Bm Fi6051 Wk5 Lecturer Notes

Financial Derivatives FI6051 Finbarr Murphy Dept. Accounting & Finance University of Limerick Autumn 2009

Week 5 – The Greeks

Delta Hedging 

The delta of an option is the rate of change of the option price with respect to the underlying asset price



That is, ∆ =



Suppose the delta of an option is 0.6



As the underlying asset price changes by a small amount the option price changes by about 60% of that amount

∂c ∂S

Delta Hedging 

The following is a graphical representation of delta

Option Price

slope = ∆ = 0.6 B

A

Stock Price

Delta Hedging 

To illustrate a delta hedging strategy suppose that the stock price is $100 and the option price is $10



Suppose that an investor has sold 2000 call option contracts



Under the short call option position the investor may be obliged to sell 2000 shares at expiration



The investor can hedge this position by buying 0.6x2000 = 1200 shares

Delta Hedging 

To see this suppose that the stock price goes up by $1



In this case the option price will increase by 0.6x1 = $0.60



The long stock position therefore increases by 1200x1 = $1200



The short call position will result in a loss of 2000x0.60 = $1200

Delta Hedging 

Suppose now that the stock price goes down by $1



In this case the option price will decrease by 0.6x1 = $0.60



The long stock position therefore decreases by 1200x1 = $1200



The short call position will result in a gain of 2000x0.60 = $1200

Delta Hedging 

Note again that the delta of a single option is 0.6, i.e. ∆ = 0.6



The delta of the overall short option position is as follows

∆( − 2000) = −1200



Given a small change in the stock price δS the change in the option position is

− 1200δS

Delta Hedging 

To be exact, note that the delta of a single stock price is 1, i.e. ∆S = 1



The delta of the overall long stock position is as follows

∆ S ( + 2000) = 1200



Given a small change in the stock price δS the change in the stock position is

1200δS

Delta Hedging 

The delta of the overall position of the investor is therefore zero 

The investors position is therefore said to be delta neutral



It is important to note that the delta of an options positon changes over time



Therefore, the hedge has to be adjusted periodically 

This process is referred to as rebalancing

Delta Hedging 

Suppose that the delta value of the option increases from 0.60 to 0.65



Note that an extra 0.05x2000 = 100 shares must be purchased in order to maintain the hedge



Such delta-hedging approach just described is often referred to as a dynamic-hedging scheme



The next section gives the exact form of delta under the Black-Scholes model

The Black-Scholes Delta 

B-S derived the following formula for the price of a European call option on a non-dividend paying stock − rT ( ) c0 = S 0 N d1 − Ke N ( d 2 )

where

d1 =

(

)

ln ( S 0 / K ) + r + σ 2 / 2 T

;

σ T ln ( S 0 / K ) + ( r − σ 2 / 2 )T d2 = = d1 − σ T σ T

and N ( •) is the cumulative probability distribution for the standard normal distribution

The Black-Scholes Delta 

From this equation, it can be shown that the delta of such a European call stock option is

∂c ∆c = = N ( d1 ) ∂S



Delta hedging of a short call option position then involves buying in N ( d1 ) shares in the underlying



Delta hedging of a long call option position then involves shorting N ( d1 ) shares in the underlying

The Black-Scholes Delta 

B-S derived the following formula for the price of a European put option on a non-dividend paying stock

p 0 = Ke − rT N ( − d 2 ) − S 0 N ( − d1 )

where d1, d2, and N ( •) are as before 

From this equation, it can be shown that the delta of such a European put stock option is

∂p ∆p = = N ( d1 ) − 1 ∂S

The Black-Scholes Delta 

Noting that by definition 0 < N ( d1 ) ≤ 1 , and so the following relation holds

−1 ≤ ∆ p < 0 

Delta hedging of a short put option position then involves shorting ∆ p shares in the underlying



Delta hedging of a long put option position then involves going long ∆ p shares in the underlying

The Black-Scholes Delta 

The following graphs shows how the delta of a call option varies with the stock price

Call Delta 1

0 K

Stock Price

The Black-Scholes Delta 

The following graphs shows how the delta of a put option varies with the stock price

Put Delta K 0

-1

Stock Price

The Black-Scholes Delta 

The following graphs shows how the delta of ITM, ATM and OTM call options vary with term to maturity Call Delta

ITM ATM OTM

Term to Maturity

The Dynamics of Delta Hedging 

It is important now to consider the implementation of a delta hedging strategy



Consider a financial institution that sells a European call option on 100,000 shares of a nondividend paying stock



