. In some papers it have been remarked that derivation of the Black Scholes Equation (BSE) contains mathematical ambiguities. In particular in [2,3 ] there are two problems which can be raise by accepting Black Scholes (BS) pricing concept. One is technical derivation of the BSE and other the pricing definition of the option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach we define random market price for each market scenario. The spot price then is interpreted as a one that reflect balance between profit-loss expectations of the market participants
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.
In this short notice, we present structure of the perfect hedging. Closed form formulas clarify the fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS option price.
In this paper, we present somewhat alternative point of view on early exercised American options. The standard valuation of the American options the exercise moment is defined as one, which guarantees the maximum value of the option. We discuss the standard approach in the first two sections of the paper. The standard approach was initially presented in the papers [3] - [7]. Our idea is that the exercise moment of the American call / put options is defined by maximum / minimum value of underlying. It was shown that at this moment exercise and sell prices are equal.
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
In this paper we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach we define random market price for each market scenario. The spot price then is interpreted as a one that reflect balance between profit-loss expectations of the market participants
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.
In this short notice, we present structure of the perfect hedging. Closed form formulas clarify the fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS option price.
In this paper, we present somewhat alternative point of view on early exercised American options. The standard valuation of the American options the exercise moment is defined as one, which guarantees the maximum value of the option. We discuss the standard approach in the first two sections of the paper. The standard approach was initially presented in the papers [3] - [7]. Our idea is that the exercise moment of the American call / put options is defined by maximum / minimum value of underlying. It was shown that at this moment exercise and sell prices are equal.
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
In this short notice, we present structure of the perfect hedging. Closed form formulas clarify the fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS option price.
Last my paper equity, implied, and local volatilitiesIlya Gikhman
In this paper we present a critical point on connections between stock volatility, implied
volatility, and local volatility. The essence of the Black Scholes pricing model is based on assumption
that option piece is formed by no arbitrage portfolio. Such assumption effects the change of the real
underlying stock by its risk neutral counterpart. Market practice shows even more. The volatility of the
underlying should be also changed. Such practice calls for implied volatility. Underlying with implied
volatility is specific for each option. The local volatility development presents the value of implied
volatility.
Pricing Exotics using Change of NumeraireSwati Mital
The intention of this essay is to show how change of numeraire technique is used in pricing derivatives with complex payoffs. In the first instance, we apply the technique to pricing European Call Options and then use the same method to price an exotic Power Option.
A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.
Notes I made in June 2013 on the derivation and use of the Black-Scholes equation. If you can forgive the terseness, you can look forward to some nifty stochastic partial differential equation twirling!
Any and all corrections are welcome!
Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)Fynn McKay
Simple implementable of Numerical Analysis to solve the famous Black-Scholes P.D.E. via Finite Difference Methods for the fair price of a European option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this short notice, we present structure of the perfect hedging. Closed form formulas clarify the fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS option price.
Last my paper equity, implied, and local volatilitiesIlya Gikhman
In this paper we present a critical point on connections between stock volatility, implied
volatility, and local volatility. The essence of the Black Scholes pricing model is based on assumption
that option piece is formed by no arbitrage portfolio. Such assumption effects the change of the real
underlying stock by its risk neutral counterpart. Market practice shows even more. The volatility of the
underlying should be also changed. Such practice calls for implied volatility. Underlying with implied
volatility is specific for each option. The local volatility development presents the value of implied
volatility.
Pricing Exotics using Change of NumeraireSwati Mital
The intention of this essay is to show how change of numeraire technique is used in pricing derivatives with complex payoffs. In the first instance, we apply the technique to pricing European Call Options and then use the same method to price an exotic Power Option.
A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.
Notes I made in June 2013 on the derivation and use of the Black-Scholes equation. If you can forgive the terseness, you can look forward to some nifty stochastic partial differential equation twirling!
Any and all corrections are welcome!
Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)Fynn McKay
Simple implementable of Numerical Analysis to solve the famous Black-Scholes P.D.E. via Finite Difference Methods for the fair price of a European option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, it is shown how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
Study Analysis of H.D.A.F Relaying Protocols in Cognitive NetworksAwais Salman Qazi
The concept of cooperative communication in cognitive radio CR networks is used to economize the over utilization of spectrum in accommodating a rising demand of higher data rates for wireless services. The CR typically consists of the Primary User (PU), the Cognitive Relays (CRs), the Cognitive Controller (CC) and the Secondary User (SU). The PU sends its spectrum information to the CRs and CRs decode and re-encode the transmitted data by using Maximum Likelihood estimation and 2 x 2 Alamouti OSTBC techniques and later amplify the OSTBC (re-encoded) data at the relays. All the relays in the system forward their collective decision to CC to finalize its decision on the basis of the information provided by the relays. The CC uses the method known as the spectrum energy detection to compare the energy of PU spectrum with a predefined threshold. If it finds the energy of PU’s spectrum exceeding the threshold, it means that PU’s spectrum can be spared for SU otherwise PU is busy in its spectrum. This book is written by a Computer Scientist, a Computer Engineer and an Electrical Engineer and thus requires no prerequisite knowledge of the students but an
interest in Radio communication.
equity, implied, and local volatilitiesIlya Gikhman
In this paper we present a critical point on connections between stock volatility, implied volatility, and local volatility. The essence of the Black Sholes pricing model is based on assumption that option piece is formed by no arbitrage portfolio. Such assumption effects the change of the real underlying stock by its risk neutral counterpart. Market practice shows even more. The volatility of the underlying should be also changed. Such practice calls for implied volatility. Underlying with implied volatility is specific for each option. The local volatility development presents the value of implied volatility.
