Explain why margin accounts are only required when clients write options but not when they buy options?

Chapter1213-TheGreeks-Students.pptx

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Chapter 12 & 13 The Greek Letters

Example

For $300,000 a bank sold a European call option on 100,000 shares of a non-dividend paying stock

S0 = 49, K = 50, r = 5%, s = 20%,

T = 20 weeks = 0.3846 yrs, μ = 13%

The Black-Scholes value of the option is $240,000 ($2.40/share)

How does the bank hedge its risk to lock in a $60,000 profit?

Naked or Covered Positions

Naked position

Take no action

Covered position

Buy 100,000 shares today

What are the risks associated with these strategies?

Stop-Loss Strategy

This involves:

Buying 100,000 shares as soon as price reaches $50

Selling 100,000 shares as soon as price falls below $50

Stop-Loss Strategy

What are we overlooking?

Delta (Δ)

Delta (D or df/dS) is the rate of change of the option price with respect to the underlying

Call option

price

Slope = D = 0.6

Stock price

Hedge

Trader would be hedged with the position:

short 1000 options

buy 600 shares

Gain/loss on the option position is offset by loss/gain on stock position

Delta changes as stock price changes and time passes

Hedge position must therefore be rebalanced

Hedge

Option Position Stock Position

Long Call Short Δ Shares

Short Call Long Δ Shares

Long Put Long Δ Shares

Short Put Short Δ Shares

Delta Hedging

This involves maintaining a delta neutral portfolio

The delta of a European call on a non-dividend paying stock is N(d 1)

Long European Call  Short e-δtN(d 1)

The delta of a European put on a non-dividend paying stock is N(d 1) – 1

Long European Put  Long e-δt [N(d 1) – 1]

Calculate Δ of call and put for Example (Slide 2)

Delta for Call Options

The Costs in Delta Hedging

Delta hedging a written option involves a “buy high, sell low” trading rule

As the price of the stock goes up Δ increases and when the price of the stock goes down Δ decreases

First Scenario for the Example:

Week	Stock price	Delta	Shares purchased	Cost (‘$000)	Cumulative Cost ($000)	Interest
0	49.00	0.522	52,200	2,557.8	2,557.8	2.5
1	48.12	0.458	(6,400)	(308.0)	2,252.3	2.2
2	47.37	0.400	(5,800)	(274.7)	1,979.8	1.9
.......	.......	.......	.......	.......	.......	.......
19	55.87	1.000	1,000	55.9	5,258.2	5.1
20	57.25	1.000	0	0	5263.3

Second Scenario for the Example

Week	Stock price	Delta	Shares purchased	Cost (‘$000)	Cumulative Cost ($000)	Interest
0	49.00	0.522	52,200	2,557.8	2,557.8	2.5
1	49.75	0.568	4,600	228.9	2,789.2	2.7
2	52.00	0.705	13,700	712.4	3,504.3	3.4
.......	.......	.......	.......	.......	.......	.......
19	46.63	0.007	(17,600)	(820.7)	290.0	0.3
20	48.12	0.000	(700)	(33.7)	256.6

Theta (Θ)

Theta (Θ) of a derivative is the rate of change of the value with respect to the passage of time

The theta of a call or put is usually negative

This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of a long call or put option declines

No point to hedge against time as we have no uncertainty

Traders use Θ as a proxy for Γ (gamma)

They are inversely related

Gamma (Γ)

Gamma (Γ) is the rate of change of delta (Δ) with respect to the price of the underlying asset

Gamma is greatest for options that are close to the money

Higher Γ means portfolio needs to be rebalanced more frequently

Δ changes faster

Gamma Addresses Delta Hedging Errors Caused By Curvature

Stock price

Call

price

C''

Relationship Between Delta, Gamma, and Theta

For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q it follows from the Black-Scholes-Merton differential equation that

Delta & Gamma Neutral

Need to utilize stock and a 2nd derivative

Stock is Γ neutral (i.e. Γ=0)

Use 2nd derivative to make portfolio Γ neutral and then use stock to make portfolio Δ neutral

Vega (𝝼)

Vega (𝝼) is the rate of change of the value of a derivatives portfolio with respect to volatility

Managing Delta, Gamma, & Vega

Delta can be changed by taking a position in the underlying asset

To adjust gamma and vega it is necessary to take a position in options or other derivatives

Example

	Delta	Gamma	Vega
Portfolio	0	−5000	−8000
Option 1	0.6	0.5	2.0
Option 2	0.5	0.8	1.2

What position in option 1 and the underlying asset will make the portfolio delta and gamma neutral?

What position in option 1 and the underlying asset will make the portfolio delta and vega neutral?

Example

	Delta	Gamma	Vega
Portfolio	0	−5000	−8000
Option 1	0.6	0.5	2.0
Option 2	0.5	0.8	1.2

What position in option 1, option 2, and the asset will make the portfolio delta, gamma, and vega neutral?

Rho

Rho is the rate of change of the value of a derivative with respect to the interest rate

Hedging in Practice

Traders usually ensure that their portfolios are delta-neutral at least once a day

Whenever the opportunity arises, they improve gamma and vega

There are economies of scale

As portfolio becomes larger hedging becomes less expensive per option in the portfolio

Greek Equations

I have not provided the equations for each Greek but it is easily displayed in the textbook (Appendix 12.B)

Just take the derivative with respect to the designated variable

Review for non-dividend and dividend paying stocks

Scenario Analysis

A scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities

Deltas for Futures & Forwards

Remember price of Future contract is:

F0 = S0e(r-q)T

The delta of a futures contract is e(r−q)T times the delta of a spot contract

Remember value of Forward contract is:

ƒ = S0e-qT – Ke-rT

The delta of a forward contract is e−qT times the delta of a spot contract and if q = 0 then delta of a forward contract is 1

Future price is Soe(r-q)T so derivative with respect to ∆So is e(r-q)T as they settle daily

Long Forwards however are valued So - Ke-rT so derivative with respect to ∆So is 1

Soe-qT - Ke-rT

Delta with Futures not Stock

The position required in futures for delta hedging is e−(r−q)T times the position required in the corresponding spot contract

Hedging vs Creation of a Synthetic Option

When we are hedging we take positions that offset delta, gamma, vega, etc

When we create an option synthetically we take positions that match delta, gamma, vega, etc

Delta for Options with Different T Values

K=50, Sigma=.50, r=.05

0.2

0.4

0.6

0.8

1.2

020406080100

Stock Price

Delta

T=.25

T=1

T=3