The document summarizes key aspects of the Black-Scholes model for pricing options. It discusses:
1) How the Black-Scholes model can be derived as the limiting case of the binomial model as the number of time periods approaches infinity.
2) How the Black-Scholes formula provides the same intuition as the binomial model, with the stock price weighted by the hedge ratio and strike price weighted by probability of being in-the-money.
3) Important properties of the Black-Scholes formulas, such as how they relate to forward contracts in the limits of high/low stock prices, approach a maximum value as volatility approaches zero, and can be interpreted using risk-
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Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)Fynn McKay
Simple implementable of Numerical Analysis to solve the famous Black-Scholes P.D.E. via Finite Difference Methods for the fair price of a European option.
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This presentation, given by Antoine Savine, one of Bruno Dupire's original alumni and a lecturer in Volatility at Copenhagen University, celebrates Dupire's most influential contributions to mathematical finance and puts in perspective the history and main results of volatility modeling.
Option Pricing under non constant volatilityEcon 643 Fina.docxjacksnathalie
Option Pricing under non constant volatility
Econ 643: Financial Economics II
Econ 643: Financial Economics II Non constant volatility 1 / 21
Department of Economics
Introduction
Attempts have been made to fix option pricing puzzles: How to be
consistent with volatility smile and smirk.
The Gram-Charlier expansion is one of then but volatility is constant
which is inconsistent with asset return’s dynamics
We review thre approaches that aim at integrating information
embedded in past returns:
GARCH type of approach,
Stochastic volatility models: Hull and White (1987),
Stochastic volatility models: Heston (1993),
Econ 643: Financial Economics II Non constant volatility 2 / 21
The GARCH option pricing
Let St be the asset price at time t and rt = ln(St/St−1) be the log-return
process. Assume that the process rt is a (G)GARCH(1,1) process:
rt = ln
(
St
St−1
)
= µt−1 + σt−1zt, zt ∼ NID(0, 1)
σ2t = ω + α(σt−1zt − θσt−1)2 + βσ2t−1.
(1)
In this model,
µt−1 = E(rt|Jt−1) is a known function of past returns.
Ex: µt = 0, µt = µ = cst, µt = µ + λσt, µt = r + λσt − 12σ
2
t , etc.
σ2t−1 = Var(rt|Jt−1) is the conditional variance of rt given the
information Jt−1 available at t − 1.
Econ 643: Financial Economics II Non constant volatility 3 / 21
GARCH: How to price options on S?
We can rely on the risk-neutral approach:
C = e−rτ E∗ (max(ST − X, 0)) ,
where E∗ is the expectation under risk-neutral dynamics.
What is the risk-neutral dynamics of St if ln(St/St−1) is a
GARCH(1,1)?
Under risk-neutral dyn., E∗
(
St
St−1
)
= er and Var∗(rt|Jt−1) = σ2t−1
(same as under historical measure). Hence, if rt ∼ GARCH(1, 1)
under risk-neutral, the corresponding mean has to be
µ∗t−1 = r −
σ2
t−1
2
. That is:
rt = r −
σ2
t−1
2
+ σt−1z
∗
t , z
∗
t ∼ NID(0, 1)
σ2t = ω + α(σt−1z
∗
t + r −
σ2
t−1
2
− µt−1 − θσt−1)2 + βσ2t−1.
(2)
Econ 643: Financial Economics II Non constant volatility 4 / 21
GARCH: Simulating the option price
To obtain the price C by simulation:
Simulate B paths of stock price using the risk-neutral dynamics (2):
(S
(b)
t+1, S
(b)
t+2, . . . , S
(b)
T
) for b = 1, . . . , B (e.g. B = 5000).
Obtain the simulated price as
Ĉ = e−rτ Ê(max(ST − X, 0)),
with
Ê(max(ST − X, 0)) =
1
B
B
∑
b=1
max(S
(b)
T
− X, 0).
Econ 643: Financial Economics II Non constant volatility 5 / 21
Option pricing under stochastic volatility
GARCH option pricing is convenient but evidence are out that
volatility is more likely stochastic.
Option pricing under SV is quite challenging because of the extra
source of uncertainty brought by the volatility equation.
The induced PDE (by SV) for option pricing can be derived but is
hard to solve.
The most common SV option pricing models are from Hull and White
(1987) and Heston (1993).
Econ 643: Financial Economics II Non constant volatility 6 / 21
Hull and White (1987)
Consider the price process St and its instantaneous variance process
V − t = σ2t obeying the dynamics:
dS.
Black-Scholes Model
Introduction
Key terms
Black Scholes Formula
Black Scholes Calculators
Wiener Process
Stock Pricing Model
Ito’s Lemma
Derivation of Black-Sholes Equation
Solution of Black-Scholes Equation
Maple solution of Black Scholes Equation
Figures
Option Pricing with Transaction costs and Stochastic Volatility
Introduction
Key terms
Stochastic Volatility Model
Quanto Option Pricing Model
Key Terms
Pricing Quantos in Excel
Black-Scholes Equation of Quanto options
Solution of Quanto options Black-Scholes Equation
In this paper we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach we define random market price for each market scenario. The spot price then is interpreted as a one that reflect balance between profit-loss expectations of the market participants
A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.
