Equity Option Pricing & Hedging

Question 1

What are the different types of volatility

QFIQ 114 17

Answer 1

Realized/Historical: Associated with two periods, typically the volatility estimate if we don’t have a more sophisticated model. As a backward looking estimate we assume historical data has some insight into future behavior
Actual volatility: No time scale, instantatnous noise in stock price return. Can be modelled as time dependent, stock and time, stochastic, jump, or even just uncertain
Implied volatiliity: Market volatility
Forward volatility refers to volatility at a future time period

Question 2

How robust is the BSM model? What does loosening assumptions do

Answer 2

It is very robust:
* Continuous hedging: discrete hedging still has BSM apply to the average of all wins/losses
* Transaction costs: the model can adjust the underlying price to include it
* Constant volatility: We can use the average volatility over the option life, even if volatility is stochastic
* Lognormal asset: Interest rates are not lognormal, but BSM is still used in those markets
* No short sell costs: adjust the RN drift rate
* Normal return dist: Thanks to CLT, we only need finite variance

Question 3

How do you calculate implied volatility

Answer 3

We can invert the BSM equation to find the constant volatility implied by the price. To find the smile, we can plot implied volatility as a function of different strikes

Question 4

Give 2 reasons for the volatility smile existing

Answer 4

• Smile incorporates kurtosis seen in returns
• Consequence of supply and demand

Question 5

List 3 models/methods for incorporating smile effect into option prices

Answer 5

• Deterministic volatility surface
• Stochastic volatility
• Jump Diffusion models

Question 6

What are 4 shapes of volatility smiles and describe features

Answer 6

• Smile: heavier tails than lognormal, common in FX, jumps make tails heavier, effect is more pronounced for short term options
• Frown: Lighter tails than GBM, BS gives high prices for OTM/ITM options, common with government intervention to floor/cap prices
• Positive skew: heavier right tail, common with futures, vol. increases with futures price. Ex is constant elasticity of variance with a>1
• Negative skew: heavier left tail, common for equity. Ex is constant elasticity of variance with a< 1. caused by leverage and crashophobia

Question 7

How can we make money if our expectation of volatility differs from the market?
## Footnote

QFIQ 115 17

Answer 7

Use a volatility arbitrage strategy. If σ_actual > σ_implied then the market is underpricing options. We can buy them and delta hedge the stock exposure away. We can do the opposite if σ_actual < σ_implied

Question 8

Describe the mechanics of delta hedging a volatility arbitrage case using σ_implied

Answer 8

If we hedge with σ_implied we are guaranteed to make a dailty profit, but the amount is random |1/2 (σ²_actual - σ²_implied) ∫(0,T) e^-r(t)S²Γ^implieddt|
We balance the fluctuations in the m2m option value with fluctuations in the stock price. This P&L has lower fluctuations than heding under σ_actual, so it is good if we are constrained to mark-to-market accounting.
We do not need to know σ_actual

Question 9

What is the PV of total profit when hedging with another volatility

Answer 9

Hedging with some volatility σ_h, then the PV of total profits is
V(S, t, σ_h - V(S, t, σ_implied + 1/2 (σ²_actual - σ²_h ∫(t₀, T) e^-r(t-t₀S²Γ^hdt * sign of the position

Question 10

What are the pros and cons of hedging with actual and implied volatilities

Answer 10

Actual: Pros: Expiration profit is guaranteed, this is the best choice in a classical risk/reward framework
Cons: The P&L fluctuating over the option life is scary, and we are not inherently confident in our volatility of actual volatility
Implie: Pros: P&L does not locally fluctuate, we are continuously making money. We just need to be on the right side of actual volatility to profit. The volatility used to calculate delta is kown
Cons: We don’t know for certain how much money we will make, just that it will be positive

Question 11

What is the Delta of a European option

Answer 11

Long Call Δ = ∂C/∂S = e^-DTN(d₁) > 0
We can use Put-Call parity or the BS equation to arrive at the Delta for a put
Long Put Δ = ∂P/∂S = e^-DT(N(d₁) - 1) < 0
Short positions have the negative delta of the long ones

