CSFB Long Dated Options

8/8/2019 CSFB Long Dated Options

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Equity-Based Insurance Guarantees Conference

November 1-2, 2010

New York, NY

Black Holes or Black Scholes: Modeling

Long-dated Options

Paul Staneski


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--

Issues in modeling longIssues in modeling long--dated optionsdated options

ovem er ovem er

13301330 1415 hrs1415 hrs

Paul Staneski, Ph.D.

Credit Suisse

Head of Derivatives Solutions & Training

Society of Actuaries Equity-Based Guarantees Conference New York


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Its a Model!

All models are wrong, some are useful.

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Some Issues

The role of Gamma

Rho vs. Ve a

Relative impacts over time

Variance Swaps

Short-term vs. long-term replication

Realized vs. Implied Vol

How do you price?

How do you hedge?

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Black-Scholes

The Black-Scholes model rivals CAPM (the Capital Asset Pricing Model) asthe most important result in the history of finance.

Fischer Black and Myron Scholes, The Pricing of Options and Corporate Liabilities,Journal of Political Economy, 81 (May-June 1973), pp. 637-659.

Scholes and Robert Merton won the 1997 Nobel Prize in Economics for this work(Black died in 1995 and the prizes are not awarded posthumously).

Historical note: Ed Thorp actually first derived this result in 1967!

Read Fortunes Formula by William Poundstone, Hill & Wang, 2005.

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Black-Scholes in a Nutshell

Option Stock

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(balance with financing)


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The Greeks are Key!

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Delta Hedging & Gamma

Value Long CallHedge (Short Stock)

Gain on Long CallGain on Hedge

Loss on HedgeLoss on Call

Stock Price

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, .


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Capturing Gamma

Hedging & re-hedging a long call

Stock Price

Initial Premium

Gamma captured

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Capturing (More) Gamma

Hedging & re-hedging a long call

Stock Price

Initial Premium

Gamma ca tured

Gamma captured

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Model Assumptions

Under all the models assumptions (there are many!) the application of themodel to a 10-day option is not at all different than its application to a 10-year

option.

However, in the real world this time invariance is not the case!

Constant interest rates (and dividends)

Constant volatility

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Question

Suppose you buy a 90-day Call option on a stock at an implied volatility of35% and dynamically hedge it (perfectly, as per the Black-Scholes

assumptions) to expiration. If realized volatility over the 90 days is 40%, willyou ma e money

Answer: ____________

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Answer

Not necessarily!

You ex ect to make mone but iven the articular rice ath of the stockyou might not.

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Wheres your gamma?

These two price paths have exactly the same vol

A B

but gamma is not constant over time.

Which one would rather be hed in over? Answer:

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_________


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Gamma and Time

0.0

5

X = 100, vol = 16%, r = 4%

a 0.0

3

0.0

4 t = 0.25

G

am

1

0.0

2

t = 1

60 80 100 120 140

0.0

0.

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Spot


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Non-Constant Gamma

The path-dependency problem induced by non-constant gamma over timeis not a serious issue for short-dated options.

Large volumes generally ensure that the expected nature of hedging profitsprevails.

, -

out may not occur.

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More Greeks: Vega and Rho

The change in an options value given a change in volatility is known as vega

Cueoption valinchg.Vega

Also known as tau or kappa.

.

Unit of change in vol is an absolute 1% (e.g., 25% to 26%).

The change in an options value given a change in rates is known as rho

Unit of change in rates is usually taken to be an absolute 1% (100 bps).

r

C

ratesinchg.

ueoption valinchg.Rho

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Vega and Rho

Slide 17Question: Why does the put value rise then begin to decline?


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Vega vs. Rho over Time

Rho increases absolutely with time but vega actually starts to decline

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Vega/Put vs. Rho/Put

Rho will eventuall exceed the value of the ut!

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Hedging Rho

For short-dated options, and especially in this low rate and low rate-volenvironment, rho is not a big concern for equity-option traders.

Rho risk that is generated by trading equity options can be hedged with FRAsand Eurodollar Futures.

- , ,

long-dated rate risk is not as easily hedged. A 100 bp change in rates is very unlikely in the next 3 months, but is virtually certain

over the next 10 years.

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Stochastic Interest Rates

Interest-rate sensitive derivatives can be priced with a model that allows for theevolution of rates (stochastic interest rates) over time.

Modelling rates is fundamentally more complex than modelling equities.

What rate to model?

Short rate (LIBOR)?

s - ree e.g., reasury

Forward rate(s)?

What distributional assumption?

Lognormal?

This has lead to a plethora of models.

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More Questions

Do rates mean revert?

Is vol de endent on the level of rates?

Are the vols of spot rates and forward rates different?

How important is fitting the market?

What is to be fit? Zero-coupon bond prices?

Forward rates?

, ,

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Basic Model Structure

Is a model a model of what is (that is, calibrated to market prices) or what

should be (equilibrium models)?

Equilibrium models: term structure of rates is an output.

Market (no-arbitrage) models: term structure is an input.

Models that fit the market all suffer from the same problems as Black-Scholes: hedging long-dated instruments is difficult.

