dt + σS(t)dW(t) (8) dB(t) = rB(t)dt (9) Then we introduce the discounted portfolio Z(t) = S(t)/B(t), leading to dZ(t) = (α − r)Z(t)dt + σZ(t)dW(t)

◮ Two measures P(A) and Q(A) are called

equivalent (written Q ∼ P) if and only if P(A) = 0 ⇐ ⇒ Q(A) = 0. (10)

◮ 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 martingale.

Some definitions

◮ Any portfolio is given by

V(t, h) = h1(t)B(t) + h2(t)S(t) (11)

◮ A portfolio strategy h is self-financing if

V(t, h) = V(0, h) + ∫ t h1(u)dB(u) + ∫ t h2(u)dS(u). (12) This reflects that no money is added or subtracted from the portfolio.

It then follows that the discounted portfolio is given by VZ(t, h) = V(t,h)

B(t) = h1(t) + h2(t)Z(t) ◮ A portfolio strategy h is self-financing if and

• nly if

dVZ(t, h) = h2(t)dZ(t) (13)

◮ If h is a self-financing portfolio, the wealth

process VZ(t, h) is a Q-martingale. Proof: Lemma 9.1

Arbitrage

◮ An arbitrage is a portfolio with zero initial

investment V(0) = 0 and P(V(T) ≥ 0) = 1, P(V(T) > 0) > 0 (14)

◮ Theorem: Assume that there exist a martingale

measure Q. Then the model is free from arbitrage Proof: Assume that h is an arbitrage portfolio wth V T h 1 and V T h 0. Then since we also have VZ T h 1 and VZ T h 0 and consequently V 0 h VZ 0 h E VZ T h (15) which contradicts the arbitrage condition V 0 0.

Arbitrage

◮ An arbitrage is a portfolio with zero initial

investment V(0) = 0 and P(V(T) ≥ 0) = 1, P(V(T) > 0) > 0 (14)

◮ Theorem: Assume that there exist a martingale

measure Q. Then the model is free from arbitrage

◮ 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(VZ(T, h) ≥ 0) = 1 and Q(VZ(T, h) > 0) > 0 and consequently V 0 h VZ 0 h E VZ T h (15) which contradicts the arbitrage condition V 0 0.

Arbitrage

◮ An arbitrage is a portfolio with zero initial

investment V(0) = 0 and P(V(T) ≥ 0) = 1, P(V(T) > 0) > 0 (14)

◮ Theorem: Assume that there exist a martingale

measure Q. Then the model is free from arbitrage

◮ 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(VZ(T, h) ≥ 0) = 1 and Q(VZ(T, h) > 0) > 0 and consequently V(0, h) = VZ(0, h) = EQ[VZ(T, h)] > 0 (15) which contradicts the arbitrage condition V(0) = 0.

Change of measure - Girsanov

◮ Let WP(t) be a P Brownian Motion. ◮ Then X(t) =

∫ t

0 g(s)ds + WP(t) is a Q Brownian

Motion

◮ Example. Find Q when the market is made up of

dS(t) = µS(t)dt + σS(t)dW(t) (16) dB(t) = rB(t)dt (17) [Solve on the Blackboard] We find that the measure is unique. This implies that the market is complete, i.e. that we can perfectly hedge all options [Meta theorem in the book by Björk]

Change of measure - Girsanov

◮ Let WP(t) be a P Brownian Motion. ◮ Then X(t) =

∫ t

0 g(s)ds + WP(t) is a Q Brownian

Motion

◮ Example. Find Q when the market is made up of

dS(t) = µS(t)dt + σS(t)dW(t) (16) dB(t) = rB(t)dt (17) [Solve on the Blackboard]

◮ We find that the Q measure is unique. This

implies that the market is complete, i.e. that we can perfectly hedge all options [Meta theorem in the book by Björk]

Back to pricing

The contingent claim price is given by π(0, S(0)) = e−rTEQ [φ(S(T))|F(0)] (18) = e−rT ∫ φ(S(T))qS(T)|S(0)(S(T))dS(T) = e−rT

• discounting

∫ φ(S(T))

Contract function

qS(T)|S(0) pS(T)|S(0) (S(T))

• Risk adjustment

pS(T)|S(0)dS(T)

• Real world dynamics

◮ Analytical (approximations) ◮ Fourier methods (FFT/COS/GL) (Chapter 9.6.1) ◮ Monte Carlo, Multi Level Monte Carlo, Quasi

Monte Carlo (Chapter 12)

◮ PDEs (FD/FE/RBFs)

See BenchOp paper on home page for comparisons.

Black & Scholes formula

The Q−market is given by dS(t) = rS(t)dt + σS(t)dW(t) (19) dB(t) = rB(t)dt, (20) making it complete and free from arbitrage. A Europan Call option is given by C(K) = e−rTEQ [max(ST − K, 0)|F(0)] (21) Solved in Section 9.2.1

Connections to PDEs

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

Connections to PDEs

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

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

Extensions

Many extensions possible (Chapt 9.5), including

◮ Stochastic volatility ◮ Jump models ◮ Local volatility ◮ Uncertain volatility

All but local volatility models can use the Fourier framework.

Extra: Carr-Madan framework

◮ Let

ψ(u) = E [ eius(T)] = ∫ eiusdQ(s). (25) characteristic function of the log price s(T) = log S(T), cf Chapt 7.5.

◮ Then it follows that

C(k) = e−αk π ∫ ∞ e−ivk e−rTψ(v − (α + 1)i) α2 + α − v2 + i(2α + 1)vdv. (26)

◮ Often easier to compute an integral than solving

a PDE!

References

◮ Björk, T. (2009). Arbitrage theory in continuous

• time. Oxford university press.

◮ von Sydow, L., Josef Höök, L., Larsson, E.,

Lindström, E., Milovanovi฀, S., Persson, J., ... & Waldén, J. (2015). BENCHOP–The BENCHmarking project in option pricing. International Journal

• f Computer Mathematics, 92(12), 2361-2379.

http: //dx.doi.org/10.1080/00207160.2015.1072172