Hedging Nonlinear Risk

Hedging Nonlinear Risk1Linear and Nonlinear HedgingLinear hedging forwards and futures values are linearly related to the underlying risk factors Because linear combinations of normal random variables are also normally distributed, linear hedging maintains normal distributions

Nonlinear hedging OptionsSince options can be replicated by dynamic trading of the underlying instruments, this also provides insights into the risks of active trading strategies

Bahattin Buyuksahin, Celso Brunetti22Options: NotationSt = current spot price of the asset in dollarsFt = current forward price of the assetK = exercise price of the option (also called strike price)ft = current value of the derivative instrumentrt = domestic risk-free rater*t = foreign risk free rate (also written as y: income produced by the asset)t = annual volatility of the rate of change (returns) in S = T t = time to maturity

For most options: ft = f(St , rt , r*t , t , K , )Bahattin Buyuksahin, Celso Brunetti33Taylor ExpansionOption pricing is about finding f Option hedging uses the partial derivativesRisk management is about combining those with the movements in the risk factors

Remember: the Taylor approximation works only for small movements in the underlying risk factorBahattin Buyuksahin, Celso Brunetti4

4A Simple Binomial Model

A stock price is currently $20In three months it will be either $22 or $18Bahattin Buyuksahin, Celso Brunetti5Stock Price = $22

Stock Price = $18Stock price = $205A Call OptionA 3-month call option on the stock has a strike price of 21. Bahattin Buyuksahin, Celso Brunetti6Stock Price = $22Option Price = $1Stock Price = $18Option Price = $0Stock price = $20Option Price=?6Setting Up a Riskless PortfolioConsider the Portfolio: long position in D shares of the stock and short 1 call option

Portfolio is riskless when 22D 1 = 18D or D = 0.25Bahattin Buyuksahin, Celso Brunetti722D 118D7Valuing the Portfolio(Risk-Free Rate is 12%)The riskless portfolio is: long 0.25 sharesshort 1 call optionThe value of the portfolio in 3 months is 22*0.25 1 = 4.50=18*0.25The value of the portfolio today is 4.5e 0.12*0.25 = 4.3670Bahattin Buyuksahin, Celso Brunetti88Valuing the OptionThe portfolio that is long 0.25 sharesshort 1 option is worth 4.367 todayThe value of the portfolio today will be 20*0.25-f=4.367where f represent option price today.The value of the option is therefore f=5.000 4.367 f=0.633Bahattin Buyuksahin, Celso Brunetti99GeneralizationA derivative lasts for time T and is dependent on a stockBahattin Buyuksahin, Celso Brunetti10uS0 uS0d dS010Generalization(continued)Consider the portfolio that is long D shares and short 1 derivative

The portfolio is riskless when S0uD u = S0d D d orBahattin Buyuksahin, Celso Brunetti11

S0 uD uS0dD dS0 f11Generalization(continued)Value of the portfolio at time T is S0u D uValue of the portfolio today is (S0u D u )erTAnother expression for the portfolio value today is S0D fHence = S0D (S0u D u )erT Bahattin Buyuksahin, Celso Brunetti1212Generalization(continued)Substituting for D we obtain = [ p u + (1 p )d ]erT

where Bahattin Buyuksahin, Celso Brunetti13

13Risk-Neutral Valuation = [ p u + (1 p )d ]e-rTThe variables p and (1 p ) can be interpreted as the risk-neutral probabilities of up and down movementsThe value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rateBahattin Buyuksahin, Celso Brunetti14S0u uS0d dS0p(1 p )14Irrelevance of Stocks Expected ReturnWhen we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevantThis is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevantBahattin Buyuksahin, Celso Brunetti1515Original Example Revisited

Since p is a risk-neutral probability20e0.12 *0.25 = 22p + 18(1 p ); p = 0.6523Alternatively, we can use the formulaBahattin Buyuksahin, Celso Brunetti16

S0u = 22 u = 1S0d = 18 d = 0S0 p(1 p )16Valuing the Option

The value of the option is e0.12*0.25 [0.6523*1 + 0.3477*0] = 0.633Bahattin Buyuksahin, Celso Brunetti17S0u = 22 u = 1S0d = 18 d = 0S00.65230.347717A Two-Step ExampleFigure 11.3, page 242

Each time step is 3 monthsK=21, r=12%Bahattin Buyuksahin, Celso Brunetti1820221824.219.816.218Valuing a Call Option