Question 12

How does delta change wrt implied volatility

Answer 12

For an OTM option: lower volatility lowers delta (less likely to enter money), and higher volatility raises delta
For an ITM option: lower volatility increases delta (less likely to leave money) and higher volatility lowers delta

Question 13

Sensitivity of Gamma wrt underlying price

Answer 13

Gamma is sensitive to S.
* Γ tends to 0 as S diverges from the strike price
* Γ is maximized when S ~ K
* Γ for vanilla calls and puts cannot be negative
* As a function of S, Γ for calls and puts has a bell shaped curve

Question 14

Sensitivity of Vega wrt underlying price (OTM option)

Answer 14

With a low volatility, the probability the underlying goes ITM is low, so we should not expect a strong sensitivity of option price wrt volatility
When volatility increases, the dependence of the option wrt volatility increases. Thus the sensitivity of Vega to volatility also increases like delta and gamma

Question 15

How do we see implied volatility not flat across different strikes

Answer 15

When we invert the BSM to solve for implied volatility, we get a measure of volatility consistent with market prices. For options on the same underlying, implied volatility as a function of strike is not flat. The curbe exhibits a smile or a downward slope, both examples of volatility smiles

Question 16

Compare the shapes of volatility smile and volatility skew

Answer 16

A smile has the lowest value of Σ for ATM options, and it increases on either side
A skew continually decreases, perhaps rising a small amount for high strikes

Question 17

What is the sticky delta approach for estimating Σ

Answer 17

The implied volatility for each expiry is expressed in terms of the delta of the option. This method is common in FX and commodities. Implied volatilities are quoted for 10, 25, and 50 delta calls/puts for a specific time to expiry
Σ is a function of the option’s delta and time to expiration

Question 18

What is the sticky strike approach for estimating Σ

Answer 18

The implied volatility for each expiry is expressed in terms of the option strike. This is more common for equity options
Each strike has a certain implied volatility associated with it regardless of whether the stock moves up or down

Question 19

What are three types of movements affecting the volatility skew

Answer 19

Much like the yield curve, we can see parallel shifts, changes in slope, and changes in curvature

Question 20

How does the delta of a long call spread behave wrt K

Answer 20

The delta of a call spread converges to 0 when S is on either side of the strike range, but it is maximized in between K₁ and K₂

Question 21

What are pros and cons of empirical interpolation and functional forms for the smile

Answer 21

Functional: Pro: The smile is naturally smooth which has convenience
Con: It may not reproduce market quotes exactly
Empirial: Pro: Always matches market quotes exactly
Con: Requires effort to handle extrapolation, and it may introduce discontinuities in the smile’s shape to perfectly reproduce all quotes

Question 22

What are 5 common characteristics of foward based asset volatility curves

Answer 22

• The long ends of price curves tend to have flattening volatilities
• Short ends usually have higher volatilities
• Short and long ends move in a partially correlated way
• A skew correction must be part of the model so all vanilla options available in the market data are priced within the bid-ask spread
• Some assets require a more sophisticated model and extra factors

Question 23

List 4 approaches of volatility modeling

Answer 23

• Forward starting volatility
• Local volatility model
• Stochastic volatility models
• Local-Stochastic volatility model

Question 24

What are some drawbacks of a stochastic volatility model

Answer 24

The drawbacks are significant:
* European option prices cannot be reproduced perfectly. These models are good for exotic trades, but not plain vanilla ones
* Calibratiing these models can be unstable, leading to jumps in mark-to-market profits
* Calibrating with vanillas can give good prices for exotics, but calibrating with exotics does not guarantee accurate pricing for vanilla options

Question 25

What is the principle of replication

VS 2

Answer 25

• If 2 securities have identical payoffs under all possible scenarios, they should have the same price
• If they had different prices, you could create riskless profits
• This concept is related to the law of one price
• It also connect to the principle of no riskless arbitrage

