A note on the Black-Scholes implied volatility with default risk

2010 ◽  
Vol 2 (3) ◽  
pp. 155-170 ◽  
Author(s):  
Shuichi Ohsaki ◽  
Takaaki Ozeki ◽  
Yuji Umezawa ◽  
Akira Yamazaki
Keyword(s):  
Default Risk ◽  
Implied Volatility ◽  
Black Scholes

Arbitrage Pricing

2019 ◽  
pp. 95-118
Author(s):  
Tomas Björk
Keyword(s):  
Continuous Time ◽  
Implied Volatility ◽  
Futures Contracts ◽  
Martingale Measures ◽  
Bank Account ◽  
No Arbitrage ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Financial Derivative ◽  
Delta Hedging

The chapter starts with a detailed discussion of the bank account in discrete and continuous time. The Black–Scholes model is then introduced, and using the principle of no arbitrage we study the problem of pricing an arbitrary financial derivative within this model. Using the classical delta hedging approach we derive the Black–Scholes PDE for the pricing problem and using Feynman–Kač we also derive the corresponding risk neutral valuation formula and discuss the connection to martingale measures. Some concrete examples are studied in detail and the Black–Scholes formula is derived. We also discuss forward and futures contracts, and we derive the Black-76 futures option formula. We finally discuss the concepts and roles of historic and implied volatility.

MARKOV MARKET MODEL CONSISTENT WITH CAP SMILE

2000 ◽  
Vol 03 (02) ◽  
pp. 161-181 ◽  
Author(s):  
P. BALLAND ◽  
L. P. HUGHSTON
Keyword(s):  
Normal Distribution ◽  
Implied Volatility ◽  
Market Model ◽  
Interest Rate Models ◽  
Market Models ◽  
Forward Measure ◽  
Black Scholes ◽  
Log Normal ◽  
Bermudan Swaptions ◽  
Log Normal Distribution

New interest rate models have emerged recently in which distributional assumptions are made directly on financial observables. In these "Market Models" the Libor rates have a log-normal distribution in the corresponding forward measure, and caps are priced according to the Black–Scholes formula. These models present two disadvantages. First, Libor rates do not in reality have a log-normal distribution since the implied volatility of a cap depends typically on the strike. Second, these models are difficult to use for pricing derivatives other than caps. In this paper, we extend these models to allow for a broader class of Libor rate distributions. In particular, we construct multi-factor Market Models that are consistent with an initial cap smile surface, and have the useful feature of exhibiting Markovian Libor rates. We show that these Markov Market Models can be used relatively easily to price complex Libor derivatives, such as Bermudan swaptions, captions or flexi-caps, by construction of a tree of Libor rates.

Valuation of Covered Warrant Subject to Default Risk

2003 ◽  
Vol 06 (01) ◽  
pp. 21-44 ◽  
Author(s):  
Shen-Yuan Chen
Keyword(s):  
Default Risk ◽  
Market Price ◽  
Option Price ◽  
Option Value ◽  
Valuation Model ◽  
Black Scholes ◽  
Covered Warrant ◽  
The Difference ◽  
Vulnerable Option ◽  
Settlement Mechanism

There is no margin settlement mechanism for existing covered warrants in Taiwan, thus the credit risk of the warrant issuer must be considered when investors evaluate the price of a covered warrant. This paper applies the vulnerable option valuation model to empirically study the difference in the theoretical value of a vulnerable warrant, Black–Scholes option price and the market price of warrant by using the Taiwan warrant data. Empirical results show that the theoretical value of a vulnerable warrant is lower than the Black–Scholes non-vulnerable option value and its market value.

An Empirical Research on the Informational Efficiency of Model Free Implied Volatility

2008 ◽  
Vol 16 (2) ◽  
pp. 67-94
Author(s):  
Byung Kun Rhee ◽  
Sang Won Hwang
Keyword(s):  
Implied Volatility ◽  
Realized Volatility ◽  
Informational Efficiency ◽  
Option Prices ◽  
Index Options ◽  
Model Free ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Korean Market ◽  
Better Than

Black-Scholes Imolied volatility (8SIV) has a few drawbacks. One is that the model Is not much successful in fitting the option prices. and It Is n야 guaranteed the model is correct one. Second. the usual tradition in using the BSIV is that only at-the-money Options are used. It is well-known that IV's of In-the-money or Qut-of-the-money ootions are much different from those estimated from near-the-money options. In this regard, a new model is confronted with Korean market data. Brittenxmes and Neuberger (2000) derive a formula for volatility which is a function of option prices‘ Since the formula is derived without using any option pricing model. volatility estimated from the formula is called model-tree implied volatillty (MFIV). MFIV overcomes the two drawbacks of BSIV. Jiang and Tian (2005) show that. with the S&P index Options (SPX), MFIV is suoerlor to historical volatility (HV) or BSIV in forecasting the future volatllity. In KOSPI 200 index options, when the forecasting performances are compared, MFIV is better than any other estimated volatilities. The hypothesis that MFIV contains all informations for realized volatility and the other volatilities are redundant is oot rejected in any cases.

A note on Black–Scholes implied volatility

2006 ◽  
Vol 370 (2) ◽  
pp. 681-688 ◽  
Author(s):  
Jesús Chargoy-Corona ◽  
Carlos Ibarra-Valdez
Keyword(s):  
Implied Volatility ◽  
Black Scholes

AN EXPLICIT IMPLIED VOLATILITY FORMULA

2017 ◽  
Vol 20 (07) ◽  
pp. 1750048 ◽  
Author(s):  
DAN STEFANICA ◽  
RADOŠ RADOIČIĆ
Keyword(s):  
Relative Error ◽  
Implied Volatility ◽  
Black Scholes ◽  
Uniformly Bounded

We show that an explicit approximate implied volatility formula can be obtained from a Black–Scholes formula approximation that is 2% accurate. The relative error of the approximate implied volatility is uniformly bounded for options with any moneyness and with arbitrary large or small option maturities and volatilities, including for long dated options and options on highly volatile underlying assets. For options within a large trading range, such as options with maturity less than five years and implied volatility less than 150%, the error of the approximate implied volatility relative to the Black–Scholes implied volatility is less than 10% points.

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