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### DerivativesInside Black Scholes

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

Lessons from the binomial model

- Need to model the stock price evolution
- Binomial model:
- discrete time, discrete variable
- volatility captured by u and d
- Markov process
- Future movements in stock price depend only on where we are, not the history of how we got where we are
- Consistent with weak-form market efficiency
- Risk neutral valuation
- The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate

Derivatives 08 Inside Black Scholes

Black Scholes differential equation: assumptions

- S follows the geometric Brownian motion: dS = µS dt + S dz
- Volatility constant
- No dividend payment (until maturity of option)
- Continuous market
- Perfect capital markets
- Short sales possible
- No transaction costs, no taxes
- Constant interest rate
- Consider a derivative asset with value f(S,t)
- By how much will f change if S changes by dS?
- Answer: Ito’s lemna

Derivatives 08 Inside Black Scholes

Ito’s lemna

- Rule to calculate the differential of a variable that is a function of a stochastic process and of time:
- Let G(x,t) be a continuous and differentiable function
- where x follows a stochastic process dx =a(x,t) dt + b(x,t) dz
- Ito’s lemna. G follows a stochastic process:

Drift

Volatility

Derivatives 08 Inside Black Scholes

Ito’s lemna: some intuition

- If x is a real variable, applying Taylor:
- In ordinary calculus:
- In stochastic calculus:
- Because, if x follows an Ito process, dx² = b² dt you have to keep it

An approximation

dx², dt², dx dt negligeables

Derivatives 08 Inside Black Scholes

Lognormal property of stock prices

- Suppose: dS= S dt + S dz
- Using Ito’s lemna: d ln(S) = ( - 0.5 ²) dt + dz
- Consequence:

ln(ST) – ln(S0) = ln(ST/S0)

Continuously compounded return between 0 and T

ln(ST) is normally distributed so that SThas a lognormal distribution

Derivatives 08 Inside Black Scholes

Derivation of PDE (partial differential equation)

- Back to the valuation of a derivative f(S,t):
- If S changes by dS, using Ito’s lemna:
- Note: same Wiener process for S and f
- possibility to create an instantaneously riskless position by combining the underlying asset and the derivative
- Composition of riskless portfolio
- -1 sell (short) one derivative
- fS = ∂f /∂S buy (long) DELTA shares
- Value of portfolio: V = - f + fS S

Derivatives 08 Inside Black Scholes

Here comes the PDE!

- Using Ito’s lemna
- This is a riskless portfolio!!!
- Its expected return should be equal to the risk free interest rate:

dV = r V dt

- This leads to:

Derivatives 08 Inside Black Scholes

Understanding the PDE

- Assume we are in a risk neutral world

Expected change of the value of derivative security

Change of the value with respect to time

Change of the value with respect to the price of the underlying asset

Change of the value with respect to volatility

Derivatives 08 Inside Black Scholes

Black Scholes’ PDE and the binomial model

- We have:
- BS PDE : f’t + rS f’S + ½² f”SS = r f
- Binomial model: p fu + (1-p) fd = ert
- Use Taylor approximation:
- fu = f + (u-1) Sf’S + ½ (u–1)² S² f”SS + f’tt
- fd = f + (d-1) Sf’S + ½ (d–1)² S² f”SS + f’tt
- u = 1 + √t + ½ ²t
- d = 1 – √t + ½ ²t
- ert = 1 + rt
- Substituting in the binomial option pricing model leads to the differential equation derived by Black and Scholes

Derivatives 08 Inside Black Scholes

And now, the Black Scholes formulas

- Closed form solutions for European options on non dividend paying stocks assuming:
- Constant volatility
- Constant risk-free interest rate

Call option:

Put option:

N(x) = cumulative probability distribution function for a standardized normal variable

Derivatives 08 Inside Black Scholes

Understanding Black Scholes

- Remember the call valuation formula derived in the binomial model:

C = S0 – B

- Compare with the BS formula for a call option:
- Same structure:
- N(d1) is the delta of the option
- # shares to buy to create a synthetic call
- The rate of change of the option price with respect to the price of the underlying asset (the partial derivative CS)
- K e-rT N(d2) is the amount to borrow to create a synthetic call

N(d2) = risk-neutral probability that the option will be exercised at maturity

Derivatives 08 Inside Black Scholes

A closer look at d1 and d2

2 elements determine d1 and d2

A measure of the “moneyness” of the option.The distance between the exercise price and the stock price

S0 / Ke-rt

Time adjusted volatility.The volatility of the return on the underlying asset between now and maturity.

