This document discusses quantifying risks associated with equity-based guarantees. It covers:
1) Risk management considerations for variable annuities, including financial modeling to quantify guarantee risks.
2) Basics of dynamic hedging, where the price of an option is the discounted value of dynamically hedging the exposure to expiration.
3) Gamma loss, which occurs when dynamically hedging a negative gamma position, leading to losses from buying high and selling low when re-hedging deltas as the underlying changes value.
Long horizon simulations for counterparty risk Alexandre Bon
The Challenges of Long Horizon Simulations in the context of Counterparty Risk modeling : CVA, PFE and Regulatory reporting.
This joint presentation reviews the key decisions that need making regarding the choice of risk factor evolution models and calibration methods. In particular, we will analyse the performance of classical historical calibration methods (such as Maximum Likelihood and the Efficient Method of Moments) in estimating the volatility and drift terms of the Hull & White class of Interest Rate models ; both in terms of convergence and stability.
As most methods perform satisfactorily for volatility but disappoint on the mean reversion estimation, we propose a new modified Variance Estimation method that significantly outperform the classical approaches.
Lastly, after reviewing historical economic evidence of mean-reversion dynmics in high interest rate regime, we propose modifying classical models by making mean reversion non-linear and accelerating for high rates - that can be referred as "+R" models.
This model address unrealistically large and persistent interest rates values often observed at high quantile in PFE and CVA simulations.
Capital Market Line graphically represents all portfolios with an optimal combination of risk and return.
https://efinancemanagement.com/investment-decisions/capital-market-line
Stabilise risks of discontinuous payoffs with Fuzzy Logic by Antoine SavineAntoine Savine
Antoine Savine's Global Derivatives 2016 Talk
We demonstrate that smoothing, a technique derivatives traders use to stabilise the risk management of discontinuous exotics, is a particular use of Fuzzy Logic.
This realisation leads to a general, automated smoothing algorithm.
The algorithm is extensively explained and its implementation in C++ is provided as a part of our book with Wiley (2018):
Modern Computational Finance: Scripting for Derivatives and xVA
Welcome to the world of Rohan Kritika, Each of these picture-perfect homes is set in a collage of amenities and framed with a splattering of lush green. This is where artistry and architecture first fell in love.2, 3 & 4BHK Luxurious Apartments
Long horizon simulations for counterparty risk Alexandre Bon
The Challenges of Long Horizon Simulations in the context of Counterparty Risk modeling : CVA, PFE and Regulatory reporting.
This joint presentation reviews the key decisions that need making regarding the choice of risk factor evolution models and calibration methods. In particular, we will analyse the performance of classical historical calibration methods (such as Maximum Likelihood and the Efficient Method of Moments) in estimating the volatility and drift terms of the Hull & White class of Interest Rate models ; both in terms of convergence and stability.
As most methods perform satisfactorily for volatility but disappoint on the mean reversion estimation, we propose a new modified Variance Estimation method that significantly outperform the classical approaches.
Lastly, after reviewing historical economic evidence of mean-reversion dynmics in high interest rate regime, we propose modifying classical models by making mean reversion non-linear and accelerating for high rates - that can be referred as "+R" models.
This model address unrealistically large and persistent interest rates values often observed at high quantile in PFE and CVA simulations.
Capital Market Line graphically represents all portfolios with an optimal combination of risk and return.
https://efinancemanagement.com/investment-decisions/capital-market-line
Stabilise risks of discontinuous payoffs with Fuzzy Logic by Antoine SavineAntoine Savine
Antoine Savine's Global Derivatives 2016 Talk
We demonstrate that smoothing, a technique derivatives traders use to stabilise the risk management of discontinuous exotics, is a particular use of Fuzzy Logic.
This realisation leads to a general, automated smoothing algorithm.
The algorithm is extensively explained and its implementation in C++ is provided as a part of our book with Wiley (2018):
Modern Computational Finance: Scripting for Derivatives and xVA
Welcome to the world of Rohan Kritika, Each of these picture-perfect homes is set in a collage of amenities and framed with a splattering of lush green. This is where artistry and architecture first fell in love.2, 3 & 4BHK Luxurious Apartments
Il Piano di Sviluppo 2016-2019 è stato approntato dal Distretto Produttivo 'Puglia Creativa' dopo un percorso di ascolto e confronto con imprese e istituzioni durato oltre tre anni.
Nel documento viene tracciato un percorso di evoluzione e di sviluppo ambizioso e di ampio respiro, per un progetto che, in linea con la strategia Europa 2020, intende contribuire attivamente ad una crescita intelligente, sostenibile ed inclusiva del territorio pugliese.
http://heiswed.tistory.com/entry/WPguide-3-Installation
워드프레스를 설치하기 위한 기본적인 서버 환경을 살펴보고,테마와 플러그인의 설치 방법을 배우고자 합니다.
