scholarly journals MOST-LIKELY-PATH IN ASIAN OPTION PRICING UNDER LOCAL VOLATILITY MODELS

2018 ◽  
Vol 21 (05) ◽  
pp. 1850029 ◽  
Author(s):  
LOUIS-PIERRE ARGUIN ◽  
NIEN-LIN LIU ◽  
TAI-HO WANG
Keyword(s):  
Large Deviation ◽  
Time Window ◽  
Brownian Bridge ◽  
Option Price ◽  
Transition Density ◽  
Small Time ◽  
Option Prices ◽  
Local Volatility ◽  
Volatility Models ◽  
Time Price

This paper addresses the problem of approximating the price of options on discrete and continuous arithmetic averages of the underlying, i.e. discretely and continuously monitored Asian options, in local volatility models. A “path-integral”-type expression for option prices is obtained using a Brownian bridge representation for the transition density between consecutive sampling times and a Laplace asymptotic formula. In the limit where the sampling time window approaches zero, the option price is found to be approximated by a constrained variational problem on paths in time-price space. We refer to the optimizing path as the most-likely path (MLP). An approximation for the implied normal volatility follows accordingly. The small-time asymptotics and the existence of the MLP are also rigorously recovered using large deviation theory.

REPLICATION SCHEME FOR THE PRICING OF EUROPEAN OPTIONS

2021 ◽  
pp. 2150014
Author(s):  
HIDEHARU FUNAHASHI
Keyword(s):  
Diffusion Process ◽  
Density Function ◽  
Numerical Experiments ◽  
Analytical Formula ◽  
Option Price ◽  
Option Prices ◽  
European Options ◽  
European Option ◽  
Volatility Models ◽  
Replication Scheme

This paper proposes an efficient method for calculating European option prices under local, stochastic, and fractional volatility models. Instead of directly calculating the density function of a target underlying asset, we replicate it from a simpler diffusion process with a known analytical solution for the European option. For this purpose, we derive six functions that characterize the density function of a diffusion process, for both the original and simpler processes and match these functions so that the latter mimics the former. Using the analytical formula, we then approximate the option price of the target asset. By comparison with previous works and numerical experiments, we show that the accuracy of our approximation is high, and the calculation is fast enough for practical purposes; hence, it is suitable for calibration purposes.

scholarly journals DUPIRE'S EQUATION FOR BUBBLES

2012 ◽  
Vol 15 (06) ◽  
pp. 1250041 ◽  
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Option Price ◽  
Local Martingale ◽  
Option Prices ◽  
Uniqueness Of Solutions ◽  
Underlying Asset ◽  
Put Options ◽  
Price Process ◽  
Usual Form ◽  
Volatility Models ◽  
Strict Local Martingale

We study Dupire's equation for local volatility models with bubbles, i.e. for models in which the discounted underlying asset follows a strict local martingale. If option prices are given by risk-neutral valuation, then the discounted option price process is a true martingale, and we show that the Dupire equation for call options contains extra terms compared to the usual equation. However, the Dupire equation for put options takes the usual form. Moreover, uniqueness of solutions to the Dupire equation is lost in general, and we show how to single out the option price among all possible solutions. The Dupire equation for models in which the discounted derivative price process is merely a local martingale is also studied.

MONOTONICITY IN THE VOLATILITY OF SINGLE-BARRIER OPTION PRICES

2006 ◽  
Vol 09 (06) ◽  
pp. 987-996 ◽  
Author(s):  
JONATAN ERIKSSON
Keyword(s):  
Option Price ◽  
Rate Of Return ◽  
Barrier Options ◽  
Option Prices ◽  
Barrier Option ◽  
Volatility Models ◽  
Risk Free Rate ◽  
Log Normal ◽  
Free Rate ◽  
Normal Stock

We generalize earlier results on barrier options for puts and calls and log-normal stock processes to general local volatility models and convex contracts. We show that Γ ≥ 0, that Δ has a unique sign and that the option price is increasing with the volatility for convex contracts in the following cases: • If the risk-free rate of return dominates the dividend rate, then it holds for up-and-out options if the contract function is zero at the barrier and for down-and-in options in general. • If the risk-free rate of return is dominated by the dividend rate, then it holds for down-and-out options if the contract function is zero at the barrier and for up-and-in options in general. We apply our results to show that a hedger who misspecifies the volatility using a time-and-level dependent volatility will super-replicate any claim satisfying the above conditions if the misspecified volatility dominates the true (possibly stochastic) volatility almost surely.

scholarly journals OPTIONS WRITTEN ON STOCKS WITH KNOWN DIVIDENDS

2004 ◽  
Vol 07 (07) ◽  
pp. 901-907
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Stock Price ◽  
Option Price ◽  
Option Prices ◽  
Strike Price ◽  
Call Options ◽  
Alternative Approach ◽  
Black Scholes ◽  
Market Practice ◽  
The Difference ◽  
Risk Free Rate

There are two common methods for pricing European call options on a stock with known dividends. The market practice is to use the Black–Scholes formula with the stock price reduced by the present value of the dividends. An alternative approach is to increase the strike price with the dividends compounded to expiry at the risk-free rate. These methods correspond to different stock price models and thus in general give different option prices. In the present paper we generalize these methods to time- and level-dependent volatilities and to arbitrary contract functions. We show, for convex contract functions and under very general conditions on the volatility, that the method which is market practice gives the lower option price. For call options and some other common contracts we find bounds for the difference between the two prices in the case of constant volatility.

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