Value at node B is e0.120.25(0.65233.2 + 0.34770) = 2.0257Value at node A ise0.120.25(0.65232.0257 + 0.34770) = 1.2823

Bahattin Buyuksahin, Celso Brunetti19201.2823221824.23.219.80.016.20.02.02570.0ABCDEF19A Put Option Example

K = 52, time step =1yrr = 5%Bahattin Buyuksahin, Celso Brunetti20504.1923604072048432201.41479.4636ABCDEF20What Happens When an Option is American Bahattin Buyuksahin, Celso Brunetti21505.0894604072048432201.414712.0ABCDEF21DeltaDelta (D) is the ratio of the change in the price of a stock option to the change in the price of the underlying stockThe value of D varies from node to nodeBahattin Buyuksahin, Celso Brunetti2222Choosing u and dOne way of matching the volatility is to set

where s is the volatility and Dt is the length of the time step. This is the approach used by Cox, Ross, and RubinsteinBahattin Buyuksahin, Celso Brunetti23

23The Probability of an Up Move

Bahattin Buyuksahin, Celso Brunetti2424The Black-Scholes-Merton Model25The Stock Price AssumptionConsider a stock whose price is SIn a short period of time of length Dt, the return on the stock is normally distributed:

where m is expected return and s is volatility

Bahattin Buyuksahin, Celso Brunetti2626The Lognormal Property

It follows from this assumption that

Since the logarithm of ST is normal, ST is lognormally distributedBahattin Buyuksahin, Celso Brunetti27

27The Lognormal Distribution Bahattin Buyuksahin, Celso Brunetti28

28Continuously Compounded ReturnIf x is the continuously compounded returnBahattin Buyuksahin, Celso Brunetti29

29The Expected ReturnThe expected value of the stock price is S0emTThe expected return on the stock is m s2/2 not m

This is because

are not the same

Bahattin Buyuksahin, Celso Brunetti3030m and ms2/2Suppose we have daily data for a period of several monthsm is the average of the returns in each day [=E(DS/S)]ms2/2 is the expected return over the whole period covered by the data measured with continuous compounding (or daily compounding, which is almost the same)

Bahattin Buyuksahin, Celso Brunetti3131The Concepts Underlying Black-ScholesThe option price and the stock price depend on the same underlying source of uncertaintyWe can form a portfolio consisting of the stock and the option which eliminates this source of uncertaintyThe portfolio is instantaneously riskless and must instantaneously earn the risk-free rateThis leads to the Black-Scholes differential equationBahattin Buyuksahin, Celso Brunetti3232The Derivation of the Black-Scholes Differential EquationBahattin Buyuksahin, Celso Brunetti33

33The Derivation of the Black-Scholes Differential Equation continuedBahattin Buyuksahin, Celso Brunetti34

34The Derivation of the Black-Scholes Differential Equation continuedBahattin Buyuksahin, Celso Brunetti35

35The Differential EquationAny security whose price is dependent on the stock price satisfies the differential equationThe particular security being valued is determined by the boundary conditions of the differential equationIn a forward contract the boundary condition is = S K when t =T The solution to the equation is = S K er (T t )Bahattin Buyuksahin, Celso Brunetti3636The Black-Scholes Formulas

Bahattin Buyuksahin, Celso Brunetti37

37The N(x) FunctionN(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than xBahattin Buyuksahin, Celso Brunetti3838Properties of Black-Scholes Formula

As S0 becomes very large c tends to S0 Ke-rT and p tends to zero

As S0 becomes very small c tends to zero and p tends to Ke-rT S0 Bahattin Buyuksahin, Celso Brunetti3939Risk-Neutral ValuationThe variable m does not appearin the Black-Scholes equationThe equation is independent of all variables affected by risk preferenceThe solution to the differential equation is therefore the same in a risk-free world as it is in the real worldThis leads to the principle of risk-neutral valuationBahattin Buyuksahin, Celso Brunetti4040Applying Risk-Neutral Valuation