Question 26

Name the two kinds of replication

Answer 26

Static and Dynamic

Question 27

What is static replication

Answer 27

Static replication reproduces the payoffs of the target secuity/portfolio with an initial portfolio whose weights will never need to be changed
Once the replicating portfolio is made, we will not need to touch it

Question 28

What is dynamic replication

Answer 28

The components of the replicating portfolio and the weights are subject to adjustments during the life of the target portfolio
We continuously buy and sell securities as time passes
It can be very complex in theory/practice, but wehn it works, we can value a wide range of securities that may be impossible to value otherwise

Question 29

Name 3 ways to alter or avoid risk

Answer 29

Dilution
Diversification
Hedging

Question 30

What is the time t value of a Collar

VS 3

Answer 30

A collar consist of buying a put with a low strike L and selling a call with a high strike U
The value at time t of the collar is
Collar(t) = S_t + P_L(S, t) - C_U(S, t)
By put call parity, this is the same as Collar(t) = PV(L) + C_L(S, t) - C_U(S, t)

Question 31

How can we replicate any general payoff using calls and puts?

Answer 31

Suppose the time T payoff of a derivative has initial value I and a series of slopes {λ_i} that change at various strike prices {K_i}
We know calls and puts have a slope of +/- 1 after their strike value. So we just need to buy/sell enough calls at the various strike points to match the slopes of the general payoff
Using calls, the value at time t would be
V(t) = I e^-r(T-t) + λ₀S_t + (λ₁ - λ₀)C(K₀) + …

Question 32

What is the net profit of a delta hedged call over a time step dt (r=0)

Answer 32

The hedged portfolio follows the fundemental hedged PDE
∂C/∂t + 1/2 Σ²S²Γ = 0
The amount we expect to lose to time decay during a step is 1/2 Σ²S²Γ
If the stock moves by an amount +/- σSsqrt(dt) with realized volatility σ, the gain from convexity is 1/2 σ²S²Γ
Thus the net profit after time dt is 1/2 S²Γ(σ² - Σ²)dt

Question 33

What are the Vega and Kappa of a call option

VS 4

Answer 33

Let C(S, K, ν) = SN(d₁) - KN(d₂)
Then Vega = ∂C/∂σ = Ssqrt(τ)/sqrt(2π) e^{-1/2 d₁2
and Kappa = ∂C/∂σ2 = Ssqrt(τ)/[2σsqrt(2π)] e-1/2 d₁2
Vega and Kappa for plain vanilla puts are equal to the call ones
For any option, Vega is highest when S ~ K}

Question 34

Define a volatility swap contract

Answer 34

• It is a contract that gives you exposure to volatility and no exposure to the level of the underlying stock
• It is a forward contract on realized volatility
• At inception, the coparties agree on a forward/strik volatility σ_K
• At maturity, they calculate the realized volatility of the instrument σ_R

Question 35

What is the payoff of a volatility swap

Answer 35

For a notional amount N (called notional Vega) the long postion pays out π = N(σ_R - σ_K)
Sometimes the payoff for these contracts is expressed in terms of variance π = N(σ_R² - σ_K²).
Here N is the notional variance

Question 36

What challenges are there for replicating a volatility swap

Answer 36

We could dynamically replicate a volatility swap by trading variance swaps, but they are illiquid so this may be too challenging.
Determining the dynamic replication of the volatility swap requires a model for the volatility of variance

Question 37

How can we replicate a variance swap

Answer 37

Consider a portfolio of plain vanilla options s.t.
K = variable strike; ρ(K) = pdf of K; C(S, K, v) = value of call option at strike K
The value of the portfolio is
π(S) = ∫(0, inf) ρ(K)C(S, K, v)dK
A continuous density of options whose weights ρ(K) decrease like 1/K² will have a variance sensitivity independent of S, thus replicating a variance swap

