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Option Greeks
1 Introduction
Option Greeks
1 Introduction
Set-up
- (^) Assignment: Read Section 12.3 from McDonald.
- (^) We want to look at the option prices dynamically.
- (^) Question: What happens with the option price if one of the inputs (parameters) changes?
- (^) First, we give names to these effects of perturbations of parameters to the option price. Then, we can see what happens in the contexts of the pricing models we use.
Set-up
- (^) Assignment: Read Section 12.3 from McDonald.
- (^) We want to look at the option prices dynamically.
- (^) Question: What happens with the option price if one of the inputs (parameters) changes?
- (^) First, we give names to these effects of perturbations of parameters to the option price. Then, we can see what happens in the contexts of the pricing models we use.
Vocabulary
Notes
- Ψ is rarer and denotes the sensitivity to the changes in the dividend yield δ
- vega is not a Greek letter - sometimes λ or κ are used instead
- The “prescribed” perturbations in the definitions above are problematic...
- (^) It is more sensible to look at the Greeks as derivatives of option prices (in a given model)!
- (^) As usual, we will talk about calls - the puts are analogous
Notes
- Ψ is rarer and denotes the sensitivity to the changes in the dividend yield δ
- vega is not a Greek letter - sometimes λ or κ are used instead
- The “prescribed” perturbations in the definitions above are problematic...
- (^) It is more sensible to look at the Greeks as derivatives of option prices (in a given model)!
- (^) As usual, we will talk about calls - the puts are analogous
Notes
- Ψ is rarer and denotes the sensitivity to the changes in the dividend yield δ
- vega is not a Greek letter - sometimes λ or κ are used instead
- The “prescribed” perturbations in the definitions above are problematic...
- (^) It is more sensible to look at the Greeks as derivatives of option prices (in a given model)!
- (^) As usual, we will talk about calls - the puts are analogous
The Delta: The binomial model
- Recall the replicating portfolio for a call option on a stock S: ∆ shares of stock & B invested in the riskless asset.
- So, the price of a call at any time t was
C = ∆S + Bert
with S denoting the price of the stock at time t
- Differentiating with respect to S, we get
∂ ∂S
C = ∆
- And, I did tell you that the notation was intentional ...
The Delta: The binomial model
- Recall the replicating portfolio for a call option on a stock S: ∆ shares of stock & B invested in the riskless asset.
- So, the price of a call at any time t was
C = ∆S + Bert
with S denoting the price of the stock at time t
- Differentiating with respect to S, we get
∂ ∂S
C = ∆
- And, I did tell you that the notation was intentional ...
The Delta: The binomial model
- Recall the replicating portfolio for a call option on a stock S: ∆ shares of stock & B invested in the riskless asset.
- So, the price of a call at any time t was
C = ∆S + Bert
with S denoting the price of the stock at time t
- Differentiating with respect to S, we get
∂ ∂S
C = ∆
- And, I did tell you that the notation was intentional ...
The Delta: The Black-Scholes formula
- (^) The Black-Scholes call option price is
C (S, K , r , T , δ, σ) = Se−δT^ N(d 1 ) − Ke−rT^ N(d 2 )
with
d 1 =
σ
T
[ln(
S
K
) + (r − δ +
σ^2 )T ], d 2 = d 1 − σ
T
- Calculating the ∆ we get...
∂S
C (S,... ) = e−δT^ N(d 1 )
- This allows us to reinterpret the expression for the Black-Scholes price in analogy with the replicating portfolio from the binomial model
The Delta: The Black-Scholes formula
- (^) The Black-Scholes call option price is
C (S, K , r , T , δ, σ) = Se−δT^ N(d 1 ) − Ke−rT^ N(d 2 )
with
d 1 =
σ
T
[ln(
S
K
) + (r − δ +
σ^2 )T ], d 2 = d 1 − σ
T
- Calculating the ∆ we get...
∂S
C (S,... ) = e−δT^ N(d 1 )
- This allows us to reinterpret the expression for the Black-Scholes price in analogy with the replicating portfolio from the binomial model
The Delta: The Black-Scholes formula
- (^) The Black-Scholes call option price is
C (S, K , r , T , δ, σ) = Se−δT^ N(d 1 ) − Ke−rT^ N(d 2 )
with
d 1 =
σ
T
[ln(
S
K
) + (r − δ +
σ^2 )T ], d 2 = d 1 − σ
T
- Calculating the ∆ we get...
∂S
C (S,... ) = e−δT^ N(d 1 )
- This allows us to reinterpret the expression for the Black-Scholes price in analogy with the replicating portfolio from the binomial model