Stochastic Local Volatility Models: Theory and ImplementationVolatility
1) Hedging and volatility
2) Review of volatility models
3) Local volatility models with jumps and stochastic volatility
4) Calibration using Kolmogorov equations
5) PDE based methods in one dimension
5) PDE based methods in two dimensions
7) Illustrations
An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model...Volatility
1) Analyse the distribution of the profit&loss (P&L) of delta-hedging strategy for vanilla options in Black-Scholes-Merton (BSM) model and an extension of the Merton jump-diffusion (JDM) model assuming discrete trading and transaction costs
2) Examine the connection between the realized variance and the realized P&L
3) Find approximate solutions for the P&L volatility and the expected total transaction costs
4) Apply the mean-variance analysis to find the trade-off between the costs and P&L variance given hedger's risk tolerance
5) Consider hedging strategies to minimize the jump risk
Turin Startup Ecosystem 2024 - Ricerca sulle Startup e il Sistema dell'Innov...Quotidiano Piemontese
Turin Startup Ecosystem 2024
Una ricerca de il Club degli Investitori, in collaborazione con ToTeM Torino Tech Map e con il supporto della ESCP Business School e di Growth Capital
how to sell pi coins in South Korea profitably.DOT TECH
Yes. You can sell your pi network coins in South Korea or any other country, by finding a verified pi merchant
What is a verified pi merchant?
Since pi network is not launched yet on any exchange, the only way you can sell pi coins is by selling to a verified pi merchant, and this is because pi network is not launched yet on any exchange and no pre-sale or ico offerings Is done on pi.
Since there is no pre-sale, the only way exchanges can get pi is by buying from miners. So a pi merchant facilitates these transactions by acting as a bridge for both transactions.
How can i find a pi vendor/merchant?
Well for those who haven't traded with a pi merchant or who don't already have one. I will leave the telegram id of my personal pi merchant who i trade pi with.
Tele gram: @Pi_vendor_247
#pi #sell #nigeria #pinetwork #picoins #sellpi #Nigerian #tradepi #pinetworkcoins #sellmypi
how to sell pi coins on Bitmart crypto exchangeDOT TECH
Yes. Pi network coins can be exchanged but not on bitmart exchange. Because pi network is still in the enclosed mainnet. The only way pioneers are able to trade pi coins is by reselling the pi coins to pi verified merchants.
A verified merchant is someone who buys pi network coins and resell it to exchanges looking forward to hold till mainnet launch.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
What price will pi network be listed on exchangesDOT TECH
The rate at which pi will be listed is practically unknown. But due to speculations surrounding it the predicted rate is tends to be from 30$ — 50$.
So if you are interested in selling your pi network coins at a high rate tho. Or you can't wait till the mainnet launch in 2026. You can easily trade your pi coins with a merchant.
A merchant is someone who buys pi coins from miners and resell them to Investors looking forward to hold massive quantities till mainnet launch.
I will leave the telegram contact of my personal pi vendor to trade with.
@Pi_vendor_247
how can I sell my pi coins for cash in a pi APPDOT TECH
You can't sell your pi coins in the pi network app. because it is not listed yet on any exchange.
The only way you can sell is by trading your pi coins with an investor (a person looking forward to hold massive amounts of pi coins before mainnet launch) .
You don't need to meet the investor directly all the trades are done with a pi vendor/merchant (a person that buys the pi coins from miners and resell it to investors)
I Will leave The telegram contact of my personal pi vendor, if you are finding a legitimate one.
@Pi_vendor_247
#pi network
#pi coins
#money
how can I sell pi coins after successfully completing KYCDOT TECH
Pi coins is not launched yet in any exchange 💱 this means it's not swappable, the current pi displaying on coin market cap is the iou version of pi. And you can learn all about that on my previous post.
RIGHT NOW THE ONLY WAY you can sell pi coins is through verified pi merchants. A pi merchant is someone who buys pi coins and resell them to exchanges and crypto whales. Looking forward to hold massive quantities of pi coins before the mainnet launch.
This is because pi network is not doing any pre-sale or ico offerings, the only way to get my coins is from buying from miners. So a merchant facilitates the transactions between the miners and these exchanges holding pi.
I and my friends has sold more than 6000 pi coins successfully with this method. I will be happy to share the contact of my personal pi merchant. The one i trade with, if you have your own merchant you can trade with them. For those who are new.
Message: @Pi_vendor_247 on telegram.
I wouldn't advise you selling all percentage of the pi coins. Leave at least a before so its a win win during open mainnet. Have a nice day pioneers ♥️
#kyc #mainnet #picoins #pi #sellpi #piwallet
#pinetwork
Resume
• Real GDP growth slowed down due to problems with access to electricity caused by the destruction of manoeuvrable electricity generation by Russian drones and missiles.
• Exports and imports continued growing due to better logistics through the Ukrainian sea corridor and road. Polish farmers and drivers stopped blocking borders at the end of April.
• In April, both the Tax and Customs Services over-executed the revenue plan. Moreover, the NBU transferred twice the planned profit to the budget.
