Greeks under normal model

Thinking, Fast and Slow

Illusions you cannot avoid

Probably mentioning this book in 2017 is a bit out of fashion, but there are yet loads of takeaways for understanding systematic biases people inevitably have. The author – Daniel Kahneman, who won Nobel Prize in 2002 for his work on human’s judgement and decision making – introduces those biases with realistic examples throughout the book (some of which are deliberately designed to fool you and you cannot escape even if you know it!).

Systematic biases and risk premia

For financial practitioners, those biases are sometimes keys to understand the nature of risk premia because often they are the very reasons why it exists in the first place. Furthermore in this way, they can provide us some guideline to believe whether the premia is persistence or not in the foreseeable future.

There are currently various ways to access to “risk premia strategy” or “smart beta strategy” via ETFs or other formats especially for institutional investors. Asking if the premia is coming from systematic biases or not could be a reasonable indicator to distinguish between genuine premia and transitory market anomaly. If that is a mere market anomaly, and you believe that a systematic bias is not the one causing it, then it may be arbitraged away anytime soon when large money were put into that factor. Or even worse the factor you rode on might be just the result of the statistical trap like the size factor.

Among these biases, what I found interesting were probability effect and certainty effect.

Probability effect

Probability effect is a bias where people put higher “probability” to rare events than they deserve. This is why so many people keep buying lottery tickets and the reason why stocks with higher skewness underperform the peers with lower skewness.

Certainty effect

This is where people value sure things too much than they should from the statistical point of view. People pay more money to shift the probability to 100% from 95% to get relieved. Effectively insurance companies make money from this bias by providing clients the sure protection. Similarly this can explain why put options on S&P500 is far too expensive compared to realised volatility, why there exists the volatility risk premia, and why long VIX futures is always bleeding.

These two are just examples of the fallacy Daniel Kahneman revealed. And by the end of the book, you would realise how stupid we are!

Thinking, Fast and Slow by Daniel Kahneman

PutWrite versus BuyWrite: Yes, Put-Call Parity Holds Here Too


PutWrite versus BuyWrite: Yes, Put-Call Parity Holds Here Too


Roni Israelov: AQR Capital Management, LLC


The CBOE PutWrite Index has outperformed the BuyWrite Index by approximately 1.1 percent per year between 1986 and 2015. That is pretty impressive. But troubling. Yes – troubling – because the theory of put-call parity tells us that such outperformance should be almost impossible via a compelling no-arbitrage restriction. This paper explains the mystery of this outperformance, which has implications for portfolio construction.


Historically PutWrite index outperformed BuyWrite index even though the put-call parity suggests that such a discrepancy should not exist.
The author lay outs three constructional differences that could have caused the outperformance.

  1. Naked beta position: By construction, BuyWrite index is a portfolio of short call option and long S&P500. Once the option expires, BuyWrite index still has a naked beta position which PutWrite does not. The author shows that this transitory equity exposure which lasts 4 hours explains most of the mismatches.
  2. Delta: BuyWrite sells the most closest strike higher than the spot while PutWrite sells the lower strike. The author estimates the two nearest strikes have delta difference of 0.03.
  3. Cash position: Collateral of BuyWrite is invested in 1 month T-bill while PutWrite invests in both 1 month and 3 month T-bill.

Decompose option PnL into its greeks


– Decomposition option PnLs into its greeks by using Taylor expansion
– Major greeks attribute to daily PnL are delta, gamma, theta and vega

Taylor expansion

An arbitrary function \( f \left( x \right) \) can be written by below formula:

f \left( x \right) = f \left( a \right) + f^{\left( 1 \right)} \left( a \right) \left( x – a \right) + \frac{f^{\left( 2 \right)} \left( a \right)}{2} \left( x – a \right)^2 + {\rm higher}.

