The greek letters
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The Greek Letters. ILLUSTRATION. The financial institution has sold for $300,000 a European call option on 100,000 shares of a non-dividend-paying stock. S 0 = 49, K = 50, r = 5%, σ= 20%, T = 20 weeks (0.3846 years), μ=13% The Black-Scholes price of the option is about $240,000.

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The Greek Letters

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The greek letters

The Greek Letters


Illustration

ILLUSTRATION

  • The financial institution has sold for $300,000 a European call option on 100,000 shares of a non-dividend-paying stock.

  • S0 = 49, K = 50, r = 5%, σ= 20%, T = 20 weeks (0.3846 years), μ=13%

  • The Black-Scholes price of the option is about $240,000.

  • The financial institution has therefore sold the option for $60,000 more than its theoretical value, but it’s faced with the problem of hedging the risk.


Naked covered positions

NAKED & COVERED POSITIONS

  • Naked Position

    One strategy open to the financial institution is to do nothing.

  • Covered Position

    Buying 100,000 shares as soon as the option has been sold.

  • Neither a naked position nor a covered position provides a good hedge.


A stop loss strategy

A STOP-LOSS STRATEGY


Delta hedging

DELTA HEDGING

  • Delta (D) is the rate of change of the option price with respect to the underlying asset.

  • c is the price of the call option.

  • S is the stock price


Delta hedging1

DELTA HEDGING


Delta hedging2

DELTA HEDGING

  • For a European call option on a non-dividend-paying stock

  • For a European put option on a non-dividend-paying stock


Delta hedging3

DELTA HEDGING


Delta hedging4

DELTA HEDGING


Delta hedging5

DELTA HEDGING

  • Π is the value of the portfolio

  • The delta of a portfolio of options or other derivatives dependent on a single asset whose price is S.

  • A portfolio consists of a quantity wi of option i (1≦i≦n)

  • Δi is the delta of the ith option.


Theta

THETA

  • The theta (Q) of a portfolio of options is the rate of change of the value of the portfolio with respect to the passage of time with all else remaining the same.

  • Theta is sometimes referred to as the time decay of the portfolio.


Theta1

THETA

  • Because N(-d2)=1-N(d2), the theta of a put exceeds the theta of the corresponding call by rKe-rT


Theta2

THETA


Theta3

THETA


Gamma

GAMMA

  • The gamma (G) is the rate of change of the portfolio’s delta (D) with respect to the price of the underlying asset.


Gamma1

GAMMA


Gamma2

GAMMA

  • Making a portfolio gamma neutral

  • A delta-neutral portfolio has a gamma equal to Γ

  • A traded option has a gamma equal to ΓT

  • The number of traded options added to the portfolio is wT

  • Calculation of Gamma


Gamma3

GAMMA


Gamma4

GAMMA


Relationship between delta theta and gamma

RELATIONSHIP BETWEEN DELTA, THETA, AND GAMMA


The greek letters

VEGA

  • The vega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility of the volatility of the underlying asset.


The greek letters

VEGA

  • For a European call or put option on a non-dividend-paying stock


The greek letters

RHO

  • The rho(ρ) of a portfolio of options is the rate of change of the value of the portfolio with respect to the interest rate.

  • It measures the sensitivity of the value of a portfolio to a change in the interest rate when all else remains the same.


The realities of hedging

THE REALITIES OF HEDGING

  • When managing a large portfolio dependent on a single underlying asset, traders usually make delta zero, or close to zero, at least once a day by trading the underlying asset.

  • Unfortunately, a zero gamma and a zero vega are less easy to achieve because it is difficult to find options or other nonlinear derivatives that can be traded in the volume required at competitive prices.


Scenario analysis

SCENARIO ANALYSIS

  • The analysis involves calculating the gain or less on their portfolio over a specified period under a variety of different scenarios.

  • The time period chosen is likely to depend on the liquidity of the instruments.

  • The scenarios can be either chosen by management or generated by a model.


Extension of formulas

EXTENSION OF FORMULAS

Delta of Forward Contracts


Extension of formulas1

EXTENSION OF FORMULAS

Delta of Futures Contracts

  • The underlying asset is a non-dividend-paying stock

    HF=e-rTHA

  • The underlying asset pays a dividend yield q

    HF=e-(r-q)THA

  • A stock index, q: the divided yield on the index

  • A currency, q: the foreign risk-free rate, rf

    HF=e-(r-rf)THA

  • T: Maturity of futures contract

  • HA: Required position in asset for delta hedging

  • HF: Alternative required position in futures contracts for delta hedging


Portfolio insurance

PORTFOLIO INSURANCE

  • A portfolio manager is often interested in acquiring a put option on his or her portfolio.

  • The provides protection against market declines while preserving the potential for a gain if the market does well.

  • Options markets don’t always have the liquidity to absorb the trades required by managers of large funds.

  • Fund managers often require strike prices and exercise dates that are different from those available in exchange-traded options markets.


Portfolio insurance1

PORTFOLIO INSURANCE

Use of Index Futures

  • The dollar amount of the futures contracts shorted as a proportion of the value of the portfolio

  • The portfolio is worth A1 times the index

  • Each index futures contract is on A2 times the index

  • The number of futures contracts shorted at any given time


Stock market volatility

STOCK MARKET VOLATILITY

  • Portfolio insurance strategies such as those just described have the potential to increase volatility.

  • When the market declines, they cause portfolio managers either to sell stock or to sell index futures contracts.

  • When the market rises, the portfolio insurance strategies cause portfolio managers either to buy stock or to buy futures contracts.


Stock market volatility1

STOCK MARKET VOLATILITY

  • Whether portfolio insurance trading strategies (formal or informal) affect volatility depends on how easily the market can absorb the trades that are generated by portfolio insurance.

  • If portfolio insurance trades are a very small fraction of all trades, there is likely to be no effect.

  • As portfolio insurance becomes more popular, it is liable to have a destabilizing effect on the market.


Summary

SUMMARY

+:Indicates that an increase in the variable causes the option price to increase

-:Indicates that an increase in the variable causes the option price to decrease

?:Indicates that the relationship is uncertain


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