Problem 18.3.

Calculate the delta of an at-the-money six-month European call option on a non-dividend-paying stock when the risk-fiee interest rate is 10% per annum and the stock price volatility is 25% per annum.

In this case, S₀=K, r = 0. 1 , σ=0.25 , and T = 0.5 .

Also,
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The delta of the option is N(d₁=0.3712) or 0.6448.

Problem 18.5.

What is meant by (he gamma of an option position?

What are the risks in the situation where the gamma of a position is large and negative and the delta is zero?

The gamma of an option position is the rate of change
of the delta of the position with respect to the asset price.

For example, a gamma of 0. 1 would indicate that
when the asset price increases by a certain small amount
delta increases by 0.1 of this amount.

When the gamma of an option writer’s position is large
and negative and the delta is zero, the option writer will lose
significant amounts of money if there is a large movement
(either an increase or a decrease) in the asset price.

Problem 18.25.

A financial institution has the following portfolio of over-the-counter options on sterling:

A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8.

(a)What position in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral?

(b)What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral?  

 The delta of the portfolio is

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The gamma of the portfolio is

[image]
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The vega of the portfolio is

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 (a) A long position in 4,000 traded options will give
a gamma-neutral portfolio since the long position
has a gamma of (-Σγ/γ×γ)=4,000×1.5=+6,000. The delta
of the whole portfolio (including traded options) is then:

(-Σγ/γ×γ)×Δ-ΣγΔ=4,000×0.6-450=1,950

Hence, in addition to the 4,000 traded options,
a short position of 1,950 in sterling is necessary so that
the portfolio is both gamma and delta neutral.

(b) A long position in 5,000 traded options will give a
vega-neutral portfolio since the long position has
a vega of (-Σν/ν)×ν=5,000×0.8=+4,000. The delta of the whole
portfolio (including traded options) is then

(-Σν/ν)×Δ-ΣγΔ=5,000×0.6-450=2,550

Hence, in addition to the 5,000 traded options, a short
position of 2,550 in sterling is necessary so that the portfolio
is both vega and delta neutral.

Problem 18.26.

Consider again the situation in Problem 18.25.

Suppose that a second traded option with a delta of 0.1, a gamma of 0.5, and a vega of 0.6 is available.

How could the portfolio be made delta, gamma, and vega neutral?

Let w₁be the position in the first traded option and w₂ be
the position in the second traded option.

We require:

(-Σγ)×γ=γw₁+γw₂

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(-Σν)=νw₁+νw₂

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The solution to these equations can easily be seen
to be . The whole portfolio then has a delta of
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The solution to these equations can easily be seen to be
w₁=3,200, w₂=2,400. The whole portfolio then has a delta of

Δ+w₁×ν+w₂×Δ=

[image]

Therefore the portfolio can be made delta, gamma and vega
neutral by taking a long position in 3,200 of the first traded
option, a long position in 2,400 of the second traded
option and a short position of 1,710 in sterling.