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Risk Management using Index Options and Futures. Outline. Introduction Using options Using futures contracts Dynamic hedging. Introduction. Portfolio protection involves adding components to a portfolio in order to establish a floor value for the portfolio using:

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Risk Management using Index Options and Futures


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outline
Outline
  • Introduction
  • Using options
  • Using futures contracts
  • Dynamic hedging
introduction
Introduction
  • Portfolio protection involves adding components to a portfolio in order to establish a floor value for the portfolio using:
    • Equity or stock index put options
    • Futures contracts
    • Dynamic hedging
hedging
Hedging
  • Hedging removes risk. Hedging involves establishing a second position whose price behavior will likely offset the price behavior of the original portfolio.
  • The objective of portfolio protection is the temporary removal of some or all the market risk associated with a portfolio. Portfolio protection techniques are generally more economic in terms of commissions and managerial time than the sale and eventual replacement of portfolio components.
using options
Using Options
  • Introduction
  • Equity options with a single security
  • Index options
introduction6
Introduction
  • Options enable the portfolio manager to adjust the characteristics of a portfolio without disrupting it
  • Knowledge of options improves the portfolio manager’s professional competence
equity options with a single security
Equity Options with A Single Security
  • Importance of delta
  • Protective puts
  • Protective put profit and loss diagram
  • Writing covered calls
importance of delta
Importance of Delta
  • Delta is a measure of the sensitivity of the price of an option to changes in the price of the underlying asset:
importance of delta cont d
Importance of Delta (cont’d)
  • Delta enables the portfolio manager to figure out the number of option contracts necessary to mimic the returns of the underlying security. This statistic is important in the calculation of many hedge ratios.
importance of delta cont d11
Importance of Delta (cont’d)
  • Delta:
    • Equals N(d1) in the Black-Scholes Call price.
    • Equals -N(-d1) in the Black-Scholes Put price.
    • Allows us to determine how many options are needed to mimic the returns of the underlying security
    • Is positive for calls and negative for puts
    • Has an absolute value between 0 and 1
protective puts
Protective Puts
  • A protective put is a long stock position combined with a long put position
  • Protective puts are useful if someone:
    • Owns stock and does not want to sell it
    • Expects a decline in the value of the stock
protective put profit and loss diagram
Protective Put Profit and Loss Diagram
  • Assume the following information for ZZX:
protective put profit loss diagram cont d
Protective Put Profit & Loss Diagram (cont’d)
  • Long position for ZZX stock:

Profit or Loss

0

Stock Price at

Option Expiration

$50

-50

protective put profit loss diagram cont d15
Protective Put Profit & Loss Diagram (cont’d)
  • Long position for SEP 45 put ($1 premium):

Profit or Loss

Maximum

Gain = $44

44

$45

0

-1

Stock Price at

Option Expiration

Maximum

Loss = $1

protective put profit loss diagram cont d16
Protective Put Profit & Loss Diagram (cont’d)
  • Protective put diagram:

