# Risk Management Homework

Spreads

A trade is referred to as a “spread” when it involves the purchase of one option and the sale of the other on the same underlying asset.

Buying a spread entails a cash outflow i.e. COF > CIF called a debit spread

Selling a spread entails a cash inflow i.e. CIF > COF called a credit spread

Spreads limit risk while offering a potential for small profits.

The reduced risk is a consequence of being both in a long and a short position simultaneously.

Types of Spreads:

Vertical Spread or Money Spread

- Options have different exercise prices.

Horizontal or calendar spreads

- Options have different time to expiration

Bull Money Spreads

A bull money spread is a combination of options designed to profit if the price of the UA rises.

1. Bull money spread with calls

Same underlying asset, same time to maturity.

Long call E1 ( E1 < ST at t=0 )

Short call E2 ( E2 > ST at t=0 )

E1 < E2

Remember a trade is referred to as a “spread when it involves buying an option and selling an option on the same UA.

S<E1 | E1 < S < E2 | S > E2 | |

Long Call E1 | S – E1 | S – E1 | |

Short Call E2 | -(S – E2) | ||

S – E1 | E2 – E1 |

Note: Both options payoff when S > E2

Bull money spread the stock price > E1 +C1 – C2

Bull money spread limits the traders risk in comparison to a long position in the UA

Bull money spreads also limit the profit to E2 – E1

Holding Period:

- Early exercise – not a problem

- Short holding period has the lowest range of profits

- When the stock price is low:
- Short holding period has the lowest loss.

- Long HP produces the higher loss
- The E1 option is always worth more than the E2 option however the relative time values differ

- Note that the option’s time value is greatest when the call is at the money. In other words when E = ST.

- When the stock price is high, E2 will have the greatest time value.

- When the stock price is low, E1 will have the greatest time value.

- If we close out the spread prior to expiration i.e. sell the call with E1 and buy the call with E2,The long call E1 will sell for more than the short call E2
- At a high stock price:

C1 – C2 diminishes. The longer call with E1 has the higher intrinsic value but the short call with E2 has the greater time value.

Profit is lower with early close

- At a low stock price:

The long call will have the greater time value.

Profit is higher than holding to expiration.

Which Holding period should an investor choose?

If investor bet correctly, the longer the position is held the greater the profit.

Longer holding periods gives P more time to move.

If investor bet incorrectly, the shorter the HP the lower the loss, but S has less time to move

Conclusion: It all depends

Example:

The following information is available to a trader:

The stock is currently trading at $105

A call with E1 = 100, C1 = $7.0 Long Position

A call with E2 = 110, C2 = $3.0 Short Position

Bull money spread with call options: Total outlay $4, Max profit $6

S<100 | 100 < S < 110 | S > 110 | |

Long Call E1 | S – 100 | S – 100 | |

Short Call E2 | -(S – 110) | ||

S – 100 | 10 |

- You pay for the spread now with calls.

2. Bull money spread with Puts.

Same underlying asset, same time to maturity.

Long put E1 ( E1 < ST at t=0 )

Short put E2 ( E2 > ST at t=0 )

E1 < E2

Remember a trade is referred to as a “spread when it involves buying an option and selling an option on the same UA.

S<E1 | E1 < S < E2 | S > E2 | |

Long Put E1 | E1 – S | ||

Short Put E2 | -( E2 – S ) | -( E2 – S ) | |

E1 – E2 | S – E2 |

- You get paid now with puts
- There is a risk with early exercise
- Loss occurring before expiration has a higher PV than at expiration.

