Practical Hedging: From Theory to Practice. OSU Financial Mathematics Seminar May 5, 2008

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1 Practical Hedging: From Theory to Practice OSU Financial Mathematics Seminar May 5, 008

2 Background Dynamic replication is a risk management technique used to mitigate market risk We hope to spend a certain amount of money up-front that will cover uncertain future cash flows Mathematical models are idealized so that replication may be done without error. The real world is not, of course, so well behaved.

3 Review of Terminology Below, the strike K is a specified value and S is the stock price at option expiry Futures contract obligation to buy or sell at price K Put option, but not obligation, to sell at a predetermined price Payoff = Max(0, K S) Call option, but not obligation, to buy at a predetermined price Payoff = Max(0, S K) Long position exchange cash for an asset (buy the asset profit from increases to asset price) Short position receive cash for selling an asset not yet owned (profit from decreases to asset price) 3

4 Market Assumptions Stock price returns are random and normally distributed: ds = µ Sdt + σsdx We may sell the stock short with full use of proceeds. There are no transaction costs or taxes. There are no dividends. We may borrow or lend at a constant riskfree rate, 4

5 More on Market Assumptions Continuous Trading Traditional derivations require We will derive the pricing equation while allowing trading at discrete intervals Volatility Assumptions At first, volatility will be constant and known. Then, we ll allow it to be constant but unknown. Volatility may also be random (beyond scope of this presentation). 5

6 Traditional Black-Scholes Derivation We start with a portfolio with a long option position and a short position in quantity _ of the underlying: Π = V ( S, t) ΔS The change in the portfolio value for a change in S is: dπ = dv ( S, t) ΔdS 6

7 Black-Scholes Derivation (Continued) From Itô s Lemma, V V 1 V dv = dt + ds + σ S t S S dt Thus the portfolio changes by V V 1 V dπ = dt + ds + σ S t S S dt ΔdS 7

8 Black-Scholes Derivation (Continued) We eliminate risk by carefully choosing _: Δ = dv ds The remainder is completely riskless: V 1 V dπ = + σ S t S dt By the no arbitrage principle, dπ = rπdt 8

9 9 Black-Scholes Derivation (Concluded) Substituting, Finally, rearranging and dividing by dt, dt S V S V r dt S V S t V = + 1 σ 0 1 = + + rv S V S S V S t V σ

10 The Accountant s Derivation of Black-Scholes We can derive the same pricing equation in expectation by examining the income generated by the hedge portfolio To illustrate, we will sell a put on 10,000 times the index. S = 1,505 K = 1,500 T = 1 year Sale Price = $818K The sale price corresponds to a risk free rate equal to 5% and implied volatility equal to 0% We then dynamically replicate our position 10

11 Derivation of Black-Scholes with Discrete Time Trading Consider the hedge portfolio We start with zero cash We sell a put option for $818K We sell some stock short (how much?) Any cash left (positive or negative) is put into a risk-free account earning interest This is the same portfolio that we used in the traditional derivation. 11

12 How much stock should we short? Let s suppose that we are given option prices as a function of the stock price and time to maturity. We can take the derivative of the option value with respect to stock price We then take an offsetting position Value of a Put 5,000,000 4,000,000 Current stock Option Value 3,000,000,000,000 1,000, ,000,000 -,000,000 1,060 1,000 1,10 1,180 1,40 1,300 1,360 1,40 1,480 1,540 1,600 1,660 1,70 1,780 1,840 1,900 1,960 Stock Price Market OV Option Delta 1

13 How does this work? We sold the put for $818K but hedge it with a model value of only $540K. A three day simulation is below. Stock Stock Option Shares BOP EOP Net Price Value (100s) Bank Bank Cash Flow 1, , ,558,16 5,559,9 77,709 1, , ,93,65 5,94, , , ,99,90 5,994,110-4,971 1, , ,753,548 5,754,

14 Three reasons why our total wealth changes from today to tomorrow 1. The option price curve changes due to the passage of time. There is an interest payment 3. The stock price changes 14

15 Time Value Change of the Option The option changes in value for the passage of time: Θ δt 15

16 Interest Payment We received Δ S from the sale of stock and V from the option sale. The money in the bank is ΔS V So the interest payment is r( ΔS V ) 16

