▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r s♦♠❡ ❢❛st st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ❜② ✈✐s❝♦s✐t② ♠❡t❤♦❞s ❉❛r✐❛ ●❤✐❧❧✐ ❯♥✐✈❡rs✐t② ♦❢ P❛❞✉❛ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ▼❛rt✐♥♦ ❇❛r❞✐ ❛♥❞ ❆♥♥❛❧✐s❛ ❈❡s❛r♦♥✐ ❚❯ ❇❡r❧✐♥✱ ❙❛t✉r❞❛② 26 ❖❝t♦❜❡r✱ ✷✵✶✹ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s
P❧❛♥ ✶ ❙t♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ✷ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❜② ✈✐s❝♦s✐t② ♠❡t❤♦❞s ✸ ❆s ❛♣♣❧✐❝❛t✐♦♥s✱ ❛s②♠♣t♦t✐❝ ❡st✐♠❛t❡s ❢♦r ❊✉r♦♣❡❛♥ ♦✉t✲♦❢✲t❤❡✲♠♦♥❡② ♦♣t✐♦♥ ♣r✐❝❡s ♥❡❛r ♠❛t✉r✐t② ❛♥❞ ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛ ❢♦r ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②✳ ✹ ❊①t❡♥s✐♦♥ t♦ t❤❡ ♥♦♥✲❝♦♠♣❛❝t ❝❛s❡ ✭✐✳❡✳ ✇❤❡♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ st♦❝❤❛st✐❝ s②st❡♠ ❛r❡ ♥♦t ♣❡r✐♦❞✐❝✮✱ ✇♦r❦ ✐♥ ♣r♦❣r❡ss✳ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s
❙t♦❝❤❛st✐❝ s②st❡♠ ✇✐t❤ ❢❛st ♦s❝✐❧❧❛t✐♥❣ r❛♥❞♦♠ ♣❛r❛♠❡t❡r ❲❡ ❝♦♥s✐❞❡r ❛ st♦❝❤❛st✐❝ s②st❡♠ ✐♥ R n ✇✐t❤ r❛♥❞♦♠ ❝♦❡✣❝✐❡♥ts✱ ✐♥ ♣❛rt✐❝✉❧❛r ✇✐t❤ ❝♦❡✣❝✐❡♥ts ❞❡♣❡♥❞❡♥t ♦♥ r❛♥❞♦♠ ♣❛r❛♠❡t❡r Y t ✳ √ X 0 = x 0 ∈ R n . dX t = φ ( X t , Y t ) dt + 2 σ ( X t , Y t ) dW t , ❆ss✉♠♣t✐♦♥✿ ✇❡ ♠♦❞❡❧ t❤✐s ♥❡✇ ♣❛r❛♠❡t❡r ❛s ❛ ♠❛r❦♦✈ ♣r♦❝❡ss ❡✈♦❧✈✐♥❣ ♦♥ ❛ ❢❛st❡r t✐♠❡ s❝❛❧❡ τ = t δ ✿ � dY t = 1 2 Y 0 = y 0 ∈ R m . δ b ( Y t ) dt + δ τ ( Y t ) dW t , ◆♦t❛t✐♦♥✿ X t ❛r❡ t❤❡ s❧♦✇ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ s②st❡♠✱ ❛♥❞ Y t ❛r❡ t❤❡ ❢❛st ❝♦♠♣♦♥❡♥ts✳ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s
❆ss✉♠♣t✐♦♥s ❚②✐❝❛❧ ❆ss✉♠♣t✐♦♥s✿ ❚❤❡ ❢❛st ✈❛r✐❛❜❧❡ ❛r❡ ❝♦♥str❛✐♥❡❞ ✐♥ ❛ ❝♦♠♣❛❝t s❡t✱ s❛②✿ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ♣r♦❝❡ss❡s ❛r❡ Z m ✲♣❡r✐♦❞✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✈❛r✐❛❜❧❡ y ✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✿ Y t ✐s ❛ r❡❝✉rr❡♥t ♣r♦❝❡ss✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ❝❛♥ ❣❡♥❡r❛❧✐③❡ t❤❡ r❡s✉❧ts ♣r❡s❡♥t❡❞ ✉♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s t❤❛t Y t ✐s ❡r❣♦❞✐❝✱ t❤✐s ♠❡❛♥s t❤❛t Y ❢♦r❣❡ts t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❢♦r ❧❛r❣❡ t✐♠❡ ✭✐✳❡✳ ❛s δ → 0 ✮ ❛♥❞ ✐ts ❞✐str✐❜✉t✐♦♥ ❜❡❝♦♠❡s st❛t✐♦♥❛r②✳ ❋♦r t❡❝❤♥✐❝❛❧ s✐♠♣❧✐❝✐t② ❢r♦♠ ♥♦✇ ♦♥ ✇❡ ❛ss✉♠❡ t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ ♣❡r✐♦❞✐❝✐t② ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts✳ ❚❤✐s ❝♦♥❞✐t✐♦♥ ❝❛♥ ❜❡ r❡❧❛①❡❞ t♦ ❡r❣♦❞✐❝✐t② ❛♥❞ ✇✐❧❧ ❜❡ tr❡❛t❡❞ ✐♥ ❛♥ ❛rt✐❝❧❡ ✐♥ ♣r❡♣❛r❛t✐♦♥✳ ❋✉rt❤❡r ❛ss✉♠♣t✐♦♥✿ t❤❡ ❞✐✛✉s✐♦♥ ♠❛tr✐① τ ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s