Assume the following Black-Scholes parameter values

S 0 = 49 ; K = 50 ; r = 5% ; σ = 20% ; T = 20 / 52 = 0.3846

The Dynamics of Delta Hedging 

The link below leads to an Excel workbook detailing the dynamic delta hedging of this short option position



The first table assumes that the option expires ITM at expiration



The second table assumes that the option expires OTM at expiration



Excel link:

FI6051_DynamicDeltaHedging_Example_HullTable14-2-3

The Delta of a Portfolio 

The delta of a portfolio of options dependent on a single asset whose price is S is given by ∂Π ∂S

where Π is the value of the portfolio 

The delta of the portfolio can be calculated from the deltas of the individual options



Suppose that the porfolio consists of n options

The Delta of a Portfolio 

Denote the proportion of investment in option i by wi



Denote the delta of option i by ∆ i



Denote the delta of the portfolio by ∆ P, and note n that

∆ P = ∑ wi ∆ i i =1

Theta 



The theta of an option is the rate of change of the option price with respect to time That is, Θ = ∂c ∂t



Assume the Black-Scholes equation for the price of a European call option on a non-dividend paying stock



In this case, Θc = −

S 0 N ′( d1 )σ 2 T

− rKe − rT N ( d 2 )

Theta 

In the above equation for theta note that

N ′( d1 ) =

1 2π

e

− d12 / 2



Assume now the Black-Scholes equation for the price of a European put option on a non-dividend paying stock



In this case, Θp = −

S 0 N ′( d1 )σ 2 T

+ rKe − rT N ( − d 2 )

Theta 

To illustrate, consider a 4-month put option on a non-dividend paying stock



Assume the following details S 0 = 305 ; K = 300 ; r = 0.08 ; σ = 0.25



In this case



Therefore,

d1 = 0.3714; d 2 = 0.2271; N ′( d1 ) =

1 2π

e −0.3714 / 2 = 0.3723

Theta 

Also, − 0.2271

1

−∞

2π

N( − d2 ) = ∫ 

e − x / 2 dx = 0.4102

And so value of theta is as follows Θp = − =−

S 0 N ′( d1 )σ

+ rKe − rT N ( − d 2 )

2 T 305 N ′( 0.3714) 0.25

2 0.33 = −15.0028

+ 0.08( 300) e −0.08( 0.33) N ( − 0.2271)

Theta 

The following graph illustrates the manner in which theta for a European call option changes with respect to the underlying stock price Call Theta K 0

Stock Price

Theta 

The following graph illustrates the manner in which theta for ITM, ATM and OTM European call options change with respect to term to maturity Call Theta Term to Maturity 0 OTM

ITM

ATM

Gamma 

Consider again a portfolio of options the value of which is denoted Π



Gamma is defined as the rate of change of the portfolio’s delta with respect to the underlying asset price



That is,

∂∆ P ΓP = ∂S

Gamma 

Recall that by definition

∂Π ∆P = ∂S 

Therefore



That is, Г is the second derivative of the portfolio’s value with respect to the underlying asset price

∂ 2Π ΓP = ∂S 2

Gamma 

Note that if Г is small (in absolute terms) then this means that delta will change only slowly



In this case adjustments to a delta-neutral portfolio need only be made infrequently



If Г is large (in absolute terms) then this means that delta will change quite quickly



In this case adjustments to a delta-neutral portfolio need to be made quite frequently

Gamma 



Consider a discrete time period δt and denote a change in the underlying asset price by δS The change in the value of the portfolio δΠ is 1 given by 2

δΠ = Θδt + ΓδS 2

where Θ is the theta of the portfolio 

The following graphs illustrate this relation for various values of gamma

Gamma 

The graph below assumes a small negative gamma

δΠ

δS

Gamma 

The next graph assumes a large negative gamma

δΠ

δS

Gamma 

The graph below assumes a small positive gamma

δΠ

δS

Gamma 

The graph below assumes a large positive gamma

δΠ

δS

Creating a Gamma Neutral Portfolio 

Note that the gamma of the underlying asset is zero



The gamma of a forward contract on the underlying asset is also zero 



The reason for this is that a forward contract is linearly dependent on the underlying asset

The gamma of an options portfolio can therefore be changed with an intrument that is nonlinearly dependent on the asset 

An example of such an instrument is another traded option

Creating a Gamma Neutral Portfolio 



Consider a delta-neutral portfolio with gamma Γ

Γˆ

wˆ Consider a traded option with gamma and assume that is the number of options added to the portfolio wˆ Γˆ + Γ