• The European side approved the Ukraine Plan, which the government adopted to determine indicators for the Ukraine Facility. That approval will allow Ukraine to receive a EUR 1.9 bn loan from the EU in May. At the same time, the EU provided Ukraine with a EUR 1.5 bn loan in April, as the government fulfilled five indicators under the Ukraine Plan.
• The USA has finally approved an aid package for Ukraine, which includes USD 7.8 bn of budget support; however, the conditions and timing of the assistance are still unknown.
• As in March, annual consumer inflation amounted to 3.2% yoy in April.
• At the April monetary policy meeting, the NBU again reduced the key policy rate from 14.5% to 13.5% per annum.
• Over the past four weeks, the hryvnia exchange rate has stabilized in the UAH 39-40 per USD range.
NO1 Uk Rohani Baba In Karachi Bangali Baba Karachi Online Amil Baba WorldWide...Amil baba
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
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Poonawalla Fincorp and IndusInd Bank Introduce New Co-Branded Credit Cardnickysharmasucks
The unveiling of the IndusInd Bank Poonawalla Fincorp eLITE RuPay Platinum Credit Card marks a notable milestone in the Indian financial landscape, showcasing a successful partnership between two leading institutions, Poonawalla Fincorp and IndusInd Bank. This co-branded credit card not only offers users a plethora of benefits but also reflects a commitment to innovation and adaptation. With a focus on providing value-driven and customer-centric solutions, this launch represents more than just a new product—it signifies a step towards redefining the banking experience for millions. Promising convenience, rewards, and a touch of luxury in everyday financial transactions, this collaboration aims to cater to the evolving needs of customers and set new standards in the industry.
The secret way to sell pi coins effortlessly.DOT TECH
Well as we all know pi isn't launched yet. But you can still sell your pi coins effortlessly because some whales in China are interested in holding massive pi coins. And they are willing to pay good money for it. If you are interested in selling I will leave a contact for you. Just telegram this number below. I sold about 3000 pi coins to him and he paid me immediately.
Telegram: @Pi_vendor_247
NO1 Uk Divorce problem uk all amil baba in karachi,lahore,pakistan talaq ka m...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
how to sell pi coins in all Africa Countries.DOT TECH
Yes. You can sell your pi network for other cryptocurrencies like Bitcoin, usdt , Ethereum and other currencies And this is done easily with the help from a pi merchant.
What is a pi merchant ?
Since pi is not launched yet in any exchange. The only way you can sell right now is through merchants.
A verified Pi merchant is someone who buys pi network coins from miners and resell them to investors looking forward to hold massive quantities of pi coins before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
what is the future of Pi Network currency.DOT TECH
The future of the Pi cryptocurrency is uncertain, and its success will depend on several factors. Pi is a relatively new cryptocurrency that aims to be user-friendly and accessible to a wide audience. Here are a few key considerations for its future:
Message: @Pi_vendor_247 on telegram if u want to sell PI COINS.
1. Mainnet Launch: As of my last knowledge update in January 2022, Pi was still in the testnet phase. Its success will depend on a successful transition to a mainnet, where actual transactions can take place.
2. User Adoption: Pi's success will be closely tied to user adoption. The more users who join the network and actively participate, the stronger the ecosystem can become.
3. Utility and Use Cases: For a cryptocurrency to thrive, it must offer utility and practical use cases. The Pi team has talked about various applications, including peer-to-peer transactions, smart contracts, and more. The development and implementation of these features will be essential.
4. Regulatory Environment: The regulatory environment for cryptocurrencies is evolving globally. How Pi navigates and complies with regulations in various jurisdictions will significantly impact its future.
5. Technology Development: The Pi network must continue to develop and improve its technology, security, and scalability to compete with established cryptocurrencies.
6. Community Engagement: The Pi community plays a critical role in its future. Engaged users can help build trust and grow the network.
7. Monetization and Sustainability: The Pi team's monetization strategy, such as fees, partnerships, or other revenue sources, will affect its long-term sustainability.
It's essential to approach Pi or any new cryptocurrency with caution and conduct due diligence. Cryptocurrency investments involve risks, and potential rewards can be uncertain. The success and future of Pi will depend on the collective efforts of its team, community, and the broader cryptocurrency market dynamics. It's advisable to stay updated on Pi's development and follow any updates from the official Pi Network website or announcements from the team.
1. 1
BLACK SCHOLES PRICING CONCEPT.
Ilya I. Gikhman
6077 Ivy Woods Court Mason,
OH 45040, USA
Ph. 513-573-9348
Email: ilyagikhman@yahoo.com
Classification code
Key words. Black Scholes, option, derivatives, pricing, hedging.
Abstract. In some papers it have been remarked that derivation of the Black Scholes Equation (BSE)
contains mathematical ambiguities. In particular in [2,3 ] there are two problems which can be raise by
accepting Black Scholes (BS) pricing concept. One is technical derivation of the BSE and other the
pricing definition of the option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on
market risk weighting. In such approach, we define random market price for each market scenario. The
spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market
participants.
BLACK SCHOLES WORLD.
We highlight two popular derivations of the BSE. One is the original derivation [1] and other is a popular
derivation represented in [5]. Following [1] let us first recall original derivation of the BSE. Next, we will
present original derivation in stochastic processes form.