목차는
- Background (배경)
- Installation (설치)
-- WordPress (워드프레스 설치)
-- Theme (테마 설치)
-- Plugin (플러그인 설치)
- Finalization (마무리)
총 세번의 강의가 계획되어있습니다.
1강. 워드프레스 개념 및 사용
2강. 워드프레스 관리 +@
3강. 워드프레스 설치 +@
As part of the YellowCats innovation leader program Willem Sodderland and Robbert Cornelissen organised a presentation and workshop around Open Source Marketing, Word of Mouth and Storytelling
For more information contact: emailus@marcusevans.com
Roger Gray, the Chief Investment Officer at USS Ltd. shared his presentation entitled "Implementing a Long-Term Investment Philosophy" at the European Pensions and Investment Summit.
Join the 2015 Summit along with leading regional pension investors and global asset managers in an intimate environment for a focused discussion of key new drivers shaping institutional investment strategies today.
For more information contact: emailus@marcusevans.com
Gnosis Company Limited was formed in Y2004 by the partners who have strong experiences in the banking and finance industry and international advisory firms. Our clients could find their corporate solutions with our solid experienced advisory team who are ready to advise and walk along with them to make their business sustainable wealth and reach their vision. We also provide the proven training programs to transform you and your company to the next level of success.
Impact of Valuation Adjustments (CVA, DVA, FVA, KVA) on Bank's Processes - An...Andrea Gigli
The talk hold in London on September 10th at the 5th Annual XVA Forum on Funding, Capital and Valuation. It covered some implications of Valuation Adjustments like CVA, DVA, FVA and KVA (XVAs) in the Pricing of Derivatives, Data Model Definition, Risk Management, Accounting, Trade Workflow processing.
Similar to Soa Equity Based Insurance Guarantees Conference 2008 (6)
3. Equity-Based Guarantees
• Equity-Based Guarantees, namely: -
Fixed Annuities
Variable Annuities
• Variable Annuities
= Unit-linked Investment Product
with given Fund Choices (Unit Trusts / Mutual Funds rather than
Market Indices); and
Investment Allocation / Fund Switching made by Customer;
(may be on the advice of Distributor / Financial Planner);
(can be on an unrestricted manner / frequency);
Plus
Investment Return / Benefit Guarantees offered
at a given fee for the specific guarantee;
which is normally fixed (less dependent on recent investment condition); (and
may not be changeable, regardless of changes in investment conditions in the
future).
3
Risk Management
• Other than Strategic, Operations, Skill-set, Reputation, Litigation,
Counterparty Risks, the Key Risk for managing Variable Annuities is Financial
(Balance Sheet) Risk
• If not Outsourcing the VA management, typical Financial Risk Management
will be: -
Asset Liability
Security Investment ← → Risk Management
Static Hedging ← → Pricing
Trading
Dynamic Hedging ← Desks / Asset ALM Financial → Valuation
Models
Static & Dynamic Hedging ←
Management ↔ → Financial Reporting
Arm
Capital Management (Naked) ← → Capital Planning
4
4. Risk Management
• Financial Modeling to quantify Guarantee Risk: -
Black Box Judgemental
← Historical Data
Quantification Results ← Models ← Assumptions ← Empirical Experiences
← Allowance for Unknown
← Unknown of Unknown
• Nested Stochastic Projections (Number of Scenario Runs)
• Partial Differential Functions to provide Analytical Results
• Heuristic Approach to save Computation Time
• Inter-dependence of Parameters / Path-dependent Scenarios
• Risk on Risk (i.e. Risk Management on Financial Modelling to make sense
for Quantification Results).
5
Risk Management
• Risk Management typically involves Identification, Assessment, Response,
Control, Monitoring
• Practically, Quantification of Risks by Financial Models require
- Understanding of Basics
- Systematic Building of Blocks and Comparing Relativity
- Using Boundary Conditions to help checking reasonableness of Outputs /
Signals
- Identifying Exotic Payoff Movements for small changes in assumptions /
underlier values
- Tradeoff of Risk Alternatives to assess Model Relativity / Consistency
- Reasoned Comparisons of Outputs across different Financial Market
disciplines
- Due Diligence / Disciplined Processes / Actions
6
5. Back to Basics
Assume No Transaction Costs/Lapses/Mortalities
PV (Guarantee Fee Income
≅
Black-Scholes Option Premium (10-
on Single Premium year Vanilla Put with Strike 100)
10-Year GMAB (Point-to-Point))
PV (Guarantee Fee Income
≅ ∑ Discrete Probability Function (K) *
K ≥100
on Single Premium Black-Scholes Option Premium
(10-Year Vanilla Put with Strike K)
10-Year GMAB (Ratchet))
7
Option Pricing
• Under the conditions of Constant Volatility and No Transaction Costs,
Black-Scholes when publishing the option pricing formula, asserted that
the Price of an Option should be the Discounted Value of the cost of
Dynamically Hedging the exposure to Expiration.
