Zura Kakushadze Leiden 02062017

Path Integral and Asset Pricing

Zura Kakushadze

Quantigicr Solutions LLC, Stamford, CT, USABusiness School & School of Physics, Free University of Tbilisi, Georgia

[email protected]

Keynote Talk Presented at the Workshop“Path Integration in Complex Dynamical Systems” (February 6-10, 2017)

Lorentz Center, Leiden, The Netherlands

February 6, 2017

Zura Kakushadze (Quantigic & FreeUni) Path Integral and Asset Pricing February 6, 2017 1 / 16

Preamble

Story

28th bday, Physics → Finance (???)

Baxter & Rennie, “Financial Calculus” (derivative pricing)

Recast into path integral → “Phynance” (Physics + Finance)

Taught “Phynance”, Spring’02, PhDs

SSRN (Social Science Research Network), May’14

Talk @ Morgan Stanley, Oct’14

Application: bond prices in short-rate models


Motivation

Why use path integral in asset pricing?

Feynman, 1948:“The formulation is mathematically equivalent to the more usualformulations. There are, therefore, no fundamentally new results.However, there is a pleasure in recognizing old things from a newpoint of view. Also, there are problems for which the new point ofview offers a distinct advantage.”

Path integral ⇔ Schrodinger’s/Heisenberg’s formulations

Feynman diagrams, QED, fine structure constant, . . .

Asset pricing: no panacea, no “fundamentally new results” (yet. . . )

Intuitive, clearer view of pathway toward solution, . . .


Derivatives a.k.a. contingent claims

What are derivatives?

Derivative: payout contingent on uncertain future event

Forward contract: deliver e.g. oil at future time T at preset price k

European call option: right to buy stock (index) at time T at price k

European put option: right to sell stock (index) at time T at price k

American options: exercise at any time t ≤ T

How to price derivatives?

No arbitrage pricing: arbitrage = risk-free profit

Assume: no transaction costs, zero interest rates, etc.

Forward at t = 0 to deliver stock at t = T at price k: k = S0k > S0: @t = 0 borrow $S0, buy stock @S0 . . . @t = T sell stock @kk < S0: @t = 0 short-sell stock @S0 . . . @t = T buy stock @k


How to Price Derivatives?

Pricing a claim

Underlying: e.g. stock St

Cash: savings account (cash bond), zero interest rate

Claim: payout fT at expiration time T

E.g.: European call option with strike k ; fT = (ST − k)+

Need: price Vt of claim at t < T

Toy model

Two time points: t = 0, t = T

t = 0: stock price S0

t = T : stock price S+ or S− (S− < S0 < S+ or else arbitrage)

Claim fT : f+ or f− depending on ST


How to Price Derivatives?


Replicating Strategy

Can we hedge a claim?

Hedging strategy: replicate claim fT

Instruments: stock and cash

Portfolio Vt : φ units of stock and ψ units of cash (dollars)

VT = fT (replication):

ST = S+ : φS+ + ψ = f+

ST = S− : φS− + ψ = f−

Claim price at t = 0

V0 = φS0 + ψ = qf+ + (1− q)f−

Risk-neutral measure: Q = {q, 1− q} [q = (S0 − S−)/(S+ − S−)]

Claim price = expectation: V0 = 〈fT 〉QStock price: 〈ST 〉Q = S0 (martingale)


Discrete Models and Brownian Motion

Binary tree

Random walk: εt = ±1 w/ prob. 1/2 [t = kδt, k = 0, 1, 2 . . . ]

Price: St+δt = St exp(σ√δtεt + µδt)

Pricing: replicating strategy

Vt = 〈fT 〉Q,Ft [Ft = history (filtration) up to t]

Risk-neutral measure: 〈St〉Q,Fτ = Sτ [0 ≤ τ ≤ t]

Brownian motion

Random walk: Wt =√δt∑t/δt−1

k=0 εkδt

Continuous limit: δt → 0

Wt : Brownian motion

St = S0 exp(σWt + µt) [log-normally distributed]

Vt = 〈fT 〉Q,Ft


Path integral

Expectation = Euclidean path integral

Previsible process: At = A(Ft)

Expectation: 〈AT 〉Q,Ft

Discretize: [t,T ]→ [t0, . . . , tN ]

FT = Ft ∪ {(x1, t1), . . . , (xN , tN)}:

〈AT 〉Q,Ft = limN∏

i=1

∫dxi√

2π∆tiexp

(−

∆x2i2∆ti

)A(FT )

=

∫x(t)=x0

Dx exp(−S) A(FT )

Action: S [x ] =∫ T

t dτ x2(τ)2 [m = ~ = 1]


Application: Short-rate Models

Nonzero interest rates

Cash bond Bt : continuous compounding, constant interest rate r

dBt = rBtdt: Bt = B0 exp(rt)

Short-rate: rt

Bt = B0 exp(∫ t

0 dτ rτ)

Time value of money: $1 @t = T is worth B−1T @t = 0 (discounting)

Claim pricing: Vt = Bt〈B−1T fT 〉Q,Ft [B−1t Vt = 〈B−1T fT 〉Q,Ft ]

How to price zero-coupon bonds?

Zero-coupon bond: pays $1 at maturity T

fT = 1

Vt =⟨

exp(−∫ T

t dτ rτ)⟩

Q,Ft

=∫Dx exp(−Seff )


Application: Short-rate Models

Short-rate = potential

Action: Seff [x ] =∫ T

t dτ[

x2(τ)2 + rτ

]Short-rate: potential energy

Short-rate model: posit risk-neutral measure Q and process rt

Ho and Lee model: rt = σWt + µt [Wτ → x(τ) in path integral]

Short-rate rt < 0: continuous spectrum, ill-behaved bond prices

Fix: e.g., rt = r0 exp(σWt + µt) [Black-Karasinski model]

Not analytically solvable

Path integral: Semiclassical approximation (Gaussian)

Corrections: Feynman diagrams


Volatility Smiles

Gaussian distribution

Black-Scholes model: St = S0 exp(σWt + µt) [log-normally distr.]

Call option: fT = (ST − k)+ [strike price k]

Empirically: implied volatility σimplied 6= const.


Volatility Smiles

Non-Gaussian Distributions

Volatility smiles: fat tails (non-Gaussian)

Gaussian distrib.: P(x , t) =∫

dk2π exp(ikx) exp(−Ht)

Non-relativistic Hamiltonian: H = k2/2 [m = ~ = 1]

Relativistic model

Relativistic ext.: H =√k2c2 + c4 − c2 [cf. rest energy]

Space-time: Euclidean (not Minkowski)

Relativistic distrib.: P(x , t) = c2tπ√

x2+c2t2K1(c

√x2 + c2t2) exp(c2t)

Large x : P(x , t) ∼ x−3/2 exp(−c |x |) [softer than exp(−x2/2t)]

Volatility smile = “relativistic effect”


Volatility Smiles


Outlook

Jumps

Levy processes: jumps (unless Brownian w/ drift)

Imperfect hedging: no previsibility (pricing OK)

Finance: incomplete market

Physics: wrong DOFs, e.g. quantizing relativistic particle

QFT: application to finance, path integral, . . .


References

For comprehensive lists of references, see:

ZK (2015) Path Integral and Asset Pricing. Quantitative Finance 15(11):1759-1771; http://ssrn.com/abstract=2506430.

ZK (2016) Volatility Smile as Relativistic Effect;http://ssrn.com/abstract=2827916.

Thank you!


Documents

Zura Kakushadze Leiden 02062017