Addressing Financial Risks volume
underlyings
products
models
users
regions Over the past 20 years, intense development of Derivatives
in terms of:

To buy or not to buy?:

$ K To buy or not to buy? Call Option: Right to buy stock at T for K
$ K $ K TO BUY NOT TO BUY CALL

Vanilla Options:

Vanilla Options European Call:
Gives the right to buy the underlying at a fixed price (the strike) at some future time (the maturity)
European Put:
Gives the right to sell the underlying at a fixed strike at some maturity

Option prices for one maturity:

Option prices for one maturity

Risk Management:

Risk Management Client has risk exposure Buys a product from a bank to limit its risk Risk Not Enough Too Costly Perfect Hedge Vanilla Hedges Exotic Hedge Client transfers risk to the bank which has the technology to handle it Product fits the risk

Risk Neutral Pricing:

Risk Neutral Pricing

Price as discounted expectation:

Price as discounted expectation Option gives uncertain payoff in the future
Premium: known price today
Resolve the uncertainty by computing expectation: Transfer future into present by discounting

Application to option pricing:

Application to option pricing Risk Neutral Probability Physical Probability

Basic Properties:

Basic Properties Price as a function of payoff is:
- Positive: - Linear: Price = discounted expectation of payoff

Toy Model:

gives 1 in state Option A gives Toy Model 1 period, n possible states in state If , 0 in all other states, where is a discount factor is a probability:

FTAP:

FTAP Fundamental Theorem of Asset Pricing
NA There exists an equivalent martingale measure
2) NA + complete There exists a unique EMM Cone of >0 claims Claims attainable from 0 Separating hyperplanes

Risk Neutrality Paradox:

Risk Neutrality Paradox Risk neutrality: carelessness about uncertainty?
1 A gives either 2 B or .5 B1.25 B
1 B gives either .5 A or 2 A1.25 A
Cannot be RN wrt 2 numeraires with the same probability Sun: 1 Apple = 2 Bananas Rain: 1 Banana = 2 Apples 50% 50%

Stochastic Calculus:

Stochastic Calculus

Modeling Uncertainty:

Modeling Uncertainty Main ingredients for spot modeling
Many small shocks: Brownian Motion (continuous prices)
A few big shocks: Poisson process (jumps)

Brownian Motion:

Brownian Motion 10 100 1000 From discrete to continuous

Stochastic Differential Equations:

Stochastic Differential Equations At the limit:
continuous with independent Gaussian increments
SDE:
drift noise

Ito’s Dilemma:

Ito’s Dilemma Classical calculus:
expand to the first order
Stochastic calculus:
should we expand further?

Ito’s Lemma:

Ito’s Lemma At the limit for f(x), If

Black-Scholes PDE:

Black-Scholes PDE Black-Scholes assumption
Apply Ito’s formula to Call price C(S,t)
Hedged position is riskless, earns interest rate r
Black-Scholes PDE
No drift!

P&L of a delta hedged option:

P&L of a delta hedged option

Black-Scholes Model:

Black-Scholes Model If instantaneous volatility is constant :
Then call prices are given by :
No drift in the formula, only the interest rate r due to the hedging argument.

Pricing methods:

Pricing methods

Pricing methods:

Pricing methods
Analytical formulas
Trees/PDE finite difference
Monte Carlo simulations

Formula via PDE:

Formula via PDE The Black-Scholes PDE is
Reduces to the Heat Equation
With Fourier methods, Black-Scholes equation:

Formula via discounted expectation:

Formula via discounted expectation Risk neutral dynamics
Ito to ln S:
Integrating:
Same formula

Finite difference discretization of PDE:

Finite difference discretization of PDE Black-Scholes PDE
Partial derivatives discretized as

Option pricing with Monte Carlo methods:

Option pricing with Monte Carlo methods An option price is the discounted expectation of its payoff:
Sometimes the expectation cannot be computed analytically:
complex product
complex dynamics
Then the integral has to be computed numerically

Computing expectationsbasic example:

Computing expectations basic example You play with a biased die
You want to compute the likelihood of getting Throw the die 10.000 times
Estimate p( ) by the number of over 10.000 runs

Option pricing = superdie:

Option pricing = superdie Each side of the superdie represents a possible state of the financial market
N final values
in a multi-underlying model
One path
in a path dependent model
Why generating whole paths?
- when the payoff is path dependent
- when the dynamics are complex running a Monte Carlo path simulation

Expectation = Integral:

Expectation = Integral Unit hypercube Gaussian coordinates trajectory Gaussian transform techniques discretisation schemes A point in the hypercube maps to a spot trajectory
therefore

Generating Scenarios:

Generating Scenarios

Low Discrepancy Sequences:

Low Discrepancy Sequences

Hedging:

Hedging

To Hedge or Not To Hedge :

To Hedge or Not To Hedge Daily Position Daily P&L Full P&L Big directional risk Small daily amplitude risk

The Geometry of Hedging:

The Geometry of Hedging Risk measured as
Target X, hedge H
Risk is an L2 norm, with general properties of orthogonal projections
Optimal Hedge:

The Geometry of Hedging:

The Geometry of Hedging

Slide42:

Super-replication Property:
Let us call:
Which implies:

Slide43:

A sight of Cauchy-Schwarz

Volatility:

Volatility

Volatility : some definitions:

Volatility : some definitions
Historical volatility :
annualized standard deviation of the logreturns; measure of uncertainty/activity
Implied volatility :
measure of the option price given by the market

Historical Volatility:

Historical Volatility
Measure of realized moves
annualized SD of logreturns

Historical volatility:

Historical volatility

Implied volatility:

Implied volatility Input of the Black-Scholes formula which makes it fit the market price :

Market Skews:

Market Skews Dominating fact since 1987 crash: strong negative skew on
Equity Markets
Not a general phenomenon
Gold: FX:
We focus on Equity Markets

A Brief History of Volatility:

A Brief History of Volatility

Evolution theory of modeling:

Evolution theory of modeling
constant deterministic stochastic nD

A Brief History of Volatility:

A Brief History of Volatility : Bachelier 1900
: Black-Scholes 1973
: Merton 1973
: Merton 1976

Local Volatility Model:

Local Volatility Model
Dupire 1993, minimal model to fit current volatility surface

The Risk-Neutral Solution:

The Risk-Neutral Solution But if drift imposed (by risk-neutrality), uniqueness of the solution

From simple to complex:

European prices Local volatilities Exotic prices From simple to complex

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