Uncertain Volatility with Static Hedge

Abukar Ali

Uncertain Volatility with Static Hedge

We begin with the Black-Scholes equation for pricing a vanilla option contract which is a

parabolic PDE with two variables, S and t and with constant parameters such as r, D, and sigma

(volatility). Instead of pricing the option with constant volatility, this assumption can be relaxed

and we can assume volatility to be parameter that lies within certain range.

Assume that volatility lies within the band

𝜎− < 𝜎 < 𝜎+

Construct a portfolio consisting of the value of a long call option V(S,t), and hedge it by shorting

-∆ of the underlying asset. The value of the portfolio becomes

Π = 𝑉 − Δ𝑆 and 𝑑𝑠 = 𝑢𝑠𝑑𝑡 + 𝜎𝑆𝑑𝑋

The change in the value of the portfolio is

𝑑Π = (𝜕𝑉

𝜕𝑡+ 0.5𝜎2 𝜕2𝑉

𝜕2𝑆) 𝑑𝑡 + (

𝜕𝑉

𝜕𝑆− ∆) 𝑑𝑠, 𝑑Π = (

𝜕𝑉

𝜕𝑡+ 0.5𝜎2 𝜕2𝑉

𝜕2𝑆) 𝑑𝑡

Since volatility is unknown, we now assume long vanilla positions to take the lowest value of

the volatility range while short position takes the highest volatility value. The return of the

portfolio is set equal to the risk-free rate.

min𝜎−<𝜎<𝜎+

(dΠ) = rΠdt

We set

min𝜎−<𝜎<𝜎+

(𝜕𝑉

𝜕𝑡+ 0.5𝜎2 𝜕2𝑉

𝜕2𝑆) 𝑑𝑡 = (𝑉 − 𝑆

𝜕𝑉

𝜕𝑆) 𝑑𝑡

The value that the volatility takes depends sign of Gamma (Γ) and the worst-case value

satisfies1

𝜕𝑉−

𝜕𝑡+ 0.5𝜎 (Γ)2 𝜕2𝑉−

𝜕2𝑆+ 𝑟𝑆

𝜕𝑉−

𝜕𝑆− 𝑟𝑉− = 0 (2.1)

Where

Γ =𝜕2𝑉−

𝜕2𝑆

and

𝜎(Γ) = 𝜎+ 𝑖𝑓 Γ < 0

1 Equation 2.1 was originally derived by Avellaneda, Levy & Paras and Lyons. This is also the same as the Hoggard – Whalley – Wilmott transaction cost model. See P.Wilmott on Quantitative Finance, 2 edition, Chapter 48.

𝜎(Γ) = 𝜎− 𝑖𝑓 Γ > 0 The worst-case scenario for a delta hedged long vanilla option is if volatility is low while the

opposite is true for a delta hedged short vanilla option.

The best option value, and hence the range of possible values can be found by solving

𝜕𝑉−

𝜕𝑡+ 0.5𝜎 (Γ)2 𝜕2𝑉−

𝜕2𝑆+ 𝑟𝑆

𝜕𝑉−

𝜕𝑆− 𝑟𝑉− = 0 (2.2)

Where

Γ =𝜕2𝑉−

𝜕2𝑆

and

𝜎(Γ) = 𝜎+ 𝑖𝑓 Γ < 0

𝜎(Γ) = 𝜎− 𝑖𝑓 Γ > 0

Static Hedging Unlike delta hedging, static hedging does not require dynamic rebalancing of the hedged

portfolio. For example, suppose we have long position of a digital option which we would like

to hedge. To hedge the position, we can use call spread and discrete hedge the position from

time to time., However, instead of continues rebalancing, we can set up a one time hedge with

the same instrument as above (call spread) but with an optimum number of call spread to use

for the static hedge.

Consider the following example; suppose we want to buy a call option with strike price 𝐾1with

the following payoff:

𝑃 = 𝑀𝑎𝑥(𝑆 − 𝐾1, 0)

To hedge the above call position, we short another call with strike 𝐾2 and a

payoff 𝑃2 = 𝑀𝑎𝑥(𝑆 − 𝐾2, 0). The difference in the payoffs is then

𝐷𝑖𝑓𝑓 = 𝑀𝑎𝑥(𝑆 − 𝐾1) − 𝜆𝑀𝑎𝑥(𝑆 − 𝐾2, 0)

Were 𝜆 is the number of short positions we need to hedge the long call position. This portfolio

is said to be statically hedged and it has smaller payoff then the unhedged call.