Equilibrium models often yield unsatisfactory prices.

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One-Factor Interest Rate Models

All of these models do a reasonable job of pricing, but alone do not provide.

Multi-factor models also exist.

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Variance Swaps

Variance Swap: a contract that pays the difference between the realizedvariance and a fixed level of variance (the strike).

= *

The multiplier M = notional size of the contract (Variance Units)

Really just a Forward.

Note 1: The strike is quoted in vol terms (20%, for example) even though thecontract is valued as the difference between variances.

Note 2: Despite how the strike is quoted, M is chosen to achieve a specificvega notional or vega amount (change in value of swap given a 1% change in

vol).

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Replicating/Hedging a Variance Swap

The payoff of a variance swap is replicated, and hence hedged, by a weightedportfolio of out-of-the-money (with respect to the Forward) puts and calls.

Imp. Vol The weights are in inverseproportion to the square of strike.

Buy all these options

StrikeOut-of-the-money Puts F Out-of-the-money Calls

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Short vs. Long-Dated Variance Swaps

Given the replicating portfolio of a variance swap, it is clear that a reasonablywide, liquid strip of options is necessary.

For short-dated swaps ( < 1 year), such options trade with sufficient liquidity onmajor indices.

- ,

to match buyers with sellers.

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Valuingan Option: Inputs

In order to value an option, we need to know

Spot price (S)

Strike Price X Interest rates (r) (and dividend yield, if applicable)

Time to expiration (t)

(Expected) Volatility ( )

XS

t

(Black-Scholes?)Option a ue

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Option Pricing Models?

Models yield values markets give us prices!

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Implied Volatility

The character of the 5 inputs is different: S, X, r, and t are all observable(and essentially unequivocal), volatility is not.

What we can observe in the market, however, is the priceof an option.

We can reverse-engineer the model to solve for the volatility that would

Valuation Model

(Black-Scholes)Implied Vol Option Price

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Realized Volatility

Realized volatility is a statistical calculation; it is the observed (annualized)standard deviation of historical returns

2

Where ri are log periodic returns (usually daily); e.g. ri = log(Pi/Pi 1)

VolPeriodicAN

Ai

R

N is the number of returns in the calculation.

A is the number of periods in a year (A = 252 for daily data).

Realized vol can be different with different choices of the above, especiallythe periodicity.

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What was realized volatility?

For the year ended August 31, 2010 we get

Daily realized = 18.9%

= .

Monthly realized = 17.4%

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Question

Is implied volatility the expectation of future realized volatility?

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Realized vs. Implied

Slide 34Source: Credit Suisse Equity Derivatives Research/Locus


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Implied Vol is More than Vol

Implied volatility, as the only number you can change in the Black-ScholesModel, embeds all the deviations from the assumptions of the model

Skewness (out-of-the-money put vol is usually > otm call vol) Aka volatility smiles

Kurtosis (fat tails).

Inability to hedge continuously.

We own options continuously but calculate realized vol at a discrete periodicity (youcan buy and sell an option intra-day, making day-to-day realized vol immaterial).

mp e vo s con am na e y a o e a ove.

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Questions

1. If the 2500 largest realized vol days from the last 50 years wereconcatenated into a single 10-year period, what would be the realized vol ofthis wild decade?

Answer: ____________

2. What is the realized vol of the past decade (which includes two of the most

Answer: ____________

. v - v z v

Answer: ____________

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Answers

1. If the 2500 largest realized vol days from the last 50 years wereconcatenated into a single 10-year period, what would be the realized vol ofthis wild decade?

Answer: _____29%____

2. What is the realized vol of the past decade (which includes two of the most

Answer: _____22%_____

. v - v z v

Answer: _____16%_____

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Pricing on a 29 Vol

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Pricing on a 22 Vol

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Pricing on a 16 Vol

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2005: C3P2-Driven Rise in 10-year Implied Vol

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10-Year SPX Vol

1999 -------------------------------------------------------------------------- 2006

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Constant Volatility

As an alternative to the constant vol input to Black-Scholes, it has often beenproposed that volatility be treated as stochastic.

The standard geometric Brownian motion process for a stock price, S, asassumed by Black-Scholes is

Where

mu is the drift (growth rate) of the stock.

sigma is the (constant) volatility.

dt is an increment of time.

dW is a random normal process (Weiner Process).

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Geometric Brownian Motion

We can view the equation on the previous slide

dS/S

(deterministic component)

(random component, which is normal)+time

dt

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Stochastic Volatility: Heston Model

Stochastic volatility process

1tSdWSdtdS Now have a time subscript on vol.

2t

2

t

2

t dW)dt(d The volatility process.

ere a e se as e ore

lambda is the mean reversion rate for the variance. theta is the long-run level to which variance reverts.

eta is the volatility of variance.

dW1 and dW2 are correlated Weiner Processes.

Heston, S. L., A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bondand Currency, The Review of Financial Studies, 1993.

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Contact Information

Paul G. Staneski, Ph.D.

Head of Derivatives Solutions & Training

212-325-2935

[email protected]

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