In the case of the geometric mean, we derive a closed-form solution for the Asian option. Most modern option pricing is derived from the ideas of the Black-Scholes theory.
Derivative
- Futures/Forward
- Options
- Swap
- Ito’s Lemma
If at the time the option expires (three months from now) the current price of the underlying stock is ST and ST > K, then the option will be exercised. The option is said to be in the money: the asset with value ST can be bought for justK.
Option pricing on stock paying no dividend
Pricing formula for option that pays no dividend
Vega is the sensitivity of the option price to changes in volatility, and theta is the change in the option price due to the passage of time. Thus, the fair price of an option is the value of its self-replicating portfolio.
Black-Scholes Partial Differential Equation (PDE) for options
Portfolio S and B are formed to replicate the behavior of the at=nr derivative of the stock shares. In the above equation we can see that the principle of risk-neutral valuation is clearly satisfied since Black Scholes is independent of μ, the expected rate of increase of the underlying price of the paper.
Option pricing on stock paying dividend
Pricing formula for option on stock paying dividend
In the time mark ∆t the holder of the portfolio earns capital gain equal to the ∆Π and dividend equal to qS∂V∂S∆t, and if ∆Q is the change in the wealth of the portfolio, we have. Since this expression is independent of the Weiner process, the portfolio is immediately risk-free, we have.
Currency option
Pricing formula for a foreign option
St is the unit value of the foreign currency in US dollars, σ is the volatility of the exchange rate and the risk-free interest rate abroad. This is also called S, being the currency price of one unit of the foreign currency.
Futures options
Pricing formula for option on Futures
A PDE for Futures option
Martingale
Option on stock paying no dividend
Proof From the basic properties of Brownian motion, it is clear that they are given by the expression. Then the arbitrage price in the Black-Scholes market of claim X which settles at time T is given by the equality πt(X) = v(St, t), where the function v : R+×[0, T] → R solves the differential equation partial Black-Scholes.
Vanilla option under stochastic volatility
Single-factor stochastic volatility
The stochastic volatility OU, under the risk-neutral probability P∗, can be written in terms of the small parameter. The parameters V2 and V3, calibrated based on the a- and b-term structure of the implied volatility surface, are given by.
Multiscale Stochastic Volatility Models
The derivation of the governing difference equation for arithmetic and geometric averages of Asian options. Consider a portfolio containing one unit Asian option and −∆ units of the underlying asset.
Fixed strike Asian option of a continuous arithmetic averaging
Arithmetic Rate Approximation (Turnbull and Wakeman)
Turnbull & Wakeman's (T&W) approach makes use of the fact that the distribution among arithmetic means is approximately lognormal. It adjusts the mean and the variance to be consistent with exact moments of the arithmetic means. They presented the first and second moments of the mean to price the option.
For averaging options that have already begun their averaging period, then T2 is simply T the original time to expiration, if the averaging period has not yet begun, then T2 is T −τ. Assuming that the averaging period has not yet begun, the first and second moments are given as
Arithmetic Rate Approximation (Levy)
Monte Carlo Simulation
Floating strike Asian option with continuous arithmetic averaging
The above equation cannot be converted into a constant coefficient equation by lognormal transformation of the independent variable. A reduced one-dimensional equation by Rogers and Shi see [9], showing the payoff at maturity. Using the fact that a geometric Brownian motion is a time-varying square Bessel process and the stability due to the additivity of this process, they provide the Laplace transform of the Asian call option.
Bouaziz et al (1996) provided an analytical approximation formula for option value in a more general setting, where the averaging period may cover only a portion of the option's life near the expiration date.
Fixed strike Asian option with continuous geometric averaging
Geometric Closed Form (Kemna and Vorst)
In 1990, Kemna and Vorst proposed a closed pricing solution for geometric mean options by changing the volatility and cost of carrying term. As we said before, that in the limit, the discrete sample means become the continuous sample means. We can now see how Yue-Kuen Kwok in [34] derived the analytical pricing formula for a continuous geometric mean Asian option using the discrete mean geometric mean Asian option formula.
Analytical value of a continuous fixed strike geometric option . 61
In the risk-neutral discounted expectations approach, the price of the discrete geometric mean European fixed strike option is given by By substituting the above bounds into the equation for the discrete geometric mean call option, we can obtain the price of the continuous fixed knitting geometric mean call option. Then the governing equation (1.5) for the price of the above European call, and using the following transformation of variable with finite condition.