Let w ( x , t ) denote the value of the call option which is a function of the stock price x and time t. The
hedge position is defined by the number
w 1 ( x , t ) =
x
)t,x(w
(1.1)
2. 2
of the options that would be sold short against one share of long stock. First order approximation of the
ratio of the change in the option value to the change in the stock price is w 1 ( x , t ). Indeed, if the stock
price changes by an amount x , the option price will change by an amount w 1 ( x , t ) x , and the
number of options given by expression (1.1) will be change by an amount of x . Thus, the change in the
value of long position in the stock will be approximately offset by the change in value of a short position
in 1 / w 1 options. The hedged position that contains one share of stock long and 1 / w 1 options short is
definrd by the formula
x – w / w 1 (1.2)
The change in the value of the hedged position over a short interval time period t is
x – w / w 1 (1.3)
Using stochastic calculus we expend note that
w = w ( x + x , t + t ) – w ( x , t ) = w 1 x +
2
1
w 11 v 2
x 2
t + w 2 t (1.4)
Here
w 11 = 2
2
x
)t,x(w
, w 2 =
t
)t,x(w
and v 2
is the variance of the return on stock. Substituting (1.4) into expression (1.3), we find that the
change in the value of the equity in hedged position is:
– (
2
1
w 11 v 2
x 2
+ w 2 ) t / w 1 (1.5)
Since return on the equity in the hedged position is certain, the return must be equal to r t . Thus the
change in the hedge position (1.5) must equal the value of the equity times r t
– (
2
1
w 11 v 2
x 2
+ w 2 ) t / w 1 = ( x – w / w 1 ) r t (1.6)
From (1.6) we arrive at Black Scholes equation
w 2 = r w – r x w 1 –
2
1
v 2
x 2
w 11 (1.7)
Boundary condition to equation (1.7) is defined by the call option payoff, which is specified at the
maturity of the option date T
w ( x , T ) = ( x – c ) χ ( x c ) (1.8)
Here χ ( x ) denotes indicator function. This formula must be the option valuation formula.
3. 3
Remark. Using modern stochastic calculus we can represent Black Scholes derivation in the next form.
Let S ( t ) denote a security price at the moment t ≥ 0 and suppose that
dS ( t ) = S ( t ) dt + σ S ( t ) dw ( t ) (1.9)
European call option written on security S is a contract, which grants buyer of the option the right to buy
a security for a known price K at a maturity T of the contract. The price K is known as the strike price of
the option. According to call option contract the payoff of the European call option is
max { S ( T , ω ) – K , 0 }
In order to buy the option contract at t buyer of the option should pay option premium at t. The option
premium is also called option price. The pricing problem is the problem of finding option price at any
moment t prior to maturity. Following [5] consider a hedged position, consisting of a long position in the
stock and short position in the number Δ ( t )
Δ ( t ) = [
S
))t(S,t(C
] – 1
of the options. Hence, hedge position (1.2) can be represented as
Π ( t ) = x – w / w 1 = S ( t ) – [
S
))t(S,t(C
] – 1
C ( t , S ( t ) ) (1.2′)
The change in the value of the hedged position in a short interval t is equal to
S ( t + t ) – [
S
))t(S,t(C
] – 1
C ( S ( t + t ) , t + t ) (1.3′)
Note that in latter formula number of options at the next moment t + t does not change and equal to
[
S
))t(S,t(C
] – 1
Taking into account Ito formula (1.4) can be rewritten as
Δ C = C ( t + t , x + x ) – C ( t , x ) = C ( t + t , S ( t + t ; t , x ) ) – C ( t , S ( t + t ; t , x ) ) +
+ C ( t , S ( t + t ; t , x )) – C ( t , x ) = C ( t + t , x ) – C ( t , x ) + C ( t , S ( t + t ; t , x ) ) –
– C ( t , x ) + o ( t ) = C /
x ( t , x ) S +
2
1
C
//
xx ( t , x ) σ 2
x 2
t + C /
t ( t , x ) t + o ( t )
where o ( t ) is the random variable defined by Taylor formula taking in the integral form and
0t
l.i.m
( t ) – 1
o ( t ) = 0. Then the formula (1.5) representing the change in the value of the hedged
portfolio can be rewritten as
4. 4
S – C [ C /
x ] – 1
= S – [ C /
x ] – 1
[ C /
x ( t , x ) S +
2
1
C
//
xx ( t , x ) σ 2
x 2
+ C /
t ( t , x ) ] t =
= – [ C /
x ] – 1
[
2
1
C
//
xx ( t , x ) σ 2
x 2
+ C /
t ( t , x ) ] t (1.5′)
The rate of return of the portfolio at t does not contain risky term of the ’white’ noise type. To avoid
arbitrage opportunity the rate of return of the portfolio at t should be proportional to risk free bond rate r.
Hence, we arrive at the BSE (1.7′) which can be represented in the form
C /
t ( t , x ) + r x C /
x ( t , x ) +
2
1
C
//
xx ( t , x ) σ 2
x 2
– r C ( t , x ) = 0 (BSE)
with boundary condition C ( T , x ) = max { x – K , 0 }.
In modern handbooks [5] one usually consider derivation of the BSE by construction hedged position by
using one option long and a portion of stocks short. This derivation is similar to original derivation [1].
The only difference between two derivations is the value
Δ ( t ) = [ C /
S ( t , S ( t ) ) ] – 1
of options in hedged portfolio in original derivation and the number of stocks
N ( t , S ( t ) ) = C /
S ( t , S ( t ) ) (1.10)
in hedged portfolio
Π ( t , S ( t ) ) = − C ( t , S ( t ) ) + N ( t , S ( t ) ) S ( t ) (1.11)
in alternative derivation [5].