• Dynamic Hedging refers to: -
- Delta Hedging of a non-linear position with linear investment(s) of the
underlying
- The deltas of the non-linear position and linear position offset, yielding
a zero delta for the hedged portfolio (π)
- The non-linear position f can generally be expressed in the parabolic
form of the underlying (S): -
f ( S ) = cS + bS + a
2
⎛∂ f ⎞ ⎛ ∂f ⎞
2
where Gamma ⎜ ⎟ = 2 c, Delta ⎜ ⎟ = b, a = f ( S = 0 )
⎝ ∂S ⎠ 2
⎝ ∂S ⎠ 8
6. Dynamic Hedging
δ line
Long Put
f f Short Put f
S S S0 S
π = f −δ * S
Zero Delta
Short δ
f stock S to f
nullify
Delta
S0 S S
9
Dynamic Hedging
• As the underlying (S) changes value, the delta of the non-linear position
changes, but not for the linear instrument. The deltas no longer offset. Thus,
the linear hedge has to be adjusted (increase or decrease exposure) to
restore the delta hedge. The continual adjusting of the linear position to
maintain a zero delta is Dynamic Hedging.
• Writing a VA is equivalent to a Short Put. Short Put creates Negative Gamma
on the non-linear exposure ( curvature opens downward) of the underlying.
• Dynamic Hedging for a Negative Gamma position will lead to Gamma Loss.
10
7. Gamma Loss
At time 0, for the delta hedged portfolio,
∂f
π = f(s ) − δ * S and δ = S = S0
0 0 0 ∂S
At time 1,
a) If S moves up to S + Δ S , π 1 = f ( S + Δ S ) − δ * ( S + Δ S )
0 0 0 0
∂f ∂f
Given Negative Gamma, S =S0 + ΔS < S < s< s0 + Δs < δ
∂S ∂S 0
> π − π = f (S + ΔS ) − δ * (S + Δ S ) − f (S ) − δ * S
[ ( ) ]
1 0 0 0 0 0
= f S + ΔS − f (S ) − δ * ΔS
0 0
⎡ ∂f ⎤
= ⎢ S0 <S <S0 +ΔS − δ ⎥ * Δ S = Negative Value * Δ S = Loss;
⎣ ∂S ⎦
b) If S moves down to S − Δ S , π = f (S − ΔS ) − δ * (S − ΔS )
1 0 1 0 0
∂f ∂f
Given Negative Gamma, S =S0 −ΔS > S >S >S0 −ΔS > δ
∂S ∂S 0
> π − π
1 0
[
= f (S − Δ S ) − f (S ) − δ * ΔS
0 0
]
⎡ ∂f ⎤
= ⎢ S 0 > S > S 0 − Δ S − δ ⎥ * ( − Δ S ) = Positive Value * (- Δ S ) = Loss
⎣ ∂S ⎦
11
Gamma Loss
• For a Negative Gamma Portfolio, it always loses value on delta re-hedge due to
Buy High / Sell Low phenomenon. It never gains it.
• For each re-hedge, the Loss will equal to ½ * Gamma * (Change of Underlying
Value)2.
• The sum of Gamma Loses, till Expiration, is the actual cost of the option (before
Transaction Costs).
• The cash balance on dynamic hedging a Short Put (negative Г) reduces as
follows: -
1 N −1 2 s −s
= ∑ (r − σ 2Δt ) Γ⎛ ⎞ s 2, where r = i + 1 i ,σ = Implied Volatility
2 i =1 i i ⎜ i, s ⎟ i
⎝ i⎠
i s i
i
← Experienced Volatility < Implied Volatility
← Experienced Volatility = Implied Volatility
12
Option Sold Option Expired
8. Pricing Considerations
• As can be seen, at a higher volatility, the underlying will fluctuate more, and the delta
hedge needs to be adjusted more frequently. The cash balance (arisen from the
premium / fee income received) will lose more rapidly when dynamically hedging the
non-linear exposure.