▼♦t✐✈❛t✐♦♥✿✿ ❛♥❛❧②s✐s ♦❢ ❢✐♥❛♥❝✐❛❧ ♠♦❞❡❧s ✇✐t❤ st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ❇❧❛❝❦✲❙❝❤♦❧❡s ♠♦❞❡❧✿ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r✐❝❡ ♦❢ ❛ st♦❝❦ ❙ ✐s ❞❡s❝r✐❜❡❞ ❜② dlogS t = γdt + σdW t , t❂t✐♠❡ , W t = ❲✐❡♥❡r ♣r♦❝✳ , ❛♥❞ t❤❡ ❝❧❛ss✐❝❛❧ ❇❧❛❝❦✲❙❝❤♦❧❡s ❢♦r♠✉❧❛ ❢♦r ♦♣t✐♦♥ ♣r✐❝✐♥❣ ✐s ❞❡r✐✈❡❞ ❛ss✉♠✐♥❣ ♣❛r❛♠❡t❡rs ❛r❡ ❝♦♥st❛♥ts✳ ■♥ r❡❛❧✐t② t❤❡ ♣❛r❛♠❡t❡rs ♦❢ s✉❝❤ ♠♦❞❡❧s ❛r❡ ♥♦t ❝♦♥st❛♥ts✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ✈♦❧❛t✐❧✐t② σ ✱ ❛ ♠❡❛s✉r❡ ❢♦r ✈❛r✐❛t✐♦♥ ♦❢ ♣r✐❝❡ ♦✈❡r t✐♠❡✱ ✐s ♥♦t ❝♦♥st❛♥t ❜✉t ❡①❤✐❜✐ts r❛♥❞♦♠ ❜❡❤❛✈✐♦✉r✳ ❚❤❡r❡❢♦r❡ ✐t ❤❛s ❜❡❡♥ ♠♦❞❡❧❡❞ ❛s ❛ ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ σ = σ ( Y t ) ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss Y t ✇✐t❤ ✶ ♥❡❣❛t✐✈❡ ❝♦rr❡❧❛t✐♦♥ ✭♣r✐❝❡s ❣♦ ✉♣ ✇❤❡♥ ✈♦❧❛t✐❧✐t② ❣♦❡s ❞♦✇♥✮ ✷ ♠❡❛♥ r❡✈❡rs✐♦♥ ✭t❤❡ t✐♠❡ ✐t t❛❦❡s ❢♦r ❛❣❡♥ts t♦ ❛❞❥✉st t❤❡✐r t❤r❡s❤♦❧❞s t♦ ❝✉rr❡♥t ♠❛r❦❡t ❝♦♥❞✐t✐♦♥s✮ ❘❡❢s✳✿ ❍✉❧❧✲❲❤✐t❡ ✽✼✱ ❍❡st♦♥ ✾✸✱ ❋♦✉q✉❡✲P❛♣❛♥✐❝♦❧❛♦✉✲❙✐r❝❛r ✷✵✵✵✱✳✳✳ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s
Multiscale stochastic volatility
❋❛st st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ■t ✐s ❛r❣✉❡❞ ✐♥ t❤❡ ❜♦♦❦ ❋♦✉q✉❡✱ P❛♣❛♥✐❝♦❧❛♦✉✱ ❙✐r❝❛r✿ ❉❡r✐✈❛t✐✈❡s ✐♥ ✜♥❛♥❝✐❛❧ ♠❛r❦❡ts ✇✐t❤ st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t②✱ ✷✵✵✵✱ t❤❛t Y t ❛❧s♦ ❡✈♦❧✈❡s ♦♥ ❛ ❢❛st❡r t✐♠❡ s❝❛❧❡ t❤❛♥ t❤❡ st♦❝❦ ♣r✐❝❡s✱ ♠♦❞❡❧❧✐♥❣ ❜❡tt❡r t❤❡ t②♣✐❝❛❧ ❜✉rst② ❜❡❤❛✈✐♦r ♦❢ ✈♦❧❛t✐❧✐t②✱ s❡❡ ♣r❡✈✐♦✉s ♣✐❝t✉r❡✳ ❋♦r t❤✐s r❡❛s♦♥ ✇❡ ♣✉t ♦✉rs❡❧✈❡s ✐♥t♦ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ♠✉❧t✐♣❧❡ t✐♠❡ s❝❛❧❡ s②st❡♠s ❛♥❞ s✐❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ❛♥❞ ✇❡ ♠♦❞❡❧ Y t ✇✐t❤ t❤❡ ❢❛st st♦❝❤❛st✐❝ ♣r♦❝❡ss ❢♦r δ > 0 � dY t = 1 2 Y 0 = y 0 ∈ R m . δ b ( Y t ) dt + δ τ ( Y t ) dW t P❛ss✐♥❣ t♦ t❤❡ ❧✐♠✐t ❛s δ → 0 ✐s ❛ ❝❧❛ss✐❝❛❧ s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♣r♦❜❧❡♠✱ ✐ts s♦❧✉t✐♦♥ ❧❡❛❞s t♦ t❤❡ ❡❧✐♠✐♥❛t✐♦♥ ♦❢ t❤❡ st❛t❡ ✈❛r✐❛❜❧❡ Y t ❛♥❞ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛♥ ❛✈❡r❛❣❡❞ s②st❡♠ ❞❡✜♥❡❞ ✐♥ R n ♦♥❧②✳ ❚❤❡r❡ ✐s ❛ ❧❛r❣❡ ❧✐t❡r❛t✉r❡ ♦♥ t❤❡ s✉❜❥❡❝t ✭ ❇❡♥s♦✉ss❛♥✱ ❑✉s❤♥❡r✱ ❍❛s♠✐♥s❦✐✐✱ P❛r❞♦✉①✱ ❇♦r❦❛r✱ ●❛❧ts❣♦r②✱ ❆❧✈❛r❡①✱ ❇❛r❞✐✳✳✳ ✮ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s
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