authorised and regulated by the financial conduct

PRIVATE AND CONFIDENTIAL. NOT TO BE DISTRIBUTED

AUTHORISED AND REGULATED BY THE FINANCIAL CONDUCT AUTHORITY

MODELLING IMPLIED VOLATILITY FOR FINANCIAL OPTIONS Samuel Jones

This presentation is for discussion purposes only and is not an offering or solicitation for investment

© Cantab Capital Partners LLP. Authorised and Regulated by the Financial Conduct Authority.

Financial options

■ Right to buy (“call”) or sell (“put”) at fixed price

■ Commodities, equities, currencies, bonds

■ Users: Hedgers, speculators

■ Valuation: Black-Scholes (1973)

𝑑𝑆𝑡𝑆𝑡

= 𝜇𝑑𝑡 + 𝜎𝑑𝑊𝑡

𝐶 𝑡, 𝑆𝑡 = 𝑆𝑡𝑁 𝑑1 − 𝐾𝑒𝑟 𝑇−𝑡 𝑁 𝑑2

𝑑1/2 =𝑙𝑛

𝑆𝐾

+ (𝑟 +12𝜎2)(𝑇 − 𝑡)

𝜎√(𝑇 − 𝑡)

2


Implied volatility

■ Different options have different “implied” volatility

3


Implied volatility


■ More complex models generate different “surfaces”

Heston model:

𝑑𝑆𝑡 = 𝜇𝑆𝑡 + 𝜎𝑡𝑆𝑡𝑑𝑊𝑡 𝑑𝜎𝑡

2 = 𝜅 𝜃 − 𝜎𝑡 𝑑𝑡 + 𝜉𝜎𝑡𝑑𝑍𝑡 𝑑𝑊𝑡 , 𝑑𝑍𝑡 = 𝜌𝑑𝑡

Implied volatility surface:

𝜎𝑖𝑚𝑝𝑙𝑖𝑒𝑑 = 𝜎(𝐾, 𝑇, 𝜎0, 𝜉, 𝜌, 𝜅, 𝜃)

4


Implied volatility



■ Increasingly complex models to fit observed prices

Inferred probability density:

𝐶 𝐾, 𝑇 = 𝐸 [max(𝑆𝑇 − 𝐾, 0) ]

𝑑2𝐶

𝑑𝐾2𝐾, 𝑇 = 𝑃(𝑆𝑇 = 𝐾)

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Implied volatility



■ Increasingly complex models to fit observed prices

■ And there are infinite models with the same prices!

■ Real world dynamics often don’t follow any nice model.

Gyongy’s Theorem:

Processes 𝑑𝑆𝑡

𝑆𝑡= 𝜎𝑡𝑑𝑊𝑡 and

𝑑𝑆 𝑡

𝑆 𝑡= 𝜎 𝑡𝑑𝑊𝑡

same marginals if 𝐸 𝜎𝑡2 𝑆𝑡 = 𝐸[𝜎 𝑡

2|𝑆 𝑡] In particular (Dupire):

𝐸 𝜎𝑇 𝑆𝑇 = 𝐾 = 𝜎𝑙𝑜𝑐 𝐾, 𝑇 =𝐾2𝐶𝐾𝐾

2𝐶𝑇

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A data driven approach

■ Nonparametric fits to the data must satisfy constraints

■ Data-driven approach within constrained setting

Minimise 𝐿(𝜎𝑓𝑖𝑡 − 𝜎𝑑𝑎𝑡𝑎) s.t.

𝜎𝐾 𝐾, 𝑇 < 𝑓 𝐾, 𝑇 𝜎𝐾𝐾 𝐾, 𝑇 < 𝑔 𝐾, 𝑇 𝜎𝑇 𝐾, 𝑇 < ℎ(𝐾, 𝑇)

lim𝐾→±∞

𝜎2′ 𝐾 ∈ [0, ±2]

7


A data driven approach

■ Nonparametric fits to the data must satisfy constraints

■ Data-driven approach within constrained setting

■ Empirical/statistical risk factors > complex modelling

8


Understanding the returns of an options portfolio

■ Selling options is profitable if the market overprices volatility (and conversely…)

■ In a constant volatility world, we can explain the returns as a function of implied and realised volatility

■ But volatility is not constant….

Estimated real world volatility (GBM) 𝑑𝑆 = 𝜎𝑆𝑡𝑑𝑊𝑡

Implied market volatility (BS PDE) 𝑑𝐶

𝑑𝑡+

𝑑2𝐶

𝑑𝑆2𝑣2𝑆2 = 0

Option portfolio profit (Ito’s lemma)

𝑑 𝐶 − Δ𝑆 = 1

2𝜎2 − 𝑣2

𝑑2𝐶

𝑑𝑆2𝑆𝑡2𝑑𝑡

9


Putting it all together

■ Returns on an option portfolio depends on all moving parts!

□ Realised volatility

□ Level/shape of implied volatility

□ Forecast dynamics of implied volatility

□ Price/volatility correlations

□ Risk profile (“Greeks”) of each option

■ With all this, we can make optimal investment decisions…

Asset dynamics: 𝑑𝑆𝑡 = 𝜎𝑡𝑆𝑡𝑑𝑊𝑡

Volatility dynamics: 𝜎𝑡 = 𝜎 𝐾, 𝑇 = 𝛼 𝐾, 𝑇 + ∑𝛽𝑖 𝐾, 𝑇 𝑍𝑡𝑖

Option price dynamics: 𝑑𝐶𝑡 =𝑑𝐶

𝑑𝑡𝑑𝑡 +

𝑑𝐶

𝑑𝑆𝑑𝑆𝑡 +

𝑑𝐶

𝑑𝜎𝑑𝜎𝑡 +

1

2

𝑑2𝐶

𝑑𝑆2

+𝑑2𝐶

𝑑𝑆𝑑𝜎𝑑𝑆𝑡 , 𝑑𝜎𝑡 +

1

2

𝑑2𝐶

𝑑𝜎2⟨𝑑𝜎𝑡⟩

10


Conclusions

■ Financial options are widely traded assets with many uses in both managing risk and improving investment returns

■ Use of options in systematic investment involves lots of data and various numerical and mathematical techniques

■ Lots of intersection between current practice & new research!

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Thank you for listening

■ Contact details:

□ Samuel Jones

□

□ Cantab – Part of GAM Systematic

□ City House, 126-130 Hills Road

□ Cambridge, CB2 1RE, UK

□ D: +44 (0)1223 755 827

□ [email protected]

□ Authorised and regulated by the Financial Conduct Authority

12

This presentation is for discussion purposes only and is not an offering or solicitation for investment

mailto:[email protected]

authorised and regulated by the financial conduct

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