The terminal payoff function of the value option with continuous geometric average of the asset price is. By Jamshidian, the governing equation for the above American Asian option is obtained by modifying the above governing equation as. If we integrate this, the integrated representation of the early exercise premium is found to be
Numerical methods
The fractional step method
It is used with the time evolution of the derivative and is affective to avoid numerical diffusion and spurious oscillations. The IDO method is used for the diffusion equation, it is an extension of the CIP method. Although the diffusion method can be solved by the standard finite difference method with accuracy, IDO is used because it can provide the time evolution of the derivatives that the CIP method needs.
First, with the solutions Vcip, VIcip and VScip of the advection equation according to the CIP method with Vtn, VI,tn and VS,tn as the initial value. By then using the solutions of the advection equation as the initial value, one obtains the solutions Vido, VIido and VSido of the diffusion equation according to the IDO method. Since it is an extension of the CIP method, it shows the time evolution of the value and the derivative.
Binomial models
The advantage of the martingale approach is that a lower and upper bound can be obtained. Pis the standard risk-neutral probability measure, where the prices of the basic risky asset (the stock) in the economy evolve according to the stochastic differential equation. The risk-free asset in the economy, the money market account, has dynamics dBt=rBtdt, B(0) = 1.
If we denote the value expressed in the stock price of the Asian American-style floating strike option by V(x(t), t), then the following theorem can be stated. It can be easily proven that using similar economic arguments, but working under the risk-neutral measure P, is an alternative characterization of the early exercise premium. In the stop = region, the value of the option (expressed by the stock price) is simply the intrinsic value of the option.
Floating strike Asian option of a geometric averaging
The value of the American floating geometric call option can also be obtained by choosing ρ= 1. An exact analytical evaluation of the expected expression of the equation of the arithmetic valuation formula (1.2) is not possible because the distribution xA(.) is unknown. Since we are dealing with an American-style option, we need more than just one approximate distribution of the variables at expiration.
And approximating at any time t the remainder of the stochastic process (xA(u))u≥t by the geometric Brownian motion (ˆxA(u))u≥t augmented by the corresponding time-varying coefficients. By approximating the logarithm of the average divided by the logarithm of the current stock price. Then the log of the fraction between these two variables is also normally distributed, the reverse is generally not true.
Floating strike Asian option of an Arithmetic averaging
To be more specific for u > t, consider ln ˆxA(t, u)|=t, which is normal with meanαA(t, u) and variance parameters βA2(t, u). Now the American style based on arithmetic can be deduced, we can now state the following theorem. Also, let ˜CAe(t) be an approximate early exercise value of the call option and ˜PAe(t) be the approximate early exercise price of a put option with the American-style option.
Numerical result To calculate the early strike price in principle, x∗(u) must be known for u∈[t, T], with the boundary condition. When x∗(0) is determined at t= 0, the early strike price can be determined by a final numerical evaluation of the integral in question. In the case of the arithmetic mean, the values were compared with the values of a finite difference method.
Fixed strike Asian option
Fixed strike Asian option of a Geometric Averaging
At point in time, the value of the European average is obtained by calculating the appropriate expectations of the terminal payoffs under the equivalent martingale measure and discounted at the risk-free rate. Since the distribution of the underlying geometric mean is lognormal, the price of European geometric mean option can be readily determined on the geometric mean. It provides the value of the mean option on the geometric mean with continuum of reset points.
It can be interpreted as the Black-Scholes formula for a call, but where the volatility parameter is ν ≡ σ/√. An analytical approach for European options on the arithmetic mean. Under the assumption that the price process is proportional (lognormal), the distribution of the arithmetic mean is the convolution of the finite number of lognormal distributions. Analytical approaches are useful because explicit expressions can be developed for comparative static measurements such as delta, gamma, theta, and vega values.
Fixed strike Asian option of an Arithmetic averaging
If we take µ(du) = δt(du), then we have a classical European call option, and if we take µ( du) = TI(u)du−δT(du) together with K = 0, where δ is the delta function, then we have a floating Asian option priced at time zero. Asian Option PDE Assume that the probability measure µ has density ρt in (0, T) and the maturity of the option T is fixed. The boundary conditions depend on the problem, in the case of the fixed Asian option.
In the case that µ is uniform at [0, T], so is the price of the Asian option with maturity T, fixed strike price K and initial price S0. By Ingersoll, the value of an Asian option is given by the following PDE in two dimensions in terms of the running sum (I), the solution to the above equation is shown as. When K1 = 0 we have a fixed strike Asian option and when K2 = 0 we have the floating strike Asian option.
And the price of an arithmetic average Asian option with multi-scale stochastic volatility is given by. The price of an Asian call at time t is given by. where we denote τ =T−tthe time to maturity, and the updated values ˆK1 = TτK1 and ˆK2 = TτK2 +1τI. For a put call Asian option parity we have.