Profit or Loss

Maximum

Gain is unlimited

$45

0

-6

Stock Price at

Option Expiration

Maximum

Loss = $6

protective put profit loss diagram cont d17
Protective Put Profit & Loss Diagram (cont’d)
  • Observations:
    • The maximum possible loss is $6
    • The potential gain is unlimited
protective put profit loss diagram cont d18
Protective Put Profit & Loss Diagram (cont’d)
  • Selecting the striking price for the protective put is like selecting the deductible for your stock insurance
    • The more protection you want, the higher the premium
writing covered calls
Writing Covered Calls
  • Writing covered calls is an alternative to protective puts
    • Appropriate when an investor owns the stock, does not want to sell it, and expects a decline in the stock price
    • An imperfect form of portfolio protection
writing covered calls cont d
Writing Covered Calls (cont’d)
  • The premium received means no cash loss occurs until the stock price falls below the current price minus the premium received
  • The stock price could advance and the option could be called
differences
Differences
  • Protective puts provide protection against large price declines, whereas covered calls provide only limited downside protection. Covered calls bring in the option premium, while the protective put requires a cash outlay.
index options
Index Options
  • Investors buying index put options:
    • Want to protect themselves against an overall decline in the market or
    • Want to protect a long position in the stock
index options cont d
Index Options (cont’d)
  • If an investor has a long position in stock:
    • The number of puts needed to hedge is determined via delta (as part of the hedge ratio)
    • He needs to know all the inputs to the Black-Scholes OPM and solve for N(d1)
index options cont d24
Index Options (cont’d)
  • The hedge ratio is a calculated value indicating the number of puts necessary:
portfolio insurance example 1 s p 100 index options oex
Portfolio Insurance Example #1: S&P 100 index Options (OEX)
  • The OEX contract is the tool of choice for many professional portfolio risk managers.
  • While the S&P 100 futures contract is similar to traditional agricultural futures, delivery does not occur, nor does it need to occur for this to be an effective hedging tool.
index options cont d26
Index Options (cont’d)

Example

OEX 315 OCT puts are available for premium of $3.25. The delta for these puts is –0.235, and the current index level is 327.19.

How many puts are needed to hedge a portfolio with a market value of $150,000 and a beta of 1.20?

index options cont d27
Index Options (cont’d)

Example (cont’d)

Solution: You should buy 23 puts to hedge the portfolio:

portfolio insurance example 2
Portfolio Insurance Example #2
  • Suppose the manager must protect a $500,000 portfolio with a beta of 2.
  • Current riskless rate is 12%.
  • Dividend yield (on both the portfolio and the market index) is 4%.
  • S&P 500 index is currently 1,000.
  • The manager must hedge the value that the portfolio will take three months from now.
example 2 cont d
Example #2 (Cont’d)
  • Suppose that we want to insure $450,000 of the portfolio, i.e. the manager is willing to let the portfolio go down in value by $50,000 but no further.
  • What strike price for the protective put option should we choose?
example 2 cont d30
Example #2 (Cont’d)
  • Suppose that we want to insure $450,000 of the portfolio, i.e. the manager is willing to let the portfolio go down in value by $50,000 but no further.
  • What strike price for the protective put option should we choose?
example 2 cont d31
Example #2 (Cont’d)
  • Terminal Portfolio Value “that we can live with”: $450,000.
  • Change in value = (450-500)/500 = -10%
  • Dividends Earned = (3/12)(4) = 1%
  • Total Portfolio Return = -10+1 = -9%
  • Note that 3-month Rf = (3/12)(12) = 3%
  • CAPM: Rportfolio = Rf+beta(Rindex-Rf)
example 2 cont d32
Example #2 (Cont’d)
  • Thus: Rindex = (Rportfolio-Rf)/beta + Rf
  • So Rindex = (-9-3)/2+3 = -3%
  • Dividends from index = (3/12)(4) = 1%
  • Change in index value = -3-1 = -4%
  • Terminal index value = 1000(1-.04) = 960
  • Therefore we need an index put option with a strike price of 960.
example 2 cont d33
Example #2 (Cont’d)
  • And the number of put option contracts that need to be purchased is:
  • N = (2)500,000/[(1,000)(100)]=10
  • Note that we do not care about “delta” this time because we hold the option until maturity.
example 2 end
Example #2 (end)
  • Also note that if the value of the index falls to 880 for example, we can compute the corresponding terminal portfolio value (using CAPM) as about $370,000.
  • The options pay (960-880)(10)(100) = $80,000
  • Adding $80,000 to $370,000 brings the net terminal value to $450,000: our required level.
using futures contracts
Using Futures Contracts
  • Importance of financial futures
  • Stock index futures contracts
  • S&P 500 stock index futures contract
  • Hedging with stock index futures
importance of financial futures
Importance of Financial Futures
  • Financial futures are the fastest-growing segment of the futures market
  • The number of underlying assets on which futures contracts are available grows every year
stock index futures contracts
Stock Index Futures Contracts
  • A stock index futures contract is a promise to buy or sell the standardized units of a specific index at a fixed price by a predetermined future date
  • Stock index futures contracts are similar to the traditional agricultural contracts except for the matter of delivery
    • All settlements are in cash
hedging with stock index futures
Hedging with Stock Index Futures
  • With the S&P 500 futures contract, a portfolio manager can attenuate the impact of a decline in the value of the portfolio components
  • S&P 500 futures can be used to hedge most broad-based portfolios (ex: mutual funds).
hedging with stock index futures cont d
Hedging with Stock Index Futures (cont’d)
  • To hedge using S&P stock index futures:
    • Take a position opposite to the stock position
      • E.g., if you are long in stock, short futures
    • Determine the number of contracts necessary to counteract likely changes in the portfolio value using:
      • The value of the appropriate futures contract
      • The dollar value of the portfolio to be hedged
      • The beta of your portfolio
hedging with stock index futures cont d40
Hedging with Stock Index Futures (cont’d)
  • Determine the value of the futures contract
    • The CME sets the size of an S&P 500 futures contract at 250 times the value of the S&P 500 index
    • The difference between a particular futures price and the current index is the basis
calculating a hedge ratio
Calculating A Hedge Ratio
  • Computation
  • The market falls
  • The market rises
  • The market is unchanged
computation
Computation
  • A futures hedge ratio indicates the number of contracts needed to mimic the behavior of a portfolio
  • The hedge ratio has two components:
    • The scale factor
      • Deals with the dollar value of the portfolio relative to the dollar value of the futures contract
    • The level of systematic risk
      • I.e., the beta of the portfolio
computation cont d
Computation (cont’d)
  • The futures hedge ratio is:
computation cont d44
Computation (cont’d)

