Finance - Alexandru

7/31/2019 Finance - Alexandru

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Denitions & Black - Scholes - Merton model

Alexandru Bratu

Term Denition

share single unit of ownership in a corporation, mutual fund, or other organization

commodity generic term for any marketable item produced to satisfy wants or need, e.g. oil, gas

fungible property of a good or a commodity whose individual units are capable of mutual substitution

instrument tradable asset of any kind, either cash; evidence of an ownership interest in an entity

bond negotiable certicate that acknowledges the indebtedness of the bond issuer to the holder

debenture document that either creates a debt or acknowledges it, and it is a debt without collateral

equity stock or any other security representing an ownership interest

ommon stocks form of corporate equity ownership, a type of security

derivative contract between two parties that species conditions under which payments are to be made betweenthe parties

forward non-standardized contract between two parties to buy or sell an asset at a specied future time at aprice agreed upon( delivery price ) today

futures standardized contract between two parties to buy or sell a specied asset of standardized quantity

and quality for a price agreed todayoption instrument that species a contract between two parties for a future transaction on an asset at a

reference price (the strike)

swaps derivative in which counterparties exchange cash ows of one party’s nancial instrument for thoseof the other party’s nancial instrument

stock original capital paid into or invested in the business by its founders, it serves as a security for thecreditors of a business

security fungible, negotiable nancial instrument representing nancial value, i.e. debt - banknotes(bill, papermoney), bonds and debentures, equity - common stocks, derivative - forwards, futures, options andswaps

underlying price or rate of an asset or liability but is NOT the asset or liability itself

arbitrage borrowing money to lend out again at a higher rate of interest

w of one price the same asset does not trade at the same price on all markets

ination swap the linear form of an ination derivative, an over-the-counter and exchange-traded derivatives thatis used to transfer ination risk from one counterparty to another

volatility measure for variation of price of a nancial instrument over time

dividend payments made by a corporation to its shareholder members

call option a.k.a. call - a company makes a call when it asks buyers of its new shares to pay some or all of theshare price

put option contract between two parties to exchange an asset (the underlying), at a specied price (the strike),by a predetermined date (the expiry or maturity)



1 Black - Scholes - Merton

denition mathematical model of a nancial market containing certain derivative investment instruments, givingprice of European-style options

S stock price

V (S, t ) price of a derivative as a function of time and stock price

C (S, t ) price of a European call option

P (S, t ) price of a European put option

K the strike of the option

r annualized risk-free interest rate, continuously compounded

µ the drift rate of S, annualized

σ the volatility of the stock’s returns, this is the square of the quadratic variation of the stock’s logprice process

t a time in years, generally with tnow = 0 , t expiry = T

Π the value of a portfolio

N (x) standard normal cumulative distribution function, N (x) = 1√ 2π

x

−∞e−

z2

2 dz

N (x) probability density function, N (x) = e− x

2

2

√ 2π

Assumption the price of the underlying asset(stock) follows a geometricBrownianmotion (continuous-time stochas-tic process in which the logarithm of the randomly varying quantity follows a Brownian motion, i.e.random motion within a specic medium) ∂S

S = µdt + σdW W = Brownian −motion , with the equationgiving the innitesimal return on the stock has an expected value of µdt and a variance of σ2dt

SM derivation the payoff of an option V (S, T ) at maturity is known. To nd its value at an earlier time we need toknow how V evolves as a function of S and t . By It o’s lemma for two variables we have

dV = µS ∂V ∂S

+∂V ∂t

+12

σ 2S 2∂ 2V ∂S 2

dt + σS ∂V ∂S

dW

Consider delta-hedge portfolio, with one option and long ∂V ∂S shares at time t. The value of these

holdings is Π = − V + ∂V ∂S S . Over the time period [t, t + ∆ t ] the total prot or loss from changes in

the values of the holdings is: ∆Π = − ∆ V + ∂V ∂S ∆ S . Now making the continuous model into more

discrete parts we get

∆ S = µS ∆ t + σS ∆ W

∆ V = µS ∂V ∂S + ∂V

∂t + 12 σ2S 2 ∂ 2 V

∂S 2 ∆ t + σS ∂V ∂S ∆ W

with ∆Π = − ∂V ∂t

− 12 σ2S 2 ∂ 2 V

∂S 2 ∆ t giving

∂V ∂t

+12

σ 2S 2∂ 2V ∂S 2

+ rS ∂V ∂S

− rV = 0

Call option the value of a call option for a non-dividend paying underlying stock in terms of BSM parameters

C (S, t ) = N (d1) S − N (d2) Ke −r(

T

−t)

