Properties of Plain Vanilla Options & Implied Vola
Ilya Dergunov
Quantitative Finance
Higher School of Economics
Module 2, November 2023
Outline
Outline
Non-BS specific properties of Plain Vanilla Options
Factors affecting option prices
Bounds on calls and puts
Put-Call parity
BS specific properties of Plain Vanilla Options
Implied Volatility
The Greeks
Reading list:
Hull, J. C: Options, Futures, and other Derivatives,
Prentice-Hall, New York. – Ch. 11, 19 & 20
Properties of Plain Vanilla Options & Implied Vola
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Non-BS specific properties of Plain Vanilla Options
Factors affecting option prices
current price of the underlying S0
volatility σ
time to maturity T
strike price K
interest rate r
dividend payments q
Properties of Plain Vanilla Options & Implied Vola
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Non-BS specific properties of Plain Vanilla Options
Factors affecting option prices
Hull, J. C: Options, Futures, and other Derivatives – Ch. 11
Properties of Plain Vanilla Options & Implied Vola
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Non-BS specific properties of Plain Vanilla Options
No dividends: Bounds on European calls
Let Cdenote the price of a European call on a non-dividend
paying asset. Then it must hold
upper bound: C≤S0
lower bound:
C≥S0−exp−rT K(1)
Proof of (1): At time 0, set up the two portfolios:
Portfolio (A): one call, and exp−rT Kunits of money,
Portfolio (B): one share of the underlying.
portfolio value at 0 value at T
(A) C+ exp−rT Kmax(ST,K)
(B) S0ST
Since at T, the value of (A) exceeds the value of (B), it must
hold: C+ exp−rT K≥S0. Else the market admits arbitrage.
Properties of Plain Vanilla Options & Implied Vola
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Non-BS specific properties of Plain Vanilla Options
No dividends: Bounds on European calls
Let Pdenote the price of a European put on a non-dividend paying
asset. Then it must hold
upper bound: P≤K
lower bound:
P≥exp−rT K−S0(2)
Proof of (2): At time 0, set up the two portfolios:
Portfolio (A): one put, and one share of the underlying,
Portfolio (B): exp−rT Kunits of money.
portfolio value at 0 value at T
(A) P+S0max(ST,K)
(B) exp−rT K K
Since at T, the value of (A) exceeds the value of (B), it must
hold: P+S0≥exp−rT K. Else the market admits arbitrage.
Properties of Plain Vanilla Options & Implied Vola
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Non-BS specific properties of Plain Vanilla Options
Put-call parity for non-dividend paying assets
Theorem
Suppose that a European call option with strike Kand expiration
Tis traded at a market price of C. Then the only arbitrage free
price for a put with same strike Kand expiration Tis given by
P=C−S0+ exp−rT K(3)
Properties of Plain Vanilla Options & Implied Vola
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Non-BS specific properties of Plain Vanilla Options
Put-call parity in the Black-Scholes model
C0−P0=S0−Kexp−rT
⇒P0=C0−S0+Kexp−rT
=S0N(d1)−exp−rT KN(d2)−S0+Kexp−rT
=S0(N(d1)−1) −exp−rT K(N(d2)−1)
= exp−rT KN(−d2)−S0N(−d1)
Properties of Plain Vanilla Options & Implied Vola
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BS specific properties of Plain Vanilla Options
Implied vola
Definition:Implied vola = vola for which the Black-Scholes price
coincides with the market price.
Let CMbe the market price of an actively traded call option. Then
σimp is the vola such that
Calle[S,K,t,T,σimp] = CM
Properties of Plain Vanilla Options & Implied Vola
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BS specific properties of Plain Vanilla Options
Uniqueness of implied vola
Recall that prices of Plain Vanilla Options are monotone increasing
in σ. Therefore the implied vola is uniquely defined!
Vola-sensitivity of Plain Vanilla options in the BS model
Definition: the derivative of an option value wrt the volatility is
called vega and is denoted by V
The vega of a call is given by
V=Sϕ(d1)p(T−t),
where ϕ(x) = 1
√2πexp−x2/2
Remark: V>0 for t<T.
Properties of Plain Vanilla Options & Implied Vola
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BS specific properties of Plain Vanilla Options
The vega of a call
Proof:
∂C
∂σ =S∂N(d1)
∂σ −exp−rT K∂N(d2)
∂σ
=Sϕ(d1)∂d1
∂σ −exp−rT Kϕ(d2)∂d2
∂σ
Note that
∂d1
∂σ =−d2
σ
∂d2
∂σ =−d1
σ
and ϕ(d2) = ϕ(d1) exp−d1σ√(T−t)−1
2σ2(T−t)=ϕ(d1)S
Kexpr(T−t)
From this we get V=Sϕ(d1)p(T−t)
Properties of Plain Vanilla Options & Implied Vola
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BS specific properties of Plain Vanilla Options
The vega of a put
Pute[S,K,t,T,σimp] = exp−rT KN(−d2)−S0N(−d1)
The vega of a put is given by
V=Sϕ(d1)p(T−t),
√2πexp−x2/2