• Thus, the initial key consideration in assessing the sufficiency of VA (i.e. GMB)
charge is: -
- to evaluate the Г of the guarantee with change in value of the underlying till
expiration
-to consider the potential volatility of the underlying (as the ultimate hedging cost
depends on the experienced volatility)
• In addition, for the non-linear exposure f=f(S,σ,r,t)
- By Taylor Expansion
∂f ∂f ∂f ∂f 1 ∂2 f
Δf = Δt + ΔS + Δσ + Δr + (ΔS ) 2 + .....
∂t ∂S ∂σ ∂r 2 ∂S 2
- By Delta Hedging using linear instruments
2
∂f ∂f ∂f ∂f 1 ∂ f 2
Δf − ΔS = 0 = Δt + Δσ + Δr + ( ΔS ) + .....
∂S
2
∂S ∂t ∂σ ∂r 2
Theta Vega Rho └ Gamma Management ┘
Decay 13
Pricing Considerations
• For VA, the non-linear exposure F
~ ~
σ
F = F(f( S i , i , r, t), Si , Ti , Transaction Cost, Profit / Capital Charge)
~
for all Fund Choices Si
where S i is the proxy investment vehicle which can be used to dynamically
~
hedge Si
~ ~
Ti is the period under which S i is selected by the Customer for F to be subject
to.
• The additional pricing considerations or quantifications, therefore, include:
- Basis / Gap Risk ( S i → s i ≅ 10 % extra cost)
~
- Policyholder Behavior (Product Design (MVA) or Fixed Penalty to address?)
-Transaction Costs for Hedging
- Fixed VA Charge vs Variable Theta Decay (Profitability & Profit Variation)
on top of cost considerations for Vega, Rho, Gamma management,
Underlying Volatility implications and Frequency of Hedging.
14
9. Vega
• May estimate the Fair Strike Value of Variance Swap, based on raw option prices.
2
ΔK ΔK
i erT Call ⎡ K ⎤ − 1 ⎜ FT −1⎟
⎛ ⎞
σ =2 ∑
2 i erT put ⎡ K ⎤ + 2 ∑
⎢ ⎥ ⎢ i⎥ ⎜ ⎟
⎣ i⎦ T ⎦ T ⎜K
K ⎣ ⎟
T K ≤F K 2 K >F
K 2 ⎜
⎝ 0
⎟
⎠
i T i i T i
S
where F = e rT , K = i
T i S
0
K is the first strike below the forward F
0 T
• Issue:
- Not sufficient option prices available in the market for longer term T.
- Work best for shorter term T
- Hong Kong: Warrant Transactions (20%-40% daily stock market turnover of US$10-
20bn), but mainly 6-9 months for T. 15
Volatility Surface
• Volatility is not constant (per Black-Scholes) and changes with St, k, T
• When Market falls, price movements are usually sharp. Volatility will shoot
up. Issuer of in-the-money VA (or GMB) will feel a double-hit.
• Implied Volatility
Volabtility Smile
Volatility Skew
Strike Strike
• Volatility skew in equities reflects investors’ fear of market crashes which
would potentially bid up the prices of options at strike below current market
levels.
• Volatility increases with reduction in expiration time. 16
10. Volatility Surface
• The hedge ratio will be off if the non-linear position is dynamically hedged without
incorporating the effect of Volatility Surface into the delta calculation.
• Standard Models to allow these effects include:
- Jump-Diffusion Model adds random / Poisson jumps to the GBM that the
underlying assumes
- Regime-Switching Model probabilistically selects different volatility bases in the
modelling process
- Stochastic Volatility Model models the underlying’s value & its volatility as
stochastic processes
17
Delta-Gamma Hedging
• To subscribe another non-linear instrument (g) in addition to the linear
instrument to hedge against the exposure on non-linear instrument (f).
∂f ∂f 1 ∂2 f 2
Δf = Δt + ΔS + (ΔS )
∂t ∂S 2 ∂S 2
∂g ∂g 1 ∂2g
Δg = Δt + ΔS + (Δ S )2
∂t ∂S 2 ∂S 2
• Hedged Portfolioπ = f + α 1S + α 2 g
⎛ ∂ f ∂ g ⎞
Δ π = ⎜ + α ⎟ Δ t
∂ t ∂ t
2
⎝ ⎠
⎛ ∂ f ∂ g ⎞
+ ⎜ + α + α ⎟ Δ S
∂ S ∂ S
1 2
⎝ ⎠
⎛ ∂ 2 f ∂ 2 g ⎞
⎟ (Δ S )
1 2
+ ⎜
⎜ + α ⎟
∂ S ∂ S 2
2 2
2 ⎝ ⎠
• To Make Terms for Δ S & (Δ S )2 = 0
∂2 f ∂2g ⎛ ∂f ∂g ⎞
+α = 0;⎜ + α1 + α ⎟ = 0
∂S 2 ∂S 2 ⎝ ∂S ∂S ⎠
2 2
∂ g 2
∂ f 2
∂f ∂g ∂2g ∂2 f
α = - ;α 1 = −( − * )
∂S 2 ∂S 2 ∂S ∂S ∂S 2 ∂S 2
2
• f=Negative Gamma → g = Positive Gamma ; with g = Positive Gamma,
direct δ exposure to S reduces 18
11. Delta-Gamma Hedging
• g=Positive Gamma means Long Volatility Option
• Option Illiquid Higher Transaction Costs
• Delta-Gamma Hedging = Dynamic + Static Hedging
• The continuous dynamic hedging (including use of static hedges) will incur
an infinite amount of transaction costs, no matter how small it is.