To value the residual payoff (Diff) in the uncertain parameter framework, one needs to solve

𝜕𝑉−

𝜕𝑡+ 0.5𝜎 (Γ)2 𝜕2𝑉−

𝜕2𝑆+ 𝑟𝑆

𝜕𝑉−

𝜕𝑆− 𝑟𝑉− = 0

With the final condition

𝑉(𝑆, 𝑇) = 𝑀𝑎𝑥(𝑆 − 𝐾1) − 𝜆𝑀𝑎𝑥(𝑆 − 𝐾2, 0)

To find the optimum hedge, we solve for the optimum number of calls to short. So, given

market price for the original call, the statically hedged portfolio with the final condition and

with the optimum hedge should produce more tighter spread using the volatility scenario under

the static hedge and uncertain parameter framework.

We initially price up the digital call option and produce its payoff chart at expiry. To hedge the

digital call option we use a portfolio of long and short call options. As stated above, we price

up these options individually and show their payoff at expiry.

Figure 2.1 Digital Call

To hedge the digital call option with strike price of 100, we use initially a short call option with

a quantity of 0.05 and long call position with quantity of 0.05. Assume the calls have market

prices as given in table below

Table 2.1

Option Type Strike Maturity Bid Ask Quantity

Target Option Digital Call 100 0.5 0 0 1 Hedging Option

1 Call 90 0.5 14.42 14.42 -0.05 Hedging Option

2 Call 110 0.5 4.22 4.22 0.05

The cost of setting up the hedge is

Hedging Cost = -0.05 x 14.42 + 0.05 x 4.22 = -0.51

We solve

𝜕𝑉−

𝜕𝑡+ 0.5𝜎 (Γ)2

𝜕2𝑉−

𝜕2𝑆+ 𝑟𝑆

𝜕𝑉−

𝜕𝑆− 𝑟𝑉− = 0

With the final condition

𝑉(𝑆, 𝑇) = 𝑀𝑎𝑥(𝑆 − 𝐾1) − 𝜆𝑀𝑎𝑥(𝑆 − 𝐾2, 0)

0

0.2

0.4

0.6

0.8

1

1.2

0

7.2

14

.4

21

.6

28

.8 36

43

.2

50

.4

57

.6

64

.8 72

79

.2

86

.4

93

.6

10

0.8

10

8

11

5.2

12

2.4

12

9.6

13

6.8

14

4

15

1.2

15

8.4

16

5.6

17

2.8

18

0

We deduct the result from the cost of setting up the hedge.

We repeat this process but this time we use the opposite scenario for the volatility and we

deduct the result from the cost of setting up the hedge.

The difference from both results should produce tighter price spread then what would have

been quoted for the original unhedged digital call option. To find the optimum number of calls

to sell and buy, we can use solver in excel.

Figure 2.2

Technical Problem.

Due to the copy office package currently running on my PC, the solver is NOT available and

downloading this add-in is also still causing some technical problems. See screen grab below.

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0

8.8

17

.6

26

.4

35

.2 44

52

.8

61

.6

70

.4

79

.2 88

96

.8

10

5.6

11

4.4

12

3.2

13

2

14

0.8

14

9.6

15

8.4

16

7.2

17

6

18

4.8

19

3.6

20

2.4

21

1.2

22

0

References M. Avellaneda, A. Levy, A. Paras, “Pricing and hedging derivative securities in markets with uncertain

volatilities”, Journal of Applied Finance, Vol 1, 1995

Paul Wilmott, P.Wilmott on Quantitative Finance”, 2nd Edition, Volume 3, Wiley.

CQF Lecture 2010/2011 – Heath, Jarrow and Morton Model

CQF Lecture 2010/ 2011 - Advanced Volatility Modelling in Complete Markets

P. Wilmott, A. Oztukel, “