Question 38

What is the volatility of a call under BSM

VS 5

Answer 38

dS = μ_SSdt + σ_SSdZ
From Ito’s Lemma, we get
dC = [∂C/∂t + (∂C/∂S)μ_SS + 1/2 (∂²C/∂S²)(σ_SS)²]dt + (∂C/∂S)σ_SSdZ
Extracting the volatility term (dZ) we see σ_CC = (∂C/∂S)σ_SS
In other terms, σ_C = Δ_Cσ_SS/C

Question 39

What is the P&L of hedging a long call using realized volatility under BSM

Answer 39

The PV of total P&L generated over the option life is
PV(P&L(I,R)] = V(S, τ, σ_R) - V(S, τ, Σ)

Question 40

What is the P&L of hedging a long call using implied volatility under BSM

Answer 40

The evolution of the P&L over time has no random component
d[P&L(I,I)] = 1/2 Γ_IS²(σ_R² - σ_I²)dt = 1/2 Γ_IS²(σ_R² - Σ²)dt
The PV is given as
PV(P&L(I,I)) = 1/2 ∫(t₀, T) e^-r(t-t₀)Γ_IS²(σ_R² - Σ²)dt

Question 41

What is the total Hedging Error when discrete hedging with n time steps

VS 6

Answer 41

Total HE = Σ(i=1,n) 1/2 Γ_iσ_i²S_i²(Z_i² - 1)dt
The mean HE is 0 because Z ~N(0,1)
The variance of this is σ_HE² = E[Σ(i=1,n) 1/2 (_iS_i²)²(σ_i²dt)²]
For an ATM option, we can express the volatility of Hedging error as
σ_HE ~ σ/sqrt(n) ∂C/∂σ

Question 42

What are some insights about Hedging Error over n time steps

Answer 42

• The Hedging Error can be viewed as arising from the uncertainty in BSM option value induced by uncertainty in volatility
• Hedging discretely introduces HE, but if we hedge at the correct realized volatility the HE decreases as n increases
• Hedging reduces with the sqrt(n)
• If you estimate future volatility correctly and hedge ctsly at it, your P&L will capture the exact option value
• Hedging discretely at the correct volatility introduces random components proportional to (Δ_I - Δ_R)dS

Question 43

What are some insights about Hedging Error when you use implied volatility

Answer 43

If implied volatility differs from realized and you hedge ctsly, the P&L is path dependent. P&L is maximized with Gamma when S ~ K
Discrete hedging at implied volatility also gets the random components from the hedge at realized volatility
Traders are more likely to hedge at implied volatility
The more implied volatility differs from realized, the more traders lose on benefits of increased hedging frequency

Question 44

What is the impact of transaction costs on hedging

VS 7

Answer 44

Longing the option requires extra cash to hedge so the option is worth less
Using the BSM formula, this corresponds to a lower implied volatility
The opposite is true for a short option, spending extra cash to hedge so you need to increase the premium to offset this
Transaction costs introduce a bid-ask spread into valuation

Question 45

Describe the shape of implied volatility for Equity indexes and why we see it

VS 8

Answer 45

We see a skew for 2 reasons:
* There is an asymmetry in the way equity index options work. Large negative returns are much more common than large positive returns
* It is difficult to hedge market crashes. Investors who want to hedge against this are willing to pay the higher premiums to get the overpriced OTM puts

Question 46

What are some major characteristics of the Equity Index skew

Answer 46

• Implied volatility and index returns are negatively correlated
• Equity crises are always characterized by high volatility, so the term structure is downward sloping
• The negative skew is generally steeper for shorter expiration options
• Implied volatility tends to be above realized volatility due to market frictions and other factors. It can be viewed as the market expectation plus a premium
• When markets are tranquil, short term vol is lower than long term
• When an index sharply drops, short term vol spikes up and the short term negative skew steepens.

