MF Presentation - Final (Revision II)
17
Extensions to Classic Black- Scholes Asset Pricing Model Ali Nabbi Elliott Bingham Mathematical Finance 15 th March 2016
Transcript of MF Presentation - Final (Revision II)
Extensions to Classic Black-Scholes Asset Pricing Model
Ali Nabbi Elliott Bingham
Mathematical
Finance15th March 2016
Black-Scholes Model Assumptions
No dividend payments.
Options are European.
No transaction costs.
Log of returns are Normally distributed with known and
constant parameters.
The volatility and risk-free rate of the underlying model are
known and constant.
2
Black-Scholes Model with Dividends
Dividends (Jumps in the model) – Predictable/Unpredictable.
Assume dividend yields are deterministic (known function of time – Short horizon). Let be the continuous payout of the underlying commodity:
Where
3
Black-Scholes Model For American Options
AM Option = EU Option + Early Exercise Premium.
Can be found as a boundary solution in a stochastic integral – cf. Kwok K.Y. 1998
Solution to the stochastic integral often unmanageable.
Therefore, analytical approximations used.
where is the critical price above which Call should be exercised and is the critical price below which Put should be exercised.
4
Black-Scholes Model For American Options
Estimation of and :
Where
cf. Efficient Analytic Approximation of American Options - MacMillan L. W. 1986
5
Black-Scholes Model including Transaction Costs
Transaction costs applied each time hedging position is changed.
Introducing transaction costs causes.
□ for in-the-money (ITM) , ,
□ conversely for out-of-the-money (OTM) Volatility skewness.
Replicating portfolio no longer produces the cheapest strategy for hedging.
6
Black-Scholes Model including Transaction Costs
At each time , set an interval for the amount of stock to hold.
If the amount to hold remains inside the interval, don’t change the hedging, else adjust the amount of stock to the closest bound.
What size bounds should be chosen?
No set size as rebalancing hedging portfolio and minimizing hedging error are in conflict.
A trade off must occur and as such there are no strategies that can perfectly replicate a contingent claim in the presence of transaction costs.
cf. Derivative Asset Pricing with Transaction Costs – Bensaid B., Lesne J.P., Pavès H. 1992 On Leland’s Option Hedging Strategy with Transaction Costs – Zhao Y., Ziemba W. 2003
7
Black-Scholes Model - Returns Distributions
Empirical investigation has shown that the returns are not normal distributed.
They show fat tails and large concentration close to the mean – leptokurtosis.
Black-Scholes model with a constant volatility will underprice OTM puts relative to those ATM.
Equivalently, “correctly” priced OTM puts will display higher implied volatility than ATM puts.
Solution: Introduce non-constant volatility.
cf. Some Drawbacks of Black Scholes - Hurvich C. 2000 Option Pricing: A Review - Sundaram R. 2008 8
Black-Scholes Model with non-constant Volatility
Smile effect: plot volatility against moneyness or maturity.
□ Symmetric smile Symmetric distribution.
□ Flat smile low Kurtosis.
Volatility smirk → e.g. High ROI / Market crash.
9
𝜎
Moneyness/MaturityATM
C: ITMP: OTM
C: OTMP: ITM
𝜎
Moneyness/Maturity
Black-Scholes Model with non-constant Volatility
Hull-White option pricing model:
Where two sources of risk have correlation of :
Approximated by:
10
Black-Scholes Model with non-constant Volatility
Trading activities are not equally distributed → Daily reestimation of the option pricing model is meaningless.
Sliding window technique:
□ Parameter estimation with minimization of SPRE.
where
11
Black-Scholes Model with non-constant Volatility
GARCH Option Pricing Model
where
Distribution of can be approximated by Monte-Carlo simulations.
Black-Scholes Model with non-constant Volatility
Hull-White more complex specification and still gives volatility smile.
and but .
Both Black-Scholes and Hull-White biased OTM in short and long term.
GARCH outperforms in terms of median, mean and standard deviation.
Also, flatter error structure across strike price.
For all models, ARPE lower for longer maturities and increased moneyness.
cf. GARCH vs. SV: Option Pricing and Risk Management – Schittenkopf C.
2002
Black-Scholes Model with non-constant Interest Rate
Same approach as in Hull-White model:
Where two sources of risk have correlation of :
cf. Pricing Stock Options with Stochastic Interest Rate - Abudy M. 2011
14
Why are these extensions not included?
Every extension you include increases the complexity of the
model and the parameters of estimation.
Don’t know what will happen in the future!
□ Will volatility / interest rate be constant or varying?
□ Will there be jumps in the model?
Even including these extensions, reality will not be matched
so “better” to use a more simplistic model.
Q&A
Thank You for Your Attention