Relationship between calls and puts in a Bull Money Spread

S<E1 | E1 < S < E2 | S > E2 | |

Call Spread | S – E1 | E2 – E1 | |

Put Spread | E1 – E2 | S – E2 | |

Difference X | E2 – E1 | E2 – E1 | E2 – E1 |

Vc1 – Vc2 = Vp1 – Vp2 + X

X = ( Vc1 – Vc2 ) – ( Vp1 – Vp2 )

Where Vc, Vp are the call and put values at expiration

Arbitrage Relationship #2

Vc1 – Vc2 = Vp1 – Vp2 + E2 – E1

Where E2 – E1 is a riskless asset paying E2 – E1 at maturity

Then today the following relationship should hold

C1 – C2 = P1 – P2 + PV(E2 – E1)

Arbitrage:

You can only have 2 possible situations:

Bull money spread where E1 < E2

1. C1 – C2 < P1 – P2 +( E2 – E1 ) eRF x t

Long Call 1

Short Call 2

Short Put 1

Long Put 2

Borrow PV ( E2 – E1 )

2. C1 – C2 > P1 – P2 +( E2 – E1 ) eRF x t

Short Call 1

Long Call 2

Long Put 1

Short Put 2

Invest PV ( E2 – E1 )

Example:

Price | ||

Call1 | $10 | 100 |

Call 2 | $5 | 110 |

Put 1 | $3 | 100 |

Put2 | $6 | 110 |

RF = 10% for the period

PV (E2 – E1 ) = 10 / ( 1+ 0.1) = 9.09

C1 – C2 = P1 – P2 + ( E2 – E1 ) eRF x t

10 – 5 ≠ 3 – 6 + 9.09

5 < 6.09 Arbitrage exists

To take advantage:

Long Call 1 Long Put 2

Short Call 2 Short Put 1

Borrow $ 9.09 at the risk free rate

In other words:

C1 + P2 = P1 + C2 + ( E2 – E1 ) eRF x t

Now | Later | |

Long Call 1 | -10 | Vc1 |

Short Call 1 | (Vc2) | |

Long Put 2 | -6 | Vp2 |

Short Put 1 | (Vp1) | |

Borrow | 9.09 | -10 |

1.09 | Must be 0 |

Bear Money Spreads

Long put E2 ( E2 > ST at t=0 )

Short put E1 ( E1 < ST at t=0 )

E1 < E2

- Bear money spreads are the mirror image of a bull money spread.

- Bear money spreads limits the profits in bear markets to E2 – E1

- Limits the loss in bull markets

S<E1 | E1 < S < E2 | S > E2 | |

Long Put E2 | E2 – S | E2 – S | |

Short Put E1 | -(E1 – S) | ||

Portfolio Value | E2 – E1 | E2 – S |

Profits: When S<E1 → E2 – E1+ P1 – P2

When E1 < S < E2 → E2 – S + P1 – P2

When S<E1 → P1 – P2

Break even point: E2 + P1 – P2

Holding Period:

To close position early: Buy back put E1and short put E2

- Longer holding period produces higher profits in a bear market.

- Longer holding period produces larger losses in bull market due to time value.
- When stock price is high, Put with E2 has more time value.
- As the stock price drops the long position loses time value faster than the short position
- When stock price is low, Put with E1 has more time value.
- As time passes investor will gain from that affect.

Collars

Is an investment strategy that protects the long stock position.

Long Stock

Long Put E1

Short Call E2

- Similar to a bull spread

- Long Stock position can be protected by a long put

- Must pay premium for that protection; P (premium of the put)

- Collar reduces the outlay for the put premium by shorting a Call.
- EC > E P ; E1 < S0 ; E2 > S0
- A collar is equivalent to a bull spread plus a RF bond paying E1 at expiration.

S<E1 | E1 < S < E2 | S > E2 | |

Long Stock | |||

Long Put E1 | E1 – S | ||

Short Call E2 | ( S – E2) | ||

Portfolio Value | E1 | E2 |

Profits: When S<E1 → E1– S0 – P1 + C2

When E1 < S < E2 → ST – S0 – P1 + C2

When S<E1 → E2 – S0 – P1 + C2

- Since E1 < S0 & E2 > S0 profit on the stock is either E1 – S0– P +C or E2 – S0– P +C

- Note: E1 – S0– P +C is negative or a loss.