Example

You are managing a $90 million portfolio with a beta of 1.50. The portfolio is well-diversified and you want to short S&P 500 futures to hedge the portfolio. S&P 500 futures are currently trading for 353.00.

How many S&P 500 stock index futures should you short to hedge the portfolio?

computation cont d45
Computation (cont’d)

Example (cont’d)

Solution: Calculate the hedge ratio:

computation cont d46
Computation (cont’d)

Example (cont’d)

Solution: The hedge ratio indicates that you need 1,530 S&P 500 stock index futures contracts to hedge the portfolio.

the market falls
The Market Falls
  • If the market falls:
    • There is a loss in the stock portfolio
    • There is a gain in the futures market
the market falls cont d
The Market Falls (cont’d)

Example

Consider the previous example. Assume that the S&P 500 index is currently at a level of 348.76. Over the next few months, the S&P 500 index falls to 325.00.

Show the gains and losses for the stock portfolio and the S&P 500 futures, assuming you close out your futures position when the S&P 500 index is at 325.00.

the market falls cont d49
The Market Falls (cont’d)

Example (cont’d)

Solution: For the $90 million stock portfolio:

-6.81% x 1.50 x $90,000,000 = $9,193,500 loss

For the futures:

(353 – 325) x 1,530 x 250 = $10,710,000 gain

the market rises
The Market Rises
  • If the market rises:
    • There is a gain in the stock portfolio
    • There is a loss in the futures market
the market rises cont d
The Market Rises (cont’d)

Example

Consider the previous example. Assume that the S&P 500 index is currently at a level of 348.76. Over the next few months, the S&P 500 index rises to to 365.00.