1. Assume that the expected return from the stock price is the risk-free rate2. Calculate the expected payoff from the option3. Discount at the risk-free rateBahattin Buyuksahin, Celso Brunetti4141Valuing a Forward Contract with Risk-Neutral ValuationPayoff is ST KExpected payoff in a risk-neutral world is S0erT KPresent value of expected payoff is e-rT[S0erT K]=S0 Ke-rTBahattin Buyuksahin, Celso Brunetti4242Implied VolatilityThe implied volatility of an option is the volatility for which the Black-Scholes price equals the market priceThere is a one-to-one correspondence between prices and implied volatilitiesTraders and brokers often quote implied volatilities rather than dollar prices Bahattin Buyuksahin, Celso Brunetti4343DeltaDelta (D) is the rate of change of the option price with respect to the underlying Bahattin Buyuksahin, Celso Brunetti44 OptionpriceABSlope = DStock price44Delta HedgingThis involves maintaining a delta neutral portfolioThe delta of a European call on a non-dividend paying stock is N (d 1)The delta of a European put on the stock is N (d 1) 1

Bahattin Buyuksahin, Celso Brunetti4545Delta HedgingcontinuedThe hedge position must be frequently rebalancedDelta hedging a written option involves a buy high, sell low trading rule

Bahattin Buyuksahin, Celso Brunetti4646DeltaKey concept:The delta of an at-the-money call option is close to 0.5. Delta moves to 1 as the call goes deep in-the-money. It moves to 0 as the call goes deep out-of-the-money

Key concept:The delta of an at-the-money put option is close to -0.5. Delta moves to 1 as the put goes deep in-the-money. It moves to 0 as the put goes deep out-of-the-moneyBahattin Buyuksahin, Celso Brunetti47

47

Bahattin Buyuksahin, Celso Brunetti4848ThetaTheta (Q) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of timeThe theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of a long option declines

Bahattin Buyuksahin, Celso Brunetti4949Theta measures the sensitivity of an option to timeUnlike other factors the movement in remaining maturity is perfectly predictable Time is not a risk factor

is generally negative for long positions in both calls and puts the option loses value as time goes byAt-the-money options lose a lot of value when the maturity is near

For American options is always negative shorter-term American options are unambiguously less valuable than longer-term optionsBahattin Buyuksahin, Celso Brunetti50

50Gamma is the second derivative of the option price with respect to the price of the underlying asset

is derivative of with respect to S it measures how stable is is driven by the bell shape of the normal density function is identical for a call and put with identical characteristicsAt-the-money options have the highest gamma moves very fast as S movesIn-the-money options & out-of-the-money options have low gammas because their is constant, close to 1 or 0, respectively As the maturity nears, the option gamma increasesNonlinearities are most pronounced for short term at the money optionsBahattin Buyuksahin, Celso Brunetti51

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Bahattin Buyuksahin, Celso Brunetti5252Gamma for options is similar to the concept of convexity for bonds Bonds usually have positive convexity, whereas options can create positive or negative

Positive convexity or is beneficial the value of the asset drops more slowly and increase more quickly when the market risk factor moves

Negative convexity and can be dangerous because it implies faster price falls and slower price increasesLong positions in options, whether calls or puts, create positive convexity Short positions create negative convexity In exchange for assuming the harmful effect of this negative convexity, option sellers receive the premiumBahattin Buyuksahin, Celso Brunetti5353Gamma Addresses Delta Hedging Errors Caused By Curvature

Bahattin Buyuksahin, Celso Brunetti54SCStock priceS'CallpriceC''C'54Interpretation of GammaFor a delta neutral portfolio, DP Q Dt + GDS 2 Bahattin Buyuksahin, Celso Brunetti55DPDS Negative GammaDPDS Positive Gamma55Relationship Between Delta, Gamma, and ThetaBahattin Buyuksahin, Celso Brunetti56For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q

56VegaVega (n) is the rate of change of the value of a derivatives portfolio with respect to volatilityVega tends to be greatest for options that are close to the moneyBahattin Buyuksahin, Celso Brunetti5757Vega measures the sensitivity of an option to volatilityVolatility is very important in option pricing

has the bell shape of the normal density function (similar to ) is identical for calls and puts > 0 for long position in optionsAt-the money option are most sensitive to volatility decreases with maturityBahattin Buyuksahin, Celso Brunetti58

58Managing Delta, Gamma, & VegaD can be changed by taking a position in the underlyingTo adjust G & n it is necessary to take a position in an option or other derivativeBahattin Buyuksahin, Celso Brunetti5959RhoRho is the rate of change of the value of a derivative with respect to the interest rateBahattin Buyuksahin, Celso Brunetti6060Rho measures the sensitivity of an option to domestic interest rate r

An increase in the rate of interest increases the value of the call the underlying asset grows at a higher rate, which increases the probability of exercising the call in the limit, for an infinite interest rate, the probability of exercise is 1 and the call option is equivalent to the stock itself