Put option the price of a corresponding put option based on put-call parity is

P (S, t ) = Ke −r (T −t ) − S + C (S, t ) = N (− d2) Ke −r (T −t ) − N (− d1) S



Greeks (nance)

ef:= measure the sensitivity to change of the option price under a slight change of a single parameter while holding theother parameters xed

Measures What Calls Putsvalue e−qτ S Φ (d1) − e−rτ K Φ (d2) e−rτ K Φ (−d2) − e−qτ S Φ (−d1)delta ∂V

∂S e−qτ Φ (d1) −e−qτ Φ (−d1)vega ∂V

∂σ Se −qτ φ (d1) √ τ = Ke −rτ φ (d2) √ τ Se −qτ φ (d1) √ τ = Ke −rτ φ (d2) √ τ theta −

∂V ∂τ −e−qτ Sφ (d 1 )σ

2√ τ − rKe −rτ Φ (d2) + qSe −qτ Φ (d1) −e−qτ Sφ (d 1 )σ2√ τ + rKe −rτ Φ (d2) + qSe −qτ Φ (d1)

rho ∂V ∂r Kτe −rτ Φ (d2) −Kτ e −rt Φ (−d2)

gamma ∂ 2

V ∂S 2 e−qτ φ(d1

)Sσ √ τ e−qτ φ(d1

)Sσ √ τ

vanna ∂ 2 V ∂S∂σ −e−qτ φ (d1) d 2

σ = ν S 1 −

d 1

σ√ τ −e−qτ φ (d1) d 2

σ = ν S 1 −

d 1

σ√ τ

charm −∂ 2 V

∂S∂τ qe−qτ Φ (d1) − e−qτ 2( r −q)τ −d 2 σ√ τ 2τσ √ τ −qe−qτ Φ (−d1) − e−qτ φ (d1) 2( r −q)τ −d 2 σ√ τ

2τσ √ τ

speed ∂ 3 V ∂S 3 −e−qτ φ(d 1 )

S 2 σ√ τ d 1

σ√ τ + 1 = −ΓS

d 1

σ√ τ + 1 −e−qτ φ(d 1 )S 2 σ √ τ

d 1

σ√ τ + 1 = −ΓS

d 1

σ√ τ + 1

zomma ∂ 3 V ∂S 2 ∂σ e−qτ φ(d 1 )( d 1 d 2 −1)

Sσ 2 √ τ = Γ d 1 d 2 −1σ e−qτ φ(d 1 )( d 1 d 2 −1)

Sσ 2 √ τ = Γ d 1 d 2 −1σ

color ∂ 3 V ∂S 2 ∂τ −e−qτ φ(d 1 )

2Sτ σ √ τ 2qτ + 1 + 2( r −q)τ −d 2 σ√ τ σ√ τ d1 −e−qτ φ(d 1 )

2Sτ σ √ τ 2qτ + 1 + 2( r −q)τ −d 2 σ√ τ σ√ τ d1

DvegaDtime ∂ 2 V ∂σ∂τ Se −qτ φ (d1) √ τ q + (r −q)d 1

σ√ τ −1+ d 1 d 2

2τ Se −qτ φ (d1) √ τ q + (r −q)d 1

σ √ τ −1+ d 1 d 2

2τ

vomma ∂ 2 V ∂σ 2 Se −qτ φ (d1) √ τ d 1 d 2

σ = ν d 1 d 2

σ Se −qτ φ (d1) √ τ d 1 d 2

σ = ν d 1 d 2

σ

Ultima ∂ 3 V ∂σ 3 −ν

σ 2 d1d2 (1 − d1d2) + d21 + d22 −ν σ 2 d1d2 (1 − d1d2) + d21 + d22

dual delta ∂V ∂S −e−rτ Φ (d2) e−rτ Φ (−d2)

dual gamma ∂ 2 V ∂S 2 e−rτ φ(d 2 )

Kσ √ τ e−rτ φ(d 2 )Kσ √ τ

where

d1 = 1σ√ τ ln S

K + r − q + σ 2

2 τ

d2 = 1σ√ τ ln S

K + r − q −σ 2

2 τ = d1 − σ√ τ

φ (x ) = 1√ 2π

e− x

2

2

Φ (x ) = 1√ 2π x

−∞e − y

2

2 dy = 1 −1√ 2π ∞

−x e−y

2

2 dy

with

S stock price

K strike price

r risk free rate

q annual dividend yield

τ T - t = time to maturity

σ volatility

φ standard normal probability density function

Φ standard normal cumulative distribution function

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Finance - Alexandru