• In the presence of transaction costs, the absence of arbitrage argument is
invalid, market is incomplete, which leads to many solutions.
• There is no definitive solution on VA management. The success in offering
VA will therefore depend on the availability of the right skill sets, integrated
processes, risk management capabilities by the underwriter to generate
viable and sustained solutions.
19
Hedging Methods
• Finally, the desirable hedging method will, inter alia, depend on Transaction Costs
and the Risk Tolerance / Appetite of the VA underwriter.
• Assume a sale of δ shares of the underlying incurs transaction costs λ/δ/s (λ≥0).
Below are 6 common hedging methods.
a) The Black-Scholes Hedging at Fixed Regular Intervals.
The balance account adjusted on reinstating the target hedge ratio: -
⎡⎛ ∂f ∂f ⎞ ⎛ ∂f ∂f ⎞⎤
⎢⎜ − ⎟ − λ ⎜ − ⎟⎥S
⎜ ⎟ ⎜ ∂S ∂S t ⎟ t+ h
⎢⎝ ∂ S t+
⎣ h ∂S t ⎠ ⎝ t+ h ⎠⎥⎦
b) The Leland Hedging at Fixed Regular Intervals
As per (a) using a modified volatility in the model.
(σ 2
m = σ 2 [1 − λ * Constant * Γ ] )
c) The Delta Tolerance Strategy
∂f
Δ − > h (a given constant) ; Re - hedge to Target Hedge Ratio
∂S
d) The Asset Tolerance Strategy
S (t + Δ t ) − S (t )
> h (a given constant) ; Re - hedge to Target Hedge Ratio
S (t )
20
12. Hedging Methods (con’t)
e) Hedging to a Fixed Bandwidth around Delta
∂f
Δ = ± h per the Delta Tolerance Strategy ;
∂S
Re-hedge to the Hedge Ratio to the nearest boundary of the Hedging Bandwidth
f) The Asymptotic Analysis of Whalley and Wilmott
∂f
1
Δ = ± h ( e − r (T − t ) S Γ 2 ) 3 ;
∂S
Similar to (e), the Size of the Hedging Bandwidth depends on the price of the
underlying and the option gamma
• h depends on the risk aversion of the VA underwriter
• Empirical studies suggest e) & f) are outperforming methods. e) outperforms f) if
the risk tolerance of the VA underwriter is higher.
21
Concluding Remarks
• The above quantification considerations will lead to a relevant Variable Theta Decay
to breakeven the related risks & costs.
• Assuming the risks associated with Policyholder Behavior to be mitigated by
relevant Product Design, it still leaves the issue of Fixed Guarantee Fee for Variable
Theta Decay in the ultimate pricing.
• There are variances in practice within the financial service industry:
Insurer Investment Bank
Use of Capital Capital at Risk(Naked) → Hedging Hedging → Capital for Extremes of Models
r=μ μ −σ
Girsanov’s Theorem r=
θ
Risk Premium Shareholder / Customer to share Customer to bear
• Reasonable Check: How the Customer values the net results across different
financial instruments / products.
22
13. Concluding remarks (con’t)
• The challenge when structuring VA in Hong Kong
US Variable Annuity Hedging via Investment Bank
Long Term Volatility ~ 15% p.a. Implied Volatility ~ 50% p.a.
(Term slope not very deep)
+ Basis / Gap Risk
+ Profit Load
• Implications: -
- Product Design → To find means to stabilize the σ for hedging
→ To reduce anti-selection Policyholder Behavior (Distributor Advice)
- Skill-set → Risk Management and Capital Planning
→ Operations and Integration with Trading Desks
• VA is a very Valued Product for Customer & for Retirement Planning / Use
• Actuary’s Social Responsibility: -
To research, develop, implement and risk manage the Product in a professional manner
23