Hedging in the BOPM References: Neftci, Chapter 7 Hull, Chapter 11 1

The Binomial Model (from previous lecture example) A stock price is currently $20 In three months it will be either $22 or $18 Stock Price = $22 Stock Price = $18 Stock price = $20

Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=0.633 The Call Option Option tree:

Option Replication Since we have two traded assets and (by assumption) there are only two states of the world, we should be able to replicate the option payoffs That is, we should be able to form a risk-less portfolio made up of the stock, the option and the risk-free asset How?

Consider the Portfolio:long  shares short 1 call option Portfolio is riskless when 22  – 1 = 18  or  =  – 1 18  Setting Up a Risk-less Portfolio

Valuing the Portfolio (Risk-Free Rate is 12%) The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22 Δ -1 ( = 22  0.25 – 1) = 4.50 Or, 18 Δ ( = 18  0.25) = 4.50 The value of the portfolio today must be PV of 4.50 at risk-free rate, i.e. 4.50e – 0.12  0.25 =

Hedging and Valuation The hedged portfolio is worth This portfolio is long 0.25 shares short 1 option The value of the shares is (= 0.25  20 ) By NA, the value of the option is therefore (= – )

Generalization Consider the portfolio that is long  shares and short 1 derivative The portfolio is riskless when S 0 u  – f u = S 0 d  – f d or S 0 u  – ƒ u S 0 d  – ƒ d ΔS 0 – f

Delta-Hedging Delta (  ) is the ratio of the change in the price of an option to the change in the price of the underlying asset More in-the-money, more delta.. and vice versa Delta is a sort of RN probability of exercise

Delta For a call: Option price A B Slope =  Stock price

Generalization (continued) Value of the delta-hedged portfolio at time T is S 0 u  – f u Value of the portfolio today is (S 0 u  – f u )e –rT Another expression for the portfolio value today is S 0  – f Hence S 0  – f  = (S 0 u  – f u )e –rT

Delta Hedging and RNV Substituting for , (see algebraic steps of proof next slide)

…Proof…

Multi-period The value of  varies from node to node More in-the-money, more delta.. and vice versa Need to replicate option at each node of multi-period tree Dynamic replication using self- financing LOCALLY risk-less portfolios

Dynamic Replication and Pricing In multi-period setting, delta-hedging leads to dynamic replication This nails multi-period option prices down to NA values Given the price of the underlying asset with which the option is dynamically replicated, the NA option price can be obtained using RN valuation of multi-period payoffs ◦ i.e. taking their RN expectation and discounting at the risk-free rate, but first we need to specify how one-period distributions integrate to multi-period ones (e.g., i.i.d. assumption on typical BOPM)

Problems with Dynamic Replication LOCALLY risk-less portfolios does not mean GLOBALLY risk-less What could go wrong? ◦ Hedge cannot be adjusted fast enough (underlying asset moves too fast, e.g. price jumps) or ‘cheaply’ enough (when liquidity “dries out”) There are more risk-factors than we are modeling ◦ e.g., interest rates are stochastic, volatility as well as returns is stochastic, etc. The consequence is a possibly poor replication and hence poor pricing and hedging Interesting and important topic but to be left to more advanced courses

A Word on the Famous Black & Scholes Model B&S is simply the continuous time version of the BOPM with i.i.d. returns at each node, as the length of the time steps between nodes becomes infinitesimally small and thus as the number of nodes tends to infinite In that case, the underlying asset is log- normally distributed