Fair valuation of participating life insurance contracts with jump riskAlex Kouam
A C++ based program which prices the fair value of a participating life insurance whereby the underlying follows a Kou process and the insurer's default occurs only at contract's maturity.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
. In some papers it have been remarked that derivation of the Black Scholes Equation (BSE) contains mathematical ambiguities. In particular in [2,3 ] there are two problems which can be raise by accepting Black Scholes (BS) pricing concept. One is technical derivation of the BSE and other the pricing definition of the option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
60 Years Birthday, 30 Years of Ground Breaking Innovation: A Tribute to Bruno...Antoine Savine
The RiO 2018 conference in mathematical finance was held in Buzios, Rio de Janeiro, Brazil, 24-28 November 2018, to celebrate the 60th birthday of Bruno Dupire, one of the most influential figures in the history of financial derivatives.
This presentation, given by Antoine Savine, one of Bruno Dupire's original alumni and a lecturer in Volatility at Copenhagen University, celebrates Dupire's most influential contributions to mathematical finance and puts in perspective the history and main results of volatility modeling.
Option Pricing under non constant volatilityEcon 643 Fina.docxjacksnathalie
Option Pricing under non constant volatility
Econ 643: Financial Economics II
Econ 643: Financial Economics II Non constant volatility 1 / 21
Department of Economics
Introduction
Attempts have been made to fix option pricing puzzles: How to be
consistent with volatility smile and smirk.
The Gram-Charlier expansion is one of then but volatility is constant
which is inconsistent with asset return’s dynamics
We review thre approaches that aim at integrating information
embedded in past returns:
GARCH type of approach,
Stochastic volatility models: Hull and White (1987),
Stochastic volatility models: Heston (1993),
Econ 643: Financial Economics II Non constant volatility 2 / 21
The GARCH option pricing
Let St be the asset price at time t and rt = ln(St/St−1) be the log-return
process. Assume that the process rt is a (G)GARCH(1,1) process:
rt = ln
(
St
St−1
)
= µt−1 + σt−1zt, zt ∼ NID(0, 1)
σ2t = ω + α(σt−1zt − θσt−1)2 + βσ2t−1.
(1)
In this model,
µt−1 = E(rt|Jt−1) is a known function of past returns.
Ex: µt = 0, µt = µ = cst, µt = µ + λσt, µt = r + λσt − 12σ
2
t , etc.
σ2t−1 = Var(rt|Jt−1) is the conditional variance of rt given the
information Jt−1 available at t − 1.
Econ 643: Financial Economics II Non constant volatility 3 / 21
GARCH: How to price options on S?
We can rely on the risk-neutral approach:
C = e−rτ E∗ (max(ST − X, 0)) ,
where E∗ is the expectation under risk-neutral dynamics.
What is the risk-neutral dynamics of St if ln(St/St−1) is a
GARCH(1,1)?
Under risk-neutral dyn., E∗
(
St
St−1
)
= er and Var∗(rt|Jt−1) = σ2t−1
(same as under historical measure). Hence, if rt ∼ GARCH(1, 1)
under risk-neutral, the corresponding mean has to be
µ∗t−1 = r −
σ2
t−1
2
. That is:
rt = r −
σ2
t−1
2
+ σt−1z
∗
t , z
∗
t ∼ NID(0, 1)
σ2t = ω + α(σt−1z
∗
t + r −
σ2
t−1
2
− µt−1 − θσt−1)2 + βσ2t−1.
(2)
Econ 643: Financial Economics II Non constant volatility 4 / 21
GARCH: Simulating the option price
To obtain the price C by simulation:
Simulate B paths of stock price using the risk-neutral dynamics (2):
(S
(b)
t+1, S
(b)
t+2, . . . , S
(b)
T
) for b = 1, . . . , B (e.g. B = 5000).
Obtain the simulated price as
Ĉ = e−rτ Ê(max(ST − X, 0)),
with
Ê(max(ST − X, 0)) =
1
B
B
∑
b=1
max(S
(b)
T
− X, 0).
Econ 643: Financial Economics II Non constant volatility 5 / 21
Option pricing under stochastic volatility
GARCH option pricing is convenient but evidence are out that
volatility is more likely stochastic.
Option pricing under SV is quite challenging because of the extra
source of uncertainty brought by the volatility equation.
The induced PDE (by SV) for option pricing can be derived but is
hard to solve.
The most common SV option pricing models are from Hull and White
(1987) and Heston (1993).
Econ 643: Financial Economics II Non constant volatility 6 / 21
Hull and White (1987)
Consider the price process St and its instantaneous variance process
V − t = σ2t obeying the dynamics:
dS.
Black-Scholes Model
Introduction
Key terms
Black Scholes Formula
Black Scholes Calculators
Wiener Process
Stock Pricing Model
Ito’s Lemma
Derivation of Black-Sholes Equation
Solution of Black-Scholes Equation
Maple solution of Black Scholes Equation
Figures
Option Pricing with Transaction costs and Stochastic Volatility
Introduction
Key terms
Stochastic Volatility Model
Quanto Option Pricing Model
Key Terms
Pricing Quantos in Excel
Black-Scholes Equation of Quanto options
Solution of Quanto options Black-Scholes Equation
In this paper we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach we define random market price for each market scenario. The spot price then is interpreted as a one that reflect balance between profit-loss expectations of the market participants
A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.
Fair valuation of participating life insurance contracts with jump riskAlex Kouam
A C++ based program which prices the fair value of a participating life insurance whereby the underlying follows a Kou process and the insurer's default occurs only at contract's maturity.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
. In some papers it have been remarked that derivation of the Black Scholes Equation (BSE) contains mathematical ambiguities. In particular in [2,3 ] there are two problems which can be raise by accepting Black Scholes (BS) pricing concept. One is technical derivation of the BSE and other the pricing definition of the option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
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