- In other words the potential gain or loss is limited.
- When E1 = S0 & E2 > S0 profit on the stock is either C – P or E2 – S0– P +C

- C – P is usually negative, loss is minimized

- Only in the middle range is there uncertainty

- Investors give up upward gains ( selling @ a max of E2 ) for the reassurance that the stock cannot be sold for less than E1.
- Collars are usually used with index options to protect portfolios not individual stocks.

- Notice that a collar is similar to bull money spread.

- In actuality a collar is equivalent to a bull money spread with calls plus a RF bond paying E1 at expiration.

Holding Period:

- Before expiration we recoup more time value on the long put than we pay to buy back the short call.
- As S↓ the longer we hold the less we gain.
- As S ↑ the earlier we closeout the position the more time value we must pay for than we receive from selling the put.

Because E2 > E1:

- As S↓ time value is greater on the long put
- As S ↑ time value is greater on the short call

Butterfly Spread

- Combination of a bull and bear money spread

- Involves 3 exercise prices E1, E2 and E3. Where E2 is half way between E1 and E3.

Call bull spread: Call Bear Spread:

Long Call E1 Long Call E3

Short Call E2 Short Call E2

Put them together:

Long Call E1

2 Short Calls E2

Long Call E3

S<E1 | E1 < S < E2 | E2 < S < E3 | S > E3 | |

Long Call E1 | S – E1 | S – E1 | S – E1 | |

2 short Calls E2 | -2(S – E2) | 2(S – E2) | ||

Long Call E3 | S – E3 | |||

Portfolio value | S – E1 | 2E2 – E1 – S | 2E2 – E1 – E3 |

Profit:

When S<E1: 2C2 – C1 – C3 → (C2 – C1) + (C2 – C3)

When E1 < S < E2: ST – E1 + 2 C2 – C1 – C3

When E2 < S < E3 : 2E2 – E1 – ST +2C2 – C1 – C3

When S > E3: 2E2 – E1 – E2 + 2C2 – C1 – C3

Two Break even points:

St – E1 – C1 +2C2 – C3 = 0 → ST = E1+ C1 – 2 C2 + C3

In other words:

A butterfly spread will be profitable if the S > E1 by at least C1 – 2C2 + C3

Butterfly spreads have a limited loss which occurs when ST moves away from E2 and a limited gain which peaks at ST = E2

Low risk transaction

Straddles

Long position in a call E

Long position in a put E

Both put and the call expire on the same day.

- Has 2 break even points.

- Holding both a put and a call with the same exercise price and expiration date, allows the investor to capitalize on stock price movement in either direction.
- Investors should invest in a straddle to profit from expected large swings in share prices in either direction.

When would that be appropriate?Example:

An investor is interested in investing in Pox Pharmaceutical which is currently trading at $25 a share. Pending an FDA approval, Pox share price is expected to rise sharply. If however approval is denied share price is expected to tumble.

Available to the investor are :

March Call; E = $25, C = $5

March Put; E = $25, P = $3

Long call, Long Put ; Same E, same maturity, total cost C + P

S<25 | S>25 | |

Long Call | S – 25 | |

Long Put | 25 – S | |

Portfolio Payoff | 25 – S | S – 25 |

Net-profit-loss diagram with breakeven points and maximum loss.

25

17

0 Stock Price

17 25 33

-8

This is called a straddle

- Maximum loss would be $8
- Potential of unlimited gain on the up side
- Break even points are $17 and $33
- Straddle tend to provide higher potential profit but also the greatest

potential loss.

- For a given stock price, a straddle losses value as it nears expiration. This is due to loss of time value on both options.
- If FDA delays approval before expiration investor will lose C + P

What happens if the options available to the investors are:

March call, E = $30, C = $3

March Put, E = $20, P = $2

How would the investor’s strategy change?

S<20 | 20<S<30 | S>30 | |

Long Put 20 | 20 – S | ||

Long Call 30 | S – 30 | ||

Portfolio payoff | 20 – S | S – 30 |

20

15

015 20 30 35 Stock Price

-5

This is called a Strangle.

It involves a low max loss of $5, but a greater likelihood of occurrence.

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