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Risk Management

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Dario Bianchi
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Dario Bianchi
This flashcard set delves into advanced risk management concepts for university-level finance students. It covers the valuation of options using models like Black-Scholes and binomial trees, focusing on key terms such as payoff, delta-hedging, and replication strategies. The set also explores the assumptions and formulas behind these models, as well as the differences between trading on stock exchanges and over-the-counter markets. It is particularly useful for those looking to understand the mathematical and practical aspects of option pricing and risk management in financial markets.
Cartes-fiches
24
Utilisateurs
2
Langue
Anglais
Catégorie
Finances
Niveau
Université
Créé / Mis à jour
29.12.2013 / 02.03.2015
  • 1  / 24

Cartes-fiches

Why risk management?

reducing systematic risk by changing operations? -> enhancing value, no impact on cashflow Risk management which reduces bankruptcy probability can create value in the amount of the present value of bankruptcy costs. Hedging to reduce the risk of large undiversified shareholders (LUS). Hedging to eliminate fluctuations of value due to forces that are not under management’s control Reduction of information asymmetries; better assessment of management performance Tax arguments: to smoothen income

Two forms of trading

Stock Exchange (Standardized contracts, No credit risk incurred for option buyer, High liquidity, high trading volumes, low transaction costs) Over the Counter (Individualized contracts, Offsetting hardly possible Credit risk for option buyer. Higher transaction costs, lower flexibility.)
Put-Call-Parity

S+P=C+X*e^(-rT)

BT formula for V

V=(q*Vu+(1-q)*Vd)/(e^r_t)

BT formula for q

q=(e^r_t-d)/(u-d)

BT formula for u

e^(sigma*__t)
BT formula for d

1/u

BT formula for _t

T=day/252 _t=T/steps in BT

Delta-Hedge:

Option delta shows the change in the option's price implied by a marginal change in the stockprice. To hedge a Long option position, we have to sell a number of stocks equal to the option's delta.

Delta-Hedge a Long Call

Sell D* Shares
Delta-Hedge a Long Put

Buy |D*| Shares

Delta-Hedge a Short Call

Buy |D*| Shares

Delta-Hedge a Short Put

Sell D* Shares

Replicate a Long Call

Buy D* Shares + Borrowing
Replicate a Long Put

Sell |D*| Shares + Lending

Replicate a Short Call

Sell |D*| Shares + Lending

Replicate a Short Put

Buy D* Shares + Borrowing

Lessons from the Binominal Model

Option valuation with the binomial model is very flexible. The model illustrates the building blocks of derivative valuation: hedging and no-arbitrage (risk-neutral pricing). the binomial model should not be regarded as an accurate description of reality, but as a numerical method for approximating the value of options option values calculated with the binomial model converge to values obtained by continuous time models
The Black-Scholes-Model: Assumptions

Frictionless capital markets, no taxes nor transaction costs. Unrestricted short sales. Stock prices follow a geometric Brownian motion. Riskfree rate is constant and non-stochastic. Stocks do not pay dividends during lifetime of the option. European Option.

BS formula

C=S*N(d1)-X*e^(-rT) *N(d2)

BS formula for d1

(ln(S/X)+r+0.5*sigma^2*T)/(sigma*_T)

BS formula for d2

d1-sigma*_T
Monte Carlo Steps for European Options

1. Simulate 1 path for the stock price in a risk neutral world 2. Calculate the payoff from the stock option 3. Repeat steps 1 and 2 many times to get many sample payoffs 4. Calculate mean payoff 5. Discount mean payoff at risk free rate to get an estimate of the value of the option

Application of MC pros and cons

• MCS can deal with path dependent options, options dependent on several underlying state variables, and options with complex payoffs • It cannot easily deal with American-style options

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