Comment. In some papers, authors expressed a confusion raised from the use value of the hedged
position (portfolio) and its difference, differential, or financial change in the value in definition of the
hedged portfolio. These notions are in general similar to each other. Misunderstanding comes from the
use one time parameter tin definition of the portfolio value differently in dynamics of the portfolio. Such
drawback can be easily corrected by introducing hedged portfolio by the function
Π ( u , t ) = S ( u ) – [ C /
S ( t , S ( t ) ) ] – 1
C ( u , S ( u )) (1.12)
of the variable u, u t where t, t 0 is a fixed parameter. Formula (1.12) defines value of the portfolio at
u , u, u t constructed at t, t 0. Then differential of the function Π ( u , t ) with respect to variable u is
defined by the formula
d Π ( u , t ) = d S ( u ) – [ C /
S ( t , S ( t ) ) ] – 1
d C ( u , S ( u ))
We arrive at the hedged position by putting variable u = t
5. 5
Π ( u , t ) | u = t = Π ( t , t ) , d Π ( u , t ) | u = t = d u Π ( t , t )
Hence, instead of defining two separate equations for portfolio value and its dynamics we present one
equation, which covers two equations, which are used in Black Scholes pricing concept. In doing such
correction we expand original coordinate space ( t , Π ) to ( t , u , Π ) , 0 ≤ t ≤ u. Correction makes enable
to present an accurate derivation of the BS pricing concept.
Black-Scholes pricing concept. For a fixed moment of time t [ 0 , T ] there exist option price
C ( t , S ( t )) and portfolio that contains one stock in long position and a portion of options in short
position for which the change in the value of the portfolio at t is riskless.
The BS pricing concept is specified by the BS’s hedged portfolio
1) borrows S ( t ) at risk free interest rate r and
2) sell immediately Δ ( t ) call options for BS price
These transactions provide investor risk free interest r on infinitesimal interval [ t , t + dt ).
On the other hand, there is no evidence pointwise hedge price can satisfy all market participants. Such
price could satisfy a hedger and a counterparty who buys options to get high return can be either satisfied
or not by the Black-Scholes price.
In general, price is interpreted as a settlement between buyer and seller. Hence, Black-Scholes pricing
does not a price in general. It likes a date-t strategy. If we use Black-Scholes pricing then we only
guarantee instantaneous risk free return on BS’s portfolio at t. Market prices of the options can be close to
Black-Scholes model prices or not and there is no evidence or justification that market actually use Black
Scholes price.
Recall that perfect hedge provided by the option covers only initial moment of time. Black and Scholes
remarked “As the variables x , t ( here x = S ( t )) change, the number of options to be sold short to create
hedged position with one share of stock changes. If the hedged position is maintained continuously, then
the approximations mentioned above become exact, and the return on the hedged position is completely
independent of the change in the value of the stock. In fact, the return on the hedged position becomes
certain. (This was pointed out to us by Robert Merton)” [1].
Hence, seller and buyer of the option are subject to market risk at the next moment t + t , t > 0 and
can get either loss or profit by applying BS price.
Let us consider the cash flow generated by continuously maintained hedged portfolio. The date-t value of
the hedged portfolio is defined by the formula (1.3′) and equal to Π ( t + Δt , t ). On the other date-(t +
Δt) value of the BS’s portfolio is equal to
Π ( t + Δt , t + Δt ) = S ( t + Δt ) – [ C /
S ( t + Δt , S ( t + Δt )) ] – 1
C ( t + Δt , S ( t + Δt ) )
Thus in order to maintain hedged portfolio one should add
Π ( t + Δt , t + Δt ) – Π ( t + Δt , t ) = (1.13)
6. 6
= { [ C /
S ( t , S ( t ) ) ] – 1
– [ C /
S ( t + Δt , S ( t + Δt )) ] – 1
} C ( t + Δt , S ( t + Δt ) )
to Π ( t + Δt , t ) the adjustment sum at the moment t + Δt. Denote f ( t , S ( t ) ) = [ C /
S ( t , S ( t ) ) ] – 1
.
Bearing in mind relationships
f /
t ( t , S ( t ) ) = – 2/
S ]))t(S,t(C[
1
C
//
tS ( t , S ( t ) )
f /
S ( t , S ( t ) ) = – 2/
S ]))t(S,t(C[
1
C
//
SS ( t , S ( t ) )
f
//
SS ( t , S ( t ) ) = – 2/
S ]))t(S,t(C[
1
C
///
SSS ( t , S ( t ) )
C ( t + Δt , S ( t + Δt ) ) = C ( t , S ( t )) + [ C ( S ( t + Δt ) , t + Δt ) – C ( t , S ( t )) ]
one can apply Ito formula. It follows from (1.13) that date-( t + Δt) adjustment is equal to
Π ( t + Δt , t + Δt ) – Π ( t + Δt , t ) =
= – 2/
S ]))t(S,t(C[
1
{ [ C
//
tS ( t , S ( t )) + C
//
SS ( t , S ( t )) μ S ( t ) +
+
2
1
C
///
SSS ( t , S ( t ) ) σ 2
S 2
( t ) ] Δt + C
//
SS ( t , S ( t ) ) σ S ( t ) Δw ( t ) } [ C ( t , S ( t )) +
+ C /
t ( t , S ( t )) + C /
S ( t , S ( t )) μ S ( t ) +
2
1
C
//
SS ( t , S ( t ) ) σ 2
S 2
( t ) ] Δt +
+ C /
S ( t , S ( t ) ) σ S ( t ) Δw ( t ) ] = (1.14)
= – 2/
S ]))t(S,t(C[
1
{ [ C
//
tS ( t , S ( t )) + C
//
SS ( t , S ( t )) μ S ( t ) +
+
2
1
C
///
SSS ( t , S ( t ) ) σ 2
S 2
( t ) +
))(tS,t(C
))(tS,t(C))(tS,t(C //
SS
/
S
σ 2
S 2
( t ) ] Δt +
+ C
//
SS ( t , S ( t ) ) σ S ( t ) Δw ( t ) } C ( t , S ( t ))
If the value Π ( t + Δt , t + Δt ) > Π ( t + Δt , t ) then portfolio adjustment is the amount which should
be added at t + Δt. Otherwise corresponding sum should be withdrawn. Such adjustment represents mark-
to-market transactions. Using formula (1.14) we represent cash flow that corresponds to maintenance of
the hedged portfolio during lifetime of the option.
