MSING0104: Introduction to Quantitative
Finance
Dr. Wei Cui
Department of Economics
Centre for Finance
UCL
The 2nd part: Lecture 2
The Basics of Option Pricing
Option pricing 1-0
Option pricing
Payoff functions
The payoff function of a call is max(ST K , 0). The payoff
function of a put is max(K ST , 0).
Intrinsic value
Intrinsic value of a call is max(St K , 0). Intrinsic value of a put is
max(K St , 0).
Option pricing 1-1
In the money (ITM)
An option is called ITM if its intrinsic value is nonzero.
At the money (ATM)
An option is called ATM if St ≈ K .
Out of the money (OTM)
An option is called OTM if its intrinsic value is zero.
Option pricing 1-2
Elementary properties of option prices
option prices are non-negative
the price of a European option at maturity T is equal to the
price of an American option at maturity T
the price of an American option is larger or equal than its
intrinsic value (this is not true for European options)
Option pricing 1-3
for two American options Cam1 and Cam2 differing only by time
to maturity, T1 ≤ T2, the following holds:
CamK ,T1(St ,T1 t) ≤ CamK ,T2(St ,T2 t)
PamK ,T1(St ,T1 t) ≤ PamK ,T2(St ,T2 t)
Since:
CamK ,T2(St ,T2 T1) ≥ intrinsic value = max(St K , 0)
= CamK ,T1(St , 0) t
(not true in general for European options)
Option pricing 1-4
the price of an American option is always bigger than or equal
to the price of the corresponding European option
call prices are monotone decreasing functions of strike prices
put prices are monotone increasing functions of strike prices
Option pricing 1-5
Put-Call-Parity
Price of a European call and put : CK ,T (St , τ), PK ,T (St , τ)
Proposition (Put-Call-Parity)
1. If Dt is the value of all earnings and costs related to the
underlying during the time period τ = T t calculated at
time t, we have
CK ,T (St , τ) = PK ,T (St , τ) + St Ke rτ Dt (1.1)
2. If we have continuous costs of carry, b = r d , the value of
the Call option at time t is
CK ,T (St , τ) = PK ,T (St , τ) + Ste(b r)τ Ke rτ (1.2)
Option pricing 1-6
Proof:
We consider the following portfolio:
(a) long put at a price of PK ,T (St , τ)
(b) short call at a price of CK ,T (St , τ)
(c) short bond with value K at T
(d) long asset
(e) short bond with current value Dt at time t
The payoff of the portfolio in T is displayed in Table 1.
Option pricing 1-7
The payoffs of the portfolio in T are
value at time T
position K < ST K ≥ ST
(a) long put PK ,T (St , τ) 0 K ST
(b) short call CK ,T (St , τ) (ST K ) 0
(c) short bond K K
(d) long asset ST ST
(e) short bond with current value Dt at time t 0 0
sum 0 0
Table 1: Value of the portfolio at time T (proposition )
Option pricing 1-8
The first result of the theorem follows from the no arbitrage
assumption. Because the value of the portfolio at T is equal to
zero - its value at t has to be zero, too. Therefore
PK ,T (St , τ) CK ,T (St , τ) + St Dt Ke rτ = 0.
The second result follows by similar arguments where position (e)
is removed from the portfolio, position (d) is replaced by
1. buy e dτ assets and invest all dividend payments into the
asset.
Option pricing 1-9
Remark
The proof of the proposition does not hold for American options.
The reason is the opportunity to exercise the option before T .
Option pricing 1-10
Convexity
Proposition The price of an American or European option is a
convex function of its exercise price.
Proof: [for call options]
Let λ ∈ [0, 1] and K1 < K0. We consider the following portfolios A
and B at time t:
B 1 a long position in λ calls with exercise price K1
B 2 a long position in (1 λ) calls with exercise price K0
A a short position in 1 call with exercise price
Kλ def= λK1 + (1 λ)K0
Option pricing 1-11
value at time t′
position St′ ≤ K1 K1 < St′ ≤ Kλ Kλ < St′ ≤ K0 K0 < St′
B 1. 0 λ(St′ K1) λ(St′ K1) λ(St′ K1)
B 2. 0 0 0 (1 λ)(St′ K0)
A 0 0 (St′ Kλ) (St′ Kλ)
sum 0 λ(St′ K1) (1 λ)(K0 St′ ) 0
Table 2: Value of the portfolio at time t ′ (proposition 2.4 )
λ(ST K1) ≥ 0 and (1 λ)(K0 ST ) ≥ 0
Hence, the difference of the portfolio values B and A in t has to be
non-negative, implying
λCK1,T (St , τ) + (1 λ)CK0,T (St , τ) CKλ,T (St , τ) ≥ 0 (1.3)
Option pricing 1-12
Example
Consider three call options on the same underlying and with the
same maturity. The exercise and option prices are given in the
following table (with λ = 23)
exercise price option price
K1 = 190 30.6 GBP
Kλ = 200 26.0 GBP
K0 = 220 14.4 GBP
Table 3: data of the example
We know from the last proposition
2
3CK1,T (St , τ) +
1
3CK0,T (St , τ) ≥ CKλ,T (St , τ) (1.4)
But this condition is not satisfied.