The reasoning is opposite for a put option

Bahattin Buyuksahin, Celso Brunetti61

61Rho** measures the sensitivity of an option to dividend yield (or foreign interest rate) r*

An increase in the dividend yield decreases the growth rate of the underlying asset the probability to exercise the call option is lower the value of the call option goes down

The reasoning is opposite for a put option

Bahattin Buyuksahin, Celso Brunetti62

62Hedging in PracticeTraders usually ensure that their portfolios are delta-neutral at least once a dayWhenever the opportunity arises, they improve gamma and vegaAs portfolio becomes larger hedging becomes less expensiveBahattin Buyuksahin, Celso Brunetti6363Scenario AnalysisA scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilitiesBahattin Buyuksahin, Celso Brunetti6464Futures Contract Can Be Used for HedgingThe delta of a futures contract on an asset paying a yield at rate q is e(r-q)T times the delta of a spot contractThe position required in futures for delta hedging is therefore e-(r-q)T times the position required in the corresponding spot contract

Bahattin Buyuksahin, Celso Brunetti6565Hedging vs Creation of an Option SyntheticallyWhen we are hedging we take positions that offset D, G, n, etc.When we create an option synthetically we take positions that match D, G, & nBahattin Buyuksahin, Celso Brunetti6666Portfolio InsuranceIn October of 1987 many portfolio managers attempted to create a put option on a portfolio syntheticallyThis involves initially selling enough of the portfolio (or of index futures) to match the D of the put option

Bahattin Buyuksahin, Celso Brunetti6767Portfolio InsurancecontinuedAs the value of the portfolio increases, the D of the put becomes less negative and some of the original portfolio is repurchasedAs the value of the portfolio decreases, the D of the put becomes more negative and more of the portfolio must be sold

Bahattin Buyuksahin, Celso Brunetti6868Portfolio InsurancecontinuedThe strategy did not work well on October 19, 1987...Bahattin Buyuksahin, Celso Brunetti6969Itos LemmaIf we know the stochastic process followed by x, Itos lemma tells us the stochastic process followed by some function G(x, t)

Derivatives are functions of the underlying asset, x, and time, t

Implication: if we know the process for x (the underlying asset), by applying Itos lemma, we can compute the process for G(x, t)Bahattin Buyuksahin, Celso Brunetti7070Itos LemmaA Taylors series expansion of G(x, t) gives

Bahattin Buyuksahin, Celso Brunetti71

71Itos LemmaBahattin Buyuksahin, Celso Brunetti72

72Itos Lemma

Bahattin Buyuksahin, Celso Brunetti7373Itos Lemma: ApplicationBahattin Buyuksahin, Celso Brunetti74

74Itos Lemma: Option Pricing

Bahattin Buyuksahin, Celso Brunetti7575Option Pricing and the GreeksLets go back to the example where we construct a balanced risk-free portfolio composed by the underlying asset, S, and the option, f

This simplification is extremely important the terms involving dz cancel each other out The portfolio has been immunized against this source of riskAlso the terms in S also cancel each other out the trend of the underlying asset, , does not appear in the Black-Scholes formulaBahattin Buyuksahin, Celso Brunetti76

76Option Pricing and the GreeksThe portfolio = f - S has no risk to avoid arbitrage this portfolio must earn the risk free rate d = [r]dt = r(f - S)dt

if the underlying asset earns a dividend yield, we have d = [r]dt + ySdt = r(f - S)dt + ySdt (13.25)

We also know that d = (0.52S2 + )dt(13.23)By setting (13.23) = (13.25), we have: (r - y)S + 0.52S2 + = rf

This is a the PDE the solution of this equation is the BS formula for call optionsBahattin Buyuksahin, Celso Brunetti7777

Bahattin Buyuksahin, Celso Brunetti7878Table 13.195% confidence level: 1.645dS = -1.645 20% $100 / 252 = -$2.07 dS = -$1.111 (25% of the option value)

dS2 = 2.072 = 4.300.5 dS2 = 0.5 0.039 4.30 = 0.084This is a gain because gamma is positive, but much smaller than the first order effect : 1.5% worse daily movement: -1.645 1.5 = -2.5%

r: annual volatility of change in interest rate of 1%Worse daily loss: -1.645 1/252 = -0.10%

