Chapter 11 - The Greeks

Question 1

How are put & call options more risky than the underlying asset ?

Answer 1

Call options allow exposure to be gained to upside movements in the price of the underlying asset. Put options allow the downside risks to be removed.
(due to the effect of gearing, they are more riskier than the underlying asset)

Question 2

Delta (Δ)

Answer 2

Δ = ∂f/∂St = ∂f(t, St)/∂St
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where f(t, St) is the value at time t of a derivative when the price of the underlying asset at t is St. (delta of an underlying asset = 1)
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def: change of the derivative price with respect to the share price

Question 3

Whats the relationship between delta hedging and delta?

Answer 3

When we consider delta hedging, we add up the deltas for the individual assets & derivatives (taking into account the number of units held of each). If this sum is 0 and if the underlying asset prices follow a diffusion then the portfolio is instantaneously risk-free. (known as delta-hedged or delta-neutral)

Question 4

Why does one ned to rebalance and what does the extent of rebalancing depend primarily on?

Answer 4

Due to keep the sum of deltas to remain close to 0 (delta-hedging) and the extent depends on gamma.

Question 5

Gamma (Γ)

Answer 5

Γ = (∂^2f)/(∂St^2)
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for the underlying asset, whose value is St. Gamma=0
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def: rate of change of Δ with the price of the underlying asset/share price. (curvature or complexity)
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high value of Γ = require more frequent rebalancing or larger trades than one with a low value of gamma.
_
the need for rebalancing can be minimised because of transactional costs.

Question 6

Vega (ν)

Answer 6

ν=∂f/∂σ
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def: rate of change of the price of the derivative w.r.t change in the assume level of VOLATILITY (v for vega&volatility)

Question 7

Rho (ρ)

Answer 7

ρ=∂f/∂r
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def: sensitivity of the derivative price w.r.t changes in the risk-free RATE of interest. (r for rho&rate)
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(low value of rho reduces risk relative to uncertainty in the rf rate of interest)

Question 8

Lamba (λ)

Answer 8

λ=∂f/∂q
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def: the change of the derivative price with the dividend rate

Question 9

Theta (θ)

Answer 9

θ=∂f/∂t
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def: change of derivative price with time
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since time is a variable which advances with certainty, it does not make sense to hedge against changes in t in the same way as we do for unexpected changes in the price of the underlying asset

Question 10

Greeks for a call option (Δ, ν, θ, ρ, λ)

Answer 10

Δ = +
ν = +
θ = -
ρ +
λ = -

Question 11

Greeks for a put option (Δ, ν, θ, ρ, λ)

Answer 11

Δ = -
ν = +
θ = -
ρ = -
λ = +