Equity Option Pricing & Hedging Flashcards
What are the different types of volatility
QFIQ 114 17
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
How robust is the BSM model? What does loosening assumptions do
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
How do you calculate implied volatility
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
Give 2 reasons for the volatility smile existing
- Smile incorporates kurtosis seen in returns
- Consequence of supply and demand
List 3 models/methods for incorporating smile effect into option prices
- Deterministic volatility surface
- Stochastic volatility
- Jump Diffusion models
What are 4 shapes of volatility smiles and describe features
- 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
How can we make money if our expectation of volatility differs from the market?
## Footnote
QFIQ 115 17
Use a volatility arbitrage strategy. If σactual > σimplied then the market is underpricing options. We can buy them and delta hedge the stock exposure away. We can do the opposite if σactual < σimplied
Describe the mechanics of delta hedging a volatility arbitrage case using σimplied
If we hedge with σimplied we are guaranteed to make a dailty profit, but the amount is random |1/2 (σ2actual - σ2implied) ∫(0,T) e-r(t)S2Γimplieddt|
We balance the fluctuations in the m2m option value with fluctuations in the stock price. This P&L has lower fluctuations than heding under σactual, so it is good if we are constrained to mark-to-market accounting.
We do not need to know σactual
What is the PV of total profit when hedging with another volatility
Hedging with some volatility σh, then the PV of total profits is
V(S, t, σh - V(S, t, σimplied + 1/2 (σ2actual - σ2h ∫(t0, T) e-r(t-t0S2Γhdt * sign of the position
What are the pros and cons of hedging with actual and implied volatilities
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
What is the Delta of a European option
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
How does delta change wrt implied volatility
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
Sensitivity of Gamma wrt underlying price
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
Sensitivity of Vega wrt underlying price (OTM option)
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
How do we see implied volatility not flat across different strikes
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
Compare the shapes of volatility smile and volatility skew
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
What is the sticky delta approach for estimating Σ
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
What is the sticky strike approach for estimating Σ
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
What are three types of movements affecting the volatility skew
Much like the yield curve, we can see parallel shifts, changes in slope, and changes in curvature
How does the delta of a long call spread behave wrt K
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
What are pros and cons of empirical interpolation and functional forms for the smile
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
What are 5 common characteristics of foward based asset volatility curves
- 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
List 4 approaches of volatility modeling
- Forward starting volatility
- Local volatility model
- Stochastic volatility models
- Local-Stochastic volatility model
What are some drawbacks of a stochastic volatility model
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
What is the principle of replication
VS 2
- 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
Name the two kinds of replication
Static and Dynamic
What is static replication
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
What is dynamic replication
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
Name 3 ways to alter or avoid risk
Dilution
Diversification
Hedging
What is the time t value of a Collar
VS 3
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) = St + PL(S, t) - CU(S, t)
By put call parity, this is the same as Collar(t) = PV(L) + CL(S, t) - CU(S, t)
How can we replicate any general payoff using calls and puts?
Suppose the time T payoff of a derivative has initial value I and a series of slopes {λi} that change at various strike prices {Ki}
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) + λ0St + (λ1 - λ0)C(K0) + …
What is the net profit of a delta hedged call over a time step dt (r=0)
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
What are the Vega and Kappa of a call option
VS 4
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
Define a volatility swap contract
- 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 σK
- At maturity, they calculate the realized volatility of the instrument σR
What is the payoff of a volatility swap
For a notional amount N (called notional Vega) the long postion pays out π = N(σR - σK)
Sometimes the payoff for these contracts is expressed in terms of variance π = N(σR2 - σK2).