Most of the risk originates from S and in particular from Bahattin Buyuksahin, Celso Brunetti7979Dynamic HedgingThe BS formula is based on a replicating portfolio: a call option is equivalent to holding a fraction () of the underlying asset, S

is the first derivative of the option price with respect to the underlying asset, S is not constant over time to replicate a call option when the price is moving implies changing the fraction () of the underlying asset owned dynamic hedging

As the stock price increases from P1 to P2, the slope of the option curve () increases the option can be replicated by a larger position in the underlying asset When the stock price decreases, the size of the position is cutDynamic hedging buy more of the asset as its price goes up and, conversely, sells it after a fall (the same is also valid for a long position in a put Option)

In contrast, short positions in calls and puts imply the opposite pattern Dynamic hedging implies selling more of the asset after its price has gone upBahattin Buyuksahin, Celso Brunetti8080

Bahattin Buyuksahin, Celso Brunetti8181Interesting QuestionShall we rebalance our portfolio every time the price move?

Be careful!!!Bahattin Buyuksahin, Celso Brunetti8282Distribution of Option PayoffsUnlike linear derivatives such as forwards and futures, payoffs on options are asymmetric This is not necessarily because of the distribution of the underlying factor, which is often symmetric

Long positions in options, whether calls or puts, have positive gamma, positive skewness, or long right tails

Short positions in options are short gamma and hence have negative skewness or long left tails

Bahattin Buyuksahin, Celso Brunetti8383VaR for OptionsAssumption: Normal distribution ( confidence level, = 1.645 for 95% confidence level)

VaR for the underlying asset:VaR = S(dS/S)

Linear VaR for option:VaR1 (dc) = VaR(dS)

Quadratic VaR for option: VaR1 (dc) = VaR(dS)- 0.5VaR(dS)Bahattin Buyuksahin, Celso Brunetti8484VaR for OptionsIn computing VaR for options it is important the nonlinearity of the option payoffs

The degree of nonlinearity depends on the horizon With a VaR horizon of two weeks, the range of possible values for S is quite narrow If S follows a normal distribution, the option value will be approximately normal If the VAR horizon is set at two months, the nonlinearities in the exposure combine with the greater range of price movements create a heavily skewed distribution

So for plain-vanilla options, the linear approximation may be adequate as long as the VaR horizon is kept short For more exotic options, or longer VAR horizons, the risk manager needs to account for nonlinearitiesBahattin Buyuksahin, Celso Brunetti8585Chart20.00000001280.00001980480.00059680520.00448789220.01708737380.04394166240.08803292920.14882729140.22290258660.30522765210.39038981570.47346514120.55049401530.61864394370.67616827780.72225469320.75683035110.78036538310.79369711580.79788456080.79409489250.7835194340.7673147830.74656420860.72225469320.69526558680.66636554980.63621515130.60537310680.5743046580.54339101240.51293908910.48319106170.45433337950.4265050750.39980526840.37429983920.35002728340.32700379770.30522765210.2846829170.26534261540.24717136750.23012759320.21416533180.19923573310.18528826850.17227170420.16013487530.14882729140.13829960360.12850395510.11939423810.11092627120.10305791560.09574913920.08896204110.0826608440.07681186140.07138344680.06634592830.06167153320.05733430580.05331002040.04957609320.04611149210.04289664810.03991336830.03714475040.03457510160.0321898590.02997551540.02791954690.02601034630.02423715840.02259002030.0210597050.01963766820.0183159990.01708737380.01594501270.01488264020.0138944470.01297505590.01211948930.01132313930.01058174020.00989134270.00924828980.00864919540.00809092340.00757056910.00708544180.00663304840.0062110790.00581739250.00545000440.00510707510.00478689840.00448789220.00420858870.00394762630.00370374130.00347576080.00326259610.00306323590.00287674130.002702240.00253892140.00238603220.00224287210.00210879020.00198318090.0018654810.00175516640.00165174950.00155477620.00146382410.00137849930.00129843530.00122329050.00115274650.00108650660.00102429410.0009658510.00091093660.00085932640.00081081080.00076519430.00072229410.00068193990.00064397220.00060824220.00057461090.00054294830.00051313280.00048505070.00045859580.00043366860.00041017610.00038803110.00036715210.00034746260.00032889130.00031137110.00029483930.00027923690.00026450890.00025060350.00023747230.00022506970.00021335310.00020228250.00019182020.0001819310.00017258170.00016374120.00015538030.00014747160.0001399892

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