Question 47

Describe the shape of implied volatility for FX and why we see it

Answer 47

The smile for FX markets can be like an index or like a single stock. it’s roughly symmetric for equally powerful currencies and less so for unequal ones.

Question 48

What bounds are placed on implied volatility

Answer 48

There are two bounds that are independent of our chosen pricing model. We are constrained that volatility cannot be so high or low that a call would decrease or a put would increase in value when strike increases

Question 49

How can we approximate the bounds on implied volatility

Answer 49

For small changes ∂K in the strike, the constraint in implied volatility (ceiling given by calls, floor from puts) is
|∂Σ| <= 1.25/sqrt(τ) * ∂K/K

Question 50

What are 3 approaches to capturing the implied volatility observed in the market?

VS 10

Answer 50

1. Move away from traditional GBM assumption for asset prices
2. Model movements of the BSM implied volatility surface Σ(S, t, K, T) rather than the movements in the volatility itself
3. Avoid formal models of either underlying or impled volatility and use heuristics for pricing and hedging

Question 51

What are 4 ways to move away from traditional GBM assets to capture volatility?

Answer 51

• Local Volatility models
• Constant elasticity of variance (CEV) models
• Stochastic volatility models
• Jump Diffusion models

Question 52

What are stock price dynamics under LV, CEV, SV?

Answer 52

Local Volatility: dS/S = μ(S, t)dt + σ(S, t)dZ
CEV: dS/S = μ(S, t)dt + σS^β-1dZ
SV: dS/S = μdt + σdZ and dσ = pσdt + qσdW; E(dWdZ) = ρdt

Question 53

What is the continuous state valuation of an Option

VS 11

Answer 53

Let p(S, t, S_T, T) = RN PDF of S_T. Then an option value V(S, t) = e^-r(T-t) ∫(0, inf) p(S, t, S_T, T)V(S_T, T) dS_T

Question 54

How do we derive the Breeden-Litzenberger formula

Answer 54

Using the cts valuation of a call, C = e^-r(T-t) ∫(K, inf) p(S, t, S_T, T)(S_T - K) dS_T
Taking 2 partial derivatives and solving for p(S, t, S_T, T), we find
p = e^r(T-t) ∂²C/∂K²
We see the RN probability of making a transition from S at t to K at T is proprtional to the second derivative of price wrt K

Question 55

What are some characteristics/consequences of the Breeden-Litzenberger formula

Answer 55

• Given the market prices of standard Euro calls/puts, we can find the RN PDF at option maturity
• This distribution p(.) is called the implied distribution, as it is implied by market prices
• The formula lets us price any other European option maturing exactly at T
• This can NOT be used to value American options
• The idea is to express the payoff of any Euro derivative as a combination of payoffs of standard puts/calls using no arbitrage principles
• The equivalence of payoffs is model independent
• The the market price of the option may appear smooth wrt K, the implied PDF may not be. If we want a smooth PDF, we need to estimate option prices with a twice diff’able fn
• Breeden-Litzenberger is model free!

Question 56

Describe the Local Volatility model

Answer 56

In a local volatility mode, the instant stock volatility is a function of S and t. The BSM implied volatility is the average of local volatilities between S and K
Σ(S, T, K) = 1/(K-S) ∫(S, K) σ(S’)dS’
In LV, realized volatility over any time period will depend on the path of stock prices
Even if LV is accurate for representing realized volatility, we may still want to use BSM out of convention

Question 57

What is the relationship of implied volatility under Local Volatility

Answer 57

When LV σ(S) is a function of stock price alone, the implied volatility for strike K is the average between S and K. This is reminiscent of the relationship
Σ²(t, T) = 1/(T-t) ∫(t,T) σ²(s)ds
If we look at local volatility as a function of time alone, the implied variance for expiration T is the average of forward variances between t and T

Question 58

What is the rule of two

Answer 58

Local volatility function σ(S) grows approximately twice as fast with stock price as implied volatilities grows with strike. Mathematically,
∂σ/∂S ~ 2 ∂Σ/∂K