7. 7
Let t = t 0 < t 1 < … < t n = T be a part ion of the lifetime period of the option. Then applying formula
(1.14) the maintenance of the hedged position can be represented by sum
H ( t , T ) =
n
1j
[ Π ( t j , t j ) – Π ( t j , t j – 1 ) ] = –
n
1j
2
1-j1-j
/
S
1-j1-j
]))t(S,t(C[
))t(S,t(C
{ [ C
//
tS ( t j – 1 , S ( t j – 1 )) + C
//
SS ( t j – 1 , S ( t j – 1 )) μ S ( t j – 1 ) +
+
2
1
C
///
SSS ( t j – 1 , S ( t j – 1 ) ) σ 2
S 2
( t j – 1 ) +
+
))t(S,t(C
))t(S,t(C))t(S,t(C
1-j1-j
1-j1-j
//
SS1-j1-j
/
S
σ 2
S 2
( t j – 1 ) ] Δt j – 1 +
+ C
//
SS ( t j – 1 , S ( t j – 1 ) ) σ S ( t j – 1 ) Δw ( t j – 1 ) } = (1.15)
=
T
t
2/
S ]))u(S,u(C[
))u(S,u(C
[ C
//
tS ( u , S ( u )) + C
//
SS ( u , S ( u )) μ S ( u ) +
+
2
1
C
///
SSS ( u , S ( u ) ) σ 2
S 2
( u ) +
))u(S,u(C
))u(S,u(C))u(S,u(C //
SS
/
S
σ 2
S 2
( u ) ] du +
+
T
t
2/
S ]))u(S,u(C[
))u(S,u(C
C
//
SS ( u , S ( u )) σ S ( u ) dw ( u )
Formula (1.15) shows that keeping hedged position over lifetime of the option is represented by a risky
cash flow. It is clear that it can be costly to maintain hedged position over the lifetime of the option. On
the other hand additional cash flow reflects additional cost for keeping hedge portfolio. Expected PV of
the future cash flow, which will adjust hedged portfolio, should be considered as collateral option price
similar to CVA. On the other hand if investor is not going to hold hedged position over lifetime of the
option why does he think that it is reasonable by option for BS price.
Now let us look at the alternative option pricing. First recall that option price at t is looking as
deterministic smooth function. Such preliminary condition implies that market risk which actually
undefined by BS’world has no effect on option price. On the other hand profit-loss analysis shows that
no=arbitrage BS’s option price admits either loss or profit. Our approach starts with the similar
observation and prescribes admitted option values at maturity a probability distribution that is specified
by underlying stock.
ALTERNATIVE APPROACH.
In this section, we represent an alternative approach to option pricing. We have discussed some
drawbacks of the BS option-pricing concept. The alternative approach to derivatives pricing was
8. 8
introduced in [2-4]. We call two cash flows equal over a time interval [ 0 , T ] if they have equal
instantaneous rates of return at any moment during [ 0 , T ].
Introduce financial equality principle. Two investments S i ( t ) , i = 1, 2 we call to be equal at moment t
if their instantaneous rates of return are equal at this moment. If two investments are equal for any
moment of time during [ 0 , T ] then we call these investments equal on [ 0 , T ]. Applying this definition
to a stock and European call option on this stock we arrive at the equation
)t(S
)T(S
{ S ( T ) > K } =
))t(S,t(C
))T(S,T(C
(2.1)
where C ( t , S ( t )) = C ( t , S ( t ) ; T , K ), 0 t T denotes option price at t with maturity T and
strike price K and C ( T , X ; T , K ) = max { X – K , 0 }. Solution of the equation (2.1) is a random
function C ( t , S ( t ), ω ) that promises the same rate of return as its underlying S ( t ) for a scenario
ω { ω : S ( T ) > K } and C ( t , S ( t ) ; ω ) = 0 for each scenario ω { ω : S ( T ) ≤ K }.
Bearing in mind that this definition of the price depending on market scenario we call this price as the
market price. Spot price which we denote c ( t , S ( t )) is interpreted as the settlement price between
sellers and buyers of the option at t. It is deterministic function in t. Let S ( t ) = x and c ( t, x ) be a
spot call option price. The market risk of the buyer of the option is defined by the chance that buyer pays
more than it is implied by the market, i.e.
