Black Scholes Option Pricing Model Ppt – We will describe the price movement of an underlying asset through a continuous model — geometric Brownian motion. We will establish a mathematical model for option pricing (Black-Scholes PDE) and derive the pricing formula (Black-Scholes formula). We will discuss how to manage risky assets using the Black-Scholes formula and hedging techniques.
3 In 1900, Louis Bachelier published his doctoral thesis “Theorie de la Speculation”, a milestone in modern financial theory, Bachelier made the first attempt to represent the movement of stock prices in a random manner. It was also addressed in his thesis.
Black Scholes Option Pricing Model Ppt
4 History – In 1964, Paul Samuelson, winner of the Nobel Prize in Economics, modified the Bachelor model, using returns instead of stock prices in the original model. Let the stock price be there, then give it back. The SDE proposed by P. Samuelson is: This correction removes the unrealistic negative value of the stock price in the original model.
Heath-jarrow-morton Model (hjm)
5 History- Study of Call Option Pricing Problem p. Samuelson (Ć. Sprinkle (1965) and studied at the same time by J. Benes (1964). The result is given below. (V, etc. as before)
6 History — In 1973, Fisher-Black and Myron Schulz gave a lower price formula for call options compared to Samuelson’s, they no longer exist. Instead, risk-free interest enters the formula.
7 History —- The novelty of this formula is that it is independent of the risk preferences of individual investors. This places all investors in a risk-neutral world where the expected return equals the risk-free interest rate. 1997 Nobel Prize in Economics M. Sholes and R. Merton (F. Black posthumously) was credited with this great formula and a series of contributions to option pricing theory based on this formula.
8 Basic Assumptions (a) The price of the underlying asset follows a geometric Brownian motion: μ – expected rate of return (continuous) σ – volatility (continuous) – standard Brownian motion
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(c) The underlying asset pays no dividends (d) No transaction costs and no taxes (e) The market is free of arbitrage
12 Δ-Hedge Technique – If portfolio Π starts at time Π, and remains unchanged at Δ(t, t + dt), then the requirement that Π be risk-free means that the portfolio must return at t + at dt
Since the right-hand side of the equation is risk-free, the coefficient of the random term on the left-hand side must be zero. So, we choose
In turn, we obtain the following PDE: This is the Caro-Scule equation that describes the movement of option prices.
The Black-scholes Option Pricing Model (bsop Model)
16 Note The line segment is also a boundary of the domain. Since the equation breaks down at S = 0, however, according to PDE theory, the boundary value at S = 0 need not be specified.
18 Well-posed problem According to PDE theory, the above Cauchy problem is well-posed. So the real problem is well established.
19 Note that the asset’s expected return μ, a parameter in the underlying asset model, is not represented in the Black-Scholes equation. Instead, R is viewed as the risk-free interest rate. As we saw in the discrete model, according to the Δ— hedging technique, the Black-Scholes equation places investors in a risk-neutral world where price is independent of individual investors’ risk preferences. Thus the option price obtained by solving the Black-Scholes equation is a risk-neutral price.
1) Starting from the value of the discrete option obtained by BTM, by interpolation, we can define a function on the domain Σ=; 2) If a function V(S, t), such that 3) If V(S, t) has a continuous second derivative in Σ, what differential equation does V(S, t) satisfy?
Black-scholes Option Pricing Model — Intro And Call Example
21 Answer V(S, t) satisfies the Black-Scholes equation in Σ. That is, if the option price converges from the BTM to a sufficiently smooth boundary function as Δ 0, then the boundary function is a solution to the Black-Schole equation.
25 Δ-Hedging Use the Δ-hedging technique to model continuous option pricing, and find pricing formulas. Construct a portfolio choose Δ such that Π is risk neutral in [t, t + dt]. is the expected result
Let α and β be the solutions of the following ODE initial value problems: The solutions of the ODE are
P(S, t) – the price of a European put option with the same strike price K and expiration date T. Then the call-put equation is given by where r=r(t) is the risk-free interest rate, q =q(t) is the rate of return, and σ= σ(t) is the volatility.
