Derivatives - Option Valuation

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Option Valuation

Agenda

Lattice methods of valuation

Binomial model

Black Scholes Model

Extensions to Black Scholes

Why Valuation Models?

Pricing / Trading Only know terminal payoff value

Need to know interim value

Risk Management

A model will tell us how sensitive a

derivative is to changes in market factors

Option Pricing

Option payoff depends on stock price Need to model stock prices

Binomial models

Continuous time models

Option values depend on stock volatility

Binomial Model

Looks at stock price movement oversmall periods of time ∆t

Start with one period, ∆t = 1 It turns out quite easy to determine option

price in this case

Then many periods Ultimately we can model infinite number of

up/down movements

This is continuous time

Binomial Price Tree

Usual to assume u > 1 > d

One-Period Model Example

S = 100, u = 2, d = 0.5 Su = 200

Sd = 50

Interest rate: 10% Bond price 100 today, 110 in one period

Suppose call strike price is 100 What is price of call option?

One-Period Model

At Su Stock worth 200

Option pays 100

Bond worth 110

Stock worth 50 Option worth 0

Bond worth 110

One-Period Model

Up Down

Stock 200 50

Bond 110 110Call 100 0

But look at a portfolio of 0.6667 stocks and -0.30303bonds: produces the same payoff as the call!

UP: (0.6667*200) - (0.30303*110) = 100

Down:(0.6667*50) - (0.30303*110) = 0

Replication The payoff of the call can be replicated

Using the stock and the bond, combined in a portfolio

The call and the portfolio must be worth the same Because they have the same payoff

So we know the value of the call! Because we know the value of the replicating portfolio

In this case: C = (100*0.6667) - (0.30303*100) = $36.37

This tells us 2 things: How to price the call

How to “create” the call if it didn’t exist

Replication - General Formula

In general, need to solve: Cu = max{Su-X, 0} = nSu + m B(1 + rf )

Cd =max{Sd-X,0} = nSd + m B(1 + rf ) B(1 + rf ) = value of bond at end of period

n = # stocks, m = #bonds

Solution: n = (Cu-Cd)/(Su-Sd) is also called the delta of the option, will see it again

m = [SuCd

- SdCu]

/[B(1 + rf

) (Su-Sd)]

Risk Neutral Valuation

Define π = (1 + rf ) - du - d

ThenS = πSu + (1- π)Sd

(1 + rf )

C = πCu + (1- π)Cd

(1 + rf )

Risk Neutral Probability

Interpretation: π is the risk-neutral probability

S is the NPV of ‘expected’ future stock cash flows

C is the NPV of ‘expected’ future option cash flows

Risk-Neutral:

Expected future cash flows are discounted usingthe risk-free rate: DF = 1 / (1 + rf )

This is true in a risk-neutral world, where there is nopremium for risk

Two Period Model

Extension of replication idea:

Must replicatedynamically

Work backwards

Solve asequence of 1-period problems

Delta Hedging

We know how to value options Next: how to manage risk

Option Value & Stock Prices

Option value changes as stock moves: Up move: call value increases, put value falls

Down move: put value increases, call value falls

Delta measures sensitivity of option price tostock movements:

Delta = (Change in option price)(Change in stock price)

Rebalancing

Option deltas change as stock price moves So, position deltas change too

Need to rebalance portfolio periodically to

keep it delta neutral Example: Long 10 calls, delta = 5

Hedge this position by selling 5 stocks

Now stock move up, and call delta increases to 0.6

Position delta is (10x0.6) - 5 = 1

Need to sell another stock to rebalance to deltaneutral

Delta Hedging: General Case

Idea: Replicate an option positionusing stock and bonds

Suppose we have 1 call and sell δ stock

Need to solve:

- Su + Cu = Br

- Sd + Cd = Br

Then δ = (Cu - Cd) / (Su - Sd)

The Black-Scholes Model

Can be viewed as limit of the binomial model What happens as the time interval between

up/down movements goes to zero.