17 The stock price changes We matched the linear change in option value with the short stock But the change in value of our portfolio is not linear Portfolio losses for instantaneous changes in stock price Portfolio Loss 800, , , , , ,000 00, ,000 0 Stock Price ds? Portfolio 17

18 Non-linear change in option value We apply a Taylor expansion 1 δ V δs δ S This is random. The expected value is 1 δ V σ S δs δt 18

19 Putting it all together Adding all of the cash flows (and ignoring the dt that multiplies all terms), we have dv dt 1 + σ S d ds V + r( ΔS V ) This looks a lot like the Black-Scholes equation except it isn t an equation! On average, its value is zero. 19

20 What does this mean? And where do we go next? Given that stock prices follow this process ds = µ Sdt + σsdx and if we know the volatility, on average the change in value of our portfolio is zero. In theory, if we can trade continuously, we can set up a risk-less portfolio But now, let s assume that we trade only daily and that we don t really know market volatility. 0

21 Delta Hedging Example Let s suppose that we sell an option for $818K (option priced at 0% volatility) and delta hedge assuming that market volatility is 15%. If market volatility is actually 15%, on average, our delta hedge will cost about $540K. If the market volatility really is 15%, we hope to make some money. 1

22 How might our delta hedging work out? If market volatility is 15%, we are almost assured of making money. But it could be 10% (even better!) or 0% (uh oh!) or even 5% (ouch!) Realized Hedge Costs with 15% Hedge Vol Thousands,000 1,800 1,600 1,400 1,00 1, Avg Cost = $75K Std Dev = $95K Avg Cost = $540K Std Dev = $41K Avg Cost = $818K Std Dev = $139K Avg Cost = $1,100K Std Dev = $88K % Realized Vol 15% Realized Vol 0% Realized Vol 5% Realized Vol

23 Now, let s hedge with options We buy put options with strikes slightly higher than our put sold and others with strikes slightly lower The amount of delta hedging is minimized Option Payoffs 5,000,000 4,000,000 3,000,000,000,000 1,000, ,700 1,660 1,60 1,580 1,540 1,500 1,460 1,40 1,380 1,340 1,300 1,60 1,0 1,180 1,140 1,100 Stock Price Put Sold Hedged Position 3

24 How might our option strategy work out? We are relatively immune to the level of realized volatility. The realized hedge cost is in a narrow boundary. Realized Hedge Costs with 15% Hedge Vol,000 Thousands 1,800 1,600 1,400 1,00 1,000 Avg Cost = $814K Std Dev = $13K Avg Cost = $817K Std Dev = $13K Avg Cost = $816K Std Dev = $11K Avg Cost = $817K Std Dev = $16K 0 10% Realized Vol 15% Realized Vol 0% Realized Vol 5% Realized Vol 4

25 Implications on the real world There are rarely true arbitrage opportunities. At best, traders can define ranges of likely market outcomes and look for prices that fall outside those ranges. By hedging with options, traders can narrow hedging results. This gives some insight into why there are bid ask spreads. More exotic options have wider bid ask spreads in part to account for the fact that there are few traded assets with which to hedge. 5

26 How do market participants model? Rich valuation, simple hedging Value positions using a rich model (stochastic volatility, interest rates, etc.) Hedge using a simple model Uncertain parameters Another approach is to not explicitly model all variables but to treat some of them as uncertain We might say, for instance that volatility is likely to be between 10% and 0%. For contracts that can change the sign of gamma, valuation requires finite differences. Volatility assumption in hedging (Forecast or Implied?) 6

27 Summary Dynamic hedging is: Matching the first derivative of the option value with stock Putting the rest of the money into a risk free bank account The total amount invested is the model option value The math of dynamic hedging may be understood by considering the cash flows of the hedge portfolio Dynamic hedging costs are: Random even when we omnisciently know future realized volatility Even more uncertain considering that we don t know what will be realized volatility 7

28 Homework Assignments Given geometric Brownian motion and when trading discretely, what is the distribution of hedge error over a single dt? Assuming realized and implied volatility (both constant, known and not equal to each other), what is the pattern of P&L for hedging with each in pricing equations? How do the distributions of realized hedge costs compare to each other? What tradeoffs and practical considerations are there for deciding which to use? How do the considerations change when we account for the fact that neither realized nor implied volatility is known or constant? 8

29 Questions or comments? Joe Stoutenburg One Nationwide Plaza Columbus, OH 4315 (614)