Derivatives 08 Inside Black Scholes

Example

Stock price S0 = 100

Exercise price K = 100 (at the money option)

Maturity T = 1 year

Interest rate (continuous) r = 5%

Volatility = 0.15

ln(S0 / K e-rT) = ln(1.0513) = 0.05

√T = 0.15

d1 = (0.05)/(0.15) + (0.5)(0.15) = 0.4083

N(d1) = 0.6585

European call :

100 0.6585 - 100 0.95123 0.6019 = 8.60

d2 = 0.4083 – 0.15 = 0.2583

N(d2) = 0.6019

Derivatives 08 Inside Black Scholes

Relationship between call value and spot price

For call option, time value > 0

Derivatives 08 Inside Black Scholes

European put option

- European call option: C = S0 N(d1) – PV(K) N(d2)
- Put-Call Parity: P = C – S0 + PV(K)
- European put option: P = S0[N(d1)-1] + PV(K)[1-N(d2)]
- P = - S0 N(-d1) +PV(K) N(-d2)

Risk-neutral probability of exercising the option = Proba(ST>X)

Delta of call option

Risk-neutral probability of exercising the option = Proba(ST<X)

Delta of put option

(Remember: N(x) – 1 = N(-x)

Derivatives 08 Inside Black Scholes

Example

- Stock price S0 = 100
- Exercise price K = 100 (at the money option)
- Maturity T = 1 year
- Interest rate (continuous) r = 5%
- Volatility = 0.15

N(-d1) = 1 –N(d1) = 1 – 0.6585 = 0.3415

N(-d2) = 1 – N(d2) = 1 – 0.6019 = 0.3981

European put option

- 100 x 0.3415 + 95.123 x 0.3981 = 3.72

Derivatives 08 Inside Black Scholes

Relationship between Put Value and Spot Price

For put option, time value >0 or <0

Derivatives 08 Inside Black Scholes

Dividend paying stock

- If the underlying asset pays a dividend, substract the present value of future dividends from the stock price before using Black Scholes.
- If stock pays a continuous dividend yield q, replace stock price S0by S0e-qT.
- Three important applications:
- Options on stock indices (q is the continuous dividend yield)
- Currency options (q is the foreign risk-free interest rate)
- Options on futures contracts (q is the risk-free interest rate)

Derivatives 08 Inside Black Scholes

Dividend paying stock: binomial model

t = 1 u = 1.25, d = 0.80r = 5% q = 3%Derivative: Call K = 100

uS0eqtwith dividends reinvested128.81

fu25

uS0ex dividend125

S0100

dS0eqtwith dividends reinvested82.44

fd0

dS0ex dividend80

f = S0 + M

f = [ p fu + (1-p) fd] e-rt = 11.64

Replicating portfolio:

uS0eqt + M ert = fu 128.81 + M 1.0513 = 25

p = (e(r-q)t – d) / (u – d) = 0.489

dS0eqt + M ert = fd 82.44 + M 1.0513 = 0

= (fu – fd) / (u – d )S0eqt= 0.539

Derivatives 08 Inside Black Scholes

Black Scholes Merton with constant dividend yield

The partial differential equation:(See Hull 5th ed. Appendix 13A)

Expected growth rate of stock

Call option

Put option

Derivatives 08 Inside Black Scholes

Options on stock indices

- Option contracts are on a multiple times the index ($100 in US)
- The most popular underlying US indices are
- the Dow Jones Industrial (European) DJX
- the S&P 100 (American) OEX
- the S&P 500 (European) SPX
- Contracts are settled in cash
- Example: July 2, 2002 S&P 500 = 968.65
- SPX September
- Strike Call Put
- 900 - 15.601,005 30 53.501,025 21.40 59.80
- Source: Wall Street Journal

Derivatives 08 Inside Black Scholes

Options on futures

- A call option on a futures contract.
- Payoff at maturity:
- A long position on the underlying futures contract
- A cash amount = Futures price – Strike price
- Example: a 1-month call option on a 3-month gold futures contract
- Strike price = $310 / troy ounce
- Size of contract = 100 troy ounces
- Suppose futures price = $320 at options maturity
- Exercise call option
- Long one futures
- + 100 (320 – 310) = $1,000 in cash

Derivatives 08 Inside Black Scholes

Option on futures: binomial model

uF0→ fu

Futures price F0

dF0→fd

Replicating portfolio: futures + cash

(uF0 – F0) + M ert = fu

(dF0 – F0) + M ert = fd

f = M

Derivatives 08 Inside Black Scholes

Options on futures versus options on dividend paying stock

Compare now the formulas obtained for the option on futures and for an option on a dividend paying stock:

Futures

Dividend paying stock

Futures prices behave in the same way as a stock paying a continuous dividend yield at the risk-free interest rate r

Derivatives 08 Inside Black Scholes

Black’s model

Assumption: futures price has lognormal distribution

Derivatives 08 Inside Black Scholes

Implied volatility – Call option

Derivatives 08 Inside Black Scholes

Implied volatility – Put option

Derivatives 08 Inside Black Scholes

Smile

Derivatives 08 Inside Black Scholes

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