Here N is the notional variance
What challenges are there for replicating a volatility swap
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
How can we replicate a variance swap
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
What is the volatility of a call under BSM
VS 5
dS = μSSdt + σSSdZ
From Ito’s Lemma, we get
dC = [∂C/∂t + (∂C/∂S)μSS + 1/2 (∂2C/∂S2)(σSS)2]dt + (∂C/∂S)σSSdZ
Extracting the volatility term (dZ) we see σCC = (∂C/∂S)σSS
In other terms, σC = ΔCσSS/C
What is the P&L of hedging a long call using realized volatility under BSM
The PV of total P&L generated over the option life is
PV(P&L(I,R)] = V(S, τ, σR) - V(S, τ, Σ)
What is the P&L of hedging a long call using implied volatility under BSM
The evolution of the P&L over time has no random component
d[P&L(I,I)] = 1/2 ΓIS2(σR2 - σI2)dt = 1/2 ΓIS2(σR2 - Σ2)dt
The PV is given as
PV(P&L(I,I)) = 1/2 ∫(t0, T) e-r(t-t0)ΓIS2(σR2 - Σ2)dt
What is the total Hedging Error when discrete hedging with n time steps
VS 6
Total HE = Σ(i=1,n) 1/2 Γiσi2Si2(Zi2 - 1)dt
The mean HE is 0 because Z ~N(0,1)
The variance of this is σHE2 = E[Σ(i=1,n) 1/2 (iSi2)2(σi2dt)2]
For an ATM option, we can express the volatility of Hedging error as
σHE ~ σ/sqrt(n) ∂C/∂σ
What are some insights about Hedging Error over n time steps
- 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 (ΔI - ΔR)dS
What are some insights about Hedging Error when you use implied volatility
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
What is the impact of transaction costs on hedging
VS 7
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
Describe the shape of implied volatility for Equity indexes and why we see it
VS 8
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
What are some major characteristics of the Equity Index skew
- 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.
Describe the shape of implied volatility for FX and why we see it
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.
What bounds are placed on implied volatility
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
How can we approximate the bounds on implied volatility
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
What are 3 approaches to capturing the implied volatility observed in the market?
VS 10
- Move away from traditional GBM assumption for asset prices
- Model movements of the BSM implied volatility surface Σ(S, t, K, T) rather than the movements in the volatility itself
- Avoid formal models of either underlying or impled volatility and use heuristics for pricing and hedging
What are 4 ways to move away from traditional GBM assets to capture volatility?
- Local Volatility models
- Constant elasticity of variance (CEV) models
- Stochastic volatility models
- Jump Diffusion models
What are stock price dynamics under LV, CEV, SV?
Local Volatility: dS/S = μ(S, t)dt + σ(S, t)dZ
CEV: dS/S = μ(S, t)dt + σSβ-1dZ
SV: dS/S = μdt + σdZ and dσ = pσdt + qσdW; E(dWdZ) = ρdt
What is the continuous state valuation of an Option
VS 11
Let p(S, t, ST, T) = RN PDF of ST. Then an option value V(S, t) = e-r(T-t) ∫(0, inf) p(S, t, ST, T)V(ST, T) dST
How do we derive the Breeden-Litzenberger formula
Using the cts valuation of a call, C = e-r(T-t) ∫(K, inf) p(S, t, ST, T)(ST - K) dST
Taking 2 partial derivatives and solving for p(S, t, ST, T), we find
p = er(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
What are some characteristics/consequences of the Breeden-Litzenberger formula
- 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!
Describe the Local Volatility model
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
What is the relationship of implied volatility under Local Volatility
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
What is the rule of two
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
What are some pros and cons of the Local Volatility model
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
How do different market regimes of volatility and market direction
dπBSM = 1/2 S2Γloc (σR2 - σloc2(S, t))dt - εdS
High vol, Up mrkt: σR > σloc, dS > 0
High vol, Down mrkt: σR > σloc, dS < 0
Low vol, Up mrkt: σR < σloc, dS > 0
Low vol, Down mrkt: σR < σloc, dS < 0
Mathematically describe the sticky strike rule
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 = ΔBSM + ∂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
Mathematically describe the sticky delta rule
Implied volatility is a function of the BSM delta ΔBSM. 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)/[ΣATM(S, K) sqrt(τ)])
What are two approaches for introducing stochastic volatility?
VS 19
The extended BS approach and the extended Local volatility approach
What is the extended BS approach for SV?
- 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
What is the extended local volatility approach for SV?
- 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
Define the Volga and Vanna under BSM
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)]
How does Volga behave for typical levels of σ and τ
- 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
How does Vanna behave for typical levels of σ and τ
- 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
