Option Greeks
1Introduction
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Understanding Option Greeks: Delta, Vega, Theta, Gamma, and Rho, Study notes of Reasoning
An introduction to Option Greeks, which are measures of the sensitivity of an option's price to various parameters such as stock price, volatility, time to expiration, interest rates, and dividend yield. the concepts of Delta, Vega, Theta, Gamma, and Rho, and their significance in option pricing. It also includes formulas and explanations for each Greek letter. useful for students and professionals in finance, economics, and related fields.
Typology: Study notes
2021/2022
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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