Show the gains and losses for the stock portfolio and the S&P 500 futures, assuming you close out your futures position when the S&P 500 index is at 365.00.

the market rises cont d52
The Market Rises (cont’d)

Example (cont’d)

Solution: For the $90 million stock portfolio:

4.66% x 1.50 x $90,000,000 = $6,291,000 gain

For the futures:

(365 – 353) x 1,530 x 250 = $4,590,000 loss

the market is unchanged
The Market Is Unchanged
  • If the market remains unchanged:
    • There is no gain or loss on the stock portfolio
    • There is a gain in the futures market
      • The basis will deteriorate to 0 at expiration (basis convergence)
hedging in retrospect
Hedging in Retrospect
  • Futures hedging is never perfect in practice:
    • It is usually not possible to hedge exactly
      • Index futures are available in integer quantities only
    • Stock portfolio seldom behave exactly as their betas say they should
  • Short hedging reduces profits in a rising market
dynamic hedging
Dynamic Hedging
  • Definition
  • Dynamic hedging example
  • The dynamic part of the hedge
  • Dynamic hedging with futures contracts
definition
Definition

Dynamic hedging involves monitoring a portfolio's position delta and readjusting this value as it deviates from a target number.

Example:

  • Attempt to replicate a put option
  • By combining a short position with a long position
  • To achieve a position delta equal to that which would be obtained via protective puts
dynamic hedging example
Dynamic Hedging Example
  • Assume the following information for ZZX:
dynamic hedging example cont d
Dynamic Hedging Example (cont’d)
  • You own 1,000 shares of ZZX stock
  • You are interested in buying a JUL 50 put for downside protection
  • The JUL 50 put expires in 60 days
  • The JUL 50 put delta is –0.435
  • T-bills yield 8 percent
  • ZZX pays no dividends
  • ZZX stock’s volatility is 30 percent
dynamic hedging example cont d59
Dynamic Hedging Example (cont’d)
  • The position delta is the sum of all the deltas in a portfolio:
    • (1,000 x 1.0) + (1,000 x –0.435) = 565
      • Stock has a delta of 1.0 because it behaves like itself
      • A position delta of 565 behaves like a stock-only portfolio composed of 565 shares of the underlying stock
dynamic hedging example cont d60
Dynamic Hedging Example (cont’d)
  • With the puts, the portfolio is 56.5 percent as bullish as without the puts
  • You can sell short 435 shares to achieve the position delta of 565:
    • (1,000 x 1.0) + (435 x –1.0) = 565
the dynamic part of the hedge
The Dynamic Part of the Hedge
  • Suppose that one week passes and:
    • ZZX stock decline to $49
    • The delta of the JUL 50 put is now –0.509
    • The position delta has changed to:
      • (1,000 x 1.0) + (1,000 x –0.509) = 491
the dynamic part of the hedge cont d
The Dynamic Part of the Hedge (cont’d)
  • To continue dynamic hedging and to replicate the put, it is necessary to sell short 74 shares (435 + 74 = 509 shares)
the dynamic part of the hedge cont d63
The Dynamic Part of the Hedge (cont’d)
  • Suppose that one week passes and:
    • ZZX stock rises to $51
    • The delta of the JUL 50 put is now –0.371
    • The position delta has changed to:
      • (1,000 x 1.0) + (1,000 x –0.371) = 629
the dynamic part of the hedge cont d64
The Dynamic Part of the Hedge (cont’d)
  • To continue dynamic hedging and to replicate the put, it is necessary to cover 64 of the 435 shares you initially sold short
dynamic hedging with futures contract
Dynamic Hedging with Futures Contract
  • Appropriate for large portfolios
  • Stock index futures have a delta of +1.0
dynamic hedging with futures contract cont d
Dynamic Hedging with Futures Contract (cont’d)
  • Assume that:
    • We wish to replicate a particular put option with a delta of –0.400
    • We manage an equity portfolio with a beta of 1.0 and $52.5 million market value
    • A futures contract sells for 700
      • The dollar value is 250 x $700 = $175,000
dynamic hedging with futures contract cont d67
Dynamic Hedging with Futures Contract (cont’d)
  • We must sell enough futures contracts to pull the position delta to 0.600
  • The hedge ratio is:
dynamic hedging with futures contract cont d68
Dynamic Hedging with Futures Contract (cont’d)
  • If the hedge ratio is 300 contracts, we must sell 40% x 300 = 120 contracts to achieve a position delta of 0.600