Black Scholes Extension Flashcards
Do B/S assumption hold?
In the Black-Scholes model we assume that log-returns are normally distributed.
* Large returns cluster, so there are periods of high and low volatility [for example after Lehman default]
* Significant outliers, showing jumps (e.g., Black Monday)
* Negative skewness, so tail extends towards the left
* Positive excess kurtosis, indicating fat tails.
Quick stats that tells us that S&P 500 log returns are not normally distributed: Skewness -0.77 (vs. 0 normal dist), Kurtosis 24.93 (vs. 3)
Neg skewness means: tendency more extreme returns to be very negative than very positive
Volatility violation in BS
Volatility is assumed to be constant in the Black-Scholes model
Empirical evidence shows that volatility is in fact time-varying
This is a violation of one of the Black-Scholes assumptions
Mertons Model Volatility
The Merton model of option pricing is similar to that of Black and Scholes but it does not assume constant volatility. It ”only” assumes that volatility is a deterministic functions of time. The option pricing formulas are the same as those of Black and Scholes except that the volatility parameter is defined as the average volatility σ ̄ over the life of an option
Earnings announcements volatility
Earnings announcements are typical events where volatility is known to be higher: time varying but predictable
Volatility is constant on non-announcement days (=σ)
On announcement day, vol =√(σ^2+σ_a^2 ), where σ_a is extra volatility caused by the announcement
Testing announcement uncertainty:
Suppose there is only one scheduled announcement between now and maturity, then according to the Merton’s formula (where T-t is option maturity measured in number of days):
σ ̅_(t,T)^2=σ^2+σ_a^2/(T-t), if t<”announcement date”
σ ̅_(t,T)^2=σ^2,if t≥”announcement date”
Volatility Smile
In practice traders and analysts often observe a “volatility smile” pattern in the implied volatilities of options as they move away from the at-the-money (ATM) position. This means that implied volatilities tend to increase as options become deep in-the-money or deep out-of-the-money.
If the BSM model is description of the world, the implied volatilities of the calls are the same and constant. However, the ATM implied volatility is not constant over time, and the implied volatility is different across the strike prices: OTM put options have higher implied volatility than OTM call options with the same “moneyness”. We observe an IV smirk!
A classic smile shaped volatility means the BSM model is under-pricing OTM options, under-pricing ITM options and over-pricing ATM options, for both calls and puts.
A smile implies that actual OTM options are more expensive than BSM model values (left)
* The (risk-neutral) probability of reaching the tails is higher than the normal distribution
* Fat tails (leptokurtosis)
Volatility Skew
A negative skew implies that option values at low strikes are more expensive than BSM model values (right)
* The (risk-neutral) probability of downward movements is higher than the normal distribu-tion
* Negative skewness in the distribution
What can generate fat tails and skews?
Stochastic volatility
Leverage effect: When stock prices go down, the leverage increase and firms become more risky
Crashophobia: OTM put options provide protection for stock market crashes. People are willing to pay a lot for protection. OTM put prices are higher, meaning higher implied vola-tility
Correlation effects: Stocks become more correlated in down markets
Stochastic volatility model
We can add stochastic volatility to the Black-Scholes model by, e.g., assuming that (dS_t)/S_t =r dt+σ_t dB_t,dσ_t^2=κ(σ_“LR” ^2-σ_t^2 )dt+γσ_t dB_t^σ
Where B_t and B_t^σ are Brownian motions that are correlated with correlation coefficient ρ. This is the Heston model. See Heston (1993) for pricing formulas
Given that volatility is usually high after negative shocks to stock returns, the correlation ρ is likely to be negative. The term κ(σ_“LR” ^2-σ_t^2 ) is positive if σ_t^2<σ_“LR” ^2 and negative if σ_t^2>σ_“LR” ^2. Hence variance will be mean-reverting, i.e., drifting toward σ_“LR” ^2, which is the long-run or unconditional mean of σ_t^2.
The rate at which σ ̅t^2 drifts toward σ“LR” ^2 is determined by κ
Jumps
We can add jumps to the Heston model, which gives us a model with jumps and stochastic vola-tility. We can let the jump intensity be time-varying and allow different distributions for the jump sizes.
Stochastic volatility can produce smiles, skews, term-structure, etc. consistent with observed implied volatility behaviour. However, they are not very pronounced for short-dated options. It seems that to successfully match the data we also need to assume prices can jump
* Many possibilities: Jumps in stock price, jumps in volatility, correlated jumps, time-varying probability of jumps, time-varying average jump size, etc.
When choosing a model, there is a trade-off between parsimony (avoid over-fitting) and explan-atory power. We should examine out-of-sample performance of the model and check accuracy of prices
What is a variance swap?
Assume you think future volatility will be lower than implied from options (but you don’t have a view on the direction of the market).
Using a variance swap you can bet on low future volatility. In a variance swap, the variance buyer receives the realized variance on a stock between the date the contract is entered and a fixed future date in exchange for a fixed payment at that date (the variance swap rate).
Implied vs. realized volatility
On average and in most time periods implied volatility is higher than realized volatility. Selling implied volatility and buying realized volatility in a variance swap is typically profitable. This is due to:
* Volatility is stochastic (risky) therefore it has a risk premium
* Volatility tends to go up in down markets, so negative beta
* Therefore, being long volatility results in a negative risk premium: Buying variance in a vari-ance swap results in negative returns on average
To capture a positive volatility risk premium: sell options, sell variance
What can cause the volatility smile?
Jump Risk and Fat Tails: The underlying asset price might be subject to jumps or extreme changes, which are not captured by the normal distribution used in Black-Scholes.
Stochastic Volatility: Volatility might change over time and depend on the underlying asset’s price level.
Market Crashes and Psychological Factors: In times of market stress, options far from the current price (particularly put options) may be priced higher due to increased demand for hedging against declines, reflecting greater fear of tail risks.
HESTON MODEL
Heston model can capture the empirical features of financial markets, such as the “smile” effect in the implied volatility surface, more accurately than models assuming constant volatility. It adds stochastic volatility to BS by:
dSt / St = rdt + σdB
dσ^2 = κ(σ^2_LR −σ^2)dt + γσdBσ
B and Bσ are corralted with correlation coefficient p. LR = long run volatility
Positve K: if current volatility is higher than long run than whole drift becomes negative. How fast the drift is depends on K.
γ Gamma: volatility has some volatility gamma that depends on itself: implies that when volatility is high that volatility of volatility is also high and vice versa.
Understanding Heston Model
Given that volatility is usually high after negative shocks to stock returns, the correlation ρ is likely to be negative
The term κ(σ2_LR −σ2) is positive if σ2 < σ2_LR and negative if σ2_LR >σ2 ,
and hence variance will be mean-reverting – drifting toward σ2_LR which is the long-run or unconditional mean of σ2
→ The rate at which σ ̄2 drifts toward σ2 is determined by κ