Question 59

What are some pros and cons of the Local Volatility model

Answer 59

If σ(S,t) is known, the PDE for option prices is
∂C/∂t + rS(∂C/∂S) + 1/2 σ(S, t)²S² (∂²C/∂S²) = rC
Pros: The equation can be solved numerically with Monte Carlo. Once calibrated, LV provides arbitrage free option values. It is very close to original BSM. It is very popular with academics and market participants
Cons: We may periodically need to recalibrate the model. LV is unable to match the short-term skew present

Question 60

How do different market regimes of volatility and market direction

Answer 60

dπ_BSM = 1/2 S²Γ_loc (σ_R² - σ_loc²(S, t))dt - εdS
High vol, Up mrkt: σ_R > σ_loc, dS > 0
High vol, Down mrkt: σ_R > σ_loc, dS < 0
Low vol, Up mrkt: σ_R < σ_loc, dS > 0
Low vol, Down mrkt: σ_R < σ_loc, dS < 0

Question 61

Mathematically describe the sticky strike rule

Answer 61

Options with different strikes can have different implied volatilities, implied volatility will be a function of K, not S. For ATM options, we have a linear approximation
Σ(S, K) = Σ₀ - β(K - S₀)
Since we know Δ = Δ_BSM + ∂C/∂Σ ∂Σ/∂S and the sticky strike implied ∂Σ/∂S = 0, the hege ratio is the same as BSM ratio. The ATM implied vol is given by Σ(S,S) = Σ₀ - β(S - S₀), so as the market increases, implied volatility for ATM options decreases

Question 62

Mathematically describe the sticky delta rule

Answer 62

Implied volatility is a function of the BSM delta Δ_BSM. That is, Σ(S, K) = f(ln(K/S)/[Σ(S, K) sqrt(τ)])
It follows that:
Implied volatility is a function of itself, where f is the delta of the option.
This equation is nonlinear and must be solved numerically
We can linearly approximate it as Σ(S, K) = Σ₀ - β(ln(K/S)/[Σ_ATM(S, K) sqrt(τ)])

Question 63

What are two approaches for introducing stochastic volatility?

VS 19

Answer 63

The extended BS approach and the extended Local volatility approach

Question 64

What is the extended BS approach for SV?

Answer 64

• We begin with the GBM model with no implied vol skew
• We allow the underlying GBM model’s volatility to become stochastic
• This introduces volatility of volatility, which is responsible for the smile’s existence
• Hull White is an example of this

Question 65

What is the extended local volatility approach for SV?

Answer 65

• Begin with the local volatility model which already features a smile
• Allow local volatility to become stochastic by introducing a second stochastic factor
• This parameter is responsible for the volatility of the smile
• SABR is an example of this

Question 66

Define the Volga and Vanna under BSM

Answer 66

Volga = ∂²C/∂σ² = ∂Vega/∂σ
Vanna = ∂²C/∂S∂σ = ∂Vega/∂S
Under r = D = 0,
Volga = Vega/σ [ln²(S/K)/(σ²τ) - σ²τ/4]
Vanna = Vega/s [1/2 - 1/(σ²τ) ln(S/K)]

Question 67

How does Volga behave for typical levels of σ and τ

Answer 67

• BSM Volga is positive everywhere except close to ATM when ln(S/K) is close to 0
• Wherever volga is positive, stochastic volatility will increase value of the option above BSM value
• When volatility is stochastic, a hedged option position is long volatility of volatility by an amount related to its convexity in volatility: the volga
• When volatility is stochastic, convexity adds value to the option away from ATM

Question 68

How does Vanna behave for typical levels of σ and τ

Answer 68

• Vanna will be positive when the call is OTM and negative when it’s ITM
• If the stock price and its volatility are positively correlated, then the Vanna will enhance the call value relative to BSM value at high strikes and reduce it for low strikes
• The opposite is true if price and volatility are negatively correlated