P { c ( 0 , x ) > C ( t , S ( t ) ; T , K ) } (2.2′)
On the other hand, option seller’s market risk is measured by the chance of the adjacent market event, i.e.
P { c ( 0 , x ) < C ( t , S ( t ) ; T , K ) } (2.2′′)
It represents the probability of the chance that the premium received by option seller is less than it is
implied by the market.
Let us illustrate alternative pricing by using a discrete space-time approximation of continuous model
(2.1). Consider a discrete approximation of the S ( T , ω ) in the form
n
1j
S j { S ( T , ω ) [ S j – 1 , S j ) }
where 0 = S 0 < S 1 … < S n < + and denote p j = P ( j ) = P { S ( T ) [ S j – 1 , S j ) }.
Note that p j for a particular j could be as close to 1 or to 0 as we wish. We eliminate arbitrage opportunity
for each scenario ω j by putting
)ω;x,t(C
KS
x
S jj
, if S j K
and
C ( t , x ; ω ) = 0 , if S j < K
9. 9
The solution of the latter equation is
C ( t , x ; ω ) =
jS
x
( S j - K ) { S j K } { S ( T , ω ) ( S j – 1 , S j ] }
Then the market price of the call option can be approximated by the random variable
C ( t , x ; ω ) =
n
1j jS
x
( S j - K ) { S j K } { S ( T , ω ) ( S j – 1 , S j ] }
For the scenarios ω ω j = { S j ≤ K } return on the underlying security is 0 < x – 1
S j ≤ K while
option return is equal to 0. Thus, the option premium c ( 0 , x ) for the scenarios ω i = { S i > K }
should compensate losses of the security return for the scenarios ω = { S ( t , ω ) ≤ K }. Investor will be
benefitted by the option c ( t , x ) < C ( t , x ; ω ) for the scenarios ω for which { S i ( T , ωi ) > K }.
Our goal to present a reasonable estimates of the spot price choice c ( t , x ) represented by the market.
One possible estimate is BS price. It is price is formed by no arbitrage principle. The drawback of the
no arbitrage pricing is the fact that in stochastic market there is no classical arbitrage that implied by the
BS’s model. We rather have a probability distribution that prescribes a particular probability for each
option value as well as a positive probability of losing the original premium including no arbitrage BS
price. As far as market, pricing equation (2.1) represents definition of the option market price for each
market scenario the reasonable first order approximation of the settlement between buyers and sellers is
the option price that represents equal market risk as underlying stock.
Let t, t [ 0 , T ] denote a moment of time. Then the value of the stock x = S ( t ) at t is equal to the
value B – 1
( t , T ) x at T. From buyer perspective the chances of profit / loss
P { S ( T , ω ) B – 1
( t , T ) x } , P { S ( T , ω ) < B – 1
( t , T ) x }
and average profit / loss on stock at date T are defined as following
avg S , profit = E S ( T , ω ) χ { S ( T , ω ) B – 1
( t , T ) x } ,
avg S , loss = E S ( T , ω ) χ { S ( T , ω ) < B – 1
( t , T ) x }
correspondingly. Similarly define chances of underpriced and overpriced option
P { C ( T , S ( T ) ) B – 1
( t , T ) c ( t , x ) } , P { C ( T , S ( T ) ) B – 1
( t , T ) c ( t , x ) }
and average profit / loss on option at maturity
avg C , profit = E C ( T , S ( T ) ) χ { C ( T , S ( T ) ) B – 1
( t , T ) c ( t , x ) } ,
avg C , loss = E C ( T , S ( T ) ) χ { C ( T , S ( T ) ) < B – 1
( t , T ) c ( t , x ) } ,
The zero-order approximation of the option price based on market risk can be defined by the equality
10. 10
P { S ( T , ω ) B – 1
( t , T ) x } = P { C ( T , S ( T ) ) B – 1
( t , T ) c ( t , x ) } (2.3)
The same value of the spot option price can be defined by the use of equality chances of losses for stock
and option. Next order adjustment can be calculated by taking into account average loss / profit ratios for
stock and option
R S ( t , T ) =
}x)T,t(B)ω,T(S{χ)ω,T(SE
}x)T,t(B)ω,T(S{χ)ω,T(SE
1-
1-
R C ( t , T ) =
})x,t(c)T,t(B))T(S,T(C{χ))T(S,T(CE
})x,t(c)T,t(B))T(S,T(C{χ))T(S,T(CE
1-
1-
where C ( T , S ( T ) ) = max { S ( T ) – K , 0 }. Let S ( t ) be a solution of the equation (1.9). Bearing
in mind that solution of the equation (2.3) can be written in the form
S ( T ) = x exp
T
t
( μ –
2
σ 2
) dv +
T
t
σ dw ( v )
We note that for any q > 0
P { S ( T ) < q } = P { ln S ( T ) < ln q } = P { ln x +
T
t
( μ –
2
σ 2
) dv +
T
t
σ dw ( v ) < ln q }
Right hand side represents distribution of the normal distributed variable with mean and variance equal to