Option pricing 1-13
Example
Construct the following arbitrage strategy:
1. long position in 23 calls with K1
2. long position in 13 calls with K0
3. short position in 1 calls with Kλ = 23K1 +
1
3K0
The cash flow of this strategy at time t is 0.80 GBP and its value
at expiration time T is given in the following table
Option pricing 1-14
Example
value at time T
position ST ≤ 190 190 < ST ≤ 200 200 < ST ≤ 220 220 < ST
1. 0 23 (ST 190) 23 (ST 190) 23 (ST 190)
2. 0 0 0 13 (ST 220)
3. 0 0 (ST 200) (ST 200)
sum 0 23 (ST 190) 13 (220 ST ) 0
Table 4: value of the portfolio at time T
For 190 < ST < 220 the portfolio has an additional positive cash
flow, hence a riskless profit.
Option pricing 1-15
Derivatives
Proposition For two European calls (puts) on the same underlying
with the same time to expiration and exercise prices K2 > K1, the
following holds
0 ≤ CK1,T (St , τ) CK2,T (St , τ) ≤ (K2 K1)e rτ (1.5)
0 ≤ PK2,T (St , τ) PK1,T (St , τ) ≤ (K2 K1)e rτ (1.6)
If CK ,T (St , τ) and PK ,T (St , τ) are differentiable w.r.t. K , we have
1 ≤ e rτ ≤ C
K ≤ 0 respectively 0 ≤
P
K ≤ e
rτ ≤ 1
Option pricing 1-16
Proof: [for call options]
Portfolio at time t:
Portfolio A a short position in a call with exercise price K1
Portfolio B a long position in a call with exercise price K2 and a
long position in zero bonds with nominal value (K2 K1)
expiring in T .
The value of this portfolio at expiration time T is given in the
following table.
Option pricing 1-17
value a time T
position ST ≤ K1 K1 < ST < K2 K2 ≤ ST
B 1. 0 0 ST K2
B 2. K2 K1 K2 K1 K2 K1
A 0 (ST K1) (ST K1)
sum K2 K1 K2 ST 0
Table 5: Value of the portfolio at time T (for the above proposition )
value of portfolio in T ≥ 0
= value of portfolio in t ≥ 0
Portfolio Insurance 2-0
Portfolio Insurance
Portfolio insurance is a transformation of a portfolio’s risk profile.
This can be achieved by adding options to a portfolio.
Consequently,
positive yields are diminished by an insurance premium,
but the value of the portfolio cannot fall below a certain level
(floor). Why Think about cash returns and other risk-less
returns...
Portfolio Insurance 2-1
Example
10 500 GBP are invested into stocks with price S0 = 100 GBP and
Puts with strikes K = 100 and T = 1 year to expiry. Puts are
traded at prices P0 = 5 GBP.
Portfolio A buy 105 shares (not insured portfolio)
Portfolio B buy 100 shares (10 000 GBP)and buy 100 put
options (500 GBP) on that stock with K = 100 and T = 1
year to expiry (insured portfolio)
The 5 GBP for the puts may be interpreted as an insurance
premium.
Portfolio Insurance 2-2
not insured insured insured
portfolio portfolio portfolio in %
spot price ST value yield value yield of the not insured
[GBP] [GBP] % p.a. [GBP] % p.a. portfolio
50 5 250 –50 10 000 –4.8 190
60 6 300 –40 10 000 –4.8 159
70 7 350 –30 10 000 –4.8 136
80 8 400 –20 10 000 –4.8 119
90 9 450 –10 10 000 –4.8 106
100 10 500 0 10 000 -4.8 95
110 11 550 +10 11 000 +4.8 95
120 12 600 +20 12 000 +14.3 95
130 13 650 +30 13 000 +23.8 95
140 14 700 +40 14 000 +33.3 95
Table 6: Portfolio insurance
Portfolio Insurance 2-3
For portfolio insurance we need to clarify:
1. Which financial instruments are available (bonds, stocks,
convertibles, swaps)
2. Which ideas has the investor on
I the composition of his portfolio
I the availability of capital
I the insurance horizon
I his risk preference
I the floor of his portfolio. For a given floor F and invested
capital V the minimum rate is (at a one-year horizon)
r = F VV
Portfolio Insurance 2-4
Different strategies for portfolio insurance split into:
static strategies: no big portfolio changes
dynamic strategies: almost continuous changes
Possibilities of static strategies:
Strategy 1: buy stocks and puts in the same quantity.
Strategy 2: buy zero bonds in nominal value of the de-
sired floor and buy calls on the stock.
