ch08 by Salim Kasap

Finance and Derivatives: Theory and Practice SĂŠ bastien Bossu and Philippe Henrotte

Chapter 8 Dynamic hedging

1. Introduction 



Calls or puts became popular derivatives because they allow buyers to bet on the direction of asset prices while limiting losses to the option premium. This payoff asymmetry appears clearly on a graph of the final P&L (profit and loss) of e.g. a long call position: P&L K premium

K’

S : underlying price

Chapter 8 Dynamic hedging

1. Introduction (2) 



K is the strike price: the price level the underlying must reach for the option to pay some amount (‘inthe-money’) K’ is the break-even point: the price level the underlying must reach for the option position to become profitable, i.e. have a positive P&L. The difference between K and K’ is roughly equal to the option premium

Chapter 8 Dynamic hedging

1. Introduction (3) 

At first sight, flipping the previous graph shows that counterparties (typically banks) who issue options seem to face a higher risk — theoretically an infinite loss: P&L profi t loss



How do option sellers manage their risks?

Chapter 8 Dynamic hedging

2. Delta-hedging 



Delta-hedging is the most commonly used strategy on option trading desks to replicate the payoff of options they sell (or buy). The option pricing model of Black, Scholes and Merton introduced later is also based on this strategy.

Chapter 8 Dynamic hedging

2. Delta-hedging (2) 



An option’s delta is the change in the option’s value due to a unit change in the underlying price. Mathematically it is simply the first derivative with respect to the underlying price:

∂f δ= ∂S



Hedging the delta of an option position means taking an opposite position in the underlying with a quantity δ. The value of this delta-neutral portfolio is thus immune to small changes in the underlying price.

Chapter 8 Dynamic hedging

2. Delta-hedging (3) 



Delta-hedging is a dynamic trading strategy which continuously maintains a delta-neutral portfolio throughout the life of the option. This is achieved by continuously buying and selling the underlying. In theory, this strategy is sufficient for option traders to entirely eliminate the directional risk arising from price movements of the underlying, and replicate the option payoff with a profit.

Chapter 8 Dynamic hedging

3. Other risk parameters 3.1. The Greek letters 



Option buyers and sellers are exposed to many more risks than delta. The other classical risk parameters are known to practitioners as ‘sensitivities’ or ‘Greeks’:    

convexity (Γ: gamma); time decay (θ: theta); volatility risk (υ: vega); interest rate risk (ρ: rho.)

Chapter 8 Dynamic hedging

Figure p.128: Greek letters δ , ∆ (delta)

Γ (gamma)

θ (theta)

υ (vega)

ρ (rho)

∂f ∂S

∂ ²f ∂∆ = ∂ S² ∂ S

∂f ∂t

∂f ∂σ

∂f ∂r

Change in option value when the underlying price St increases by +1

Change in option’s delta when the underlying price St increases by +1

Change in option value due to the passage of time (generally converted into 1 day)

Change in option value when the volatility σ increases by +1 point (+1%)

Change in option when the interest rate r increases by 100 basis points (i.e. +1%)

Chapter 8 Dynamic hedging

3.1. The Greek letters (2) Note:  Each Greek letter is a first-order, ceteris paribus (‘all other things being equal’) approximation 



f (t , S + ∆S , σ , r ) ≈ f (t , S , σ , r ) + δ × ( ∆S )

Greek letters they can be generalized to portfolios of options on the same underlying 

The delta of a portfolio which is long 1,000 calls with delta δ1 and short 500 calls with delta δ2 is δP = 1,000δ1 – 500δ2.

Chapter 8 Dynamic hedging

3.2. Gamma 

Analytical interpretation (Taylor expansion):

1 ∆f ≈ δ × (∆S ) + Γ × (∆S )² 2



When the change in underlying price ∆S is small, the second term in (∆S)² is negligible and one can rely on the approximation ∆f ≈ δ × (∆S). For large movements the second term can be significant, especially if the Γ coefficient is high. Therefore, the gamma measures the deltahedging error: when gamma is high, the deltahedge should be rebalanced more frequently.

Chapter 8 Dynamic hedging

Figure p.129: Call value vs. delta approximation 100 Value 75

of the call

50 25

Delta = 0.62

0 0

100

150

200

SĂŠbastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright ÂŠ Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

Chapter 8 Dynamic hedging

3.2. Gamma (2) 



The second interpretation of Gamma is of more fundamental nature and shows why option traders tend to be more comfortable with a ‘long gamma’ position, meaning that their option portfolio has positive gamma. The graphs in the next two slides show the P&L of a delta-hedged call and a delta-hedged put in function of the underlying price (i.e. the change in value of a portfolio which is long the option and short δ units of the underlying asset.)

Chapter 8 Dynamic hedging

Figure p.129: P&L of a delta-hedged call 40

P&L of a deltahedged call

Gamma prediction S

0 0

100

150

200

SĂŠbastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright ÂŠ Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

Chapter 8 Dynamic hedging

Figure p.129: P&L of a delta-hedged put 40

P&L of a deltahedged put

Gamma prediction S

0 0

100

150

200

SĂŠbastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright ÂŠ Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

Chapter 8 Dynamic hedging

3.2. Gamma (3) 



The two graphs are identical: delta-hedged calls and puts have the same P&L profile. The P&L is always positive: a delta-hedged long call or put will always generate profits as the underlying price moves away from its initial price. Unsurprisingly, there is a downside which we investigate later.

Chapter 8 Dynamic hedging

3.2. Gamma (4) 

The Gamma prediction of P&L is quite accurate. Thus, the gamma is a good measure for the P&L of a delta-hedged option position caused by movements in the underlying price: 



Positive gamma means profits & negative gamma means losses. The larger the gamma and the price movement, the larger the profit or loss.

Chapter 8 Dynamic hedging

3.3. Theta vs. gamma 



The daily theta of an option measures the P&L after one day if all other parameters – in particular the underlying price S – remain unchanged. For plain vanilla options, theta is negative: as time passes and we approach maturity the option loses time value. There is an inverse relationship between theta and gamma:

Chapter 8 Dynamic hedging

3.3. Theta vs. gamma (2) 



The meaning of this relationship for an option trader following a delta-hedging strategy is crucial: as time passes, the profits on a long gamma position will be counterbalanced by losses on the theta as illustrated below, and conversely for a short gamma position. Therefore, the trader will want to be long gamma when she expects large moves in the underlying price and short gamma when she expects gains on theta to more than compensate losses on gamma.

Chapter 8 Dynamic hedging

Figure p.131: Theta vs. gamma

P&L at the beginning of the day

Φ Φ

P&L at the end of the day