The gamma of the new portfolio is



The appropriate position in the traded option to ensure a neutral portfolio is found by wˆ Γˆ + Γ =gamma 0 solving

Creating a Gamma Neutral Portfolio 

This clearly leads to

−Γ wˆ = Γˆ



Note also that by adding the traded option to the portfolio, the portfolio’s delta is changes also



In order to ensure a neutral delta it is also necessary to adjust the position in the underlying asset

Creating a Gamma Neutral Portfolio 

A delta- and gamma-neutral portfolio can be regarded as correcting for not being able to adjust delta continuously



Delta neutrality protects against relatively small changes in the underlying asset price between rebalancing



Gamma neutrality protects against larger movements in the underlying asset price between rebalancing

Calculating Gamma 

The gamma on a European call or put option on a non-dividend paying stock is given by

Γ= 

N ′( d1 )

S 0σ T

The variation of gamma with respect to time to maturity for OTM, ATM and ITM options is given in the next graph

Calculating Gamma Gamma ATM OTM

ITM

Term to Maturity

The Dynamics of Gamma Hedging 

Consider again the example used to illustrate the dynamics of delta hedging



That is, consider a financial institution that sells a European call option on 100,000 shares of a nondividend paying stock



Assume the following Black-Scholes parameter values

S 0 = 49 ; K = 50 ; r = 5% ; σ = 20% ; T = 20 / 52 = 0.3846

The Dynamics of Gamma Hedging 

For the gamma hedging of the short options position assume another traded options contract is used



That is, consider an options contract with the same details but where the strike price is different



Assume the strike price for this option is K1 = 49, (the option is ITM)

S 0 = 49 ; K 1 = 49 ; r = 5% ; σ = 20% ; T = 20 / 52 = 0.3846

The Dynamics of Gamma Hedging 

The link below leads to an Excel workbook detailing the dynamic delta and gamma hedging of this short option position



The first table assumes that the option expires ITM at expiration



The second table assumes that the option expires OTM at expiration



Excel link:

FI6051_DynamicDeltaGammaHedging_Example_HullTable14-2-3

The Relation Between Delta, Theta and Gamma 

B-S showed that the price of an option on a nondividend paying asset satisfies the following PDE

∂f t ∂f t 1 2 2 ∂ 2 f t + rS t + σ St − rf t = 0 2 ∂t ∂S t 2 ∂S t 

In a similar manner, the value of a portfolio of options must also satisfy the same PDE



That is,

∂Π ∂Π 1 2 2 ∂ 2 Π + rS t + σ St − rΠ = 0 2 ∂t ∂S t 2 ∂S t

The Relation Between Delta, Theta and Gamma 

From our definitions of delta, theta and gamma, the PDE can be rewritten as follows 1 2 2 Θ + rS t ∆ + σ S t Γ − rΠ = 0 2

Vega 

Recall that within the B-S framework volatility is assumed to be constant



In reality of course volatility will change over time



So the value of an option can change as a result of changes in volatility 

As well as a result of changes in the underlying asset and time

Vega 

Again let Π denote the value of a portfolio of options



Vega is the rate of change of Π with respect to the volatility of the underlying asset



That is,

∂Π ν = ∂σ

Vega 

If vega is high in absolute terms, the portfolio’s value is very sensitive to small changes in volatility 

The opposite is true in the case where vega is low in absolute terms



Note that a position in the underlying asset has a vega of zero



Therefore, another traded option can be used to change a portfolio’s vega

Vega 



Let νˆ be the vega of the traded option The position to be taken in the traded option to vega neutrality is given by

ν wˆ ν = − νˆ



Note that a portfolio with such a vega hedge in place will not be in general gamma neutral



To ensure both gamma and vega neutrality at least two traded options must be used – see Hull

Vega 

For a European call or put option on a nondividend paying stock, vega is given by

ν = S 0 T N ′( d1 )

Rho 

The rho of a portfolio - with value Π - is the rate of change of Π with respect to the interest rate r



That is,



Rho measures the sensitivity of an portfolio’s value to changes in the level of interest rates

∂Π ρ= ∂r

Rho 

For a European call option on a non-dividend paying stock ρ = KTe − rT N ( d 2 )



For a European put option on a non-dividend paying stock ρ = − KTe − rT N ( d 2 )

Further reading 

Hull, J.C, “Options, Futures & Other Derivatives”, 2009, 7th Ed. 

Chapter 17