ln x + ( μ –
2
σ 2
) ( T – t ) , σ 2
( T – t )
correspondingly. Therefore
P { S ( T ) < q } =
)tT(σπ2
1
2
qln
-
exp –
)tT(σπ2
])tT()
2
σ
μ(xlnv[
2
2
2
dv
Differentiation of the right hand side with respect to q brings the density distribution ρ ( t , x ; T , q ) of
the random variable S ( T )
ρ ( t , x ; T , q ) =
)tT(qσπ2
1
22
exp –
)tT(σπ2
])tT()
2
σ
μ(
x
q
ln[
2
2
2
Left and right hand sides of the equation (2.3) can be represented by formulas
11. 11
p S , profit = P { S ( T ) B – 1
( t , T ) x } = P { exp [ ( μ –
2
σ 2
) ( T – t ) +
+ σ [ w ( T ) – w ( t ) ] ] B – 1
( t , T ) } =
=
)tT(σπ2
1
2
)T,t(Bln-
exp –
)tT(σ2
])tT()
2
σ
μ(xlnv[
2
2
2
dv =
= N (
tTσ
)tT()
2
σ
μ(xln)T,t(Bln
2
)
where N ( · ) is the standard normal distribution cumulative distribution function. Then
p C , profit = P { C ( T , S ( T )) B – 1
( t , T ) c ( t , x ) } =
= P { max { S ( T ) – K , 0 } B – 1
( t , T ) c ( t , x ) } = P { S ( T ) K + B – 1
( t , T ) c ( t , x ) } =
= N (
tTσ
)tT()
2
σ
μ(xln])x,t(c)T,t(BK[ln
2
1
)
Then equation (2.3 ) can be rewritten as
N (
tTσ
)tT()
2
σ
μ(xln])x,t(c)T,t(BK[ln
2
1
) = p S , profit
The solution of the equation can be represented in closed form
c ( t , x ) = B ( t , T ) { exp – [ σ tT N – 1
( p S , profit ) +
+ ln x + ( μ –
2
σ 2
) ( T – t ) ] – K } (2.4)
Given c ( t , x ) one can calculate the value account average loss / profit ratio of the option. If the value
R C ( t , T ) is small then the use option price in the form (2.4) does not looks reasonable. In this case one
can start with numeric solution of the equation
R C ( t , T ) = R S ( t , T ) (2.5)
Right hand side of the equation (2.5) is known number and left hand side is equal to
12. 12
R C ( t , T ) =
})x,t(c)T,t(BK)T(S{χ}0,K-)T(S{maxE
})x,t(c)T,t(BK)T(S{χ}0,K-)T(S{maxE
1
1-
1
1-
=
= [ E max { S ( T ) – K , 0 } χ { S ( T ) K + B – 1
( t , T ) c 1 ( t . x ) } ] – 1
– 1
Here
E max { S ( T ) – K , 0 } χ { S ( T ) K + B – 1
( t , T ) c 1 ( t . x ) } =
=
)tT(σπ2
1
2
)x,t(c)T,t(BlnK 1
1
( v – K) exp –
)tT(σ2
])tT()
2
σ
μ(xlnv[
2
2
2
dv
Hence, equation (2.5) admits numeric approach for solution it with respect to c 1 ( t , x ). In this case it
might be that left hand side of the inequality
P { S ( T , ω ) B – 1
( t , T ) x } > P { C ( T , S ( T ) ) B – 1
( t , T ) c 1 ( t , x ) }
remarkably exceeds right hand side, i.e. chance to get profit is too small with respect to similar
characteristic on stock investment. Such situation suggests establishing option price, which is a
combination of two estimates represented by equations (2.3) and (2.5). One can define variance of the
loss and profit of the option
V 2
C , loss = E { C ( T , S ( T ) ) χ [ C ( T , S ( T ) ) < B – 1
( t , T ) c ( t , x ) ] –
– E C ( T , S ( T ) ) χ [ C ( T , S ( T ) ) < B – 1
( t , T ) c ( t , x ) ] } 2
V 2
C , profit = E { C ( T , S ( T ) ) χ [ C ( T , S ( T ) ) B – 1
( t , T ) c ( t , x ) ] –
– E C ( T , S ( T ) ) χ [ C ( T , S ( T ) ) B – 1
( t , T ) c ( t , x ) ] } 2
which will supplement to above risk characteristics of the latter estimates. Option loss and profit
variances can be compared with correspondent characteristics of the underlying asset
V 2
S , loss =
= E { S ( T , ω ) χ [ S ( T , ω ) < B – 1
( t , T ) x ] – E S ( T , ω ) χ [ S ( T , ω ) < B – 1
( t , T ) x ] } 2
V 2
S , profit =
= E { S ( T , ω ) χ [ S ( T , ω ) B – 1
( t , T ) x ] – E S ( T , ω ) χ [ S ( T , ω ) B – 1
( t , T ) x ] } 2
Conclusion. We consider Black Scholes derivatives pricing concept as oversimplified pricing. The
oversimplified pricing means that definition of the derivatives price ignores market risk of any spot option
price including no arbitrage pricing. Our definition of option price is based on weighted risk-reward or
profit-loss ratios.
13. 13
I am grateful to P.Carr for his interest and useful discussions.
References.
1. Black, F., Scholes, M. The Pricing of Options and Corporate Liabilities. The Journal of Political
Economy, May 1973.
2. Gikhman, Il., On Black- Scholes Equation. J. Applied Finance (4), 2004, p. 47-74,
3. Gikhman, Il., Derivativs Pricing. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=500303.
4. Gikhman, Il., Alternative Derivatives pricing. Lambert Academic Publishing, p.154.
5. Hull J., Options, Futures and other Derivatives. Pearson Education International, 7ed. p.814.