Derivation Of Bs Model From Binomial
However, the value of the option must be constant t=t_1: V(S(t_1-0), t_1-0)=V(S(t_1+0), t_1+0). Therefore, S and V must satisfy the boundary condition at t=t_1: V(S, t_1-0)=V(S-Q, t_1+0) to establish an option pricing model (e.g. call Take option), consider. Two periods [0, t_1], [t_1, T] separately.
In 0≤ S<∞, t_1 ≤ t ≤ T, V=V(S, t) satisfies the boundary-terminal value problem Find V=V(S, t) on t_1
I 0 ≤ S< ∞, 0 ≤ t ≤ t_1, V=V(S, t) satisfies By solving the above problems, we can find the premium V(S_0, 0) to be paid at the start date t=0 (S_0) to determine the stock price at that time).
44 Note 1– In case of assumption (c~), we used the dividend Q, which is related to the stock price itself. Therefore, on payment day t = t_1, the stock price is therefore t = t_1 as the limit condition for the option price. We should be aware of this difference when solving real problems.
Black Scholes Model: Unveiling The Theoretical Value Of Options
46 Note 2 – As in the derivation we did earlier, choose a risk-free Π in (t, t+dt). Then we get the terminal boundary problem for the arbitrary value V=V(S,t) This equation does not have a closed-form solution in general. A numerical approach is required.
In case: t = T: stock price strike price, the holder receives $1 in cash.
In case: t = T: stock price < strike price, option = 0; In case: t = T: stock price < strike price, the option pays the stock price.
Consider a vanilla call option, CONC and AONC with the same strike price K and the same expiration date T. Their values are shown as V, V_C and V_A respectively. At the expiration date t = T, these values are fulfilled
European Option Pricing For A Stochastic Volatility Lévy Model With Stochastic Interest Rates
And V(S, t), V_A(S, t) and V_C(S, T) satisfy the same Black-Schole equation. Given the line of the terminal problem, so in Σ, ie. For the life of the option, a vanilla call is a combination of AONC in a long position and K times CONC in a short position.
55 Proof Theorem 5.2 Let V_A(S, t) = Su(S, t). It is easy to verify that u(S, t) satisfies: define, then the above equation can be written
Once the CONC value V_C(S, t; r, q) is found, the AONC value V_A(S, t; r, q) can be obtained by Theorem 5.2. To fix the CONC problem, perform the conversion
Then Ori Prob. As the derivation of the Cauchy problem reduced to the Black-Scholes formula, we have
European Option Pricing Black-scholes Formula
2. At t=T_1 buy a call option on a put option; 3. At T = T_1 sell call option; 4. At t=T_1 sell option on put option;
62 Compound Options – Includes three risky assets: the underlying asset (stock), the underlying option (stock option) and the compound option. First, in the domain Σ_2, define the lower option value, which can be given by the Black-Scholes formula, defined as V(S, t).
63 Compound Choices – Then on Σ_1, set up the PDE problem for the compound choice V_(S, t). For this we again use the Δ hedging technique to derive the Black-Schole equation for V_(S, t).
If r, q, σ are constant, we can obtain the value formula for this form of compound option. For example call t=T_1.
Black Scholes Model: Forecasting Options Prices With Financial Simulation Models
70 Preferred Options — To find the value of the preferred option V_(S, t) onΣ_1, we need to solve the following threshold value problem:
71 Preferred Options —- If the following call option V_C and put option V_P have strike price K and expiration date T_2, then according to the call-put equation: then
With today’s computing power, numerical is often preferred, although there are closed-form solutions for costing European options. Especially for complex option pricing problems, such as compound options and selective options, numerical methods have special advantages.
75 questions: a. How to solve Black-Scholes equation by finite difference method (FDM)? b. What is the relationship between BDM and BTM? c. How to prove the convergence of BTM, which is a stochastic algorithm, in the framework of numerical solutions of partial differential equations?
An Investigation Of Higher Order Moments Of Empirical Financial Data And Their Implications To Risk: Heliyon
A differential method is a differential method for boundary value problems by substituting differences for partial differential equations.
77 Types of Methods There are several methods for setting up finite difference equations similar to partial differential equations. Regarding methods of solving equations, there are two main types: 1. Explicit finite difference scheme, whose solution process is obvious and can be solved directly by calculation. 2. Complementary finite difference scheme, which can be solved by solving only one system