Pricing and Hedging Path-Dependent Options Under the CEV Process
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Citations
Pricing European vanilla options under a jump-to-default threshold diffusion model
Optimal Investment Strategy for Merton's Portfolio Optimization Problem under a CEV Model
Pricing of Double Barrier Options by Spectral Theory
An Analytical Approximation for Pricing VWAP Options
Equity-linked security pricing and Greeks at arbitrary intermediate times using Brownian bridge
References
The Pricing of Options and Corporate Liabilities
Theory of rational option pricing
Brownian Motion and Stochastic Calculus
Continuous martingales and Brownian motion
Related Papers (5)
Frequently Asked Questions (11)
Q2. What are the future works mentioned in the paper "Pricing and hedging path-dependent options under the cev process" ?
The CEV model with negative elasticity exhibits convex and monotonically decreasing implied volatility smiles similar to the smiles empirically observed in the stock index options market and, thus, allows us to study the effect of volatility smiles on pricing and hedging of path-dependent options. Parameters used in the calculation: 0 = 100 = 0 25, r = 0 1, q = 0, T = 0 5. Second, the authors apply the analytical formulae to carry out a comparative statics analysis and demonstrate that, in the presence of a CEV-based volatility smile, barrier and lookback option prices and their hedge ratios can deviate dramatically from the lognormal values. Strikingly, the authors show that their deltas can have different signs under lognormal and CEV specifications, as well as CEV specifications with different elasticities. The authors find that it is much more important to have the accurate model specification for hedging barrier options that are canceled in-the-money than for standard options and barrier options that are canceled out-of-the-money.
Q3. What is the salient feature of the results?
13The salient feature of the results is that the value of 1 has a much greater impact on the prices of upand-out and double knock-out calls than on standard, capped and down-and-out calls.
Q4. What is the implied volatility of the S&P 500?
The Black-Scholes implied volatility of CEV calls exhibits a typical downward sloping volatility smile pattern (also called smirk, skew or frown), with higher implied volatilities corresponding to lower strikes (in-the-money calls) and lower implied volatilities corresponding to higher strikes (out-of-themoney calls).
Q5. What is the effect of volatility smiles on the price of options?
The CEV model with negative elasticity exhibits convex and monotonically decreasing implied volatility smiles similar to the smiles empirically observed in the stock index options market and, thus, allows us to study the effect of volatility smiles on pricing and hedging of path-dependent options.
Q6. What is the value of cash rebates included in finitely lived knock-out options?
To value cash rebates included in finitely lived knock-out option contracts with expiration T < , the authors need to value claims that pay one dollar at times = L, U , or L U , given ≤ T .
Q7. What is the probability of absorption at zero?
For 1< 0, the risk-neutral probability of absorption at zero (bankruptcy), given S0 = S, is (Cox 1975) QS ST = 0 =G 3 4/2 where G 3 x is the complementary Gamma distribution function and 4 is defined in Equation (33) below.
Q8. What is the effect of the negative elasticity values of stock index options?
The negative elasticity values are characteristic of stock index options (Reiner 1994 and Jackwerth and Rubinstein 1998 find that the prices of S&P 500 index options imply values of beta as low as 1 = −4).
Q9. What is the effect of knockout on the price of lookback options?
A slight error in estimating knockout probability can result in large pricing and hedging errors as it is multiplied by the large dollar value.
Q10. How much volatility does the S&P 500 stock index have?
The mean of their daily estimates over the post-crash 1988–1994 period is close to −4, with the mean at-the-money implied volatility of 17%.
Q11. what is the constant part of the risk-neutral density of the bessel process?
The continuous part of the risk-neutral density of ST , conditional on S0 = S, is obtained from the well known expression for transition density of the Bessel process (see Borodin and Salminen 1996, p. 115, and Revuz and Yor 1999, p. 446) and is given by (3 = 1/ 2 1 )p0 T S ST = S −21− 32 T S 1 2 02 1 T exp ( −S −21+S−21T 20212T )×