Portfolio Insurance 2-5
Example: Portfolio Insurance
Assume that our investor has already taken the decision to invest
into a stock. According to the dividend payments we distinguish
two cases:
(i) continuous rate d
(ii) known discrete dividend payments with present value D0
The data of this example can be inferred from Table 7. Stock
volatility is a measure of risk, i.e. the size of fluctuations in the
stock, which we will inspect more deeply later in class. Assume
t = 0, so τ = T t = T in t = 0.
The task is now to calculate the number of stocks n, the number of
puts necessary and the strike price K of the puts.
Portfolio Insurance 2-6
Data of example on portfolio insurance:
actual time t 0
capital V 100 000 GBP
desired floor F 95 000 GBP
time to maturity τ (τ = T t) 2 years
actual spot price S0 100 GBP
interest rate r 0.10
stock volatility σ 0.30
dividend
(i) d 0.02
(ii) D0 5 GBP
Table 7: Data of example
Portfolio Insurance 2-7
Case (i): The stock yields 0.02 p.a., which we reinvest
continuously.
Hence, after T years we have nedT stocks. Obviously, we need to
buy that many puts.
Assuming t = 0, then the time to maturity τ = T . The capital
outstanding for investment at time t = 0 is V , it must hold that
n · S0 + nedT · PK ,T (S0,T ) = V (2.1)
where PK ,T (S0,T ) denotes the price of the put.
Portfolio Insurance 2-8
The strike price K of the put is to be determined such that the
desired floor after 2 years is guaranteed. This means that at time T
with ST ≤ K the value of the portfolio is equal to the floor F :
n · ed ·TST + n · ed ·T (K ST ) = F (2.2)
n = FK e
d ·T
Combining equations (2.1) and (2.2) yields:
e d ·TS0 + PK ,T (S0,T ) VF · K = 0 (2.3)
Portfolio Insurance 2-9
The Black/Scholes option valuation formula (be patient for its
arrival!) expresses the put price as a function of its strike price K .
Solving the formula numerically yields K = 99.58.
To guarantee the floor of 95 000 GBP we need to buy
n = FK e
d ·τ = 916.6 stock and
n · ed ·τ = 954 puts with strike price K = 99.58.
Under Black/Scholes the price of the put is:
PK ,τ (S0, τ) = 8.74 GBP.
Portfolio Insurance 2-10
not insured insured insured
portfolio portfolio portfolio in %
spot price ST value yield value yield of the not insured
[GBP] [GBP] % p.a. [GBP] % p.a. portfolio
70 72 857 –27 95 000 –5 130
80 83 265 –17 95 000 –5 114
90 93 673 –6 95 000 –5 101
100 104 081 +4 95 400 –5 92
110 114 489 +15 104 940 +5 92
120 124 897 +25 114 480 +14 92
130 135 305 +35 124 020 +24 92
140 145 714 +46 133 560 +34 92
Table 8: Effect of portfolio insurance in case (i): value and yield
Portfolio Insurance 2-11
For the equivalent strategy with zero bonds and calls we invest
Fe r ·τ = 77779.42 GBP (r = interest rate) and buy 954 calls with
strike price K = 99.58 with value 23.28 GBP per call.
The strategy is equivalent since due to the put call parity.
n · S0 + ned ·TPK ,T (S0,T ) = nKe b·T + ned ·TCK ,T (S0,T )
with:
b = r d and nKe bτ = Fe rτ
Portfolio Insurance 2-12
Case (ii) Dividend of present value D0 = 5 GBP. We invest in
zero bonds and obtain at time τ : DT = D0erT = 6.107 GBP. Buy
n stock and n puts, which yields similar equations as before:
n · S0 + nPK ,T (S0 D0,T ) = V (2.4)
nK + nDT = F (2.5)
Put equation (2.5) in (2.4):
S0 + PK ,T (S0 D0,T ) VF · (K + DT ) = 0.
Solving produces
K = 96.42 and n = FK + DT
= 926.55
For the equivalent strategy we buy 926.55 calls with value 23.99
GBP per call, K = 96.92 and invest 95 000e rτ = 77 779 GBP in
zero bonds.
Portfolio Insurance 2-13
80 100 120 140
80000
100000
120000
140000
Stock Price
Po
rt
fo
lio
V
al
ue
Insured (blue) vs Non-insured (red)
Figure 1: Effect of portfolio insurance: portfolio value as function of stock
price (blue line – for insured portfolio, red line – for not insured portfolio).
Portfolio Insurance 2-14
not insured insured insured
portfolio portfolio portfolio in %
spot price ST value yield value yield of the not insured
[GBP] [GBP] % p.a. [GBP] % p.a. portfolio
70 76 107 –24 94 996 –5 125
80 86 107 –14 94 996 –5 110
90 96 107 –4 94 996 –5 99
96.42 102 527 +3 94 996 –5 93
100 106 107 +6 98 313 –2 93
110 116 107 +16 107 579 +8 93
120 126 107 +26 116 844 +17 93
130 136 107 +36 126 110 +26 93
140 146 107 +46 135 375 +35 93
Table 9: Effect of portfolio insurance in case (ii): value and yield