The option present value \( V_t \left( S_t, T, \sigma_t, r_t \right) \) is expressed with its derivatives(option greeks) in a similar manner:

V_t \left( S_t, \tau_t, \sigma_t, r_t \right) = V_{t-1} \left( S_{t-1}, \tau_{t-1}, \sigma_{t-1}, r_{t-1} \right) + \\
\frac{{\partial}V}{{\partial}S_{t-1}}{\rm d}S + \frac{{\partial}V}{{\partial}\tau_{t-1}}{\rm d}\tau + \frac{{\partial}V}{{\partial}\sigma_{t-1}}{\rm d}\sigma + \frac{1}{2} \frac{{\partial}^2V}{{\partial}S_{t-1}^2}{\rm d}S^2 + {\rm residuals},

where \( S_t \) is underlying, \( \tau \) is time to expiry, \( \sigma \) is volatility and \( r \) is interest rate respectively. Usually the effect of the interest rate sensitivity(rho, PV01 sensitivity for swaptions) is limited and included to the “residual” term as well as other higher derivatives.
Daily PnL \( R_t = V_t – V_{t-1} \) is hence written by below formula:

R_t \approx \Delta_{t-1}{\rm d}S + \theta_{t-1}{\rm d}\tau + \nu_{t-1}{\rm d}\sigma + \frac{1}{2} \Gamma{\rm d}S^2.

If you consider stochastic volatility model such as SABR, above equation also holds sensitivity terms to that parameters(volvol, beta, rho). Furthermore, other related greeks including forward-delta, forward-gamma, normal-vega can be applied above interchangeably.


Taylor series – Wikipedia, the free encyclopedia

Riding the Swaption Curve by Johan G. Duyvesteyn, Gerben J. De Zwart :: SSRN






1. ボラティリティ水準とガンマの比較
→ ATM付近ではガンマとボラティリティは逆の関係。ボラティリティが上昇するとガンマが下がる

2. ボラティリティとオプション・プレミアムの関係
→ これは簡単に1対1の関係。ボラティリティが上昇するとプレミアムも上昇

3. ATMFにおけるボラティリティとベガの関係
→ 一定レベル以上でのボラティリティの変動はベガにあまり影響を与えない
→ 低ボラティリティ環境下においては1ベガあたりのPLへの影響が大きい(本当か?)

4. OTMにおけるボラティリティとベガの関係
→ ボラティリティが上昇するとベガも上昇。DeepOTMになるほどその影響は小さくなる

5. 異なる原資産水準でのATMFの行使価格とベガの関係
→ 原資産の水準が高いほどベガは大きくなる

6. 原資産価格とボルガ(ボラティリティの2階微分)の関係
→ 3で見たように、ATM近辺ではボラティリティが上昇してもベガはあまり変化しない(ボルガ≒0)
→ 4で見たように、ATM以外の領域ではボルガが正 = ボラティリティが上昇するとベガも上昇する

7. 異なるボラティリティ水準でのボルガの変化
→ ATMではボラティリティ水準によらずボルガはほぼゼロ
→ OTMは知らん。というかボルガってオプショントレーダーでもあるまいし普通見るんかね。よう知らんけど。

8. 原資産とゾンマの関係
→ あんのかなーと思ったら案の定あった謎のGreeksゾンマ。ガンマをボラティリティで微分したもの。
→ 1で見たように、ATM付近ではゾンマが負なのでボラティリティとガンマが逆関係。ATMから外れるとゾンマがプラスになり、ボラティリティが上昇するとガンマも上昇する。

{\rm vega} = \frac{{\partial }V}{{\partial } \sigma} \\
{\rm volga} = \frac{{\partial }^2 V}{{\partial } \sigma^2} \\
{\rm zomma} = \frac{{\partial } \Gamma}{{\partial } \sigma} = \frac{{\partial }^3 V}{{\partial } S^2 {\partial } \sigma}





Greeks (finance) – Wikipedia, the free encyclopedia

Rのオブジェクト指向について(R) – script of bioinformatics