Option Dynamic Replication References: See course outline 1

Option Replication BOPM based on the idea that, since we have two traded assets (underlying stock and risk-free bond) and 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 self-financing risk-less portfolio made up of the stock, the option and the risk- free asset. How?

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:

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 the PV of 4.50 at the risk-free rate, i.e. 4.50e – 0.12  0.25 =

Hedging and Valuation So the risk-less portfolio is worth This portfolio is long 0.25 shares short 1 option The value of the shares is (= 0.25  20 ) By LOOP, the value of the option is therefore (= – )

Self-financing risk-less portfolio Because it is riskless, the portfolio can be made self-financing if we add of borrowings or short a 3-month zero-coupon bond with face value 4.50 = e 0.12  The value of the resulting self-financing replication portfolio is zero by construction and, if LOOP holds, its value remains constant, i.e. zero, regardless of the state of the world at the end of the period. ◦ The debt contracted at time 0 to fund the purchase of  units of stock is repaid, with interest, by selling the shares and using the proceeds from having invested at the risk-free rate the premium received at time t.

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, by LOOP, S 0  – f  = (S 0 u  – f u )e –rT

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

…Proof…

RN Probabilities Notice that, in order to interpret the q and 1 – q thus obtained as RN probabilities, they must be positive. That is, we can derive them using LOOP alone but, to use them to price other derivatives written on the same underlying, we must assume NA.

Multi-period The value of  varies from node to node More in-the-money, more delta.. and vice versa Need to replicate the option at each node of multi-period tree

…Multi-period In every one-period step, we can form risk-less one-period portfolios. As  changes, we rebalance. ….

…Multi-period The sequence of one-period risk-less portfolios results in a dynamically rebalanced multi-period portfolio, that is kept LOCALLY riskless by dynamic hedging of the option component. ◦ If we think of the initial one-period portfolios as self- financing, the dynamically rebalanced portfolio must be, by LOOP, self-financing too. The value of the underlying asset hedge required to ‘keep the portfolio hedged’ changes with Δ and so, as the latter changes, funding must change too. But, under LOOP, the value of the dynamically rebalanced portfolio must remain equal to zero from start to end.

The Two-Step Example (K = 21) A B C D E F

Dynamic Replication and Pricing In multi-period setting, delta-hedging leads to dynamic replication. This nails multi-period option prices down to NA values, by enforcing the LOOP alone. Conversely, 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. ◦ That is, we can take the RN expectation of the final payoffs and discount 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 of typical BOPM).

Problems with Dynamic Replication LOCALLY (i.e., only between adjacent nodes) risk-less 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

The delta of a European call on a stock paying dividends at rate d is N(d 1 )e – dT The delta of a European put is e – dT [N (d 1 ) – 1] Just like in BOPM, a position in Δ units of the underlying locally replicates the option and delta-hedging involves maintaining a delta neutral portfolio Delta and Delta Hedging in B&S

Using Futures for Delta Hedging The delta of a futures contract is e (r-q)T times the delta of a spot contract The position required in futures for delta hedging is therefore e -(r-q)T times the position required in the corresponding spot contract

RN vs Real World Parameters N(d 2 ) is the risk neutral probability of exercise The implied volatility of an option is the risk neutral volatility for which the Black-Scholes price equals the market price ◦ If Black-Scholes held true, this should be also the market expectation of future volatility (under real world probabilities) ◦ The is a one-to-one correspondence between prices and implied volatilities ◦ Traders and brokers often quote implied volatilities rather than dollar prices

Other ‘Greeks’ Consider option with price f Besides delta, there are at least two other key sensitivity parameters are ◦ Gamma (  ), the rate of change of delta (  ) with respect to the price of the underlying asset ◦ Theta (  ), the rate of change of f with respect to the passage of time ◦ But also vega, rho, etc.

Price Time value TV = price - IV Intrinsic value IV = Max(S – K, 0)

Gamma Gamma (  ) is the rate of change of delta (  ) with respect to the price of the underlying asset Curvature  the hedge position must be dynamically rebalanced By no-arbitrage: ◦ Delta hedging a written option must involve a “buy high, sell low” trading rule ◦ Delta hedging a long option must involves a “buy low, sell high” trading rule

Gamma Addresses Delta Hedging Errors Caused By Curvature S C Stock price S’ Call price C’ C’’

Theta Theta (  ) of a is the rate of change of the value with respect to the passage of time,  dS/dt

Interpretation of Gamma Neglecting funding costsgains and second and higher order terms in dt, for a delta neutral portfolio we must have d  »  dt + ½  dS 2 dd dSdS Negative Gamma dd dSdS Positive Gamma

Vega But volatility is not constant Vega ( ) is the rate of change of the value of a derivatives portfolio with respect to volatility

Implied Volatility N(d 2 ) is the risk neutral probability of exercise The implied volatility of an option is the risk neutral volatility for which the Black-Scholes price equals the market price The is a one-to-one correspondence between prices and implied volatilities Traders and brokers often quote implied volatilities rather than dollar prices

Causes of Volatility Changes Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed e.g. French & Roll (1986) For this reason time is usually measured in “trading days” not calendar days when options are valued

Rho Rho is the rate of change of the value of a derivative with respect to the interest rate For currency options there are 2 rho’s

Relationship Between Greeks With ‘plain vanilla’ options: ◦ Positions that are long delta are also long rho (long call, short put). ◦ Positions that are long gamma are also long vega and short theta (long call, put). ◦ Why?

Managing Delta, Gamma, & Vega   can be changed by taking a position in the underlying To adjust  and it is necessary to take a position in an option or other derivative

Hedging in Practice Traders usually ensure that their portfolios are delta-neutral at least once a day Whenever the opportunity arises, they improve gamma and vega As portfolio becomes larger hedging becomes less expensive

Hedging vs Creation of an Option Synthetically When we are hedging we take positions that offset , ,, etc. When we create an option synthetically we take positions that match  & 

Portfolio Insurance In October of 1987 many portfolio managers attempted to create a put option on a portfolio synthetically This involves initially selling enough of the portfolio (or of index futures) to match the  of the put option

Portfolio Insurance continued As the value of the portfolio increases, the  of the put becomes less negative and some of the original portfolio is repurchased As the value of the portfolio decreases, the  of the put becomes more negative and more of the portfolio must be sold

Portfolio Insurance continued Powerful tool... but the strategy did not work well on October 19, 1987 See notes on portfolio insurance