Take a time period t, divide it into n intervals

Let σ = volatility of the stock

u = exp{σ√t/n}, d = 1/u

This value ensures process has correct volatility σ

Black-Scholes Model In limit, stock price process has the following

properties:

a) returns are normally distributed

b) returns over different periods are independent c) stock prices over any interval are log-normally

distributed

Formally, can be represented by a continuoustime stochastic process

Continuous Time

Random Walk Model

Stochastic Differential Equation

S is stock price

µ is drift factor

σ is stock volatility

X is Weiner Process X = ε(δt)1/2

ε is standardized Normal random variate

dX dt S

dSσ µ +=

Black-Scholes Equation

Consider delta-hedged option portfolio Must grow at risk free rate, else arbitrage

Leads to following relationship:

DeltaGamma

Black-Scholes Option Pricing

Solution of Black-Scholes equation:

t r X S

d SN d N XeP

d N Xed SN C

√−=

−−−=

−=−

)2 / () / ln(

Properties of the Model

Call prices increase with S, r Put prices decrease S, r

Both prices increase with volatility Value depends on time to maturity

Called time decay

Application

IBM Options S = 107 1/2, strike = 105, Call option

Expiration:

Sat after third Fri = July 20

0.137 years

Int rate: use 6.15% Volatility 30%

Option Calculator – Pricing

Implied Volatility

Use quoted option price to back out volatility Using Newton-Raphson or other iterative method

Volatility estimate implicit in option price

Assuming Black-Scholes model used

Market’s estimate of future volatility Over the life of the option

May be higher or lower than historical volatility

Option “prices” often quoted in terms of implied

volatility

Application: Implied Volatility

Sun Microsystems Options S = 104, strike = 95

Option price $11.5

Expiration (Aug options): 0.08 years

Int rate: 6.15%

Implied Volatility: 53.29%

Option Calculator - Implied Vol

Black-Scholes Assumptions

Returns are normally distributed

In reality return distributions are non-Normal

Can modify model to accommodate other forms of

distribution (Student T, Pareto-Levy) Zero transaction costs

Bid-Offer spreads and other costs can be factored in

Volatility is static In fact volatility is stochastic

Use GARCH & other stochastic volatility models

Limitations of Black Scholes

Stocks Cannot handle dividends

American options

Options on Futures Assumes underlying is deliverable today

Foreign exchange

Two assets both paying “dividend” Domestic and foreign risk free rate

Bonds Bonds prices are constrained, unlike stocks

Validity of Black-Scholes Model

Can be extended quite easily Asset classes

Dividends

Foreign Exchange

Futures

More realistic assumptions E.g. stochastic volatility

Vanilla model is a benchmark May not use it to value options, but use for

reference quotes

Extensions of Black Scholes

Index Options Options on e.g. the S&P500

European options, but with dividends

Merton’s model

Extension of Black Scholes

Underlying asset pays continuous dividends

Good approximation for S&P500

not as good for smaller indices

Merton’s Model

q: dividend yield

Futures: q = r

Currencies q = foreign interest rate

t t qr X Sd

d N Sed N XeP

d N Xed N SeC

+−+=

−−−=

−−

)2() / ln(

Application to Index Options

S&P500 Index 1368.36 Yield: 2%

June options (mature June 22) t = 0.06

Risk free rate: 6.15%

1375 call priced at 45.25

Option trading at implied vol. of 34.46%

Option Calculator:S&P500 Index Options

Foreign Exchange Options

Garman-Kohlhagen model Identical to Merton’s model

q is foreign risk free rate

Holding cost

h = r - q

Black’s Model Simple extension of Black-Scholes

Originally developed for commodity futures Used to value caps and floors

Let F = forward price, X = strike price Value of call option:

t X F d

d XN d FN eC rt

√−=

−= −

)2 / () / ln(

)]()([

Application to Caps

Example: 1-year cap NP = notional principal

R j = reference rate at reset period j

R x = strike rate Then, get NP x Max{R j - R x,0} in arrears

But this is an option on R j, not F j Use F j as an estimator of R j and apply Black’s

model to F j Previously was a forward price, now a forward rate

Black’s Model for Caps Payments: NP x Max{R

,0} in arrears

These are a series of options:

One for each R j , the future spot interest rate

Called caplets Let F j = forward rate from j to j+1

Value of caplet j:

Discount by (1+ F j) as paid in arrears

C = NP x e-rt[F jN(d1) - R xN(d2)] / (1 + F j)

Black’s Model - Example

8% cap on 3-m LIBOR (R x = Strike = 8%) Capped for period of 3m, in 1-year’s time

f = 1-year forward rate for 3m LIBOR is 7%

R f = 1-year spot rate is 6.5%

Yield volatility is 20% pa

See Excel workbook Black’s Model - Example Spreadsheet

Black’s Model - Example

C = BSOpt (8%, 1, 0, 7%, 20%, 6.5%, 0, 0, 0)

Holding Cost

Hcost = (R f -d) for stocks, 0 for Futures

Cap Premium = 0.00211

Convert to %: C% = C x t / (1 + F * t)

0.00211 x 0.25 x 1 / (1 + 7% x 0.25)

Cap Premium % = 0.0518% (5.18bp)

So cost of capping $1000,000 loan would be $518

Strike Term Fwd

VolHCostC/PRf

Bl k’ M d l E i l t

Black’s Model - Equivalent

Formulation in Terms of Price

Cap = Put option on price Equivalent of call option on rate

Useful if know price volatility rather than yield vol.

F = 1 / (1 + f x t) is forward price F = 1 / (1 + 7% x 0.25) = 0.982801

X = 1/(1 + R x x t) is strike price X = 1 / (1 + 8% x 0.25) = 0.980392

Require price volatility Other parameters as before

Black’s Model - Price Example

C = BSOpt (.980392, 1, 0, 0.982801, 0.3702%, 6.5%, 1, 0, 0)

Cap Premium % = 0.0518% (5.18bp) So cost of capping $1000,000 loan would be $518

Premium already expressed as % of FV This time we are price a put option

Strike TermFwd

C/PRf E/A

Limitations of Black’s Model

Problems: Unbiasedness: empirically false

Option on R j not same as option on F j Discount rate: fixed - but F j variable

Rates both stochastic and fixed!

If applied to prices the additional problem Assumes prices can be any positive number

But can’t exceed value of future cash flows

Time Dependent Volatility

Black-Scholes still valid if σ is functionof time

Instead of σ, use: σ(t) =

Fit σ(t) to implied volatilities of options

of varying maturities

τ τ σ

Stochastic Volatility dσ = p(S, σ, t)dt + q(S, σ, t)dX

X is a Weiner process

Example: GARCH(1,1) model

{et}is a white noise process

Black-Scholes differential equation includesextra terms:

λ(S, s, t) is market price of volatility risk

1 t t t αε βσ ω σ ++=+

∂∂− V

q p )(

Volatility Smiles & Surfaces Implied volatilities of options with

different strikes varies Inconsistent with Black-Scholes

Implies volatilities of OTM options typicallygreater than ATM options

Smile: Plot IV vs. Strike Shows “smile” effect

Surface: Plot IV vs. Strike & Maturity

Volatility Smile – Example

Volatility Surface – Example

Lab: YAHOO Construct implied volatility smile &

surface

Solution: YAHOO

11 5 0.07

Solution: YAHOO

Non-Normal Models Significant skewness and excess

kurtosis in returns

Increases with sampling frequency

Skewness 0 for Normal distribution

Kurtosis

3 for Normal distribution

33 / ])[( σ µ τ −= X E

44 / ])[( σ µ κ −= X E

MOT 5-Minute ReturnsMOT 5-Minute Returns

-2 . 0 0 %

-1 . 0 0 %

0 . 0 0 %

1 . 0 0 %

2 . 0 0 %

Kurtosis= 27.5

Impact on Options

Option Valuation

ATM / OTM options undervalued

Volatility Curve

Volatility curve smiles & skews

Risk Measurment

Option Greeks

Modeling Non-Normality

Extreme value distributions

Models maximum values

Normal mixture models

Simulate market jumps

Extreme Value Distributions Extreme value M

n Mn = Max(X1, . . . , Xn)

Standardized: Y n = (Mn – µn) / σn

Generalized Extreme Value Distn

{ }{ }⎩⎨⎧ >+≠+− =−= −

0)1(,0)1(exp0exp)(

/ 1 yif y

if e yF

ξ ξ ξ ξ

Extreme Value Distributions Gumbel: tail index ξ = 0

Positive skew

Exponential tails

Normal or Lognormal returns Weibull: tail index ξ = <0

Uniform density in returns

Frechet: tail index ξ = >0 Returns generated by GARCH, Student-t or stable

Pareto

Gumbel DensityGumbel

Normal Mixtures Weighted sum of two Normal processes

Low volatility

High volatility (“jumps”)

)()1()()( 21 y y yg φ ρ ρφ −+=

Moments of Normal Mixture Mean

Weighted sum of means (zero)

Variance - weighted sum of variances

Skewness – zero

Kurtosis { }224 / 3 ∑∑ iiii σ ρ σ ρ

∑= 2ii M σ ρ σ

Selecting Appropriate Mixtures Three parameters

ρ, σ1 and σ2

Need three equations to solve

Match variance, kurtosis and sixth momentof empirical distribution

Applications Option Pricing

Using MCS

Value at Risk

Improves measure of tail risk

Option Pricing with Kurtosis Kurtosis will affect option value if present

over life of option Hence typically more relevant to shorter dated

option

Option value is weighted sum of price undereach density in mixture

f(σi) is option price assuming normal distributionwith volatility σi

NB: NOT price at volatility of mixture distribution

)(...)( 11 nn f f P σ ρ σ ρ ++=

Option Pricing with

Normal Mixture 45 day Call option

$105 strike, $100 stock

Normal mixture

Vol1 = 15%

Vol2 = 30%

Probability of jump = 0.5 “Expected” vol 22.5%

Normal Mixture Example

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

80 82 84 86 88 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118

Strike

Jump Diffusion Brownian Motion with jumps

Poisson process

dq = 0 with probability (1-λ)dt dq = 1 with probability λdt

λ is the inter-arrival time of jumps

dq J dX dt

dS)1( −++= σ µ

Jump Diffusion ExampleJump Diffusion

Ito with Jump Diffusion Hold portfolio

Option and –∆ of stock

Π =V(S,t) - ∆ S

If dq = 0 then ∆ = δ V/δS eliminates risk

If there is a jump the portfolio changes byan amount that cannot be hedged away

dqS J t SV t JSV dSS

V d ))1(),(),((

222 −∆−−+⎟

⎞⎜⎝

⎛ ∆−∂∂

+⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ∂∂

+∂∂

=Π σ

Pricing with Jump Diffusion Merton (1976):

if jump component is uncorrelated then jump risk should not be priced in

Since diversifiable Option value is weighted sum of n Black-

Scholes values

Weights are probability of n jumps

),;,()(

r t SV t e

σ λ λ ′′∞

∑ ])1[1( −+=′ J E λ λ

Jump Diffusion Option Pricing

Jump Diffusion vs Black-Scholes

60 70 80 90 100 110 120 130 140

Summary: Option Valuation Binomial

Option replication Black-Scholes

Limiting case of binomial

Simple extensions Merton

More complex extensions Stochastic Volatility Extreme value

Jump models

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